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TRANSACTIONS

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

ESTABLISHED NOVEMBER 15, 1819,

VOLUME THE TENTH.

CAMBRIDGE:

^tinteO at tfie sanibrnttB ^wss ;

AND SOLD BY

DEIGHTON, BELL AND CO. AND MAOMTT.T.AN AND CO. CAMBRIDGE;
BELL AND DALDY, LONDON.

1I.DCCC.LXIT.

©ambuDgc :

FEINTED BY C. J. CLAY, M.A.
AT THK UNIVERSITY HIESS.

CONTENTS.

PART I.

PAGE

Xo. I. Oil a Direct Method of estimating Velocities, Accelerations, and all similar
Qtiantities with respect to Axes moveable in any manner in Space, with
Applications. By R. B. Hayward, M.A., Fellow of St Johns College,
Reader in Natural Philosophy in the University of Durham 1

II. On the question. What is the Solution, of a Differential Equation 1 A Sup-
plem.ent to the third section of a paper, On some points of the Integral
Calculus, printed in Vol. IX. Part II. By Augustus De Morgan, of
Trinity College, Vice-President of tits Royal Astronomical Society, and
Professor of Mat/iematics in University College, London 21

III. On Faraday's Lines of Force. By J. Clerk Maxwell, B.A., Fellow of

Trinity College, Cambridge 27

IV. The Structure of the Athenian Trireme ; considered with reference to certain

difficulties of interpretation. By J. W. Donaldson, D.D., late Fellow of
Trinity College, Cambridge 84

V. Of the Platonic Theory of Ideas. By W. Whewell, D.D., Master of

Trinity College, Cambridge 94

VI. On the Discontinuity of Arbitrary Constants which appear in Divergent

Developments. By G. G. Stokes, M.A., D.C.L., Sec. U.S., Fellow of
Pembroke College, a i%d Lucasian Professor of Mathematics in the Univer-
sity of Cambridge 106

VII. On the Beats of Imperfect Consonamces. By Augustus De Morgan,
F.R.A S., of Trinity College, Professor of Matliematics in University
College, London 129

VIII. On the Genuineness of the Sophista of Plato, and on some of its philosophical
hearings. By W. H. Thompson, M.A., Fellow of Trinity College, and
Regius Professor of Greek 146

IX. On the Substitution of Methods founded on Ordinary Geometry for Metliods
based on the General Doctrine of Proportions, in the Treatment of some
Geometrical Problems. By G. B. Airy, Esq., Astronomer Royal '. 166

X. On the Syllogism, No. III., and on Logic in general. By Augustus De
Morgan, F.R.A.S., of Trinity College, Professor of Mathematics in
University College, London 173

XI. On the Status of Solon mentioned by jEschines and Demosthenes. By J. W.

Donaldson, D.D., Vice-President of the Society 231

XIL Instances of remarkable Abnormities in the Voluntary Muscles. By G. E.

Paget, M.D., F.R.C.P., late Fellow ofGonville and Cains College 240

XIII. On Organic Polarity. By H. F. Baxter, Esq., M.R.C.S.L 248

XIV. A Proof of the Existence of a Root in every Algebraic Equation: with an

examinaiion and extension of Cauchy' s Theorem on Iinaginary Roots, and
Remarks on the Proofs of tJie existence of Roots given by Argand and by
Mmirey. By Augustus De Morgan, F.R.A.S., of Trinity College,

Professor of Mathematics in University College, London 261

6

CONTENTS.

PAKT II.

PAOK

No. I. On a Chart and Diagram for facilitating Cheat-Circle Sailing. By Hugh

GoDFRAY, M.A., St John's College 271

II. Suggestion of a Proof of the Theorem that every Algebraic Equation has

a Boot. By G. B. Airy, Esq., Astronomer Royal 283

III. On tlis General Principles of which tlie Composition or Aggregation of

Forces is a Consequence. By Augustus De Morgan, F.R.A.S., of
Trinity College, Professor of Matliemaiics in University College, London 290

IV. On Platds Cosmical System as exJdbited in the Tenth Book of " The

Republic." By J. W. Donaldson, D.D., Trinity College; Vice-Presi-
dent of the Society 305

V. On the Origin and Proper Use of tihe word Argument. By J. W.

Donaldson, D.D., late Fellow of Trinity College, Cambridge 317

VI. Supplement to a Proof of the TJieorem tlwA every Algebraic Equation lias

a Root. By G. B. Airy, Esq., Astronomer Royal 327

VII. On the Syllogism, No. IV., and on the Logic of Relations. By Augustus

De Morgan, F.RA.S., of Trinity College, Professor of Maifiematics

in University College, London 331

VIII. On the Motion of Beams and thin Elastic Rods. By 3. H. Eohbs, M.A.,

Fellow of the Cambridge Philosophical Society — 359

IX. On a Metrical Latin Inscription copied by Mr Blakesley at Cirta and
pvhlished in his "Four Months in Algeria." By H. A. J. Muneo, M.A.,
Fellow of Trinity College 374

X. On the Thfiory of Errors of Observation. By Augustus De Morgan,

F.R.A.S., of Trinity College, Professor of Mathematics in University
College, London 409

XI. On the Syllogism, No. V., and on various points of the Onymatic System.

By Augustus De Morgan, F.R.A.S., of Trinity College, Professor

of Mathematics in University College, London 428

ADVERTISEMENT.

Tee Society as a body is not to be considered responsible for any
facts and opinions advanced in the several Papers, which must rest
entirely on the credit of their respective Atithoi^s.

The Society takes this opportunity of expressing its grateful
acknowledgments to the Syndics of the University Press, for their
liberality in taking upon themselves the expense of printing this Volume
of the Transactions.

TRANSACTIONS

', K OP Taifc-vS^ '

iRl.

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

VOLUME X. PART I.

CAMBRIDGE:

PRINTED BY C. J. CLAY, M. A. AT THE UNIVERSITY PRESS ;

AND SOLD BY

DEIGHTON, BELL AND CO. AND MACMILLAN AND CO. CAMBRIDGE j
BELL AND DALDY, LONDON.

M. DOCC. Lvni.

I. On a Direct Method of estimating Velocities, Accelerations, and all similar
Quantities with respect to Axes moveable in any manner in Space, with Appli-
cations. By R. B. Haywakd, M.A. Fellow of St John's College, Reader in
Natural Philosophy in the University of Durham.

[Read Fel. 25, 1856.]

"...gardons-nous de croire qu'une science soit &ite quand on I'a reduite a des fonnules analytiques. Rien ne
nous dispense d'etudier les choses en elles-memes, et de nous bien rendre compte des idees qui font I'objet de nos
speculations." Poinsot.

"...c'est une remarque que nous pouvons faire dans toutes nos recherches mathematiques ; ces quantites
auxUiaires, ces calculs longs et difiiciles ou Ton se trouve entraine, y sont presque toujours la preuve que notre esprit
n"a point, des le commencement, considere les choses en elles-memes et dune vue assez directe, puisqu'il nous faut
tant d'artifices et de detours pour y arriver ; tandis que tout s'abrege et se simplifie sitot qu'on se place au vrai
point de vue." Ibid.

The general principles, which I have endeavoured to keep in view in the investigations of
this paper, are those contained in the above quotations from Poinsot. My object is not so
much to obtain new results, as to regard old ones from a point of view which renders all
our equations directly significant, and to develop a corresponding method, by which these
equations result directly from one central principle instead of being (as is commonly the case)
deduced by long processes of transformation and elimination from certain fundamental
equations, in which that principle has been embodied.

The frequent occurrence of exactly corresponding equations, (though this correspondence is
sometimes disguised under a different mode of expression) in many investigations of Kinematics
and Dynamics suggests the inquiry whether they do not result from some common principle,
from which they may be deduced once for all. An investigation based on this idea forms the
first part of this paper, in which it will be shewn how the variations of any magnitude,
which is capable of representation by a line of definite length in a definite direction and is
subject to the parallelogrammic law of combination, may be simply and directly estimated
relatively to any axes whatever. The second part is devoted to the general problem of the
dynamics of a material system, treated in that form which the previous Calculus suggests,
together with a development of the solution in the case of a body of invariable form.

Since whatever novelty of view is contained in this paper consists rather in the relation
of the details to the general method than in the details themselves, much that is familiar to
every student of Dynamics must be repeated in its proper place, but it is hoped that such
repetition will in general be compensated by a new or fuller significance being obtained. As
regards the problem of rotation, M. Poinsot's solution in the " Theorie de la Rotation " is so
Vol. X. Part I. 1

^C:^

2 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

complete and so entirely satisfies the conditions expressed in our quotations above from that
work, as to leave nothing to be desired. But it does not appear to me that his method,
which depends essentially on the summation of the centrifugal forces, is so widely applicable
beyond the limits of this particular problem as that by which the same results are obtained
in this paper: but be this as it may, any new point of view, if, a true one ("vrai point de
vue") has its special advantages, and on this ground may claim some attention.

SECTION I.

The Method, with some kinematical Applications.

1. As we shall here be concerned only with the directions of lines in space, and not with
their absolute positions, it will be convenient to suppose them all to pass through a common
origin O, and to define the inclination of two lines as OP, OQhy the arc PQ of the great
circle, in which the plane POQ meets a sphere whose centre is O and radius constant. We
shall also suppose any linear velocity, acceleration or force, represented by a length along OP,
to tend from O towards P, and any angular velocity or the like, represented in like manner,
to tend in such a direction about OP that, if OP were directed to the north pole, the direction
of rotation would coincide with that of the diurnal motion of the heavens.

2. Let u denote any magnitude, which can be completely represented by a certain length
along the line OU, and which can be combined with a similar magnitude v along OF by means
of a parallelogram, like the parallelogram of forces or velocities. Then of course u may be
resolved in different directions by the same principles, and thus if we adopt rectangular
resolution, the resolved part of u along OP will be mcos UP, which may be denoted by Up.
We proceed to inquire how Up varies by a change in the position of OP.

3. Suppose OP to be a line moving in any manner about O, and that it shifts from OP
to a consecutive position OP' in the time dt ; and conceive that this motion arises from an
angular velocity Q about an instantaneous axis 01. Resolve Q into its components Qcos lU
about OU and Qsin/i7 about a line in the plane lOU, perpendicular to OU: and farther
resolve this latter component in the plane perpendicular to OU into the components Q sin lU
. cos lUP in the plane POU and Q sin lU . sin lUP perpendicular to the same plane.

Then the component in OU and that perpendicular to it in the plane POU produce
displacements of P perpendicular to the arc UP, and consequently do not ultimately alter the
length of the arc UP, so that Up remains ultimately unchanged so far as the motion of OP is
due to these components : but the component perpendicular to the plane POU increases UP
by the arc Q sin lU . sin lUP . dt, and therefore the increment of Up from this component (being
equal to - u . sin UP . d . UP) is

- mQ sin UP . sin lU . sin lUP . dt.
But the other increments being zero, this is the total increment of Up, wherefore we have

^^= - mQ sin lU. sin f7P. sin IUP...{A).
dt

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 3

It is well to observe that ^ vanishes, when the three axes 01, OU, OP lie in the same

at

plane, and in particular when two of them coincide, as is evident from the above equations, or

from considerations similar to those by which it was obtained.

4. In the above investigation we have supposed u to be constant both in direction and
intensity ; let us now suppose u to vary in both respects with the time {t). The change in u
in the time dt may be conceived to arise from its composition with the quantity fdt in the line
OF, and / may properly be called the acceleration of u at the time t. Now fdt may be
resolved in the plane UOF into /. dt cos FU along OU and /. dt sin FU perpendicular to OU,
and the components oi u + du will therefore be u + f. dt cos FU along OU and /. dt sin FU
perpendicular to OU; whence, if d(p denote the angle through which OU shifts in the time dt
towards OF, it will readily be seen that ultimately

^=:f cos FU, u^=fsmFU...{B),
at at

If then the acceleration / be known both as to direction and intensity at every instant, the
motion oi OU and the variation in the intensity of u may be determined by these last
equations. In fact, the point U on the sphere of reference continually follows the point F

with the velocity —-, so that the problem of determining U's path is the same as the old

problem of the path described by a dog always running towards his master who is himself in
motion, the only difference being that the path is here on a sphere instead of a plane.

5. Next for the variation of Up, when u varies with the time. It is plain that Up varies
from two causes ; first, by reason of the acceleration /, and secondly, by reason of the motion
of OP due to the angular velocity Q about 01, and that the total variation will be the sum of
these two partial variations. Now the latter has been calculated above, and the former is
obviously the resolved part oi fdt along OP or f.dt cos FP, therefore we obtain the equation*

^^^ = f cos FP - uQ sin lU. sia UP, sin lUP.. .(C).

This equation of course contains the previous equations (B) : thus, if OP and 0Z7 coincide
always, UP is always zero and the second side of (C) reduces to its first term : and again if

OP be always in the plane FOU and perpendicular to OU, Up is always zero, Q = — - , lU

and UP are quadrants, lUP a right angle, and FP the complement of FU, and therefore, as
above,

= fsinFU-u-^.

'' dt

6. We may farther illustrate the application of equation (C) by supposing OP to coincide
with certain other lines specially connected with 0?7 and OF. , ,

" It should be remarked here that the angle lUP must be I Q about 01 causes the motion of P, resolved in the arc UP, to
considered positive or negative, according as the positive rotation I be from or towards U.

1_2

Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

Let U, If, U" and F, F', F" be three consecutive positions of U and F respectively, and
K, K' those poles of FF", F'F" respectively, (considered as arcs of great circles) about which
positive rotation brings F to J", and F' to F". We know that IT lies on the arc UF between
U and F, and V" on the arc ITF' between If and
F'. Also it is plain that F' is the pole of KK', and
therefore that KK' measures the angle ofcontingence
between the consecutive elements FF', F'F": in
fact, the loci of K and F are so connected that the
elementary arcs of the one are equal to the angles
of contingence of the other, and vice versa.

Suppose the locus of F to be defined by ele-
ments, corresponding to what Dr Whewell has
named in plane curves intrinsic elements, that is,

by elements a, e such that the elementary arc FF' = da, and the angle of contingence between
FF' and F'F'' = de : and suppose the locus of U defined in like manner, so that UU' = d(p,
and the angle of contingence FU' V" = dri. Also let UF = in, and angle UFF' = v.

Now let OP coincide always with OF. Then will Up=u cos m. and / being taken to

coincide with k, Q = — , and therefore equation (3) becomes

dt

d -. . da .

J- (u cos fx) =f-u-—. sm n . sin KU.sin KUF.

dt dt

But sin kU . sin kUF = sin kFU = sin ( v] =

cos V,

and— = /cos (IX ;

/ .
■cosi/ + -smu= 0.
u

.(1).

therefore we obtain after reduction

dfi da
dt ~ dt
Again let OP coincide always with OK, then

Up'=u cos UK = M sin M • cos UFK = m sin ju , sin v,
and I may be taken to coincide with F' or ultimately with F, so that (3) becomes,
de\

(q being = ^),

^ . . de

— (m sm jui .sin v) = -u-~.sm FU . sin UK. sin FUK

dt

or after reduction

dt

de .
= — M — - Sin fj. . cos V,
dt

d/u.

dfi tdv de\
dt"^ [dt'^dtj'

tan a f .

h ■- Sin ^1 = 0.

tan V u

.(2).

The equations (l) and (2) together with the two equations (B) serve to determine (after
du d(p , , da de

dt ""

eliminating fx and v) -£ and"-^, when f, -jj, ~ are given, that is, when the intensity and

dt

dt dt

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 5

variation in direction of the acceleration of u are given for every instant. And we have also

from the triangle FU'F' ultimately

drj , da .

— . sin/ui = -r7.smi/,
at at

to determine — : and therefore we have equations to determine the intensity and variation in

direction of u itself. Hence we have obtained a solution of the problem, " Given the path of
F and the variable intensity of/, to determine the path of U and the intensity of «," the whole
being referred to intrinsic elements.

7. It will be useful to obtain results analogous to equation (C) for three rectangular
axes in a somewhat different form. Of course these might be obtained from that equation
itself, but it will be better to investigate them independently by the same kind of reasoning.

Let u„ Uy, u^ denote the resolved parts of u along the moveable rectangular axes Ox, Oy
Ox, and let Q^, Q^, Q^ and /„ f^, / denote in like manner the resolved parts of Q and /. Now
by reason of the acceleration /, u^ receives in the time dt the increment f^ dt : also Ox changes
its position by reason of the rotations Qy, Q^, the first of which shifts it in the plane of xx
through the angle Qydt from 0^, and the latter in the plane of xy through the angle Q^dt
towards Oy] and from the first of these causes u^ receives the increment

u^ cos I — \-Qydt\ + u^ cos (Qydt) - u^,

or — u^Q dt ultimately, while from the second it receives the increment

Uy COS ( - - Q^dt I + u^ cos (Q^dt) - u,,

or UyQ^dt ultimately. Hence the total increment of u^, being the sum of these partial
increments, we obtain the equation

du^

Similarly for Uy, u^ we should obtain

du^ , \ {E).

du

8. To illustrate the applicability of these last obtained equations, we will select a few
particular kinematical problems.

a. Relative velocities of a point in motion with respect to revolving axes.

From the nature of the quantity u, it will be seen that it may be taken to denote the radius
vector OP of a point P, and u^, Uy, u, may then be replaced by the co-ordinates, w,y, z: also
/, denoting the acceleration of u, will in this case denote the absolute velocity of P, and f^ fy,
ft the absolute velocities resolved in the directions of the axes, which we will denote by v„ Vy,
v^. Then by the equations above we have three equations, of which the type is

Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

and which determine the relative velocities — , — , — of the point with respect to the co-or-

dt dt dt ^ ^

dinate axes.

If the point be fixed relatively to the axes, and a?o» Dm ^o ^^ its co-ordinates, the above
equation becomes

v^ = \i^ . Zq — i2j: . y^,

one of a set of well known equations, determining the linear velocity of a point in a body
revolving with given angular velocities.

If the point lie in the axis of at, so that y, x both vanish,

dx

— - = 7)^, = u„ - wQ^, = v^ + a?Qy.
dt

In these, if iv, y, ss are in the directions of the radius vector, a perpendicular to it in the

vertical plane, and a perpendicular to this plane respectively, and if r, d, (p denote radius

vector, altitude and azimuth, then

w = r, Qj,

d0

whence

cos d, ^z = -rr,
dt dt

dr de ^ d(h

V. = — , v„ = r — - , v, = r cos —~- ,
dt " dt ' dt

the common expressions for the components, relatively to polar co-ordinates, of the velocity of
a point.

b. Accelerations, radial, transversal in the vertical plane, and perpendicular to that plane.

In our general formulae u will now denote a velocity, and f an acceleration strictly so

called. And in this case

dr dd . dd)

Mi = — , My = »■ ;t- > u,=^rcosO

dt

dt

dt '

^ dd) . „ d(b ^ de

a =--^ sine, ^=--fcos0, Q, = -j-,
dt " dt dt

wherefore, by equations {E)

d/T f
radial acceleration —f, = [r

-^ df \

d9
~dt

+ r cos'' Q — ^
dt

)■

transversal acceleration in the vertical plane =^

d [ d9\ ( d(p

-p »'-r - -v-sme. cos0-p-
dt \ dtj V dt

1 d f „d9\ . „ „dd>
-— ( r' — - + r sin . cos 0-p-
rdt\ dt) dt

dt

d9\
dt)

azimuthal acceleration =/. = — ( rcosO — | - f . -^ cos + r sin — - . -^]

''" dt\ dt) \ dt dt dt dt)

VELOCITIES. &c. WITH RESPECT TO AXES MOVEABLE IN SPACE.

1 d (

^ • I '

r cos d dt \

dt

c. Let the axes of as, y, z be always parallel to the tangent, principal normal and normal
to the osculating plane of any curve. Then

ds

Mj, = 0,

M^ = 0,

" dt

^ = 0'

where de, dr denote respectively the angle between consecutive tangents, and that between
consecutive osculating planes.

Hence

tangential acceleration =f^=-~,

... . . , , ^ ds de /ds\2 de 1 (dsY

acceleration m principal normal = r, = -—.-— =-- .-^ = --^5
^ ^ -^^ dt dt \dtl ds fj \dtj

acceleration in normal to osculating plane =f^ = 0.

SECTION II.

Dynamical Applications.

9. I propose here to consider the problem of the motion of any material system, so far as
it depends on external forces only, and to develop the solution in that case in which the entire
motion is determined by these forces, namely, in the case of an invariable system.

1 0. This problem naturally resolves itself into two : for, since every system of forces is
reducible to a single force and a single couple, we have to investigate the effects of that force,
and the effects of that couple. Now we know that the resultant force determines the motion
of the centre of gravity of the system, be the constitution of the system what it may. In like
manner the resultant couple determines something relatively to the motion of the system about
its centre of gravity, which in the case of an invariable system defines its motion of rotation
about that point, but which in other cases is not usually recognised as a definite objective
magnitude, and has therefore no received name. This defect will be remedied by adopting
momentum as the intermediate term between force and velocity, and by regarding as distinct
steps the passage from force to momentum and that from momentum to velocity. In accordance
with this idea we proceed to shew that as in our first problem we shall be concerned with the
magnitudes, force, linear momentum or momentum of translation, and linear velocity or velocity
of translation, so in the other we shall be concerned with the corresponding magnitudes, couple,
angular momentum or momentum of rotation, and angular velocity or velocity of rotation ; and
that, as all these magnitudes possess the properties characteristic of the magnitude u in the
previous section, the Calculus there developed is applicable to them.

8 Mb R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

11. Consider a material system at any instant of its motion. Each particle is moving
with a definite momentum in a definite direction, which may be resolved into components in
given directions in the same manner as a velocity or a force. Let this momentum be resolved
in the direction of a given axis OP, and its moment about that axis taken, the resolved part
may be called the linear momentum, and the moment the angular momentum, of the particle
relatively to the axis OP. Let the same be done for every particle of the system, and the
sums of their linear and angular momenta taken, these sums may then be called respectively
the linear and angular momenta of the system relatively to the axis OP.

12. Let the linear momenta relatively to the three axes Ox, Oy, 0% be denoted by u„ Uy,
u^, and the corresponding angular momenta by h^, \, h^ respectively ; then it may easily be
shewn that the linear momentum relatively to the axis, whose direction-cosines are I, m, n, is

lu^ + mUy + nu^,
and that the angular momentum relatively to the same axis is

Ih^ + mhy + nh^.
The first expression will be a maximum, and equal to \u,^ + u^ + u^\i, when

I : m : n :: Uj : Uy : u^;
and if this be denoted by m, it is plain that the linear momentum along any line inclined to the
direction of u at an angle 9 will be u co%6. Hence we may regard the whole linear momentum
of the system as equivalent to the single linear momentum u determinate in intensity and
direction.

In like manner we may conclude that the whole angular momentum is reducible to a single
angular momentum h determinate in intensity and direction.

13. Thus, just as a system of forces is reducible to a single force and a single couple, the
momenta of the several particles of a system are reducible to a single linear and a single
angular momentum, which we shall speak of as the linear and angular momenta of the system.
It is to be observed that the linear momentum u is independent of the origin O both as regards
direction and intensity, but the angular momentum h is in both respects dependent on the
position of O. Also it may be proved, as in the case of a system of forces, that the angular
momentum h remains constant, while O moves along the direction of the linear momentum u,
but changes, as O moves in any other direction; and finally, that its intensity will be a
minimum and its direction coincident with that of u, when O lies upon a certain determinate
line, which (from analogy) may be termed the central axis of momenta.

14. Now let us consider the changes in the linear and angular momenta, as the time
changes, when the system is acted on by any forces.

In the time dt any force P generates in the particle on which it acts the momentum Pdt,
and these momenta, being resolved and summed as was done above, will give rise to a linear
momentum Rdt in the direction of the resultant force R of the forces (P), and an angular
momentum Gdt relatively to the axis of the resultant couple G of the same forces. Since
however the internal forces consist of pairs of equal and opposite forces in the same straight
line, by the nature of action and reaction, the momenta produced by them will vanish in the

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 9

summation over the whole system ; we may therefore regard R and G as the resultant fprce
and resultant couple of the external forces. Then the linear momentum u along the line OU
must be compounded with the linear momentum Rdt in the line OR in order to obtain its value
at the time t + dt: and in like manner the angular momentum h relatively to the axis OH must
be compounded with the angular momentum Gdt relatively to the axis OG.

15. Hence the method of the previous section applies to momenta of both kinds, replacing
/ in one case by R and in the other case by G. Thus the equations {B) give us

du dd)

— = RcosRU, u-^= R sin RU,

dt dt

where d(b is the arc through which U moves towards R in the time dt : and

— = G cos GH, A -/^ = G sin GH,

dt dt

where d^ is the arc through which H moves towards G in the time dt.

Also for fixed rectangular axes, with respect to which the components of R and G are
X, Y, Z and L, M, N respectively, it is plain from the above reasoning that we should have

dt ~ ' dt ~ ' dt ~ '

dh^ d\ dh^

dt dt dt

which are really the six fundamental equations of motion of our works on Dynamics.

For rectangular axes moveable about 0, the equations {E) of the last section furnish
two sets of three equations, of which the types are

<iM^ „ „ „

-^ = -y + up., - u^Q,^,

16. If the system be acted on by no external forces, it follows that both u and k
are constant in intensity and invariable in direction. This result might by analogy be
named the principle of the Conservation of Momentum.

This principle, as applied to linear momentum, is obviously equivalent to the prin-
ciple of the conservation of motion of the centre of gravity : as applied to angular
momentum, the constancy of direction of the axis of h and therefore of a plane perpen-
dicular to it shews that there is an invariable axis or plane, while the constancy of its
intensity and therefore of its resolved part in any fixed direction is equivalent to the asser-
tion of the truth of the principle of the conservation of areas for any fixed axis.

It may also be noted that there is an infinite number of invariable axes, and that,
if the origin be taken on the central axis of momenta, the corresponding invariable
axis will coincide with the central axis, and the angular momentum about it will then be
Vol. X. Paet I. 2

10 mk r. b. hayward, on a direct method of estimating

a minimum: also that for any other position of the origin the direction of the invariable
axis and the intensity of the momentum about it will depend upon the position of the line,
parallel to the central axis, in which the origin lies, just as in the corresponding propositions
relative to couples,

17. Any one of the different sets of equations in § (15) may be used to determine
completely u and h, when the forces are given or vice versa. It is to be observed that
the equations involving h, refer either to -a fixed origin, or to an origin, whose motion is
always in the instantaneous direction of u the linear momentum, for, as we saw, a change of
the origin in this direction does not produce a change in h, as its change in any other
direction does. It would be easy to introduce terms depending on the motion of the origin ;
in the last set of equations, for instance, if a^apO^ denote the linear velocities of the origin
in the directions of the axes, the equation for h^ becomes

—r- = L + hyQ^ — h^Qj, + u^z — uja.y.

The equations involving u, are entirely independent of the origin, and will there-
fore not be affected, however the origin be supposed to move.

18. It appears then that the linear and angular momenta are determined solely by the
external forces acting on the system, and not on the system itself otherwise than the forces
themselves depend on it : in fact, they are simply the accumulated effects of the forces and
the initial momenta. To proceed to the determination of the actual motion of the system
from these momenta, the system must be particularised, and as one system may differ from
another both as to the quantity of matter included in it, and as to its arrangement,
we may consider separately how much farther particularisation in either respect will enable
us to carry our results.

19. If the quantity of matter or mass of the whole system be given, it is well known
that the linear momentum of the system is that of its whole mass collected at its centre

of gravity, so that, M denoting this mass, the velocity of the centre of gravity is — in

M

the direction of the linear momentum : thus the motion of a certain point definitely related
to the system is obtained, and this is usually regarded as defining its motion of trans-
lation. For any other point definitely related to the system, the motion will in general
depend also on h and the arrangement of its matter.

20. If then the translation of the system be referred to its centre of gravity, its
motion about the centre of gravity will depend solely on h and the arrangement of its mass ;
for the direction of motion of the centre of gravity being that of the linear momentum, h
referred to that point as origin will be independent of u. Now the arrangement of a system
of matter may be either permanent or variable. If the former, it is spoken of as a body

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 11

or system of invariable form*, and the investigation of its motion about the centre of
gravity requires only the determination of its axis of rotation and the intensity of rotation
about that axis.

If the arrangement be variable, the laws of its variation must be given, and according
to the number of possible laws will be the number of different solutions of the problem :
here then the problem diverges into special problems; such as that of the motion of a body
expanding or contracting according to a given law and the like, where the law of variation
is geometrically expressed ; and such as the problems of the motion of fluids, of elastic bodies,
or of systems of bodies like the solar system, where the law of variation is mechanically
expressed by defining the nature of the internal actions and reactions of the system. We
shall confine our attention to the simpler problem of the motion of a system of invariable
form, which we proceed to discuss.

21. The motion of an invariable system is always reducible to the motion of translation
of some point invariably connected with it combined with a motion of rotation about a certain
axis through that point. Let v^, v^, Vz denote the resolved velocities along Ox, Oy, 0% of
the point O, to which the translation is referred, and let w^, Wy, w, denote the resolved angular
velocities about the same lines ; then the velocity of any particle m, whose co-ordinates are
X, y, ss, is v^ + Wy% - w^y in the direction of Oof, with similar expressions for the directions
Oy, Osx. Hence summing the linear and angular momenta of the several particles of the
system, we find

u^ = 2(m) .v^ + cD^. '2(mz) - W;,'2,(my),

' I avoid the use of the terra rigid body because of the
mechanical notion conveyed in the term rigid. The pro-
positions usually enunciated with reference to a rigid body
must, if that term be retained, be understood of & geometrically, ■
not a mechanically, rigid body ; that is, of a body the disposi-
tion of whose parts is by hypothesis unaltered, not of one in
which the disposition cannot be altered or can only be insensibly
altered by force applied to it. But it is difficult (and perhaps
not desirable) to divest this term of its mechanical meaning,
as is seen in the modes of expression commonly adopted in the
case of flexible strings, fluids, &c., where it is frequently de-
manded of us to suppose our strings to become inflexible, our
fluids to become rigid, or to be enclosed in rigid envelops, and
the like — a process which must always stagger a beginner and
leave a certain want of confidence in his results, until this is
gained by familiarity with the process, or until he learns that it
simply amounts to asserting that what has been laid down to
be true of a rigid body is no less true of a non-rigid body,
while there is no change in the disposition of its parts. As
another instance of a needless limitation in our current defini-
tions, we may cite that of Statics as the science which treats of
the equilibrium of forces, whereas the truer view would be to
regard it as treating of those relations of forces which are inde-
pendent of time, and thus every dynamical problem would have
its statical part in which the state of the system and the forces
is considered at each instant, and its truly dynamical part in
which the changes effected from instant to instant are deter-

mined. This view presents Statics as a natural preparation for
Dynamics, instead of as a science of coordinate rank separated
by a gulf to be bridged over by a fictitious reduction of dy-
namical problems to problems of equilibrium through the intro-
duction oi fictitious forces. In several of our more recent works
the terms accelerating force and centrifugal force have been
rejected or explained as mere abbreviations, the one as not
being properly & force, the other as being a fictitious and not an
actual force : this it would be well to carry out still more com-
pletely, to restrict force in fact to that which is expressible by
weight and to admit only actual forces (to the exclusion oi cen-
trifugal forces, effective forces and the like) under the two
divisions of internal forces, or those whose opposite Reactions
are included within the system, and external forces, or those
whose opposite Reactions axe not so included. If then Statics
and Dynamics were defined as above, one great division of
Rational Mechanics would be formed of the Statics and Dyna-
mics of a system of given invariable form, without the par-
ticular constitution of the system being defined and there-
fore independent of Internal Forces ; while the other great
division would include the Statics and Dynamics of special
systems of defined constitutions, as flexible bodies, fluids,
elastic solids and the like, in which the laws of the internal
forces must be more or less completely known. These re-
marks are thrown out as suggestions for a more natural
system of grouping the special mechanical sciences than has
yet been commonly received.

2—2

12 mk r. b. hayward, on a direct method of estimating

and h^ = 2m(y . v^ + w^y - w,jc - z .Vy + wjs - w^ss)

= 1(my) . v^ - 2(m«) , v^ + S(m . y" + z') . w^ - '^(mxy)it)y - 2(m«*) . w^,
with similar expressions for Uy, u^ and hy, h^.

From these equations it appears that, when the linear and angular velocities of the system
are referred to an arbitrary point 0, each depends in general on both the linear and the
angular momentum. If however be the centre of gravity, the linear velocity depends on
the linear momentum only, and the angular velocity on the angular momentum only, for
in this case "^{mx), "^(my), 2(m«) all vanish, and the equations become those, of which the

types are

u^ = 2(m) . Vj:,

h^ = 2(TOy^ + z'')w^ - 1,(mxy) . Wy - '2imzx)a)^.

22. Thus the motions of translation of the centre of gravity and of rotation about it are
independent, a property which is true of no other point. Also it is to be observed that
the direction of motion of the centre of gravity coincides with that of the linear momentum,
while that of the axis of angular velocity does not in general coincide with that of the angular
momentum. This is the cause of a greater complication in the problem of rotation than in
that of translation. In the former the passage from momentum to velocity involves the
changing of the direction of the axis as well as division by a quantity of the dimensions of
a moment of inertia, whose value depends on the position of the momental axis in the
system : in the latter the corresponding step involves simply division by a constant quantity,
the mass, without change of direction. If the operation by which the step is taken from
momentum to velocity, be considered as the measure of the inertia, we may express the above
by stating that the measure of the inertia of a system relatively to translation (the centre
of gravity* being the point of reference) is the mass of the system, and that the measure of
its inertia relatively to rotation is not a simple numerically expressible magnitude, but, in
Sir W. Hamilton's language, a quaternion, dependent on the position of the axis of angular
momentum or of that of angular velocity in the system.

23. Confining our attention henceforth to the problem of rotation, we must first obtain a
more distinct idea of the relation between the axes of angular momentum and velocity.
We may obtain this from our previous equations for h„ hy, h., in their general form ; but
more simply when we consider our axes as coincident with the principal axes through
the centre of gravity. If ^, B, C denote the moments of inertia about these axes,
the equations become (substituting 1, 2, 3 as subscripts for x, y, z respectively)

hi = Awi, h-z = Bb}2, A3 = Cw^ ;
hence the axis of angular momentum OH, whose equation is

x y ss
hi hz hs'
is parallel to the normal to the central ellipsoid

• It will be observed that, if the translation be referred to any other point than the centre of gravity, the measure of
inertia relatively to translation is also a quaternion.

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 13

^a?» + 5y2 + C«- = 1,
at the point, where the axis of angular velocity 01, whose equation is

ai y as

(Ol W2 ft)3

meets it. Also reciprocally 01 is parallel to the normal to the ellipsoid, whose equation is

x^ y' %^

at the point where OH meets it.

Thus a simple geometrical construction enables us to determine 07, when O^ is given,
and vice versa. If now w be the angular velocity about 01, and / the moment of inertia
about the same line, the angular momentum about it must be Iw, since w is the total angular
velocity, and therefore the angular velocity about a line perpendicular to 0/is zero; hence

lo) = h. cos HI,
an equation connecting h and w, the quantities / and HI being known when the above con-
struction has been made.

24. If h be constant, and its direction OH invariable, it is plain from the above con-
struction that 01 will not in general remain fixed, nor m constant, for, by the motion of the
system about 01, the position of OH in the system is altered, and to this new position of OH
a new position of 01 will correspond, and then &> will change by reason of the variation of

TT T

. There is an exception however in the case where OH and 01 coincide, for then the

rotation does not change the position of OH in the system : this can only be the case
when the radius 01 of the central ellipsoid is also a normal, that is, when it coincides with one
of the principal axes. Hence the principal axes are the only permanent axes of rotation of a
body acted on by no forces (as is implied in our supposition of h being constant) : in all
other cases the axis of rotation moves in the body and in space, and the angular velocity
about it varies. ,

25. If w be constant and its axis 01 fixed in the body, OH will also be fixed in the
body, and h will be constant ; but OH will then in general move in space, and the system
must therefore be acted on by forces, whose resultant couple has its axis perpendicular to OH
and in the plane of motion of OH. Hence the plane of the couple is HOI, if 01 be fixed in
space as well as in the body, and its moment is constant, since the velocity of OH is constant ;
thus the constraining couple on a body revolving uniformly about a fixed axis through its
centre of gravity is determined.

In the exceptional case of a principal axis, OH is also fixed in space, and there is no
constraining couple.

26. Before proceeding to the solution of the problem of a body's rotation about its
centre of gravity by a method more in accordance with the plan of this paper, it will be well
to shew how readily Euler's equations may be obtained from our principles.

\i

Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

If the moveable rectangular axes in § (15) be supposed fixed in the body and coincident
with the principal axes, we must substitute

o)], («2> '^3 ^o** ^xi ^j,» ^zi and hi, Ag, A3, or Awi, Bw^, Ccos for A^, Aj,, A^,
and then we obtain three equations, of which the type is, either

or A~=L +{B- C). «>,W3.
The latter is the well known form of Euler's equations.

27. Instead of employing these equations, let us endeavour to solve our problem more
directly. Our object is to determine the motion of 01, the axis of rotation, both in the
body and in space, and the variation of w, the angular velocity about it. This may be
conceived to be due to an angular acceleration of definite intensity about a definite line ; and
this may be regarded as compounded of two similar accelerations, the one arising from the
acceleration of momentum produced by the couple G about its axis OG, the other being the
angular acceleration which would exist if no forces acted. Now the forces in the elementary
time dt produce the angular momentum Gdt about OG, and this momentum gives rise to a
corresponding angular velocity Kdt about an axis OK related to OG, just as 0/ is OH: thus
the angular acceleration *c due to the forces is determined as to direction and intensity. The
other component of the angular acceleration is in like manner due to a cori'esponding accele-
ration of momentum, which it is now necessary to determine.

28. Regard any line OP fixed in the body and moving with it by reason of the velocity
10 about 01; and apply equation (C) of section I., putting A for m ; therefore

dt

= -hw . sin IH . sin HP . sin IHP,

which determines the acceleration of momentum for any line OP. This acceleration will be zero,
if OP be in the plane HOI, and a maximum, if OP be perpendicular to HOI, when its value is
hot] sin HI: we may therefore regard the total acceleration* (/) due to the motion of the body
as being about the line OF, perpendicular to HOI, and equal to + hw sin HI, when OF is
taken on that side of HOI on which a positive rotation about OF would move OH towards
01. Now to this acceleration of momentum (/) about OF will correspond an acceleration of
angular velocity (\) about a line OL which is related to OF, just as 0/is to OH.

29. To sum up our results, we have shewn that, if OH be the axis of angular momentum
(A) and 01 that radius of the central ellipsoid at whose extremity the normal is parallel to
OH, 01 is the axis of angular velocity (w) : if OG be the axis of the impressed couple (G),
and OK the radius for which the normal is parallel to OG, OK is the axis of angular accele-

• This result is that which M. Poinsot states thus : " The
axis of the couple due to the centrifugal forces is perpendicular
at once to the axis of rotation and to that of the ' couple d'impul-

sion.'" — M. Poinsot's "couple d'impulsion" is our angular
momentum.

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE.

15

ration due to the forces (/r) : lastly, if OF be perpendicular to the plane HOI, it is the axis
of acceleration of angular momentum in the moving body, and OL, the radius for which the
normal is parallel to OH, is the axis of angular acceleration due to the motion of the body (\).
Also we have the three equations for w, k, X,

lit) = h cos HI,

Kk = G cos GK,

L\ = / cos FL,
where y= hw sin HI,
I, K, L denoting the moments of inertia about 01, OK, OL respectively. It will be observed
that 01 is the direction, to which the plane through O perpendicular to OH is diametral, and
that OL is the direction to which the plane HOI is diametral, hence OL lies in the plane
perpendicular to OH. Also if the rectangular planes HOI, FOL intersect in OM, it will
be seen that the axes* 01, OL, OM axe conjugate diameters of the central ellipsoid.

30. We will develop the solution in the simpler case of OG coinciding with OH and
therefore OK with OL In this case OH remains fixed in space, and the motion of 01 is
conveniently referred to its motion in the plane HOI and the motion of that plane about OH.

AT

Let the conjugate radii 01, OL, OMhe denoted by r, r, r", then the moments of inertia

about them are -n -f^ , -rr^ > by the property of the central ellipsoid : also let the angles HOI,

FOL be denoted by 9, 6' : then our last equations become

(1) a) = Ar*cos0, (2) k = Gr^ cos ^, (S) \ = (Aw sin 0) . r'^ cos ^.
Resolve co, k, \ along the axes OH, OM, OF; the component velocities are then w cos 6
along OH, w sin 9 along OM, and zero along OF, while the component accelerations are
K cos 9 along OH, Ksm 9 +X sin 9' along OM, and X cos 9^ along OF ; whence, by applying
either the equation (C) or the equations (£),

d

dt

(w cos 9) =KCos9 = Gr^ cos'' 9, .

.(*)

* Hence if no forces act, the instantaneous motion of the axis of rotation OJ will be towards OL, the radius with
respect to which the plane HOI is diametral. '

16 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

— (w sin 0) = K sin + X sin ff = Gr* cos sin + {Im sin 0) . r'* cos 6' sin 0', (5)

o) sin . Q = Xcosfl' = (Aw sin d) .r'^cos^ff, (6)

where Q is the angular velocity of OM {i. e. of the plane HOI) about OH.

Also we have —r- = G (7)

at

Let p, p denote the perpendiculars from O on the tangent planes to the central ellipsoid
at /, L respectively, then p = r cos 9, p = r cos Q' .

Equation (4) becomes by (1) — (Ap^) = Gp% whence by (7), jo is constant. This shews

that the tangent plane at / to the central ellipsoid is fixed, and that the central ellipsoid
therefore rolls on it as a fixed^ plane.
Also by (4) and (5)

l(!^ = iff!L!!!L^U^' = Ay^tan0.tan0', (8)

dt dt \w cos 61 ft) cos d

and from (6) Q=V' (9)

31. Now r, /, r" being conjugate radii of the central ellipsoid, there exist three
relations between them and the conjugate axes ; these are, (putting p sec Q, p sec for r, r
respectively and denoting the angle lOL by y)

p" sec' d + p^ sec' Q' + r"' °=l + ;g + ?i = -^' suppose,

y/'2 + y V" + p'i)'' sec' Q sec' e' . sin' X = ;b^'^^"*"2b"'^' ^"PP°^^'

j,2p'2/'2 ^ ___ = G, suppose,

and by reason of the rectangularity of the planes lOM, LOM, we have i

cos j^ = sin sin Q'.
Eliminating r" and ^, we obtain

C

p' sec' 6 + /' sec' 9' + -^-^j = £,

G f-^ + 4) + py'(sec' + sec^ 9'-l)^F.
From these eliminating sec' 9', we obtain

which, (remembering what E, F, G denote, and putting a, /3, 7 for the three quantities

1 1 1 .• 1 X

, , , , ,1 ^ , 1 , 1 — — - respectively)

Ap" Bp^' Cp" ^ ^'

is equivalent to

p'2 = p2(i + a/37 cot' 9);

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 17

also, since p', ff are involved in precisely the same manner as p, 6, it follows that

where a, /3', 7' are what a, /3, 7 become, when p is put for p.
From these equations we obtain

cot' 0' = - -t;^,

but a' = 1 —7- = 1 -

a'/3'7' 1 + a/37 cot' 0'

1 1 _ 1 + /37 cot= e

If" "" ' ~ Af ■ r+^^7Co?y ~ " 1 + a^7 cot= '

whence, with the corresponding expressions for /3 , 7 ,

(1 4- c.i67 cot^ e)'

cot= Q =- cot^ ^ • (1 ^ ^^ cot-' Q){\ + 7a cot- 0)(I + a^ cot^ 0) '

hence p', Q' are known in terms of p, 0.

32. Substituting now for p , 6' in terms of p, 6, we obtain from equation (8)

d(cot 0) _ , ,2 cot
~d^ " ^ coit?

= ± A/^ - (1 +|87cot^0)(l +7acot^0)(l +a/3cot^0)|i, (10)

and from equation (9)

Q = hp^ (1 + a)37 cot- 6).

If A be known by means of (7), these two equations determine completely the motion of
01 the axis of angular velocity in altitude and azimuth, since p, and therefore a, j8, 7, are
constants.

If (b denote the azimuth at any instant, -j- = Q, and dividing the last equation by the
' at

preceding, we obtain a relation involving (p and 9 only, which will therefore be the differential

equation to the conical path of 01 in space ; and it is worth notice that, this relation being

independent of h, the path of 01 is the same whether the body be acted on by a couple whose

axis coincides with OH, or whether it be acted on by no forces. The effect of the couple in

this case is in fact only to alter the velocities of the different lines, not the paths which

they describe.

Also equation (l) gives to = hp^ sec 6, from which w is known when 6 is known by means

of equation (10), and thus the velocity about 01 is known completely as well as its position at

any time.

33. If there be no forces acting, i. e. if G = 0, h is constant, as is also o) cos 0, the re-
solved angular velocity of the body about OH. Also the vis viva of the body

w

-' r cos 9

and is therefore constant; and hence — is constant, or war; both well known results. It may

r

Vol. X. Part I. 3

ig Mb R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

vis viva

also be well to note that p* = -. — , even if G do not vanish, and therefore

(angular njomentum)^

that the vis viva cc (angular momentum)^, when the angular momentum has a fixed direction.

It is needless to carry the solution farther by investigating the path of 07 in the body, the

position of the principal axes relatively to Oil, 01 at any time, 8iC., since all these questions

are discussed with the utmost completeness and elegance in M. Poinsot's Theorie de la

Rotation.

34. We will conclude this paper by solving the problems of Foucault's Gyroscope as
applied to shew the effects of the earth's rotation, as it will furnish a good illustration of the
advantages of the methods of this paper in enabling us to form our equations immediately
with respect to the most convenient axes.

The Gyroscope is essentially a body, whose central ellipsoid is an oblate spheroid by
reason of its two lesser principal moments being equal, and which is capable of moving freely
about its centre of gravity. In this case, if a rapid rotation be communicated to it about
its axis of unequal moment, that axis will evidently retain a fixed direction in space however
the centre of gravity move, and therefore relatively to a place on the surface of the earth will
alter its position just like a telescope, whose axis is always directed to the same star. But
there are two other remarkable cases, where the motion about the centre of gravity is partially
constrained ; the first, where the axis of rotation is compelled to remain in the plane of the
meridian, the second, when it is compelled to remain in the horizontal plane. These we will
now consider.

35. When the polar axis of the central spheroid always lies in the plane of the meridian,
let denote the north polar distance of its extremity A. Let OB coincide with the equato-
rial axis in the plane of the meridian, and OC with that perpendicular to the same plane, and
refer the motion to the axes OA, OB, OC. Now if Q denote the angular velocity of the earth

d9
about its axis, the motions of OA, OB, OC will be due to the velocities Q cos 0, Q sin 6, —

at

about them respectively : also the actual velocities of the body about the same axes are

respectively w, Q, sin d, — -, and the consequent angular momenta Aui,B\lsia d,B-—, where to, — ,
at at at

are reckoned positive when the motion about their axes is in the same direction as the earth's

about its axis.

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 19

It is evident that in this case the constraint is equivalent to a couple, whose axis coincides
with OB, let this be denoted by G. Then the equations (£) in the first section applied to the
case before us give

— (Jw) = 5 — . Q sin - 5Q sin . — = 0,
dt^ ^ dt dt

-(BQs\ne) = G + Aw.-'-B—.Qcose,

at dt dt

t

-{B — ] =5Qsia0.12cos0-^ft,.Qsin0;
dt\. dt)

from the first equation, w is constant, and from the last

d"e (A

«c' (A A ■ ^

-— = — — a> - Q cos y Q sin ;
dt' \B )

now in this case Q the velocity of the earth's rotation is very small compared with w, neglecting
therefore the second term of this equation,

— - = — — (oQ sm Q,
df B '

whence the motion of the axis OA is precisely similar to that of the circular pendulum, whose

S A Bs

length is I, where - = — wO, and therefore 1= — , the direction of the earth's axis taking the

I B AtoQ,

place of the direction of the force of gravity.

Also since-— = 0, when sin = 0, there are two positions of equilibrium of the axis OA,

namely, when 9 = 0, and 6 = ir: the former is stable and the latter unstable, when w and Q
have the same sign. Hence the axis of rotation will remain at rest, if originally placed in the
direction of the earth's axis, stably or unstably according as the rotation regarded from the end
directed to the north pole is in the same direction, or the contrary, with the earth's rotation re-
garded from the same pole. If placed originally in any other position, it will oscillate about
its position of stable equilibrium according to the same laws as a circular pendulum.

36. Next, let the polar axis OA always remain in the horizontal plane, and let (p denote
its azimuth from the south towards the east. Taking OB and OC as before, the latter will
now coincide with the vertical.

If c denote the co-latitude, Q may be resolved into Q cos c vertical and Q sin c horizontal
in the north direction : hence the angular velocities by which the axes move, are relatively
to OA, OB, OC respectively

. . d(b

— Q sin c cos 0, - Q sin c sin 0, -^ + Q cos c,

etc

and the corresponding angular momenta are

Am, - BQ sin c sin 0, 5 ( -^ + Q cos c j ,

3 — 2

so Mr R. B. HAYWARD, ON A DIRECT METHOD, &c.

whence as before,

^ (- jBQ sin c sin 0) = G + Aw i~y + cose) + SI -^ + Q coscj . Q sin c cos (p,

d ( dd) \ . . . . .

— I S — i- + cos c 1 = BQ. sm c sin (^ . Q sin c cos + Joj . Q sin c sin (p,

and therefore w is constant, and

d?<t) A . . . , .

-TT = t; wQ sin c sm + Q^ sin- c . sin cos 0,

dfi B ^ ^ '

or approximately

-^ = - «Q sm c . sin ;

whence, the rotation about OA being in the same direction seen from A as that of the earth

seen from the north pole, it will be in a position of stable equilibrium when directed to the

north, and of unstable equilibrium in the opposite position : also if originally directed in any

other direction, it will oscillate about its position of stable equilibrium like a circular pendulum

Bg

about the vertical whose length is ; ,

AwQ sin c

Durham, Feb. 19, 1856. H. B. H.

II. On the question. What is the Solution of a Dijirential Equation f A Sup^de-
ment to the third section of a paper, On some points of the Integral Calculus,
printed in Vol. IX. Part II. By Augustus De Morgan, of Trinity College,
Vice-President of the Royal Astronomical Society, and Professor of Ma-
thematics in University College, London.

[Read April 28, 1856.]

Trusting that it will be sufficient excuse for a very elementary paper, that writers of the
highest character are not agreed with each other on a very elementary point, I beg to offer
some remarks upon the usual solution of such an equation as dy'^ — a^dar' = 0, to which Euler
assigns the integral form (y - aa> + b) {y + ax + c) = 0, where b and c are independent
constants. Most other writers insist on the condition b = c.

Lacroix refers only to Euler and to a paper by D'Alembert (Berl. Mem. 1748) which I
have not seen. All the reasons which have been given on the subject are reducible, so far as
I have met with them, to those which I shall cite from Lacroix himself and from Cauchy.

Lacroix (ii. 280) in his explanation of this case, and in defence of the substitution of
(y - a.v + b) (y +ax + b) for (y - ax + b) (y + ax + c), makes two remarks. The first, —
chacun de ses facteurs doit etre considere isolement; the second, alluding to the form with two
constants, is — on n'en tire pas d'autres lignes que celles qui resulteraient de Tintegrale renferm-
ant une seule constante. M. Cauchy (Moigno, ii. 456) says — On ne restreindra pas la generalite
de cette integrale en designant toutes les constantes arbitraires par la meme lettre... : and
grounds the right to do this on the possibility of thus obtaining all the curves which can satisfy
the equation.

In searching out this matter, I found it by no means clearly laid down what is meant by
the solution of a differential equation : and, on looking further, I found some degree of ambi-
guity attaching to the word equation itself. The following remarks will sufficiently explain
what I mean.

A connexion between the values of letters, by which one is inevitably determined when the
rest are given, may be called a relation. But an equation is the assertion of the equality of
two expressions. Every simple explicit relation leads to an equation, to one equation : but every
equation does not imply only one relation. The object of the pi'oblem being relation between
y and x, the equation {y — x) (y — x^) = implies power of choice between the relations y = x^,
y '^ w. The equation {y - x') (a; - 1) = implies the relation y = od' with a dispensation from
all relation in the case of a; = 1.

Now I assert that in mathematical writings confusion between the equation and the simple
relation is by no means infrequent : without dwelling on instances, I think we shall find, by

22 Mk DE morgan, ON THE QUESTION,

examining approved modes of reasoning, that the confusion cannot but be seen to have existed,
so soon as the statement of what it consists in is made.

It is affirmed that the primitive of a primordinal equation cannot have two arbitrary
constants : but all that can be proved is that no such differential equation can have two related
arbitrary constants in its primitive.

Let /(a;, y, y') = involve any number of relations between at, y, y : and let (p{x, y, a,b) =
be a relation between a and b, or any number of relations. Consequently, selecting one relation
by which to satisfy (p = 0, values of a and b can be found to satisfy both (p(x, y, a, h) — 0, and
also (p(x + h,y + k, a,b) = 0, for any values of x, y, h, k. Hence, for any values of x and y,
y' may have any value whatever: and this is incompatible with /(a;, «/, y') =0. But this is no
argument against any form of (p{;v,y, a, b,) = 0, in which the constants are not in relation ; as
\^(.r, y, a) . ^(<r, y, b) = 0, For we cannot pretend to satisfy

\l/{x,y, a) . ^{x, y, b) = 0, %|,(.» + h,y + k, a) .^i^ + h,y + k, b) = 0,

for any values of w, y, h, k, except by >|/(cr, y, a) = 0, and ^(cT? + h, y + k, b) = 0, or else by
\^(.r + h, y + k, a) = 0, ^(x, y, b) = 0. And from neither set can we deduce?/'. If >|/(<if, y, a) =
be a primitive oi f{x,y,y') — 0, there appears nothing a priori to prevent our saying that
•y|/(.r, y, a) . y]/{x, y, b) = h a primitive. This point will be presently examined.

It is affirmed that a primordinal differential equation cannot have two really different
primitives with an arbitrary constant in each : but all that can be proved is that one prim-
ordinal relation cannot have two distinct primitives. If y'=/{x,y) be satisfied by different
relations <p[x, y, a) = 0, \|/(a;, y, b) = 0, then, taking a and b so as to satisfy both at a given
point (*,«/), we find, generally, two values of y at {x,y). But y'=f{x,y) may give these two
values; irreducibly connected, as in y' = 1 ± y/y, or reducibly, as in y' = 1 ± -y/y^ The great
point of algebraical interest, namely, that when the two values of y are irreducibly connected
(f) = and \|/ = are the alternatives of an equation which can be rationalised or otherwise
inverted into % = 0, where ^ is of univocal form, is foreign to the present purpose. That
purpose is, to make it clear that the common theorems about the singularity of the constant
of integration must be transferred from differential equations to differential relations, of which
one equation may contain any number.

The question whether y = or, which is certainly one relation for determination of y from
x, is to be considered as giving one or two relations for determination of x from y, ends in a
question of definition, perhaps, but ends in a question which cannot be adequately treated
without a close attention to the meaning of the word continuity. And here immediately arises
the distinction of permanence of form and continuity of value.

Form is expression of modus operandi: and permanence of form implies and is implied in
permanence of the modus operandi through all values of the quantities to be operated on.
In arithmetic, the signs + or — are of the form, and not of the value : but in algebra, the +
or — which the ^ei^er carries in its signification are of the value, so called. Accordingly,
permanence of form does not necessarily give continuity of value. The immediate passage of

00

I smxv.v'^dv from + Itt to — iT, as x passes through 0, might be discovered by the

•0

WHAT IS THE SOLUTION OF A DIFFERENTIAL EQUATION? ^

arithmetical computer, utterly ignorant of the Integral Calculus, by use of skeleton forms set
up from one form of type. Nor does discontinuity of form necessarily give discontinuity of

value. The branch of y = x which ends at x =0 joins the branch o? y = x + e"" which
begins at x = with a contact of the order co . as order of contact is usually defined. We
may even propound the question whether (— x)"- and (+ .r)^ be not different forms ?

Let continuity of no order, or non-ordinal continuity, be when and so long as infinitely
small accessions to the variable give infinitely small accessions to the function. And let the
passage from ± eo to =f co be counted under this term. I will not, on this point, give more
than an expression of my conviction that the word continuity must, by that dictation which has
turned unity into a number, and its factor into a multiplier, be extended to contain the usual
passage through infinity. Let w-ordinal continuity be when and so long as y, y ,y" ,,..y^"^ are of
non-ordinal continuity.

These definitions being premised, we have in the passage from the positive to the negative
value of <i-^ an interminable continuity, and a change of form answering to, and indeed derived
from, the change of form seen in (+ x)- and (- x)^. We have, in truth, all the quantitative
properties of one relation, and all the formal properties of two. The attainment of a reducible
case is the loss of the quantitative properties also : thus {jxr + d)^ is non-ordinally continuous,
and not so much as primordinally, when a = 0.

We are now in a condition to answer the question. What is the solution of a differential
equation ? — at least so far as having a clear view of the imperfect manner in which the
question is put. We are obliged to ask in return, what requirements as to continuity are
conveyed in the word solution ?

1. The word solution may require the most absolute notion of permanence of form, not
granting even the passage from { — xf to ( + xf. In this case we must be compelled to satisfy
the differential equation by a relation of permanence equally strict, and in so many ways as we
can do this, in so many ways can we announce a solution. Thus to y'^ = 2^«/ . y we announce
three solutions. To y = 0, any parallel to the axis of x. To y' = 2 x the positive value of -^/y,
the right hand branch, from x = a onwards, as figures are usually drawn, of any parabola
y = (,v — ay. To y = 2 X the negative value of -^'y, the left hand branch of the same
up to X = a. The change from any one of these to any other is entirely forbidden : and a
must be less in one case, and greater in the other, than any value of x which is to be employed.
Problems are frequently stated in a manner which will admit only one branch of an ordinary
solution : and the investigator, so soon as this is perceived, generally widens his enunciation,
rather than narrow his notion of a solution.

2. In a solution we may allow only such changes of form as take place in the inversions
of ordinary algebra, and no others. In this case we should say, that we have y = a and
y = (x — by, which we please, but only one, for the solution of y"' = ^\/y . y ■ In this case'
and the last we satisfy Lacroix's requirement that the factors must be considered in isolation :
but it is not correct to imply that such isolation is part of the meaning of a compound relation.
From PQ = o we only learn that one of the two factors is to vanish : the equation has no
power to deny us the use of one factor for some values of x, and of the other factor for others.
The isolation of the factors is the postulation of a certain permanence of form.

24 mh de morgan, on the question,

3. In a solution we may allow change of form, with a given kind of continuity at the
junction. If we mean to stipulate nothing whatever about continuity, we may at any value of
X leave one curve, and proceed upon another. If we require non-ordinal continuity, we can
only do this where two curves join each other. If we require ordinal continuity or continuity
of the same order as the equation, we may propound as a solution of y' = ^\/y any number
of parabolas with as much of the singular solution y = as lies between their vertices. If we
require every degree of continuity, we have, in the case before us, what is tantamount to
requiring permanence of form, in its ordinary sense.

No prepossession derived from ordinary algebra would be offended by a solution which has
a continuity of no higher order than the order of the equation itself: which would allow us,
on arriving at the singular solution, or connecting curve, to break off from the curve thitherto
employed, to proceed along any arc of the connecting curve, and to abandon this last at any
chosen point in favour of the ordinary solution which there touches it.

In the graphical method by which the possibility of a solution is established, that is, by
construction of a polygon from Ay = ;^(.r, y) . A.r, with a very small value of Aw, which may
be as small as we please in the reasoning, a solution of y = xi^' V) '® shewn to exist : but it
may be one of the kind just alluded to. The draughtsman employed to construct such a
solution, when his arc of the ordinary curve comes very near the point of contact with the
singular solution, cannot undertake to remain on that ordinary curve, without reference to
quantities of the second order. The accidents of paper and pencil are casualties of this order,
which might divert his arc of solution from the ordinary curve on to the singular solution,
might keep it there for a while, and then throw it off upon another ordinary solution. In fact,
the solution established a priori has not of necessity permanence of form, but has only
continuity of the order of the equation. And this remark applies to equations of all orders.
In the case of y = 2y/y, when once a side of the polygon ends on y = 0, the draughtsman can
never leave that line again, without constructing one side by help of Ay = (Aaty.

It may now be affirmed that (y — ax + b) (y + ax + c) = 0, b and c being perfectly
independent constants, is a solution of y'^ — a" = ; nothing in the general theory of the
primordinal differential relation in any way withstanding. It remains to examine the assertion
that the generality of this solution is not restricted by the supposition b = c.

To a certain extent this assertion is true: no more curves are obtained or included before
the limitation than after it. Beyond this point the assertion is not true. The condition b = c
belongs to one mode of grouping a solution of y = a with a solution of y = —a: but there
is an infinite number of modes in b = (pc. If ordinal continuity be held sufficient, and if
<p{x, y, b) = 0, yp{x, y,c)=0 be independent relations satisfying f(x, y, y) = 0, and if P = be
the most complete singular solution, then

P.(p{x,y,b^) . (p{x,y,b,) x|/(.r,y, cj . x/^Or, y, Cj),., =0

is the most general solution, where 6j, b2,...Ci, c^,.-. are in any number, and of any values.
This however is but equivalent to P . (p{x, y, b) . \^(a7, y, c) = with the usual addition 'for any
values whatever of b and c'.

This point will be best illustrated by reference to the biordinal equation and its theory.
A primordinal equation belongs to a group or family of curves which may be called of single

WHAT IS THE SOLUTION OF A DIFFERENTIAL EQUATION? 25

entry: a biordinal equation to a group of double entry, out of which an infinite number of
groups of single entry may be collected. Thus, b and c being in relation in <p(a!, y, b, c) = 0,
we may designate all the curves contained in ip{a;,y,fc,c) = as the group (fc,c). Generally
speaking, the curves of the group (fc,c) are different from those of (Fc, c). The unlimited
number of cases of (fc, c) is the key to the unlimited number of primordinal equations which
give rise to one and the same biordinal equation. It is then the characteristic of the biordinal
equation that it represents a group of double entry. When the constants are not in relation,
as in (h{x,y, b) . \//(a?, y, c) = 0, we have still groups of double entry, but the biordinal equation
ceases to exist : the distinction between one group and another consists in the distinct ways in
which individuals of the two groups ^ = and \|/ = are joined together. This defective
grouping — not defective in the variety of its cases, but defective in the variety of the elements
out of which cases are to be compounded — is within the compass of a primordinal equation,
into which therefore the biordinal equation degenerates.

As an instance, let (P - b) {Q - c) + R = 0, P, Q, R, being each a function of ai and y :
and let P' represent P^ + P^. y, &c.

When b = fc, the primordinal equation of the group (/c, c) is

^ R' + -^(R" + 4P'Q:R) a R'--y/{R^= + 4.PQ'R)[

y+ r^T — =J{P +

2P' •'\ "^ 2Q' /■

Let R = fxV, where F is a finite function, and fx a constant. When /x diminishes without
limit, and finally vanishes, each primordinal equation becomes either P' = or Q' = 0, for
otherwise we have only Q = fP, the algebraic result of eliminating c between (P — 6) (Q — c) = 0,
and (P —fc) Q' + {Q - c) P' = 0. And the biordinal equation is determined by differentiating

, ^ r'+^(r'^ + 4P'q:r)

6 = Q + y^^ -.

Do this fully, clear the result of fractions, and write /xV for R : it will then appear that y" is
seen only in terms multiplied by positive powers of ^t ; and so that n = gives P'Q' = in place
of a biordinal equation.

The correction which the common theory requires is as follows ; — An equation in which n
constants are in relation with a? and y, cannot have any differential equation clear of those
constants under the wth order ; and an equation of single and irreducible relation between
w,y,y ,...y^''^ must have a primitive containing n constants in relation to ir and y. But a
primitive equation in which n constants are contained in alternative relations, Mj in one relation,
w in a second, &c. does not require a differential equation of the nth order ; but has an equation
of alternative relations, one of the Wjth order, one of the n^th order, &c.

From a primitive having n constants, in relation with x and y, no constants can be
eliminated in favour of y', y", &c without one new equation of differentiation for every constant
which is to disappear. But this is by no means true of constants in relation with x,y, and one
or more of the set y',y",..., to begin with. This point is made clear enough in the section of
my former paper to which these remarks form a supplement : but the whole may be illustrated
as follows. If (p(at, y,a) =0 give a = (^{x, y,), and therefore 4>j + <I>y . y' = for a differential
equation, in which a has disappeared and y is introduced, it is easy to give this differential
Vol. X. Part I. 4

86 Ma DE MORGAN, ON THE QUESTION, WHAT IS THE SOLUTION &c.

equation a primitive containing any number of separate and independent constants. For
Aa + Ai <!>(>■», y) + A2 {^G^)^)}'' + ... = cannot give any relation in which one of these constants
disappears in favour of y except ^j + ^„ . y' = 0, in which they all disappear. But this is
merely formal ; for Ao + Ai <l>(a?,j/) + ... = is but a transformation of some case of <I>(.r, y) =
f{Afi,A^,...) or of (p[ic,y,f{Af„Ai,...)^ =0. All we have done, then, amounts to no more
than use of the obvious theorem that a single arbitrary constant is equivalent to an arbitrary
function of as many arbitrary constants as we please. Moreover, we may prove that P + y'
can only be a factor in the differential of one class of forms. If \F(.v,y)Y give M{P + y),
nothing but {\f/F(a!, y)\' can give N(P + y) : and F{ar, y) — const, and \l/F{ai,y) = const, are
the same equations.

But it is otherwise with P + y' , P being a function of x,y, y. This occurs, as previously
shewn, in the differentiations of two distinct classes of forms. Thus + y" is a factor in
\f{aiy' — y)Y ssiA \n \Fy'\'. The equation

f{xy' -y) = Ao+A, Fy' + A, {Fy']' + ...
is one which contains in every sense, formal and quantitative, as many arbitrary constants as we
please; and an alteration in the value of one of them, is an alteration in the character of the
relation subsisting between wy — y and y'. Nevertheless, it is impossible to get rid of any one
constant in favour of y" in any way except one which results in y" = 0, an equation from
which all the constants have disappeared.

Considerations similar to those which have been applied to primordinal equations might
also be applied to equations of any order,

A. DE MORGAN.

Univeesity College, London,
March 29, 1856.

III. On Faraday s Lines of Force. By J. Clerk Maxwell, B.A. Fellow of

Trinity College, Cambridge.

[Read Dec. 10, 1855, and Feb. U, 1856.]

The present state of electrical science seems peculiarly unfavourable to speculation. The
laws of the distribution of electricity on the surface of conductors have been analytically
deduced from experiment; some parts of the mathematical theory of magnetism are esta-
blished, while in other parts the experimental data are wanting ; the theory of the con-
duction of galvanism and that of the mutual attraction of conductors have been reduced
to mathematical formulae, but have not fallen into relation with the other parts of the
science. No electrical theory can now be put forth, unless it shews the connexion not
only between electricity at rest and current electricity, but between the attractions and
inductive effects of electricity in both states. Such a theory must accurately satisfy those
laws, the mathematical form of which is known, and must afford the means of calculating the
effects in the limiting cases where the known formulae are inapplicable. In order therefore to
appreciate the requirements of the science, the student must make himself familiar with a
considerable body of most intricate mathematics, the mere retention of which in the memory
materially interferes with further progress. The first process therefore in the effectual study of
the science, must be one of simplification and reduction of the results of previous investiga-
tion to a form in which the mind can grasp them. The results of this simplification may take
the form of a purely mathematical formula or of a physical hypothesis. In the first case we
entirely lose sight of tlie phenomena to be explained; and though we may trace out the
consequences of given laws, we can never obtain more extended views of the connexions of
the subject. If, on the other hand, we adopt a physical hypothesis, we see the phenomena only
through a medium, and are liable to that blindness to facts and rashness in assumption which
a partial explanation encourages. We must therefore discover some method of investigation
which allows the mind at every step to lay hold of a clear physical conception, without being
committed to any theory founded on the physical science from which that conception is
borrowed, so that it is neither drawn aside from the subject in pursuit of analytical subtleties,
nor carried beyond the truth by a favourite hypothesis.

4 — 2

28 Mr maxwell, ON FARADAY'S LINES OF FORCE.

In order to obtain physical ideas without adopting a physical theory we must make our-
selves familiar with the existence of physical analogies. By a physical analogy I mean that
partial similarity between the laws of one science and those of another which makes each of
them illustrate the other. Thus all the mathematical sciences are founded on relations between
physical laws and laws of numbers, so that the aim of exact science is to reduce the problems
of nature to the determination of quantities by operations with numbers. Passing from the
most universal of all analogies to a very partial one, we find the same resemblance in
mathematical form between two different phenomena giving rise to a physical theory of light.

The changes of direction which light undergoes in passing from one medium to another,
are identical with the deviations of the path of a particle in moving through a narrow space
in which intense forces act. This analogy, which extends only to the direction, and not to the
velocity of motion, was long believed to be the true explanation of the refraction of light; and
we still find it useful in the solution of certain problems, in which we employ it without danger,
as an artificial method. The other analogy, between light and the vibrations of an elastic
medium, extends much farther, but, though its importance and fruitfulness cannot be over-
estimated, we must recollect that it is founded only on a resemblance inform between the laws
of light and those of vibrations. By stripping it of its physical dress and reducing it to
a theory of " transverse alternations," we might obtain a system of truth strictly founded on
observation, but probably deficient both in the vividness of its conceptions and the fertility of
its method. I have said thus much on the disputed questions of Optics, as a preparation
for the discussion of the almost universally admitted theory of attraction at a distance.

We have all acquired the mathematical conception of these attractions. We can reason
about them and determine their appropriate forms or foimulte. These formula} have a
distinct mathematical .signifitance, and their results are found to be in accordance with natural
phenomena. There is no formula in applied mathematics more consistent with nature than
the formula of attractions, and no theory better established in the minds of men than that of
the action of bodies on one another at a distance. The laws of the conduction of heat in
uniform media appear at first sight among the most different in their physical relations from
those relating to attractions. The quantities which enter into them are temperature, flow of
heat, conductivity. The word /orce is foreign to the subject. Yet we find that the mathe-
matical laws of the uniform motion of heat in homogeneous media are identical in form with
those of attractions varying inversely as the square of the distance. We have only to substitute
source of heat for centre of attraction, flow of heat for accelerating effect of attraction at any
point, and temperature for potential, and the solution of a problem in attractions is transformed
into that of a problem in heat.

This analogy between the formulae of heat and attraction was, I believe, first pointed out
by Professor William Thomson in the Cambridge Math. Journal, Vol. III.

Now the conduction of heat is supposed to proceed by an action between contiguous
parts of a medium, while the force of attraction is a relation between distant bodies, and
yet, if we knew nothing more than is expressed in the mathematical formulae, there would
be nothing to distinguish between the one set of phenomena and the other.

Mr. maxwell, ON FARADAY'S LINES OP FORCE. 29

It is true, that if we introduce other considerations and observe additional facts, the two
subjects will assume very different aspects, but the mathematical resemblance of some of
their laws will remain, and may still be made useful in exciting appropriate mathematical
ideas.

It is by the use of analogies of this kind that I have attempted to bring before the
mind, in' a convenient and manageable form, those mathematical ideas which are necessary
to the study of the phenomena of electricity. The methods are generally those suggested
by the processes of reasoning which are found in the researches of Faraday *, and which,
though they have been interpreted mathematically by Prof. Thomson and others, are very
generally supposed to be of an indefinite and unmathematical character, when compared with
those employed by the professed mathematicians. By the method which I adopt, I hope
to render it evident that I am not attempting to establish any physical theory of a science
in which I have hardly made a single experiment, and that the limit of my design is to
shew how, by a strict application of the ideas and methods of Faraday, the connexion of
the very different orders of phenomena which he has discovered may be clearly placed before
the mathematical mind. I shall therefore avoid as much as I can the introduction of anything
which does not serve as a direct illustration of Faraday's methods, or of the mathematical
deductions which may be made from them. In treating the simpler parts of the subject
I shall use Faraday's mathematical methods as well as his ideas. When the complexity of the
subject requires it, I shall use analytical notation, still confining myself to the development
of ideas originated by the same philosopher.

I have in the first place to explain and illustrate the idea of "lines of force."

When a body is electrified in any manner, a small body charged with positive electricity,
and placed in any given position, will experience a force urging it in. a certain direction.
If the small body be now negatively electrified, it will be urged by an equal force in a
direction exactly opposite.

The same relations hold between a magnetic body and the north or south poles of a
small magnet. If the north pole is urged in one direction, the south pole is urged in
the opposite direction.

In this way we might find a line passing through any point of space, such that it represents
the direction of the force acting on a positively electrified particle, or on an elementary north
pole, and the reverse direction of the force on a negatively electrified particle or an elementary
south pole. Since at every point of space such a direction may be found, if we commence
at any point and draw a line so that, as we go along it, its direction at any point shall
always coincide with that of the resultant force at that point, this curve will indicate the
direction of that force for every point through which it passes, and might be called on that
account a line of force. We might in the same way draw other lines of force, till we had
filled all space with curves indicating by their direction that of the force at any assigned
point.

• See especiaUy Series XXXVIII. of the Experimental Researches, and Phil. Mag. 1852.

so Mr. maxwell, ON FARADAY'S LINES OF FORCE.

We should thus obtain a geometrical model of the physical phenomena, which would
tell us the direction of the force, but we should still require some method of indicating
the intensity of the force at any point. If we consider these curves not as mere lines,
but as fine tubes of variable section carrying an incompressible fluid, then, since the ve-
locity of the fluid is inversely as the section of the tube, We may make the velocity vary
according to any given law, by regulating the section of the tube, and in this way we might
represent the intensity of the force as well as its direction by the motion of the fluid in these
tubes. This method of representing the intensity of a force by the velocity of an imaginary
fluid in a tube is applicable to any conceivable system of forces, but it is capable of great
simplification in the case in which the forces are such as can be explained by the hypothesis
of attractions varying inversely as the square of the distance, such as those observed in elec-
trical and magnetic phenomena. In the case of a perfectly arbitrary system of forces, there
will generally be interstices between the tubes; but in the case of electric and magnetic forces
it is possible to arrange the tubes so as to leave no interstices. The tubes will then be mere
surfaces, directing the motion of a fluid filling up the whole space. It has been usual to
commence the investigation of the laws of these forces by at once assuming that the phenomena
are due to attractive or repulsive forces acting between certain points. We may however
obtain a different view of the subject, and one more suited to our more difficult inquiries,
by adopting for the definition of the forces of which we treat, that they may be represented in
magnitude and direction by the uniform motion of an incompressible fluid.

I propose, then, first to describe a method by which the motion of such a fluid
can be clearly conceived ; secondly to trace the consequences of assuming certain conditions
of motion, and to point out the application of the method to some of the less complicated
phenomena of electricity, magnetism, and galvanism ; and lastly to shew how by an extension
of these methods, and the introduction of another idea due to Faraday, the laws of the
attractions and inductive actions of magnets and currents may be clearly conceived, without
making any assumptions as to the physical nature of electricity, or adding anything to
that which has been already proved by experiment.

By referring everything to the purely geometrical idea of the motion of an imaginary
fluid, I hope to attain generality and precision, and to avoid the dangers arising from a
premature theory professing to explain the cause of the phenomena. If the results of
mere speculation which I have collected are found to be of any use to experimental
philosophers, in arranging and interpreting their results, they will have served their purpose,
and a mature theory, in which physical facts will be physically explained, will be formed
by those who by interrogating Nature herself can obtain the only true solution of the
questions which the mathematical theory suggests.

1. Theory of the Motion of an incompressible Fluid.

(1) The substance here treated of must not be assumed to possess any of the properties
of ordinary fluids except those of freedom of motion and resistance to compression. It is not

Me maxwell, ON FARADAY'S LINES OF FORCE. : 31

even a hypothetical fluid which is introduced to explain actual phenomena. It is merely a
collection of imaginary properties which may be employed for establishing certain theorems in
pure mathematics in a way more intelligible to many minds and more applicable to physical
problems than that in which algebraic symbols alone are used. The use of the word " Fluid"
will not lead us into error, if we remember that it denotes a purely imaginary substance with
the following property :

The portion ofjluid which at any instant occupied a given volume, will at any succeed-
ing instant occupy an equal volume.

This law expresses the incompressibility of the fluid, and furnishes us with a convenient
measure of its quantity, namely its volume. The unit of quantity of the fluid will therefore
be the unit of volume.

(2) The direction of motion of the fluid will in general be different at different points of
the space which it occupies, but since the direction is determinate for every such point, we
may conceive a line to begin at any point and to be continued so that every element of the line
indicates by its direction the direction of motion at that point of space. Lines drawn in such
a manner that their direction always indicates the direction of fluid motion are called lines of

jluid motion.

If the motion of the fluid be what is called steady motion, that is, if the direction and
velocity of the motion at any fixed point be independent of the time, these curves will repre-
sent the paths of individual particles of the fluid, but if the motion be variable this will not
generally be the case. The cases of motion which will come under our notice will be those of
steady motion.

(3) If upon any surface which cuts the lines of fluid motion we draw a closed curve,
and if from every point of this curve we draw a line of motion, these lines of motion will
generate a tubular surface which we may call a tube of fluid motion. Since this surface is.
generated by lines in the direction of fluid motion no part of the fluid can flow across it, so
that this imaginary surface is as impermeable to the fluid as a real tube.

(4) The quantity of fluid which in unit of time crosses any fixed section of the tube is
the same at whatever part of the tube the section be taken. For the fluid is incompressible,
and no part runs through the sides of the tube, therefore the quantity which escapes from
the second section is equal to that which enters through the first.

If the tube be such that unit of volume passes through any section in unit of time it is
called a unit tube ofjluid motion.

(5) In what follows, various units will be referred to, and a finite number of lines or
surfaces will be drawn, representing in terms of those units the motion of the fluid. Now
in order to define the motion in every part of the fluid, an infinite number of lines would have
to be drawn at indefinitely small intervals ; but since the description of such a system of lines
would involve continual reference to the theory of limits, it has been thought better to suppose

82 Mb maxwell, ON FARADAY'S LINES OF FORCE,

the lines drawn at intervals depending on the assumed unit, and afterwards to assume the unit
as small as we please by taking a small submultiple of the standard unit.

(6) To define the motion of the whole fluid by means of a system of unit tubes.

Take any fixed surface which cuts all the lines of fluid motion, and draw upon it any
system of curves not intersecting one another. On the same surface draw a second system of
curves intersecting the first system, and so arranged that the quantity of fluid which crosses
the surface within each of the quadrilaterals formed by the intersection of the two systems
of curves shall be unity in unit of time. From every point in a curve of the first system let
a line of fluid motion be drawn. These lines will form a surface through which no fluid
passes. Similar impermeable surfaces may be drawn for all the curves of the first system.
The curves of the second system will give rise to a second system of impermeable surfaces,
which, by their intersection with the first system, will form quadrilateral tubes, which will be
tubes of fluid motion. Since each quadrilateral of the cutting surface transmits unity of fluid
in unity of time, every tube in the system will transmit unity of fluid through any of its
sections in unit of time. The motion of the fluid at every part of the space it occupies is
determined by this system of unit tubes ; for the direction of motion is that of the tube
through the point in question, and the velocity is the reciprocal of the area of the section
of the unit tube at that point.

(7) We have now obtained a geometrical construction which completely defines the
motion of the fluid by dividing the space it occupies into a system of unit tubes. We have
next to shew how by means of these tubes we may ascertain various points relating to the
motion of the fluid.

A unit tube may either return into itself, or may begin and end at diff'erent points, and
these may be either in the boundary of the space in which we investigate the motion, or within
that space. In the first case there is a continual circulation of fluid in the tube, in the
second the fluid enters at one end and flows out at the other. If the extremities of the tube
are in the bounding surface, the fluid may be supposed to be continually supplied from without
from an unknown source, and to flow out at the other into an unknown reservoir ; but if the
origin of the tube or its termination be within the space under consideration, then we must
conceive the fluid to be supplied by a source within that space, capable of creating and emit-
ting unity of fluid in unity of time, and to be afterwards swallowed up by a sink capable of
receiving and destroying the same amount continually.

There is nothing self-contradictory in the conception of these sources where the fluid is
created, and sinks where it is annihilated. The properties of the fluid are at our disposal, we
have made it incompressible, and now we suppose it produced from nothing at certain points
and reduced to nothing at others. The places of production will be called sources, and their
numerical value will be the number of units of fluid which they produce in unit of time. The
places of reduction will, for want of a better name, be called sinks, and will be estimated by the
number of units of fluid absorbed in unit of time. Both places will sometimes be called
sources, a source being understood to be a sink when its sign is negative.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 33

(8) It is evident that the amount of fluid which passes any fixed surface is measured by
the number of unit tubes which cut it, and the direction in which the fluid passes is determined
by that of its motion in the tubes. If the surface be a closed one, then any tube whose ter-
minations lie on the same side of the surface must cross the surface as many times in the one
direction as in the other, and therefore must carry as much fluid out of the surface as it
carries in. A tube which begins within the surface and ends without it will carry out unity
of fluid ; and one which enters the surface and terminates within it will carry in the same
quantity. In order therefore to estimate the amount of fluid which flows out of the closed
surface, we must subtract the number of tubes which end within the surface from the number
of tubes which begin there. If the result is negative the fluid will on the whole flow inwards.

If we call the beginning of a unit tube a unit source, and its termination a unit sink, then
the quantity of fluid produced within the surface is estimated by the number of unit sources
minus the number of unit sinks, and this must flow out of the surface on account of the
incompressibility of the fluid.

In speaking of these unit tubes, sources and sinks, we must remember what was stated in
(5) as to the magnitude of the unit, and how by diminishing their size and increasing their
number we may distribute them according to any law however complicated.

(9) If we know the direction and velocity of the fluid at any point in two different cases,
and if we conceive a third case in which the direction and velocity of the fluid at any point is
the resultant of the velocities in the two former cases at corresponding points, then the
amount of fluid which passes a given fixed surface in the third case will be the algebraic
sum of the quantities which pass the same surface in the two former cases. For the rate at
which the fluid crosses any surface is the resolved part of the velocity normal to the surface,
and the resolved part of the resultant is equal to the sum of the resolved parts of the com-
ponents.

Hence the number of unit tubes which cross the surface outwards in the third case must
be the algebraical sum of the numbers which cross it in the two former cases, and the number
of sources within any closed surface will be the sum of the numbers in the two former cases.
Since the closed surface may be taken as small as we please, it is evident that the distribution
of sources and sinks in the third case arises from the simple superposition of the distributions
in the two former cases.

II. Theory of the uniform motion of an imponderable incompressible fluid through a

resisting medium.

(10) The fluid is here supposed to have no inertia, and its motion is opposed by the
action of a force which we may conceive to be due to the resistance of a medium through
which the fluid is supposed to flow. This resistance depends on the nature of the medium,
and will in general depend on the direction in which the fluid moves, as well as on its velocity.
For the present we may restrict ourselves to the case of a uniform medium, whose resistance is
the same in all directions. The law which we assume is as follows.

Vol. X. Paet I, 6

$4 Me maxwell, ON FARADAY'S LINES OF FORCE.

Any portion of the fluid moving through the resisting medium is directly opposed by a
retarding force proportional to its velocity.

If the velocity be represented by v, then the resistance will be a force equal to kv acting on
unit of volume of the fluid in a direction contrary to that of motion. In order, therefore, that
the velocity may be kept up, there must be a greater pressure behind any portion of the
fluid than there is in front of it, so that the difl^erence of pressures may neutralise the effect of
the resistance. Conceive a cubical unit of fluid (which we may make as small as we please,
by (5)), and let it move in a direction perpendicular to two of its faces. Then the resistance
will be kv, and therefore the diff^erence of pressures on the first and second faces is kv, so that
the pressure diminishes in the direction of motion at the rate of kv for every unit of length
measured along the line of motion ; so that if we measure a length equal to h units, the dif-
ference of pressure at its extremities will be kvh.

(11) Since the pressure is supposed to vary continuously in the fluid, all the points at
which the pressure is equal to a given pressure p will lie on a certain surface which we may
call the surface (p) of equal pressure. If a series of these surfaces be constructed in the fluid
corresponding to the pressures 0, 1, 2, S &c., then the number of the surface will indicate the
pressure belonging to it, and the surface may be referred to as the surface 0, 1, 2 or 3. The
unit of pressure is that pressure which is produced by unit of force acting on unit of surface.
In order therefore to diminish the unit of pressure as in (5) we must diminish the unit of force
in the same proportion.

(12) It is easy to see that these surfaces of equal pressure must be perpendicular to the
lines of fluid motion ; for if the fluid were to move in any other direction, there would be a
resistance to its motion which could not be balanced by any diff'erence of pressures. (We must
remember that the fluid here considered has no inertia or mass, and that its properties are those
only which are formally assigned to it, so that the resistances and pressures are the only things
to be considered.) There are therefore two sets of surfaces which by their intersection form
the system of unit tubes, and the system of surfaces of equal pressure cuts both the others at
right angles. Let h be the distance between two consecutive surfaces of equal pressure mea-
sured along a line of motion, then since the difference of pressures = 1,

kvh = 1,
which determines the relation of v to h, so that one can be found when the other is known.
Let s be the sectional area of a unit tube measured on a surface of equal pressure, then since
by the definition of a unit tube

vs = \,
we find by the last equation

« = kh.

(13) The surfaces of equal pressure cut the unit tubes into portions whose length is h
and section s. These elementary portions of unit tubes will be called unit cells. In
each of them unity of volume of fluid passes from a pi-essure p to a pressure {p—l) in
unit of time, and therefore overcomes unity of resistance in that time. The work spent in
overcoming resistance is therefore unity in every cell in every unit of time.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 35

(14) If the surfaces of equal pressure are known, the direction and magnitude of
the velocity of the fluid at any point may be found, after which the complete system of
unit tubes may be constructed, and the beginnings and endings of these tubes ascertained
and marked out as the sources whence the fluid is derived, and the sinks where it disappears.
In order to prove the converse of this, that if the distribution of sources be given, the
pressure at every point may be found, we must lay down certain preliminary propositions.

(15) If we know the pressures at every point in the fluid in two different cases, and
if we take a third case in which the pressure at any point is the sum of the pressures at
corresponding points in the two former cases, then the velocity at any point in the third
case is the resultant of the velocities in the other two, and the distribution of sources is
that due to the simple superposition of the sources in the two former cases.

For the velocity in any direction is proportional to the rate of decrease of the pressure
in that direction ; so that if two sj-^stems of pressures be added together, since the rate
of decrease of pressure along any line will be the sum of the combined rates, the velocity
in the new system resolved in the same direction will be the sum of the resolved parts
in the two original systems. The velocity in the new system will therefore be the resultant
of the velocities at corresponding points in the two former systems.

It follows from this, by (9), that the quantity of fluid which crosses any fixed surface
is, in the new system, the sum of the corresponding quantities in the old ones, and that
the sources of the two original systems are simply combined to form the third.

It is evident that in the system in which the pressure is the difference of pressure
in the two given systems the distribution of sources will be got by changing the sign of
all the sources in the second system and adding them to those in the first.

(16) If the pressure at every point of a closed surface be the same and equal to jo,
and if there be no sources or sinks within the surface, then there will be no motion of the
fluid within the surface, and the pressure within it will be uniform and equal to p.

For if there be motion of the fluid within the surface there will be tubes of fluid
motion, and these tubes must either return into themselves or be terminated either within
the surface or at its boundary. Now since the fluid always flows from places of greater
pressure to places of less pressure, it cannot flow in a re-entering curve ; since there are
no sources or sinks within the surface, the tubes cannot begin or end except on the surface ;
and since the pressure at all points of the surface is the same, there can be no motion
in tubes having both extremities on the surface. Hence there is no motion within the
surface, and therefore no difference of pressure which would cause motion, and since the
pressure at the bounding surface is p, the pressure at any point within it is also p.

(17) If the pressure at every point of a given closed surface be known, and the
distribution of sources within the surface be also known, then only one distribution of
pressures can exist within the surface.

For if two different distributions of pressures satisfying these conditions could be found,
a third distribution could be formed in which the pressure at any point should be the

5—2

S6 Mr maxwell, on FARADAY'S LINES OF FORCE.

difference of the pressures in the two former distributions. In this case, since the pressures
at the surface and the sources within it are the same in both distributions, the pressure
at the surface in the third distribution would be zero, and all the sources within the
surface would vanish, by (15).

Then by (l6) the pressure at every point in the third distribution must be zero;
but this is the difference of the pressures in the two former cases, and therefore these
cases are the same, and there is only one distribution of pressure possible.

(18) Let us next determine the pressure at any point of an infinite body of fluid
in the centre of which a unit source is placed, the pressure at an infinite distance from
the source being supposed to be zero.

The fluid will flow out from the centre symmetrically, and since unity of volume

flows out of every spherical surface surrounding the point in unit of time, the velocity at a

distance r from the source will be

1

The rate of decrease of pressure is therefore kv or — -, and since the pressure =

when r is infinite, the actual pressure at any point will he p = .

47rr

The pressure is therefore inversely proportional to the distance from the source.

It is evident that the pressure due to a unit sink will be negative and equal to

k

47rr

If we have a source formed by the coalition of S unit sources, then the resulting

kS

pressure will be » = , so that the pressure at a given distance varies as the resistance

47rr

and number of sources conjointly.

(19) If a number of sources and sinks coexist in the fluid, then in order to determine
the resultant pressure we have only to add the pressures which each source or sink produces.
For by (15) this will be a solution of the problem, and by (17) it will be the only one.
By this method we can determine the pressures due to any distribution of sources, as by the
method of (14) we can determine the distribution of sources to which a given distribution
of pressures is due.

(20) We have next to shew that if we conceive any imaginary surface as fixed in
space and intersecting the lines of motion of the fluid, we may substitute for the fluid
on one side of this surface a distribution of sources upon the surface itself without altering
in any way the motion of the fluid on the other side of the surface.

For if we describe the system of unit tubes which defines the motion of the fluid,
and wherever a tube enters through the surface place a unit source, and wherever a tube
goes out through the surface place a unit sink, and at the same time render the surface
impermeable to the fluid, the motion of the fluid in the tubes will go on as before.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 37

(21) If the system of pressures and the distribution of sources which produce them
be known in a medium whose resistance is measured by k, then in order to produce the
same system of pressures in a medium whose resistance is unity, the rate of production
at each source must be multiplied by k. For the pressure at any point due to a given
source varies as the rate of production and the resistance conjointly; therefore if the pressure
be constant, the rate of production must vary inversely as the resistance.

(22) On the conditions to be fulfilled at a surface which separates two media whose
coefficients of resistance are k and k'.

These are found from the consideration, that the quantity of fluid which flows out
of the one medium at any point flows into the other, and that the pressure varies con-
tinuously from one medium to the other. The velocity normal to the surface is the same
in both media, and therefore the rate of diminution of pressure is proportional to the
resistance. The direction of the tubes of motion and the surfaces of equal pressure will
be altered after passing through the surface, and the law of this refraction will be, that it
takes place in the plane passing through the direction of incidence and the normal to the
surface, and that the tangent of the angle of incidence is to the tangent of the angle of
refraction as k' is to k.

(23) Let the space within a given closed surface be filled with a medium different
from that exterior to it, and let the pressures at any point of this compound system due
to a given distribution of sources within and without the surface be given ; it is required
to determine a distribution of sources which would produce the same system of pressures
in a medium whose coefficient of resistance is unity.

Construct the tubes of fluid motion, and wherever a unit tube enters either medium
place a unit source, and wherever it leaves it place a unit sink. Then if we make the
surface impermeable all will go on as before.

Let the resistance of the exterior medium be measured by k, and that of the interior
by k'. Then if we multiply the rate of production of all the sources in the exterior medium
(including those in the surface), by k, and make the coefficient of resistance unity, the
pressures will remain as before, and the same will be true of the interior medium if we
multiply all the sources in it by k', including those in the surface, and make its resistance
unity.

Since the pressures on both sides of the surface are now equal, we may suppose it
permeable if we please.

We have now the original system of pressures produced in a uniform medium by a
combination of three systems of sources. The first of these is the given external system
multiplied by k, the second is the given internal system multiplied by k', and the third is the
system of sources and sinks on the surface itself. In the original case every source in the
external medium had an equal sink in the internal medium on the other side of the surface,
but now the source is multiplied by k and the sink by k', so that the result is for every
external unit source on the surface, a source = {k - Ic). By means of these three systems of
sources the original system of pressures may be produced in a medium for which i = 1.

88 Mr maxwell, ON FARADAY'S LINES OF FORCE.

(24) Let there be no resistance in the medium within the closed surface, that is,
let k' = 0, then the pressure within the closed surface is uniform and equal to p, and the
pressure at the surface itself is also p. If by assuming any distribution of pairs of sources
and sinks within the surface in addition to the given external and internal sources, and by
supposing the medium the same within and without the surface, we can render the pressure
at the surface uniform, the pressures so found for the external medium, together with the
uniform pressure p in the internal medium, will be the true and only distribution of pressures
which is possible.

For if two such distributions could be found by taking different imaginary distributions
of pairs of sources and sinks within the medium, then by taking the difference of the two
for a third distribution, we should have the pressure of the bounding surface constant in
the new system and as many sources as sinks within it, and therefore whatever fluid flows
in at any point of the surface, an equal quantity must flow out at some other point.

In the external medium all the sources destroy one another, and we have an infinite
medium without sources surrounding the internal medium. The pressure at infinity is zero,
that at the surface is constant. If the pressure at the surface is positive, the motion of
the fluid must be outwards from every point of the surface ; if it be negative, it must flow
inwards towards the surface. But it has been shewn that neither of these cases is possible,
because if any fluid enters the surface an equal quantity must escape, and therefore the
pressure at the surface is zero in the third system.

The pressure at all points in the boundary of the internal medium in the third case
is therefore zero, and there are no sources, and therefore the pressure is everywhere zero,
by (16).

The pressure in the bounding surface of the internal medium is also zero, and there
is no resistance, therefore it is ^ero throughout ; but the pressure in the third case is the
difference of pressures in the two given cases, therefore these are equal, and there is only
one distribution of pressure which is possible, namely, that due to the imaginary distribution
of sources and sinks.

(25) When the resistance is infinite in the internal medium, there can be no passage
of fluid through it or into it. The bounding surface may therefore be considered as
impermeable to the fluid, and the tubes of fluid motion will run along it without cutting it.

If by assuming any arbitrary distribution of sources within the surface in addition to
the given sources in the outer medium, and by calculating the resulting pressures and
velocities as in the case of a uniform medium, we can fulfil the condition of there being
no velocity across the surface, the system of pressures in the outer medium will be the
true one. For since no fluid passes through the surface, the tubes in the interior are
independent of those outside, and may be taken away without altering the external motion.

(26) If the extent of the internal medium be small, and if the difference of resistance
in the two media be also small, then the position of the unit tubes will not be much
altered from what it would be if the external medium filled the whole space.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 39

On this supposition we can easily calculate the kind of alteration which the introduction
of the internal medium will produce ; for wherever a unit tube enters the surface we must

conceive a source producing fluid at a rate — — - , and wherever a tube leaves it we must

k' — k
place a sink annihilating fluid at the rate — - — , then calculating pressures on the supposition

that the resistance in both media is k the same as in the external medium, we shall obtain
the true distribution of pressures very approximately, and we may get a better result by
repeating the process on the system of pressures thus obtained.

(27) If instead of an abrupt change from one coefficient of resistance to another we
take a case in which the resistance varies continuously from point to point, we may treat
the medium as if it were composed of thin shells each of which has uniform resistance. By
properly assuming a distribution of sources over the surfaces of separation of the shells, we
may treat the case as if the resistance were equal to unity throughout, as in (23). The
sources will then be distributed continuously throughout the whole medium, and will be
positive whenever the motion is from places of less to places of greater resistance, and negative
when in the contrary direction.

(28) Hitherto we have supposed the resistance at a given point of the medium to be
the same in whatever direction the motion of the fluid takes place ; but we may conceive
a case in which the resistance is different in diff'erent directions. In such cases the lines of
motion will not in general be perpendicular to the surfaces of equal pressure. If a, b, c
be the components of the velocity at any point, and a, ft, y the components of the
resistance at the same point, these quantities will be connected by the following system of
linear equations, which may be called " equations of conduction,'''' and will be referred to
by that name.

• a = A« + ^3/3 + -^2 7'

b = P^ft+ Q,y + R^a,
c = P37 + Q^a + R^ft.
In these equations there are nine independent coefficients of conductivity. In order to
simplify the equations, let us put

Qi + ifi = 25'i, Q^-R^ = 21T,
&c &c.

where 4 7^ = ( Q, - R,Y + {Q, - Rf + (Q3 - R,Y>

and I, m, n are direction cosines of a certain fixed line in space.
The equations then become

a = P^a-^ S,ft + S^y + (nft - my)T,
b = F^ft+S,y + S,a + (ly -na)T,
c = P37 + S^a + S,ft - {ma - I ft) T.
By the ordinary transformation of coordinates we may get rid of the coefficients marked aS".
The equations then become

4a Mr maxwell, ON FARADAY'S LINES OF FORCE.

ar.p^a+ (,n'(i-m'y)T,

h = P//3 + {I'y - n'a)T,

C = P^y+ima- t(i)T,
where ^, m, n' are the direction cosines of the fixed line with reference to the new axes.
If we make

the equation of continuity

becomes

dp dp , dp

= — , V = -T- 1 and ^ = -— ,
dw' ^ dy' dz'

da db dc

— + 1 = 0,

dx dy d%

^' d.v^ ^ ^' df ^ ^' dz^ '

and if we make a? = v P/^, y = v -^2 »/' ^ ~ V i's^t

d^p d'p d'p

the ordinary equation of conduction.

It appears therefore that the distribution of pressures is not altered by the existence
of the coefficient T, Professor Thomson has shewn how to conceive a substance in which
this coefficient determines a property having reference to an axis, which unlike the axes
of Pj, Pj, P3 is dipolar.

For further information on the equations of conduction, see Professor Stokes On the
Conduction of Heat in Crystals (Cambridge and Dublin Math. Journ.), and Professor
Thomson on the Dynamical Theory of Heat, Part V. (Transactions of Royal Society of
Edinburgh, Vol. XXI. Part I.)

It is evident that all that has been proved in (14), (15), (16), (17), with respect to the
superposition of diffiarent distributions of pressure, and there being only one distribution of
pressures corresponding to a given distribution of sources, will be true also in the case in
which the resistance varies from point to point, and the resistance at the same point is
different in different directions. For if we examine the proof we shall find it applicable
to such cases as well as to that of a uniform medium.

(29) We now are prepared to prove certain general propositions which are true in the
most general case of a medium whose resistance is different in different directions and varies
from point to point.

We may by the method of (28), when the distribution of pressures is known, construct the
surfaces of equal pressure, the tubes of fluid motion, and the sources and sinks. It is evident
that since in each cell into which a unit tube is divided by the surfaces of equal pressure
unity of fluid passes from pressure p to pressure (^ — l) in unit of time, unity of work is
done by the fluid in each cell in overcoming resistance.

The number of cells in each unit tube is determined by the number of surfaces of equal
pressure through which it passes. If the pressure at the beginning of the tube be p and at
the end p', then the number of cells in it will he p - p. Now if the tube had extended from the

Mb maxwell, ON FARADAY'S LINES OF FORCE. 41

source to a place where the pressure is zero, the number of cells would have been p, and if
the tube had come from the sink to zero, the number would have been p', and the true number
is the difference of these.

Therefore if we find the pressure at a source S from which S tubes proceed to be p, Sp
is the number of cells due to the source S; but if S" of the tubes terminate in a sink at
a pressure p', then we must cut off S'p' cells from the number previously obtained. Now if
we denote the source of S tubes by 5*, the sink of S' tubes may be written - ,S^, sinks always
being reckoned negative, and the general expression for the number of cells in the system will
be 2 iSp).

(30) The same conclusion may be arrived at by observing that unity of work is done on
each cell. Now in each source S, S units of fluid are expelled against a pressure p, so that
the work done by the fluid in overcoming resistance is Sp. At each sink in which
S' tubes terminate, ^S^ units of fluid sink into nothing under pressure p'; the work done upon
the fluid by the pressure is therefore S'p'. The whole work done by the fluid may therefore
be expressed by

TF=2Si)-25'y,
or more concisely, considering sinks as negative sources,

W='2(Sp).

(31) Let S represent the rate of production of a source in any medium, and let p be
the pressure at any given point due to that source. Then if we superpose on this another
equal source, every pressure will be doubled, and thus by successive superposition we find that
a source nS would produce a pressure np, or more generally the pressure at any point due to
a given source varies as the rate of production of the source. This may be expressed by the
equation

p = RS,

where iZ is a coeflicient depending on the nature of the medium and on the positions of the

source and the given point. In a uniform medium whose resistance is measured by k,

kS „ Ar

p = - — , .-. B = - — ,

4nrr 47rr

R may be called the coefficient of resistance of the medium between the source and the given
point. By combining any number of sources we have generally

p = ^(RS).

(32) In a uniform medium the pressure due to a source S

_k_ S
47r r
At another source (S^ at a distance r we shall have

S'p= = Sp,

if p' be the pressure at S due to S. If therefore there be two systems of sources 2(5') and
2(y), and if the pressures due to the first be p and to the second p', then

2(yp) = 2(V).

For every term S'p has a term Sp' equal to it.

Vol. X. Paet I. 6

42 Mb maxwell, ON FARADAY'S LINES OF FORCE.

(33) Suppose that in a uniform medium the motion of the fluid is everywhere parallel
to one plane, then the surfaces of equal pressure will be perpendicular to this plane. If we
take two parallel planes at a distance equal to k from each other, we can divide the space
between these planes into unit tubes by means of cylindric surfaces perpendicular to the planes,
and these together with the surfaces of equal pressure will divide the space into cells of which
the length is equal to the breadth. For if h be the distance between consecutive surfaces of
equal pressure and s the section of the unit tube, we have by (IS) s = kh.

But s is the product of the breadth and depth ; but the depth is k, therefore the breadth
is h and equal to the length.

If two systems of plane curves cut each other at right angles so as to divide the plane
into little areas of which the length and breadth are equal, then by taking another plane at
distance k from the first and erecting cylindric surfaces on the plane curves as bases, a system
of cells will be formed which will satisfy the conditions whether we suppose the fluid to run
along the first set of cutting lines or the second *.

Application of the Idea of Lines of Force.

I have now to shew how the idea of lines of fluid motion as described above may be
modified so as to be applicable to the sciences of statical electricity, permanent magnetism,
magnetism of induction, and uniform galvanic currents, reserving the laws of electro-magnetism
for special consideration.

I shall assume that the phenomena of statical electricity have been already explained by
the mutual action of two opposite kinds of matter. If we consider one of these as positive
electricity and the other as negative, then any two particles of electricity repel one another with
a force which is measured by the product of the masses of the particles divided by the square
of their distance.

Now we found in (18) that the velocity of our imaginary fluid due to a source 5" at a distance
r varies inversely as r°. Let us see what will be the efffect of substituting such a source for
every particle of positive electricity. The velocity due to each source would be proportional to
the attraction due to the corresponding particle, and the resultant velocity due to all the
sources would be proportional to the resultant attraction of all the particles. Now we may
find the resultant pressure at any point by adding the pressures due to the given sources, and
therefore we may find the resultant velocity in a given direction from the rate of decrease
of pressure in that direction, and this will be proportional to the resultant attraction of the
particles resolved in that direction.

Since the resultant attraction in the electrical problem is proportional to the decrease of
pressure in the imaginary problem, and since we may select any values for the constants in
the imaginary problem, we may assume that the resultant attraction in any direction is nume-
rically equal to the decrease of pressure in that direction, or

ao)

See Cambridge and Dublin Mathematical Journal, Vol. III. p. 286.

Mn MAXWELL, ON FARADAY'S LINES OF FORCE. 43

By this assumption we find that if V be the potential,

dV = Xdoo + Ydy + Zd% = - dp,

or since at an infinite distance V =0 andp = 0, F = —p.
In the electrical problem we have

In the fluid p = 2 ( ] ;

p-.f^).

_ 47r ,
.•. .y = — dm.
k

If A; be supposed very great, the amount of fluid produced by each source in order to
keep up the pressures will be very small.

The potential of any system of electricity on itself will be

^{pdm)= ^,-E{pS)^-^W.
47r 47r

If 2 (dm), S (dm') be two systems of electrical particles and pp' the potentials due to them
respectively, then by (32)

2(pdm') = — , 2(p5') = — , 2 (p'^ = 2 (p'dm),

47r 'tir

or the potential of the first system on the second is equal to that of the second system
on the first.

So that in the ordinary electrical problems the analogy in fluid motion is of this kind :

r=-p,

X=-^ = Am,
dx

k
dm = — S,
47r

k .

whole potential of a system = - '2Vdm = — W, where W is the work done by the fluid in over-

coming resistance.

The lines of force are the unit tubes of fluid motion, and they may be estimated numerically
by those tubes.

Theory/ of Dielectrics.

The electrical induction exercised on a body at a distance depends not only on the distri-
bution of electricity in the inductric, and the form and position of the inducteous body, but on
the nature of the interposed medium, or dielectric. Faraday * expresses this by the conception

* Series XI.

6—2

44 Mr maxwell, ON FARADAY'S LINES OF FORCE.

of one substance having a greater inductive capacity, or conducting the lines of inductive
action more freely than another. If we suppose that in our analogy of a fluid in a resisting
medium the resistance is different in different media, then by making the resistance less we
obtain the analogue to a dielectric which more easily conducts Faraday's lines.

It is evident from (23) that in this case there will always be an apparent distribution of
electricity on the surface of the dielectric, there being negative electricity where the lines enter
and positive electricity where they emerge. In the case of the fluid there are no real sources
on the surface, but we use them merely for purposes of calculation. In the dielectric there
may be no real charge of electricity, but only an apparent electric action due to the surface.

If the dielectric had been of less conductivity than the surrounding medium, we should
have had precisely opposite efl^ects, namely, positive electricity where lines enter, and negative
where they emerge.

If the conduction of the dielectric is perfect or nearly so for the small quantities of elec-
tricity with which we have to do, then we have the case of (24). The dielectric is then
considered as a conductor, its surface is a surface of equal potential, and the resultant attrac-
tion near the surface itself is perpendicular to it.

Theory of Permanent Magnets.

A magnet is conceived to be made up of elementary magnetized particles, each of which has
its own north and south poles, the action of which upon other north and south poles is
governed by laws mathematically identical with those of electricity. Hence the same applica-
tion of the idea of lines of force can be made to this subject, and the same analogy of fluid
motion can be employed to illustrate it.

But it may be useful to examine the way in which the polarity of the elements of a
magnet may be represented by the unit cells in fluid motion. In each unit cell unity of fluid
enters by one face and flows out by the opposite face, so that the first face becomes a unit
sink and the second a unit source with respect to the rest of the fluid. It may therefore be
compared to an elementary magnet, having an equal quantity of north and south magnetic
matter distributed over two of its faces. If we now consider the cell as forming part of a
system, the fluid flowing out of one cell will flow into the next, and so on, so that the source
will be transferred from the end of the cell to the end of the unit tube. If all the unit tubes
begin and end on the bounding surface, the sources and sinks will be distributed entirely
on that surface, and in the case of a magnet which has what has been called a solenoidal or
tubular distribution of magnetism, all the imaginary magnetic matter will be on the surface*.

Theory of Paramagnetic and Diamagnetic Induction.

V

Faraday •)■ has shewn that the effects of paramagnetic and diamagnetic bodies in the magnetic
field may be explained by supposing paramagnetic bodies to conduct the lines of force better,

• See Professor Thomson On the Mathematical Theory of Magnetism, Chapters III. & V. Phil. Trans. 1851.
■^ Experimental Researches (3293).

Mb maxwell, ON FARADAY'S LINES OF FORCE. 45

and diamagnetic bodies worse, than the surrounding medium. By referring to (23) and (26),
and supposing sources to represent north magnetic matter, and sinks south magnetic matter,
then if a paramagnetic body be in the neighbourhood of a north pole, the lines of force on
entering it will produce south magnetic matter, and on leaving it they will produce an equal
amount of north magnetic matter. Since the quantities of magnetic matter on the whole are
equal, but the southern matter is nearest to the north pole, the result will be attraction. If
on the other hand the body be diamagnetic, or a worse conductor of lines of force than the
surrounding medium, there will be an imaginary distribution of northern magnetic matter
where the lines pass into the worse conductor, and of southern where they pass out, so that on
the whole there will be repulsion.

We may obtain a more general law from the consideration that the potential of the whole
system is proportional to the amount of work done by the fluid in overcoming resistance. The
introduction of a second medium increases or diminishes the work done according as the resist-
ance is greater or less than that of the first medium. The amount of this increase or diminu-
tion will vary as the square of the velocity of the fluid.

Now, by the theory of potentials, the moving force in any direction is measured by the
rate of decrease of the potential of the system in passing along that direction, therefore when
k', the resistance within the second medium, is greater than k, the resistance in the sur-
rounding medium, there is a force tending from places where the resultant force v is greater to
where it is less, so that a diamagnetic body moves from greater to less values of the resultant
force *.

In paramagnetic bodies k' is less than k, so that the force is now from points of less to
points of greater resultant magnetic force. Since these results depend only on the relative values
of k and k', it is evident that by changing the surrounding medium, the behaviour of a body
may be changed from paramagnetic to diamagnetic at pleasure.

It is evident that we should obtain the same mathematical results if we had supposed that
the magnetic force had a power of exciting a polarity in bodies which is in the same direction
as the lines in paramagnetic bodies, and in the reverse direction in diamagnetic bodies -j*. In
fact we have not as yet come to any facts which would lead us to choose any one out of
these three theories, that of lines of force, that of imaginary magnetic matter, and that of
induced polarity. As the theory of lines of force admits of the most precise, and at the same
time least theoretic statement, we shall allow it to stand for the present.

Tlieory of Magnecrystallic Induction.

The theory of Faraday J with respect to the behaviour of crystals in the magnetic field
may be thus stated. In certain crystals and other substances the lines of magnetic force are

" Experimental Researches (2797), (2798). See Thom-
son, Cambridge and Dublin Mathematical Journal, May,
1847.

t Exp. Res. (2429), (3320). See Weber, PoggendorfF,
Ixxxvii. p. 145. Prof. TyndaU, Phil, Trans. 1856, p. 237.
i Exp. Res. (2836), &c.

46 Mr maxwell, ON FARADAY'S LINES OF FORCE.

conducted with different facility in different directions. The body when suspended in a
uniform magnetic field will turn or tend to turn into such a position that the lines of force shall
pass through it with least resistance. It is not difficult by means of the principles in (28)
to express the laws of this kind of action, and even to reduce them in certain cases to numerical
formulae. The principles of induced polarity and of imaginary magnetic matter are here
of little use; but the theory of lines of force is capable of the most perfect adaptation to this
class of phenomena.

Theory of the Conduction of Current Electricity,

It is in the calculation of the laws of constant electric currents that the theory of fluid
motion which we have laid down admits of the most direct application. In addition to the
researches of Ohm on this subject, we have those of M. Kirchhoff, Ann. de Chim. xli. 496, and
of M. Quincke, xlvii. 203, on the Conduction of Electric Currents in Plates. According to
the received opinions we have here a current of fluid moving uniformly in conducting circuits,
which oppose a resistance to the current which has to be overcome by the application of
an electro-motive force at some part of the circuit. On account of this resistance to the motion
of the fluid the pressure must be different at different points in the circuit. This pressure,
which is commonly called electrical tension, is found to be physically identical with the potential
in statical electricity, and thus we have the means of connecting the two sets of phenomena.
If we knew what amount of electricity, measured statically, passes along that current which
we assume as our unit of current, then the connexion of electricity of tension with current
electricity would be completed*. This has as yet been done only approximately, but we
know enough to be certain that the conducting powers of different substances differ only in
degree, and that the difference between glass and metal is, that the resistance is a great
but finite quantity in glass, and a small but finite quantity in metal. Thus the analogy
between statical electricity and fluid motion turns out more perfect than we might have
supposed, for there the induction goes on by conduction just as in current electricity, but
the quantity conducted is insensible owing to the great resistance of the dielectrics -j-.

On Electro-motive Forces.

When a uniform current exists in a closed circuit it is evident that some other forces
must act on the fluid besides the pressures. For if the current were due to difference of
pressures, then it would flow from the point of greatest pressure in both directions to the
point of least pressure, whereas in reality it circulates in one direction constantly. We

• See £jrp. Res. (371). f Exp. Res. VoL HI. p. 513.

Mr maxwell, on FARADAY'S LINES OF FORCE. 47

must therefore admit the existence of certain forces capable of keeping up a constant current
in a closed circuit. Of these the most remarkable is that which is produced by chemical
action. A cell of a voltaic battery, or rather the surface of separation of the fluid of the
cell and the zinc, is the seat of ati electro-motive force which can maintain a current in
opposition to the resistance of the circuit. If we adopt the usual convention in speaking
of electric currents, the positive current is from the fluid through the platinum, the conducting
circuit, and the zinc, back to the fluid again. If the electro-motive force act only in the
surface of separation of the fluid and zinc, then the tension of electricity in the fluid must
exceed that in the zinc by a quantity depending on the nature and length of the circuit
and on the strength of the current in the conductor. In order to keep up this diffierence
of pressure there must be an electro-motive force whose intensity is measured by that difference
of pressure. If F be the electro-motive force, / the quantity of the current or the number
of electrical units delivered in unit of time, and K a quantity depending on the length and
resistance of the conducting circuit, then

F^lK='p-'p,

where p is the electric tension in the fluid and p' in the zinc.

If the circuit be broken at any point, then since there is no current the tension of
the part which remains attached to the platinum will be p, and that of the other will
be 'p . p —p, or F affbrds a measure of the intensity of the current. This distinction of
quantity and intensity is very useful*, but must be distinctly understood to mean nothing
more than this : — The quantity of a current is the amount of electricity which it transmits
in unit of time, and is measured by / the number of unit currents which it contains.
The intensity of a current is its power of overcoming resistance, and is measured by F
or IK, where K is the resistance of the whole circuit.

The same idea of quantity and intensity may be applied to the case of magnetism "I".
The quantity of magnetization in any section of a magnetic body is measured by the
number of lines of magnetic force which pass through it. The intensity of magnetization
in the section depends on the resisting power of the section, as well as on the number of lines
which pass through it. If k be the resisting power of the material, and S the area of
the section, and / the number of lines of force which pass through it, then the whole
intensity throughout the section

When magnetization is produced by the influence of other magnets only, we may put
p for the magnetic tension at any point, then for the whole magnetic solenoid

F = I j^dx = IK^p-p'.

• Exp. Res. Vol. III. p. 519. f Exp. Res. (2870), (3293).

48 Mr maxwell, ON FARADAY'S LINES OF FORCE.

When a solenoidal magnetized circuit returns into itself, the magnetization does not
depend on difference of tensions only, but on some magnetizing force of which the intensity is F.

If i be the quantity of the magnetization at any point, or the number of lines of
force passing through unit of area in the section of the solenoid, then the total quantity
of magnetization in the circuit is the number of lines which pass through any section
/ = 2idyd«, where dydss is the element of the section, and the summation is performed over
the whole section.

The intensity of magnetization at any point, or the force required to keep up the
magnetization, is measured by hi =f, and the total intensity of magnetization in the circuit
is measured by the sum of the local intensities all round the circuit,

where dw is the element of length in the circuit, and the summation is extended round
the entire circuit.

In the same circuit we have always F = IK, where X is the total resistance of the
circuit, and depends on its form and the matter of which it is composed.

On the Action of closed Currents at a Distance.

The mathematical laws of the attractions and repulsions of conductors have been most ably
investigated by Ampere, and his results have stood the test of subsequent experiments.

From the single assumption, that the action of an element of one current upon an
element of another current is an attractive or repulsive force acting in the direction of
the line joining the two elements, he has determined by the simplest experiments the
mathematical form of the law of attraction, and has put this law into several most elegant
and useful forms. We must recollect however that no experiments have been made on
these elements of currents except under the form of closed currents either in rigid conductors
or in fluids, and that the laws of closed currents only can be deduced from such experiments.
Hence if Ampere's formulae applied to closed currents give true results, their truth is not
proved for elements of currents unless we assume that the action between two such elements
must be along the line which joins them. Although this assumption is most warrantable and
philosophical in the present state of science, it will be more conducive to freedom of investi-
gation if we endeavour to do without it, and to assume the laws of closed currents as the
ultimate datum of experiment.

Ampere has shewn that when currents are combined according to the law of the
parallelogram of forces, the force due to the resultant current is the resultant of the forces
due to the component currents, and that equal and opposite currents generate equal and
opposite forces, and when combined neutralize each other.

He has also shewn that a closed circuit of any form has no tendency to turn a
moveable circular conductor about a fixed axis through the centre of the circle perpendicular
to its plane, and that therefore the forces in the case of a closed circuit render Xdx+ Ydy+Zdz
a complete differential.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 4&

Finally, he has shewn that if there be two systems of circuits similar and similarly
situated, the quantity of electrical current in corresponding conductors being the same,
the resultant forces are equal, whatever be the absolute dimensions of the systems, which
proves that the forces are, ceeteris paribus, inversely as the square of the distance.

From these results it follows that the mutual action of two closed currents whose areas are
very small is the same as that of two elementary magnetic bars magnetized perpendicularly
to the plane of the currents.

The direction of magnetization of the equivalent magnet may be predicted by remembering
that a current travelling round the earth from east to west as the sun appears to do, would
be equivalent to that magnetization which the earth actually possesses, and therefore in the
reverse direction to that of a magnetic needle when pointing freely.

If a number of closed unit currents in contact exist on a surface, then at all points
in which two currents are in contact there will be two equal and opposite currents which
will produce no effect, but all round the boundary of the surface occupied by the currents
there will be a residual current not neutralized by any other; and therefore the result will
be the same as that of a single unit current round the boundary of all the currents.

From this it appears that the external attractions of a shell uniformly magnetized
perpendicular to its surface are the same as those due to a current round its edge, for
each of the elementary currents in the former case has the same effect as an element of
the magnetic shell.

If we examine the lines of magnetic force produced by a closed current, we shall find
that they form closed curves passing round the current and embracing it, and that the total
intensity of the magnetizing force all along the closed line of force depends on the quan-
tity of the electric current only. The number of unit lines * of magnetic force due to a
closed current depends on the form as well as the quantity of the current, but the number
of unit cells f in each complete line of force is measured simply by the number of unit
currents which embrace it. The unit cells in this case are portions of space in which
unit of magnetic quantity is produced by unity of magnetizing force. The length of a
cell is therefore inversely as the intensity of the magnetizing force, and its section is inversely
as the quantity of magnetic induction at that point.

The whole number of cells due to a given current is therefore proportional to the strength
of the current multiplied by the number of lines of force which pass through it. If by
any change of the form of the conductors the number of cells can be increased, there will
be a force tending to produce that change, so that there is always a force urging a conductor
transverse to the lines of magnetic force, so as to cause more lines of force to pass through
the closed circuit of which the conductor forms a part.

The number of cells due to two given currents is got by multiplying the number of
lines of inductive magnetic action which pass through each by the quantity of the currents
respectively. Now by (9) the number of lines which pass through the first current is the
sum of its own lines and those of the second current which would pass through the first if the

• Exp. Res. (3122). See Art. (6) of this paper. t Art. (13).

Vol. X. Paet I.

j^

Mh maxwell, on FARADAY'S LINES OF FORCE.

second current alone were in action. Hence the whole number of cells will be increased
by any motion which causes more lines of force to pass through either circuit, and therefore
the resultant force will tend to produce such a motion, and the work done by this force
during the motion will be measured by the number of new cells produced. All the actions
of closed conductors on each other may be deduced from this principle.

On Electric Currents produced hy Induction.

Faraday has shewn * that when a conductor moves transversely to the lines of magnetic
force, an electro-motive force arises in the conductor, tending to produce a current in it. If the
conductor is closed, there is a continuous current, if open, tension is the result. If a closed
conductor move transversely to the lines of magnetic induction, then, if the number of lines
which pass through it does not change during the motion, the electro-motive forces in the
circuit will be in equilibrium, and there will be no current. Hence the electro-motive forces
depend on the number of lines which are cut by the conductor during the motion. If the
motion be such that a greater number of lines pass through the circuit formed by the conductor
after than before the motion, then the electro-motive force will be measured by the increase of
the number of lines, and will generate a current the reverse of that which would have produced
the additional lines. When the number of lines of inductive magnetic action through the
circuit is increased, the induced current will tend to diminish the number of the lines, and when
the number is diminished the induced current will tend to increase them.

That this is the true expression for the law of induced currents is shewn from the fact
that, in whatever way the number of lines of magnetic induction passing through the circuit
be increased, the electro-motive effect is the same, whether the increase take place by the motion
of the conductor itself," or of other conductors, or of magnets, or by the change of intensity of
other currents, or by the magnetization or demagnetization of neighbouring magnetic bodies, or
lastly by the change of intensity of the current itself.

In all these cases the electro-motive force depends on the change in the number of lines of
inductive magnetic action which pass through the circuit ■[•.

• Exp. Res. (3077), &c.

+ The electro-magnetic forces, which tend to produce motion
of the material conductor, must be carefully distinguished
from the electro-motive forces, which tend to produce electric
currents.

Let an electric current be passed through a mass of metal
of any form. The distribution of the currents within the metal
will be determined by the laws of conduction. Now let a
constant electric current be passed through another conductor
near the first. If the two currents are in the same direction
the two conductors will be attracted towards each other, and
would come nearer if not held in their positions. But though
the material conductors are attracted, the currents (which are
free to choose any course within the metal) will not alter their
original distribution, or incline towards each other. For, since
no change takes place in the system, there will be no electro-
motive forces to modify the original distribution of currents.

In this case we have electro-magnetic forces acting on the
material conductor, without any electro-motive forces tending
to modify the current which it carries.

Let us take as another example the case of a linear con-
ductor, not forming a closed circuit, and let it be made to
traverse the lines of magnetic force, either by its own motion,
or by changes in the magnetic field. An electro-motive force
will act in the direction of the conductor, and, as it cannot pro-
duce a current, because there is no circuit, it will produce
electric tension at the extremities. There will be no electro-
magnetic attraction on the material conductor, for this attraction
depends on the existence of the current within it, and this is
prevented by the circuit not being closed.

Here then we have the opposite case of an electro-motive
force acting on the electricity in the conductor, but no attraction
on its material particles.

Mb maxwell, ON FARADAY'S LINES OF FORCE. 51

It is natural to suppose that a force of this kind, which depends on a change in the
number of lines, is due to a change of state which is measured by the number of these lines."
A closed conductor in a magnetic field may be supposed to be in a certain state arising from
the magnetic action. As long as this state remains unchanged no effect takes place, but, when
the state changes, electro-motive forces arise, depending as to their intensity and direction on
this change of state. I cannot do better here than quote a passage from the first series of
Faraday's Experimental Researches, Art. (60).

"While the wire is subject to either volfa-electric or magneto-electric induction it appears
to be in a peculiar state, for it resists the formation of an electrical current in it ; whereas, if
in its common condition, such a current would be produced ; and when left uninfluenced it has
the power of originating a current, a power which the wire does not possess under ordinary
circumstances. This electrical condition of matter has not hitherto been recognised, but it
probably exerts a very important influence in many if not most of the phenomena produced
by currents of electricity. For reasons which will immediately appear (71) I have, after
advising with several learned friends, ventured to designate it as the electro-tonic state."
Finding that all the phenomena could be otherwise explained without reference to the electro-
tonic state, Faraday in his second series rejected it as not necessary ; but in his recent
researches* he seems still to think that there may be some physical truth in his conjecture
about this new state of bodies.

The conjecture of a philosopher so familiar with nature may sometimes be more pregnant
with truth than the best established experimental law discovered by empirical inquirers, and
though not bound to admit it as a physical truth, we may accept it as a new idea by which
our mathematical conceptions may be rendered clearer.

In this outline of Faraday's electrical theories, as they appear from a mathematical point of
view, I can do no more than simply state the mathematical methods by which I believe that
electrical phenomena can be best comprehended and reduced to calculation, and my aim has
been to present the mathematical ideas to the mind in an embodied form, as systems of lines or
surfaces, and not as mere symbols, which neither convey the same ideas, nor readily adapt
themselves to the phenomena to be explained. The idea of the electro-tonic state, however, has
not yet presented itself to my mind in such a form that its nature and properties may be
clearly explained without reference to mere symbols, and therefore I propose in the following
investigation to use symbols freely, and to take for granted the ordinary mathematical
operations. By a careful study of the laws of elastic solids and of the motions of viscous
fluids, I hope to discover a method of forming a mechanical conception of this electro-tonic
state adapted to general reasoning -f-.

Part II. On Faraday's "Electro-tonic State."

When a conductor moves in the neighbourhood of a current of electricity, or of a magnet,
or when a current or magnet near the conductor is moved, or altered in intensity, then a force

" (3172) (3269). I tion of Electric, Magnetic and Galvanic Forces. Camb. and

t See Prof. W. Thomson On a Mechanical Representa- I Dub. Math. Jeur, Jan. 1847.

7 2

52 Mk maxwell, on FARADAY'S LINES OF FORCE.

acts on the conductor and produces electric tension, or a continuous current, according as the
'circuit is open or closed. This current is produced only by changes of the electric or magnetic
phenomena surrounding the conductor, and as long as these are constant there is no observed
effect on the conductor. Still the conductor is in different states when near a current or
magnet, and when away from its influence, since the removal or destruction of the current or
magnet occasions a current, which would not have existed if the magnet or current had not
been previously in action.

Considerations of this kind led Professor Faraday to connect with his discovery of the
induction of electric currents, the conception of a state into which all bodies are thrown by the
presence of magnets and currents. This state does not manifest itself by any known phenomena
as long as it is undisturbed, but any change in this state is indicated by a current or tendency
towards a current. To this state he gave the name of the " Electro-tonic State," and although
he afterwards succeeded in explaining the phenomena which suggested it by means of less
hypothetical conceptions, he has on several occasions hinted at the probability that some phe-
nomena might be discovered which would render the electro-tonic state an object of legitimate
induction. These speculations, into which Faraday had been led by the study of laws which
he has well established, and which he abandoned only for want of experimental data for the
direct proof of the unknown state, have not, I think, been made the subject of mathematical
investigation. Perhaps it may be thought that the quantitative determinations of the various
phenomena are not sufficiently rigorous to be made the basis of a mathematical theory ;
Faraday, however, has not contented himself with simply stating the numerical results of his
experiments and leaving the law to be discovered by calculation. Where he has perceived a law
he has at once stated it, in terms as unambiguous as those of pure mathematics ; and if the
mathematician, receiving this as a physical truth, deduces from it other laws capable of being
tested by experiment, he has merely assisted the physicist in arranging his own ideas, which
is confessedly a necessary step in scientific induction.

In the following investigation, therefore, the laws established by Faraday will be assumed
as true, and it will be shewn that by following out his speculations other and more general
laws can be deduced from them. If it should then appear that these laws, originally devised
to include one set of phenomena, may be generalized so as to extend to phenomena of a different
class, these mathematical connexions may suggest to physicists the means of establishing
physical connexions ; and thus mere speculation may be turned to account in experimental
science.

On Quantity and Intensity as Properties of Electric Currents.

It is found that certain effects of an electric current are equal at whatever part of the
circuit they are estimated. The quantities of water or of any other electrolyte decomposed at two
different sections of the same circuit, are always found to be equal or equivalent, however
different the material and form of the circuit may be at the two sections. The magnetic
effect of a conducting wire is also found to be independent of the form or material of the wire

Mr maxwell, ON FARADAY'S LINES OF FORCE, 58

in the same circuit. There is therefore an electrical effect which is equal at every section of the
circuit. If we conceive of the conductor as the channel along which a fluid is constrained to
move, then the quantity of fluid transmitted by each section will be the same, and we may
define the quantity of an electric current to be the quantity of electricity which passes across
a complete section of the current in unit of time. We may for the present measure quantity
of electricity by the quantity of water which it would decompose in unit of time.

In order to express mathematically the electrical currents in any conductor, we must have
a definition, not only of the entire flow across a complete section, but also of the flow at a given
point in a given direction.

Def. The quantity of a current at a given point and in a given direction is measured,
when uniform, by the quantity of electricity which flows across unit of area taken at that point
perpendicular to the given direction, and when variable by the quantity which would flow
across this area, supposing the flow uniformly the same as at the given point.

In the following investigation, the quantity of electric current at the point (.ryar) estimated
in the directions of the axes x, y, x respectively will be denoted by a^ h.^ Cj.

The quantity of electricity which flows in unit of time through the elementary area dS

= dS {la.^ + mh^ + nc^,

where Imn are the direction-cosines of the normal to dS.

This flow of electricity at any point of a conductor is due to the electro-motive forces
which act at that point. These may be either external or internal.

External electro-motive forces arise either from the relative motion of currents and
magnets, or from changes in their intensity, or from other causes acting at a distance.

Internal electro-motive forces arise principally from difference of electric tension at points of
the conductor in the immediate neighbourhood of the point in question. The other causes are
variations of cnemical composition or of temperature in contiguous parts of the conductor.

Let P2 represent the electric tension at any point, and X2 Y^ Z-^ the sums of the parts of
all the electro-motive forces arising from other causes resolved parallel to the co-ordinate axes,
then if aj, ^2 72 be the effective electro-motive forces

Oj = X2 — ,
ax

dy

ry dp.,

1- (A)

dz

Now the quantity of the current depends on the electro-motive force and on the resistance
of the medium. If the resistance of the medium be uniform in all directions and equal to k.^,

as = Asffla, /Sa = ^2625 72 = hc» (B)

but if the resistance be different in different directions, the law will be more complicated.

These quantities 02 (i^ 72 may be considered as representing the intensity of the electric
action in the directions of xyz.

64 Mr maxwell, ON FARADAY'S LINES OF FORCE.

The intensity measured along an element da of a curve

e = la + mft + ny,
where Imn are the direction-cosines of the tangent.

The integral feda taken with respect to a given portion of a curve line, represents the total
intensity along that line. If the curve is a closed one, it represents the total intensity of the
electro-motive force in the closed curve.

Substituting the values of a/3 7 from equations (A)

feda = JiXdo! + Ydy + Zdz) - p + C.
If, therefore {Xdx + Ydy + Zdz) is a complete differential, the value of feda for a closed curve
will vanish, and in all closed curves

feda = f{Xdx + Ydy + Zdz),
the integration being effected along the curve, so that in a closed curve the total intensity
of the effective electro-motive force is equal to the total intensity of the impressed electro-
motive force.

The total quantity of conduction through any surface is expressed by

fedS,
where

e = la + mb + nc,

Imn being the direction-cosines of the normal,

.*. fedS = ffadydz + ffbdzdx + ffcdxdy,
the integrations being effected over the given surface. When the surface is a closed one, then
we may find by integration by parts

If we make

da db dc

— + T- + T" = 4'r|0 (C)

dx dy dz

fedS = 4 TT Jffpdxdydz,
where the integration on the right side of the equation is effected over every part of space
within the surface. In a large class of phenomena, including all cases of uniform currents,
the quantity p disappears.

Magnetic Quantity and Intensity.

From his study of the lines of magnetic force, Faraday has been led to the conclusion that
in the tubular surface* formed by a system of such lines, the quantity of magnetic induction
across any section of the tube is constant, and that the alteration of the character of these lines
in passing from one substance to another, is to be explained by a difference of inductive
capacity in the two substances, which is analogous to conductive power in the theory of
electric currents.

• Exp. Res. 3271, definition of " Sphondyloid."

Mb maxwell, ON FARADAY'S LINES OF FORCE. SS

In the following investigation we shall have occasion to treat of magnetic quantity and
intensity in connexion with electric. In such cases the magnetic symbols will be distinguished
by the suffix 1, and the electric by the suffix 2. The equations connecting a, b, c, k, a, fi, y,
p, and p, are the same in form as those which we have just given, a, b, c are the symbols of
magnetic induction with respect to quantity ; k, denotes the resistance to magnetic induction,
and may be different in different directions; a, /3, y, are the effective magnetizing forces, con-
nected with a, b, c, by equations (B) ; jo, is the magnetic tension or potential which will be
afterwards explained ; p denotes the density of real magnetic matter and is connected with
a, b, c by equations (C). As all the details of magnetic calculations will be more intelligible
after the exposition of the connexion of magnetism with electricity, it will be sufficient here to
say that all the definitions of total quantity, with respect to a surface, and total intensity with
respect to a curve, apply to the case of magnetism as well as to that of electricity.

Electro-magnetism.

Ampere has proved the following laws of the attractions and repulsions of electric
currents :

I. Equal and opposite currents generate equal and opposite forces.

II. A crooked current is equivalent to a straight one, provided the two currents nearly
coincide throughout their whole length.

III. Equal currents traversing similar and similarly situated closed curves act with
equal forces, whatever be the linear dimensions of the circuits.

IV. A closed current exerts no force tending to turn a circular conductor about
its centre.

It is to be observed, that the currents with which Ampere worked were constant and
therefore re-entering. All his results are therefore deduced from experiments on closed
currents, and his expressions for the mutual action of the elements of a current involve the
assumption that this action is exerted in the direction of the line joining those elements. This
assumption is no doubt warranted by the universal consent of men of science in treating of
attractive forces considered as due to the mutual action of particles; but at present we
are proceeding on a different principle, and searching for the explanation of the phenomena,
not in the currents alone, but also in the surrounding medium.

The first and second laws shew that currents are to be combined like velocities or forces.

The third law is the expression of a property of all attractions which may be conceived of
as depending on the inverse square of the distance from a fixed system of points ; and the
fourth shews that the electro-magnetic forces may always be reduced to the attractions and
repulsions of imaginary matter properly distributed.

In fact, the action of a very small electric circuit on a point in its neighbourhood is
identical with that of a small magnetic element on a point outside it. If we divide any
given portion of a surface into elementary areas, and cause equal currents to flow in the
same direction round all these little areas, the effect on a point not in the surface will be the

56 Mr maxwell, ON FARADAY'S LINES OF FORCE.

same as that of a shell coinciding with the surface, and uniformly magnetized normal to its
surface. But by the first law all the currents forming the little circuits will destroy one
another, and leave a single current running round the bounding line. So that the magnetic
effect of a uniformly magnetized shell is equivalent to that of an electric current round the
edge of the shell. If the direction of the current coincide with that of the apparent motion of
the sun, then the direction of magnetization of the imaginary shell will be the same as
that of the real magnetization of the earth*.

The total intensity of magnetizing force in a closed curve passing through and embracing
the closed current is constant, and may therefore be made a measure of the quantity of the
current. As this intensity is independent of the form of the closed curve and depends only on
the quantity of the current which passes through it, we may consider the elementary case of
the current which flows through the elementary area dydss.

Let the axis of w point towards the west, ss towards the south, and y upwards. Let wyz
be the position of a point in the middle of the area dydss, then the total intensity measured
round the four sides of the element is

/ d(i,dz\

-[^^--di^n

f dy, dy\

^v'^--diYn

Total intensity = (-^ - -jAdy dz.

The quantity of electricity conducted through the elementary area dydx is a.jdydz, and
therefore if we define the measure of an electric current to be the total intensity of magnetizing
force in a closed curve embracing it, we shall have

Oa — — —- — ,

dx dy

dy, dai

bi = — -T->

< dio das

dai dfS^
dy d,v

These equations enable us to deduce the distribution of the currents of electricity whenever
we know the values of a, /3, y, the magnetic intensities. If a, jS, y be exact differentials of
a function of wyz with respect to ar, y and z respectively, then the values of aj b^ c, disappear ;

* See Experimental Researches (3265) far the relations between the electrical and magnetic circuit, considered as muiually
embracing curves.

Me maxwell, ON FARADAY'S LINES OF FORCE. 57

and we know that the magnetism is not produced by electric currents in that part of the field
which we are investigating. It is due either to the presence of permanent magnetism within
the field, or to magnetizing forces due to external causes.

We may observe that the above equations give by differentiation

da^ db, dc-i
dx dy dz '
which is the equation of continuity for closed currents. Our investigations are therefore for
the present limited to closed currents; and in fact we know little of the magnetic effects of any
currents which are not closed.

Before entering on the calculation of these electric and magnetic states it may be
advantageous to state certain general theorems, the truth of which may be established
analytically.

Theorem I.

The equation

d^V dW d'V
d^v^ + df^^^'^'P'''
(where V and p are functions of oryx never infinite, and vanishing for all points at an infinite
distance,) can be satisfied by one, and only one, value of V. See Art, (17) above.

Theoeem II.

The value of V which will satisfy the above conditions is found by integrating the expression

pd.vdydx

ffh

(x — x'l^ "^ y — y\ "^ ^ — «' 1')

where the limits of xyz are such as to include every point of space where p is finite.

The proofs of these theorems may be found in any work on attractions or electricity, and
in particular in Green's Essay on the Application of Mathematics to Electricity. See Arts.
18, 19 of this Paper. See also Gauss, on Attractions, translated in Tayl.'s-.'s Scientific Memoirs.

Theorem III.

Let U and V be two functions oi xyz, then

rrr..ldPV dW d'V\ , , , rrrldUdV dUdV dU dV\ ^ , ,

jjJ''[dx^'-df^d^r'^^'='-jjJ[-d.^v-'^

where the integrations are supposed to extend over all the space in which U and V have values

differing from (Green, p. 10.)

This theorem shews that if there be two attracting systems the actions between them are
equal and opposite. And by making (7 = F we find that the potential of a system on itself is
proportional to the integral of the square of the resultant attraction through all space; a
Vol. X. Part I. 8

58 Mr maxwell, ON FARADAY'S LINES OF FORCE.

result deducible from Art. (30), since the volume of each cell is inversely as the square of
the velocity (Arts. J 2, 13), and therefore the number of cells in a given space is directly
as the square of the velocity.

Theorem IV.

Let a, /3, y, p be quantities finite through a certain space and vanishing in the space beyond,
and let k be given for all parts of space as a continuous or discontinuous function of xyz,
then the equation in p

d If dp\ d !/_ dp\ d 1/ dp\
Txlk"- d-v)-'jyk['^~dy)^lilcV-irzr^''P=''^

has one, and only one solution, in which p is always finite and vanishes at an infinite
distance.

The proof of this theorem, by Prof. W. Thomson, may be found in the Cambridge and
Dublin Math. Journal, Jan. 1848.

If a /3 7 be the electro-motive forces, p the electric tension, and k the coefficient of resist-
ance, then the above equation is identical with the equation of continuity

da^ dbi dCi
dx dy dz ' '

and the theorem shews that when the electro-motive forces and the rate of production of
electricity at every part of space are given, the value of the electric tension is determinate.

Since the mathematical laws of magnetism are identical with those of electricity, as far as
we now consider them, we may regard a^y as magnetizing forces, p as magnetic tension, and p
as real magnetic density, k being the coefficient of resistance to magnetic induction.
The proof of this theorem rests on the determination of the minimum value of *

where V is got from the equation

rfT d'V d'V

and p has to be determined.

The meaning of this integral in electrical language may be thus brought out. If the pre-
sence of the media in which k has various values did not affect the distribution of forces, then the

"quantity" resolved in x would be simply — and the intensity ^ — . But the actual quan-

1 / dp\ dp

tity and intensity are -la - —I and a - — , and the parts due to the distribution of media

alone are therefore

1 dp\ dV dp dV

k dxj doe dco dx '

Mr maxwell, ON FARADAY'S LINES OF FORCE. M

Now the product of these represents the work done on account of this distribution of
media, the distribution of sources being determined, and taking in the terms in y and «
we get the expression Q for the total work done by that part of the whole effect at any
point which is due to the distribution of conducting media, and not directly to the presence
of the sources.

This quantity Q is rendered a minimum by one and only one value of p, namely, that
which satisfies the original equation.

Theoeem V.

If a, b, c be three functions of ir, y, z satisfying the equation

da db dc

dx dy dz

it is always possible to find three functions a, fi, 7 which shall satisfy the equations

d/3 dy

dz

dy

= a,

dy
dw

da
dz~

6,

da
dy'

dx

c.

Let A =fcdy, where the integration is to be performed upon c considered as a function
of y, treating x and % as constants. Let B =fadz, C=jbdx, A' = jbdx, B> =fcdx, C = fady,
integrated in the same way.

Then

a=A-A' + f,
dx

dy

will satisfy the given equations ; for

d(i dy rda rdc rdb f^^j

dx dy J dy J dz J dy J dy

and

rda rdb rd<i ,

= / — dx + \ — dx + / — dx;
J dx J dv J dz

dy

_, d(B dy f^^ -, r^<^ J f^^
d% dy J dx ' J dy J dz

- a.

8—2

60 Mr maxwell, ON FARADAY'S LINES OF FORCE.

In the same way it may be shewn that the values of a, /3, 7 satisfy the other given equations.
The function \^ may be considered at present as perfectly indeterminate.

The method here given is taken from Prof. W. Thomson's memoir on Magnetism
(Phil. Trans. 1851, p. 283).

As we cannot perform the required integrations when a, b, c are discontinuous functions of
a?, y, s, the following method, which is perfectly general though more complicated, may indicate
more clearly the truth of the proposition.

Let A, B, C be determined from the equations

^A d'A d^
daf dy^ dx^

d'B dTB d'B

H H h 6 = 0,

daf dy" dz^

d'C d^C d'C
da^ dy" dz"

by the methods of Theorems I. and II., so that A, B, C are never infinite, and vanish when ai, y,
or z is infinite. ^

Also let

dB dC dy^,
dz dy dx

dC dA d\l^
dx dz dy '

dA dB dv|r
dy dx dz '

d(i

dz

dy
dy

d idA dB dC\ l<
dx \dx dy dz) \

=

d IdA dB dC\
dx \dx dy dz I

then

id'A (PA drA

I + - H

\ dx' dy'^ dz

If we find similar equations in y and z, and differentiate the first by x, the second by y, and
the third by z, remembering the equation between o, h, c, we shall have

\dx' dy' dz'l \dx dy dz J

and since A, B, C are always finite and vanish at an infinite distance, the only solution of this
equation is

dA dB dC

dx dy dz

and we have finally

d^_dy
dz dy

with two similar equations, shewing that a, (3, y have been rightly determined.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 61

The function yj/ is to be determined from the condition

da d^ dy /d" £_ ^\\ .

if the left-hand side of this equation be always zero, >^ must be zero also.

Theorem VI.

Let a, b, c be any three functions of a:, y, z, it is possible to find three functions a, /3, y and
a fourth V, so that

da d3 dy
dio dy dz

d)3 dy dV

and " = :i -r + "T »

dx dy dx

dy da dV
dx dis dy '

Let

_da _d0 dV
dy dx dz

da db dc

dx dy dz "'

and let V be found from the equation

then

satisfy the condition

d^V d'V d'V

dV
'""'-di'

b'.b-'I,
dy

dV

c = c - — - ,

da db' dc
dx dy dz

and therefore we can find three functions J, B, C, and from these a, /3, y, so as to satisfy the
given equations.

Theorem VII.

The istegral throughout infinity

Q = fffidiai + bi(ii + Cxyx)dxdydz,

6» Mr maxwell, ON FARADAY'S LINES OF FORCE.

where ai b^ Cj, ui jSj 71 are any functions whatsoever, is capable of transformation into

in which the quantities are found from the equations

da, db, dcj
-—- + -—+ — -+ 47rpi = 0,
ax dy dz

da, d&x dy, ,

-J- +-r + :T^+47rp/= 0;
d.v dy dz

«oi3c7o^ are determined from a, 6, c, by the last theorem, so that

dz dy dot
a-i 62 C2 are found from a, /3i 71 by the equations

ddx dvi o

a.,= — &c.,

J dx dy

and p is found from the equation

dJ'p d-p d^p ,

doe' dy' dz' "'

For, if we put a^ in the form

dA)_^o dV
dz dy dm

and treat b^ and Cj similarly, then we have by integration by parts thA)ugh infinity, remem-
bering that all the functions vanish at the limits,

or Q= + flf\ (47r Vp) - (a^a^ + fiA + 70C2) } djcdydx,
and by Theorem III,

fffVp'dxdydz = fjfppdjsdydz,
so that finally

Q = ffll'^'^Pp - ("o«2 + ^0^2 = y^2)]da>dydx.

If Ci 61 Cj represent the components of magnetic quantity, and ai /3i 71 those of magnetic
intensity, then p will represent the real magnetic density, and p the magnetic potential or
tension. 02 62 <^2 '"'iH be the components of quantity of electric currents, and Oq /Sq 7o will be three
functions deduced from OiftjCi, which will be found to be the mathematical expression for
Faraday's Electro-tonic state.

Let us now consider the bearing of these analytical theorems on the theory of magnetism.
Whenever we deal with quantities relating to magnetism, we shall distinguish them by the
suffix ( 1 ). Thus Oi 61 Cj are the components resolved in the directions of x, y, z of the

Mr maxwell, ON FARADAY'S LINES OF FORCE. 68

quantity of magnetic induction acting through a given point, and ai/3i7i are the resolved inten-
sities of magnetization at the same point, or, what is the same thing, the components of the
force which would be exerted on a unit south pole of a magnet placed at that point without
disturbing the distribution of magnetism.

The electric currents are found from the magnetic intensities by the equations

flj = —---/- &c.
dz ay

When there are no electric currents, then

a^dx + fi^dy + y^dz = dp^,

a perfect differential of a function of *, y, z. On the principle of analogy we may call p^ the
magnetic tension.

The forces which act on a mass m of south magnetism at any point are

dpi dpi dp^

— m—- , — m —— , and — m -— ,
die dy dz

in the direction of the axes, and therefore the whole work done during any displacement of a
magnetic system is equal to the decrement of the integral

Q = fffpiPidxdydz
throughout the system.

Let us now call Q the total potential of the system on itself. The increase or decrease
of Q will measure the work lost or gained by any displacement of any part of the system,
and will therefore enable us to determine the forces acting on that part of the system.

By Theorem III. Q may be put under the form

Q = + — jjjifl\f^i + ^i/^i + Ciyi)d.vdydx,

in which a, jSj 7, are the differential coefficients of p^ with respact to a?, y, x respectively.

If we now assume that this expression for Q is true whatever be the values of oi (Bi yu we
pass from the consideration of the magnetism of permanent magnets to that of the magnetic
effects of electric currents, and we have then by Theorem VII.

So that in the case of electric currents, the components of the currents have to be multiplied
by the functions aafioyo respectively, and the summations of all such products throughout the
system gives us the part of Q due to those currents.

We have now obtained in the functions oq jS^ 70 the means of avoiding the consideration of
the quantity of magnetic induction which passes through the circuit. Instead of this artificial
method we have the natural one of considering the current with reference to quantities existing
in the same space with the current itself. To these I give the name of Electro-tonic functions,
or components of the Electro-tonic intensity.

64 Mb maxwell, ON FARADAY'S LINES OF FORCE.

Let us now consider the conditions of the conduction of the electric currents within the
medium during changes in the electro-tonic state. The method which we shall adopt is an
application of that given by Helmholtz in his memoir on the Conservation of Force*.

Let there be some external source of electric currents which would generate in the con-
ducting mass currents whose quantity is measured by a^ 63 Cj and their intensity by aj jSa 72-

Then the amount of work due to this cause in the time di is

dtjjj{a./i2 + 62/82 + c./yi)daidydz
in the form of resistance overcome, and

dt d

47r dt

Jjjiazaa + hfio + c^y^dxdydz

in the form of work done mechanically by the electro-magnetic action of these currents. If
there be no external cause producing currents, then the quantity representing the whole work
done by the external cause must vanish, and we have

dt jjf(«2a2+ ^A + c.,y2)da!dydx+ — — Jfjictiaa + b.,(i^ + c.i'y^)dxdydz,

where the integrals are taken through any arbitrary space. We must therefore have

a-ift^ + 60/82 + CjYjj = — — (a,ao + h^fi^ + c^yo)

for every point of space; and it must be remembered that the variation of Q is supposed due to
variations of oq fio 7o' ^"^ ^^^ "^ '^^ ^2 '^2- ^^ must therefore treat a.^ b^ c^ as constants, and
the equation becomes

\ A>Tr dt I V 4,7r dt J \' 'iir dt J

In order that this equation may be independent of the values of a^ h.^ c^, each of these co-
efficients must = ; and therefore we have the following expressions for the electro-motive
forces due to the action of magnets and currents at a distance in terms of the electro-tonic
functions,

°^ 4.7r dt ' 47r dt ' '^' 47r dt '

It appears from experiment that the expression --" refers to the change of electro-tonic state

of a given particle of the conductor, whether due to change in the electro-tonic functions
themselves or to the motion of the particle.

If Op be expressed as a function of x, y, x, and t, and if x, y, % be the co-ordinates of a moving
article, then the electro-motive force measured in the direction of x is

1 /dao dx da^ dy daa dz da^\
47r V dx dt dy dt dz dt dt J

• Translated in Taylor's New Scientific Memoirs, Part 1 1.

Mr maxwell, ON FARADAY'S LINES OF FORCE. ^

The expressions for the electro-motive forces in y and « are similar. The distribution of
currents due to these forces depends on the form and arrangement of the conducting media
and on the resultant electric tension at any point.

The discussion of these functions would involve us in mathematical formulae, of which this
paper is already too full. It is only on account of their physical importance as the mathema-
tical expression of one of Faraday's conjectures that I have been induced to exhibit them at
all in their present form. By a more patient consideration of their relations, and with the
help of those who are engaged in physical inquiries both in this subject and in others not
obviously connected with it, I hope to exhibit the theory of the electro-tonic state in a form
in which all its relations may be distinctly conceived without reference to analytical calcula-
tions.

Summary of the Theory of the Electro-tonic State.

We may conceive of the electro-tonic state at any point of space as a quantity determinate
in magnitude and direction, and we may represent the electro-tonic condition of a portion of
space by any mechanical system which has at every point some quantity, which may be a
velocity, a displacement, or a force, whose direction and magnitude correspond to those of the
supposed electro-tonic state. This representation involves no physical theory, it is only a kind
of artificial notation. In analytical investigations we make use of the three components of the
electro-tonic state, and call them electro-tonic functions. We take the resolved part of the
electro-tonic intensity at every point of a closed curve, and find by integration what we may
call the entire electro-tonic intensity round the curve,

Peop. I. If on any surface a closed curve be dratvn, and if the surface within it be
divided into small areas, then the entire intensity round the closed curve is equal to the sum
of the intensities round each of the small areas, all estimated in the same direction.

For, in going round the small areas, every boundary line between two of them is passed
along twice in opposite directions, and the intensity gained in the one case is lost in the other.
Every effect of passing along the interior divisions is therefore neutralized, and the whole
effect is that due to the exterior closed curve.

Law I. The entire electro-tonic intensity round the boundary of an element of surface
measures the quantity of magnetic induction which passes through that surface, or, in other
words, the number of lines of magnetic force which pass through that surface.

By Prop. I. it appears that what is true of elementary surfaces is true also of surfaces of
finite magnitude, and therefore any two surfaces which are bounded by the same closed curve
will have the same quantity of magnetic induction through them.

Law II. The magnetic intensity at any point is connected with the quantity of magnetic
induction by a set of linear equations, called the equations of conduction*.

• See Art. (28).

Vol. X. Part I.

66

Mr maxwell, ON FARADAY'S LINES OF FORCE.

Law III. The entire magnetic intensity round the boundary of any surface measures
the quantity of electric current which passes through that surface.

Law IV. The quantity and intensity of electric currents are connected by a system of
equations of conduction.

By these four laws the magnetic and electric quantity and intensity may be deduced from
the values of the electro-tonic functions. I have not discussed the values of the units, as that
will be better done with reference to actual experiments. We come next to the attraction of
conductors of currents, and to the induction of currents within conductors.

Law V. The total electro-magnetic potential of a closed current is measured by the product
of the quantity of the current multiplied by the entire electro-tonic intensity estimated in the
same direction round the circuit.

Any displacement of the conductors which would cause an increase in the potential will be
assisted by a force measured by the rate of increase of the potential, so that the mechanical
work done during the displacement will be measured by the increase of potential.

Although in certain cases a displacement in direction or alteration of intensity of tJie
current might increase the potential, such an alteration would not itself produce work, and
there will be no tendency towards this displacement, for alterations in the current are due to
electro-motive force, not to electro-magnetic attractions, which can only act on the conductor.

Law VI. The electro-motive force on any element of a conductor is measured by the
instantaneous rate of change of the electro-tonic intensity on that element, whether in magnitude

or direction. i

The electro-motive force in a closed conductor is measured by the rate of change of the
entire electro-tonic intensity round the circuit referred to unit of time. It is independent of the
nature of the conductor, though the current produced varies inversely as the resistance; and
it is the same in whatever way the change of electro-tonic intensity has been produced, whether
bv motion of the conductor or by alterations in the external circumstances.

In these six laws I have endeavoured to express the idea which I believe to be the mathe-
matical foundation of the modes of thought indicated in the Expenmental Researches. I do
not think that it contains even the shadow of a true physical theory; in fact, its chief merit as a
temporary instrument of research is that it does not, even in appearance, account for anything.

There exists however a professedly physical theory of electro-dynamics, which is so elegant,
so mathematical, and so entirely different from anything in this paper, that I must state its
axioms, at the risk of repeating what ought to be well_known. It is contained in M. W.
Weber's Electro-dynamic Measurements, and may be found in the Transactions of the
Leibnitz Society, and of the Royal Society of Sciences of Saxony *. The assumptions are,

(1) That two particles of electricity when in motion do not repel each other with the
same force as when at rest, but that the force is altered by a quantity depending on the
relative motion of the two particles, so that the expression for the repulsion at distance r is

• When this was written, I was not aware that part of M.
Weber's Memoir is translated in Taylor's Scientific Memoirs,
Vol. V. Art. XIV. The value of his researches, both experimen.

tal and theoretical, renders the study of his theory necessary to
every electrician.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 67

ee' / dr

te I dr !> , dV\

— 1 + a — +or -—]
r" V dt dt'

(2) That when electricity is moving in a conductor, the velocity of the positive fluid
relatively to the matter of the conductor is equal and opposite to that of the negative fluid.

(3) The total action of one conducting element on another is the resultant of the
mutual actions of the masses of electricity of both kinds which are in each.

(4) The electro-motive force at any point is the difference of the forces acting on the
positive and negative fluids.

From these axioms are deducible Ampere's laws of the attraction of conductors, and those
of Neumann and others, for the induction of currents. Here then is a really physical theory,
satisfying the required conditions better perhaps than any yet invented, and put forth by a
philosopher whose experimental researches form an ample foundation for his mathematical
investigations. What is the use then of imagining an electro-tonic state of which we have no
distinctly physical conception, instead of a formula of attraction which we can readily under-
stand .'' I would answer, that it is a good thing to have two ways of looking at a subject, and
to admit that there are two ways of looking at it. Besides, I do not think that we have any
right at present to understand the action of electricity, and I hold that the chief merit of a
temporary theory is, that it shall guide experiment, without impeding the progress of the true
theory when it appears. There are also objections to making any ultimate forces in nature
depend on the velocity of the bodies between which they act. If the forces in nature are to
be reduced to forces acting between particles, the principle of the Conservation of Force re-
quires that these forces should be in the line joining the particles and functions of the distance
only. The experiments of M. Weber on the reverse polarity of diamagnetics, which have been
recently repeated by Professor Tyndall, establish a fact which is equally a consequence of
M. Weber's theory of electricity and of the theory of lines of force.

With respect to the history of the present theory, I may state that the recognition of
certain mathematical functions as expressing the "electro-tonic state" of Faraday, and the use
of them in determining electro-dynamic potentials and electro-motive forces, is, as far as I am
aware, original ; but the distinct conception of the possibility of the mathematical expressions
arose in my mind from the perusal of Prof. W. Thomson's papers "On a Mechanical Represen-
tation of Electric, Magnetic and Galvanic Forces," Cambridge and Dublin Mathematical
Journal, January, 1847, and his " Mathematical Theory of Magnetism," Philosophical Transac-
tions, Part I. 1851, Art. 78, &c. As an instance of the help which may be derived from other
physical investigations, I may state that after I had investigated the Theorems of this paper
Professor Stokes pointed out to me the use which he had made of similar expressions in his
"Dynamical Theory of Diffraction," Section 1, Cambridge Transactions, Vol. I.X. Part i.
Whether the theory of these functions, considered with reference to electricity, may lead to new
mathematical ideas to be employed in physical research, remains to be seen. I propose in the
rest of this paper to discuss a few electrical and magnetic problems with reference to spheres.
These are intended merely as concrete examples of the methods of which the theory has been
given ; I reserve the detailed investigation of cases chosen with special reference to experiment
till I have the means of testing their results.

9 — z

61 Mb maxwell, ON FARADAY'S LINES OF FORCE.

Examples.

I. Theory of Electrical Images.

The method of Electrical Images, due to Prof. W. Thomson*, by which the theory of

spherical conductors has been reduced to great geometrical simplicity, becomes even more

simple when we see its connexion with the methods of this paper. We have seen that the

pressure at any point in a uniform medium, due to a spherical shell (radius = a) giving out

a^ .
fluid at the rate of '^irPa^ units in unit of time, is kP — outside the shell, and kPa inside it,

r

where r is the distance of the point from the centre of the shell.

If there be two shells, one giving out fluid at a rate 4nrPa\ and the other absorbing at the

rate 4nrP'a'^, then the expression for the pressure will be, outside the shells,

a' , a'^

p = 4:TrP 47rP — ; ,

r r

where r and r are the distances from the centres of the two shells. Equating this expression
to zero we have, as the surface of no pressure, that for which

r P'a'-'

r Pa''

Now the surface, for which the distances to two fixed points have a given ratio, is a sphere
of which the centre O is in the line joining the centres of the shells CC produced, so that

co = cc ^ —

Pa'^-Fa'Y
and its radius

_^^ Pa'.P^a'

Pa']'-P'a''Y' '

If at the centre of this sphere we place another source of the fluid, then the pressure due to
this source must be added to that due to the other two ; and since this additional pressure
depends only on the distance from the centre, it will be constant at the surface of the sphere,
where the pressure due to the two other sources is zero.

We have now the means of arranging a system of sources within a given sphere, so that
when combined with a given system of sources outside the sphere, they shall produce a given
constant pressure at the surface of the sphere.

" See a series of papers " On the Mathematical Theory of Electricity," in the Cambridge and Dublin Math. Jour., begin-
ning March, 1848.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 69

Let a be the radius of the sphere, and p the given pressure, and let the given sources be
at distances b^ b^ 8ec. from the centre, and let their rates of production be 47rPi, 'iirPs &c.

a^ a-
Then if at distances — , — &c. (measured in the same direction as 6j b^ Sec. from the
bi 62

centre) we place negative sources whose rates are

a a

- 4nrPi - , - ^trPi - &c.

the pressure at the surface r = a will be reduced to zero. Now placing a source 47r — at
the centre, the pressure at the surface will be uniform and equal to p.

The whole amount of fluid emitted by the surface r = a may be found by adding the
rates of production of the sources within it. The result is

-{^^^-}•

To apply this result to the case of a conducting sphere, let us suppose the external sources
47rPi, ^wPi to be small electrified bodies, containing e^ 62 of positive electricity. Let us also sup-
pose that the whole charge of the conducting sphere is = E previous to the action of the external
points. Then all that is required for the complete solution of the problem is, that the surface
of the sphere shall be a surface of equal potential, and that the total charge of the surface shall
he E.

If by any distribution of imaginary sources within the spherical surface we can effect this,
the value of the corresponding potential outside the sphere is the true and only one. The
potential inside the sphere must really be constant and equal to that at the surface.

We must therefore find the images of the external electrified points, that is, for every
point at distance b from the centre we must find a point on the same radius at a distance

— , and at that point we must place a quantity =—e — of imaginary electricity.
At the centre we must put a quantity E' such that

E'=E+ei- +e.,-+kc.;
bi 62

then if R be the distance from the centre, r^r^ &c. the distances from the electrified points,
and /j/g the distances from their images at any point outside the sphere, the potential at
that point will be

jE' /I a \\ /I an„

R \ri birj ' Vi-2 i2rj

E eifa 61 a\ e~>(a b^ a\

70 Mk maxwell, on FARADAY'S LINES OF FORCE.

This is the value of the potential outside the sphere. At the surface we have

so that at the surface

R=a and — = -;-,

E

= — &c.

a 61 5,

;+&c.

and this must also be the value of p for any point within the sphere.

For the application of the principle of electrical images the reader is referred to Prof.
Thomson's papers in the Cambridge and Dublin Mathematical Journal. The only case

which we shall consider is that in which —^ =/, and 6, is infinitely distant along axis of x,

"1
and £=0.

The value p outside the sphere becomes then

;>=/.( -^^).

and inside p=0.

IL On the effect of a paramagnetic or diamagnetic sphere in a uniform field of magnetic

force *.

The expression for the potential of a small magnet placed at the origin of co-ordinates
in the direction of the axis of <r is

d (m\ , X
dx \rj r'

The effect of the sphere in disturbing the lines of force may be supposed as a first
hypothesis to be similar to that of a small magnet at the origin, whose strength is to
be determined. (We shall find this to be accurately true.)

Let the value of the potential undisturbed by the presence of the sphere be

p = Ix.

Let the sphere produce an additional potential, which for external points is

p = A —x,

and let the potential within the sphere be

Pi = Bx.
Let k' be the coefficient of resistance outside, and k inside the sphere, then the
conditions to be fulfilled are, that the interior and exterior potential should coincide at the

" See Prof. Thomson, on the Theory of Magnetic Induction,
Phil. Mag, March, 1851. The inductive capacity of the sphere,
according to that paper, is the ratio of the quantity of magnetic

induction (not the intensity) within the sphere to that without.

1 k' 3jt'
It is therefore equal to j S -r- = S7 — y according to our notation.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 71

surface, and that the induction through the surface should be the same whether deduced
from the external or the internal potential. Putting x = r cos 0, we have for the external
potential

p = [ir ■{■ A~\co%Q,

and for the internal

p,= Br cos Q,

and these must be identical when r = a., or

The induction through the surface in the external medium is

k dr,^, k
and that through the interior surface is

These equations give

and .-. -;(7-2^) = - B.
k k

. k — fC _, 3k ^
^ = -1 T' ^> B= — -, /.

ih + k Zk + k

The effect outside the sphere is equal to that of a little magnet whose length is / and
moment ml, provided

ml= — rrO' !•

2k + k

Suppose this uniform field to be that due to terrestrial magnetism, then, if k is less than
k' as in paramagnetic bodies, the marked end of the equivalent magnet will be turned to the
north. If k is greater tHan k' as in diamagnetic bodies, the unmarked end of the equivalent
magnet would be turned to the north.

III. Magnetic field of variable Intensity.

Now suppose the intensity in the undisturbed magnetic field to vary in magnitude and
direction from one point to another, and that its components in ccyix are represented by a,)8i7i,
then, if as a first approximation we regard the intensity within the sphere as sensibly equal to
that at the centre, the change of potential outside the sphere arising from the presence of

78

Mr maxwell, ON FARADAY'S LINES OF FORCE.

the sphere, disturbing the lines of force, will be the same as that due to three small magnets at
the centre, with their axes parallel to a>, y, and ar, and their moments equal to

k-k'
2A; + k

, a?a.

k-k'

Zk + k

-,a'(i.

k-k'
Sk + k'

a^y.

The actual distribution of potential within and without the sphere may be conceived as the
result of a distribution of imaginary magnetic matter on the surface of the sphere ; but since
the external effect of this superficial magnetism is exactly the same as that of the three small
magnets at the centre, the mechanical effect of external attractions will be the same as if the
three magnets really existed.

Now let three small magnets whose lengths are Z, l^ I3, and strengths m^ nii m^ exist at the
point xy% with their axes parallel to the axes of wyx; then, resolving the forces on the three
magnets in the direction of J^, we have

— X = till

da l{

ai+

dx 2

dah

A-mJ

"'^dxl]

da 4"
rft/2

da la

+ m.

a +

da I3
dz2

-«! +

da , da ,

aw ay

dy2)
da

dak

dz2

Substituting the values of the moments of the imaginary magnets

~2k + k' \dx '^ dx '^ dx) 2k + &' 2 dx ^

The force impelling the sphere in the direction of x is therefore dependent on the variation
of the square of the intensity or (a^ + /3' + 7*), as we move along the direction of x, and the
same is true for y and z, so that the law is, that the force acting on diamagnetic spheres is
from places of greater to places of less intensity of magnetic force, and that in similar distri-
butions of magnetic force it varies as the mass of the sphere and the square of the intensity.

It is easy by means of Laplace's Coefficients to extend the approximation to the value of
the potential as far as we please, and to calculate the attraction. For instance, if a north or
south magnetic pole whose strength is M, be placed at a distance b from a diamagnetic sphere,
radius a, the repulsion will be

^-'<*-*')?G-^F-

3.2

4.3

Sk + 2k' W 4& + 3k' b*

+ &c.

When — is small, the first term gives a sufficient approximation. The repulsion is then as
b

the square of the strength of the pole and the mass of the sphere directly and the fifth power

of the distance inversely, considering the pole as a point.

Mb maxwell, ON FARADAY'S LINES OF FORCE.

73

IV. Two Spheres in uniform Jield.

Let two spheres of radius a be connected together so that their centres are kept at a dis-
tance ft, and let them be suspended in a uniform magnetic field, then, although each sphere by
itself would have been in equilibrium at any part of the field, the disturbance of the field will
produce forces tending to make the balls set in a particular direction.

Let the centre of one of the spheres be taken as origin, then the undisturbed potential is

p — I r cos 9,
and the potential due to the sphere is

, k-k'a^

p = I —. ;^ — cos U.

2k + k' r*

The whole potential is therefore equal to

^ f k- k a\

V 2k

1 dp
r dO

2k + K

dp , / k — k' a?\

dr V 2k + k T^j

k — k' a^

cos 9,

, /■ k- k a^\ . . dp

»-=^

dr

+

1 dp

t^ dd

r sm

dp
'9 d^

= r4i +

k — k' a?
2k + k' r^

d(p

(l-3cos=0) +

k-k'
2k +k'

:(1 +3cos'

'■e)]-

This is the value of the square of the intensity at any point. The moment of the couple
tending to turn the combination of balls in the direction of the original force

L = l

d

k-k'

T — 3 ri

^2 2k + k'

7/1 - 1 7 7' ^'1 ^l^en r = ft,
d9 \2k+k J

k-k' ^a^ ( k-k' a?\ . ^

ft-^l^-sTTl'ft-j^^"^^-

This expression, which must be positive, since ft is greater than a, gives the moment of
a force tending to turn the line joining the centres of the spheres towards the original lines of
force.

Whether the spheres are magnetic or diamagnetic they tend to set in the axial direction,
and that without distinction of north and south. If, however, one sphere be magnetic and
the other diamagnetic, the line of centres will set equatoreally. The magnitude of the force
depends on the square of {k — k'), and is therefore quite insensible except in iron *.

V. Two Spheres between the poles of a Magnet.

Let us next take the case of the same balls placed not in a uniform field but between a
north and a south pole, ± M, distant 2c from each other in the direction of x.

Vol. X. Part I.

• See Prof. Thomson in Phil. Mag. March, 18S1.

10

74f Mb maxwell, ON FARADAY'S LINES OF FORCE.

The expression for the potential, the middle of the line joining the poles being the
origin, is

1 1

\,\/c'+r^-9, cos Qcr \/c''+r^+2 cos Qcri '

From this we find as the value of P,

■: ^ 3s = - 18 — / sin 20,
dO c*

and the moment to turn a pair of spheres (radius a, distance 26) in the direction in which

9 is increased is

k-k' M'a'b^ . ^

This force, which tends to turn the line of centres equatoreally for diamagnetic and axially
for magnetic spheres, varies directly as the square of the strength of the magnet, the cube of
the radius of the spheres and the square of the distance of their centres, and inversely as the
sixth power of the distance of the poles of the magnet, considered as points. As long as
these poles are near each other this action of the poles will be much stronger than the
mutual action of the spheres, so that as a general rule we may say that elongated bodies
set axially or equatoreally between the poles of a magnet according as they are magnetic
or diamagnetic. If, instead of being placed between two poles very near to each other,
they had been placed in a uniform field such as that of terrestrial magnetism or that produced
by a spherical electro-magnet (see Ex. VIII.), an elongated body would set axially whether
magnetic or diamagnetic.

In all these cases the phenomena depend on k — k', so that the sphere conducts itself
magnetically or diamagnetically according as it is more or less magnetic, or less or more
diamJignetic than the medium in which it is placed.

VI. On the Magnetic Phenomena of a Sphere cut from a substance whose coefficient
of resistance is different in different directions.

Let the axes of magnetic resistance be parallel throughout the sphere, and let them
be taken for the axes of a?, y, ss. Let k^, k^, k^, be the coefficients of resistance in these three
directions, and let k' be that of the external medium, and a the radius of the sphere. Let /
be the undisturbed magnetic intensity of the field into which the sphere is introduced, and
let its direction-cosines be I, m, n.

Let us now take the case of a homogeneous sphere whose coefficient is A;, placed in a
uniform magnetic field whose intensity is II in the direction of a?. The resultant potential
outside the sphere would be

Mr maxwell, ON FARADAY'S LINES OF FORCE. 75

and for internal points

«i= 12 — -> a?.

So that in the interior of the sphere the magnetization is entirely in the direction of .v.
It is therefore quite independent of the coefficients of resistance in the directions of w and y,
which may be changed from k^ into k.^ and ks without disturbing this distribution of magnetism.
We may therefore treat the sphere as homogeneous for each of the three components of /,
but we must use a different coefficient for each. We find for external points

, f [k,-k' k^-k' k^-k' \aF\

and for internal points

, / Sk^ , Shi Sks \
Pi= I — —, Ix + — -, my + — -, nx .

The external effect is the same as that which would have been produced if the small
magnet whose moments are

K^—K «2"" k k^— k

— — -, lla?y — -, mlar, — -; n/a'

2*1+ A; 2*2+ ft 2*3+ A;

had been placed at the origin with their directions coinciding with the axes of <B,y, «. The
effect of the original force / in turning the sphere about the axis of x may be found by
taking the moments of the components of that force on these equivalent magnets. The
moment of the force in the direction of y acting on the third magnet is

and that of the force in x on the second magnet is

Kn-^ Hi

The whole couple about the axis of a? is therefore

, mnlV.

mnPa?,

(2*3 + *')(2*2 + *')

tending to turn the sphere round from the axis of y towards that of z. Suppose the sphere
to be suspended so that the axis of ■» is vertical, and let / be horizontal, then if 9 be the
angle which the axis of y makes with the direction oi I, m = cos 0, n = — sin 9, and the
expression for the moment becomes

I . /'^t~r^\'. /Vsin2g

^ (2&2+^)(2*3+*)

tending to increase 9. The axis of least resistance therefore sets axially, but with either
end indifferently towards the north.

Since in all bodies, except iron, the values of k are nearly the same as in a vacuum,

10—2

76

Mr maxwell, ON FARADAY'S LINES OF FORCE.

the coefficient of this quantity can be but little altered by changing the value of k' to k, the
value in space. The expression then becomes

1 Kj" K,

k

Pa? sin 9.9,

independent of the external medium *.

VII. Permanent magnetism in a spherical shell.

The case of a homogeneous shell of a diamagnetic or paramagnetic substance presents no
difficulty. The intensity within the shell is less than what it would have been if the shell
were away, whether the substance of the shell be diamagnetic or paramagnetic. When the
resistance of the shell is infinite, and when it vanishes, the intensity within the shell is zero.

In the case of no resistance the entire effect of the shell on any point, internal or external,
may be represented by supposing a superficial stratum of magnetic matter spread over the
outer surface, the density being given by the equation

p = 3l cos 0.
Suppose the shell now to be converted into a permanent magnet, so that the distribution of
imaginary magnetic matter is invariable, then the external potential due to the shell will be

p = - / - cos 0,

and the internal potential pi= — Ir cos 0.

Now let us investigate the effect of filling up the shell with some substance of which

the resistance is k, the resistance in the external medium being k'. The thickness of the

magnetized shell may be neglected. Let the magnetic moment of the permanent magnetism

be /a^ and that of the imaginary superficial distribution due to the medium k = Aa'. Then

the potentials are

a'
external p'= {I + A) -- cos 6, internal pi= (I + A) r cos 9.

T

The distribution of real magnetism is the same before and after the introduction of the
medium Ic, so that

i,/4/ = l(/+^)+|(/H-^).

or A^ — -, /.

9.k + k

The external effect of the magnetized shell is increased or diminished according as X: is
greater or less than k'. It is therefore increased by filling up the shell with diamagnetic
matter, and diminished by filling it with paramagnetic matter, such as iron.

* Taking the more general case of magnetic induction re-
ferred to in Art. (28), we find, in the expression for the moment
of the magnetic forces, a constant term depending on T, besides
those terms which depend on sines and cosines of 8. The result
is, that in every complete revolution in the negative direction
round the axis of T, a certain positive amount of work is
gained ; but, since no inexhaustible source of work can exist

in nature, we must admit that ^=0 in all substances, with
respect to magnetic induction. This argument does not hold
in the case of electric conduction, or in the case of a body
through which heat or electricity is passing, for such states are
maintained by the continual expenditure of work. See Prof.
Thomson, Phil. Mag. March, 1851, p. 186.

Mb maxwell, ON FARADAY'S LINES OF FORCE. 77

VIII. Electro-magnetic spherical shell.

Let us take as an example of the magnetic effects of electric currents, an electro-magnet
in the form of a thin spherical shell. Let its radius be a, and its thickness t, and let its
external effect be that of a magnet whose moment is la^. Both within and without the shell
the magnetic effect may be represented by a potential, but within the substance of the shell,
where there are electric currents, the magnetic effects cannot be represented by a potential.
Let p, p, be the external and internal potentials,

3

p' = /— ■ cos 9, p, = Ar cos 9,

r

dp' dpi
and since there is no permanent magnetism, — = —r- , when r = a,

A = -2l.
If we draw any closed curve cutting the shell at the equator, and at some other
point for which 9 is known, then the total magnetic intensity round this curve will be
Sla cos 9, and as this is a measure of the total electric current which flows through it,
the quantity of the current at any point may be found by differentiation. The quantity
which flows through the element td9 is - 37a sin 9d9, so that the quantity of the current
referred to unit of area of section is

-31- sin 9.
t

If the shell be composed of a wire coiled round the sphere so that the number of coils

to the inch varies as the sine of 9, then the external effect will be nearly the same as if

the shell had been made of a uniform conducting substance, and the currents had been

distributed according to the law we have just given.

If a wire conducting a current of strength I^ be wound round a sphere of radius a

2a
so that the distance between successive coils measured along the axis of a? is — , then

n

there will be n coils altogether, and the value of /j for the resulting electro-magnet will be

/i =
The potentials, external and internal, will be

ba

, ^ n a' ^ ^ n r ^

^^ = -'2 S -J cos 9, Pi = - S-^a 7. - cos 9.

or 6 a

The interior of the shell is therefore a uniform magnetic field. .

IX. Effect of the core of the electro-magnet.

Now let us suppose a sphere of diamagnetic or paramagnetic matter introduced into the

electro-magnetic coil. The result may be obtained as in the last case, and the potentials become

n Sk' a^ ^ ^ n 3k r ^

p = I„- — J — cos 9, Pi=- 2/, - -, 77 - cos 9.

'^ ^ 6 2k + k r" ^' ^6 2k+k a

The external effect is greater or less than before, according as k' is greater or less

thai) k, that is, according as the interior of the sphere is magnetic or diamagnetic with

78 Mr maxwell, ON FARADAY'S LINES OF FORCE.

respect to the external medium, and the internal effect is altered in the opposite direction,
being greatest for a diamagnetic medium.

This investigation explains the effect of introducing an iron core into an electro-magnet.
If the value of k for the core were to vanish altogether, the effect of the electro-magnet
would be three times that which it has without the core. As k has always a finite value,
the effect of the core is less than this.

In the interior of the electro-magnet we have a uniform field of magnetic force, the
intensity of which may be increased by surrounding the coil with a shell of iron. If
k' = 0, and the shell infinitely thick, the effect on internal points would be tripled.

The effect of the core is greater in the case of a cylindric magnet, and greatest of all when
the core is a ring of soft iron.

X. Electro-tonic functions in spherical electro-magnet.

Let us now find the electro-tonic functions due to this electro-magnet.

They will be of the form

Co = 0> ^0 = t>)X, 7o = - <^yi

where to is some function of r. Where there are no electric currents, we must have Ojj ^a' <^3
each = 0, and this implies

the solution of which is

/ d(i)\

(3. + r.-j=0.

d f dc

dr

c. + ^

Within the shell w cannot become infinite ; therefore w = C, is the solution, and outside
a must vanish at an infinite distance, so that

0} =

r.

3

is the solution outside. The magnetic quantity within the shell is found by last article to be

-2
therefore within the sphere

^6a 2k + k ' dr dy

Wq= -

I^n 1

2a 3k + k' ■

Outside the sphere we must determine w so as to coincide at the surface with the internal
value. The external value is therefore

•■2"
W =

2a Sh + A' r^ '

where the shell containing the currents is made up of n coils of wire, conducting a current of
total quantity I^.

Let another wire be coiled round the shell according to the same law, and let the total
number of coils be n ; then the total electro-tonic intensity EI^ round the second coil is
found by integrating

EI^ = / a)ffl sin Qds,

''a

Mb maxwell, ON FARADAY'S LINES OF FORCE. 79

along the whole length of the wire. The equation of the wire is

\ = -^
where w is a large number ; and therefore

cos 9 = -7— ,
n IT

ds = a sin 6d(p,
= - an'irsm^OdO,
„, 47r „ , Stt ,, 1 •

^ S 3 3k + k

E may be called the electro-tonic coefficient for the particular wire.

XI. Spherical electro-magnetic Coil-Machine.

We have now obtained the electro-tonic function which defines the action of the one coil on
the other. The action of each coil on itself is found by putting n? or n'' for nri. Let the
first coil be connected with an apparatus producing a variable electro-motive force F. Let us
find the effects on both wires, supposing their total resistances to be R and R', and the
quantity of the currents / and T.

Let N stand for -— — -7- , then the electro-motive force of the first wire on the second is
3 {3k+k)

That of the second on itself is

AT '^^

- Nnn -— .
dt

dt

The equation of the current in the second wire is therefore

-Nnn'~-Nn"% = R'r (l)

dt dt ^ '

The equation of the current in the first wire is

-Nn'>--Nnn'~ + F = RI. (2)

dt dt '

Eliminating the differential coeflicients, we get

nnn

, ,, /to' n"\ dl ^ F n'^ dF , .

from which to find / and T . For this purpose we require to know the value of F in terms
of t.

Let us first take the case in which F is constant and / and T initially = 0. This is the
case of an electro-magnetic coil-machine at the moment when the connexion is made with the
galvanic trough.

W Mr maxwell, ON FARADAY'S LINES OF FORCE.

Putting A T for iV — - + -— we find •

\R R I

N n

F

The primary current increases very rapidly from to — , and the secondary commences at

R

F n

- — , — and speedily vanishes, owing to the value of t being generally very small.

The whole work done by either current in heating the wire or in any other kind of action
is found from the expression

fPRdt
The total quantity of current is

/ Idt.
For the secondary current we find

Jo Rn" 4 Jg Rn 2

The work done and the quantity of the current are therefore the same as if a current

Fn
of quantity /' = ■ had passed through the wire for a time t, where

T= 2iV

In? n"^\

[-r^b)-

This method of considering a variable current of short duration is due to Weber, whose
experimental methods render the determination of the equivalent current a matter of great
precision.

Now let the electro-motive force F suddenly cease while the current in the primary wire
is /„ and in the secondary = 0. Then we shall have for the subsequent time

R n

p '

The equivalent currents axe. i /q and 4 ^o "Vv " > ^^^ their duration is t.

K n

When the communication with the source of the current is cut off, there will be a change
of R. This will produce a change in the value of r, so that if R be suddenly increased, the
strength of the secondary current will be increased, and its duration diminished. This is the
case in the ordinary coil-machines. The quantity N depends on the form of the machine,
and may be determined by experiment for a machine of any shape.

Mr maxwell, ON FARADAY'S LINES OF FORCE.

81

XII. Spherical shell revolving in magnetic field.

Let us next take the case of a revolving shell of conducting matter under the influence
of a uniform field of magnetic force. The phenomena are explained by Faraday in his
Experimental Researches, Series II., and references are there given to previous experiments.

Let the axis of z be the axis of revolution, and let the angular velocity be w. Let
the magnetism of the field be represented in quantity by /, inclined at an angle 9 to the
direction of z, in the plane of zx.

Let R be the radius of the spherical shell, and T the thickness. Let the quantities
Oo, /3o, 7o» l>6 t^^ electro-tonic functions at any point of space; a^, b„ c,, Oj, jS,, 71 symbols
of magnetic quantity and intensity; a^, b^, c^, a^, /Sj, 72 of electric quantity and intensity.
Let Pa be the electric tension at any point,

dpi
dx

/3. = |^ + A62 ^ 0)

7,= -+*C3

da^ db^ dc^

dx dy dz

2 dfi^ dy^

+ + ■

dx dy dz

da.

=V>

(2);

The expressions for a^, /3o, 70 ^^^ to the magnetism of the field are

ao = -^0 + ~ y cos 6,

(if, =' Ba + - (z sin 6 - X cos 6),

^ I • ^

yo= Co--y sm 9,

Aq, Bg, C(, being constants; and the velocities of the particles of the revolving sphere are

dx

dy
dt

= wx.

dz
di

= 0.

We have therefore for the electro-motive forces

1 doo I I
a^— — = cos y o)X,

^.= -

72 = -

47r dt

47r dt

— hi..
47r dt

4>7r 2

cos 9wy,

47r 2 ^

sin 9a)X.

47r2

Vol. X. Paet L

11

8a Mb maxwell, ON FARADAY'S LINES OF FORCE.

Returning to equations (l), we get

l [^ _ ^\ - ^ _ ^ =
\dz dyl dm dy

. (dc^ da^\ dy, da^ 1 I .
Kdic dss I dx dz 4nr x
(du^ _ dh^\ ^ da^ _ ^^' ^ p

Idu^ db^X da^ d^^
\dy dx) dy dx

From which with equation (2) we find

]_ J_ 7

A 47r 4

62 = 0,

a, = - — T- T sin Ouiss,

11/..

C2 = sin dwx,

k 4iir 4!

P2 = — — /w{ (.r* + y^) cos 6 - xz sin 6],

lOTT

These expressions would determine completely the motion of electricity in a revolving
sphere if we neglect the action of these currents on themselves. They express a system
of circular currents about the axis of y, the quantity of current at any point being
proportional to the distance from that axis. The external magnetic effect will be that

'Tips

of a small magnet whose moment is wl sin 6, with its direction along the axis of y,

^8irk

so that the magnetism of the field would tend to turn it back to the axis of x*.

The existence of these currents will of course alter the distribution of the electro-tonic

functions, and so they will react on themselves. Let the final result of this action be a

system of currents about an axis in the plane of xy inclined to the axis of x at an angle

and producing an external effect equal to that of a magnet whose moment is TB?.

The magnetic inductive components within the shell are

Ii sin — 9,1' cos (j) in a,

— 9,1' sin (p in y,

/i cos in z.

Each of these would produce its own system of currents when the sphere is in motion,

and these would give rise to new distributions of magnetism, which, when the velocity is

uniform, must be the same as the original distribution,

T ■

(/i sin 0-2/ cos (p) in m produces 2 w (/, sin - 2/* cos (pi) in y,

487rft

T

(- 9.1' sin 0) in y produces 2 a {9,1' sin 0) in a?;

487rA

/, cos 6 in X produces no currents.

* The expression for ^2 indicates a variable electric tension in the shell, so that currents might be collected by wires touching
it at the equator and poles.

Mr maxwell, ON FARADAY'S LINES OF FORCE. 88

We must therefore have the following equations, since the state of the shell is the same at
every instant,

T

L sin - 2 i* cos = /, sin + toS/ sin (b

T

— 2/' sin ^ "= w(Ji sin - 2i* cos <f>),

whence

247rA;
T
24,irk

^o^<P= - -^TZi «' -f = i ^^ .-^1 sin Q-

V.f-^

ID

2iTrk

To understand the meaning of these expressions let us take a particular case.

Let the axis of the revolving shell be vertical, and let the revolution be from north to
west. Let / be the total intensity of the terrestrial magnetism, and let the dip be 6, then
J cos 9 is the horizontal component in the direction of magnetic north.

The result of the rotation is to produce currents in the shell about an axis inclined at a

T

small angle = tan"' ^ -w to the south of magnetic west, and the external effect of these

currents is the same as that of a magnet whose moment is

1 Tw

^B?I cos 9.

2 v'247rA;]'+ Tu^
"The moment of the couple due to terrestrial magnetism tending to stop the rotation is

■mPcos^9,

2 247rAl^ + rV
and the loss of work due to this in unit of time is

247rA Tw^

R'rcos^9.

2 24^ + T'w*

This loss of work is made up by an evolution of heat in the substance of the shell, as is
proved by a recent experiment of M. Foucault, (see Comptea Bendus, xli. p. 450).

11—2

IV. The Structure of the Athenian Trireme ; considered with reference to certain
difficulties of interpretation. By J. W. Donaldson, D.D. late Fellom of
Trinity College, Cambridge.

[[Read November 6, 1856.]

The formal recognition of philology, as one of the subjects for discussion at the meetings
of the Cambridge Philosophical Society, seems to me to impose on those of the members, who
have more especially devoted themselves to this branch of academic study, the duty of sug-
gesting as soon as possible some discussion calculated to awaken an interest in this new or
rather additional department of our transactions. And as pure linguistic investigation is a
sealed book to many, and eminently uninviting to all those, who are not critical scholars by
profession, I have thought it best to take an application of philological research, on which
I have something new to offer, and which is, or ought to be, both intelligible and interesting
to all, who care for the language or the doings of the ancient Greeks.

As the Athenians, at the time when their literature assumed its distinctive form, were
pre-eminently a maritime people, it was to be expected that nautical terms would take their
place among the most usual figures of speech. Many of their best writers had either, as we
say, " served in the navy," or had become familiar with the language and habits of the sea-
ports. Even if the wealthier men had not personally served as strategi or trierarchs, or had
not made voyages for profit or pleasure, they had lounged in the dockyards and factories of
the Piraeus, and seen the triremes put to sea on some great expedition ; and if the poorer
citizens had not pulled the long oar on the upper benches, they had lived in familiar inter-
course with many whose hands were hardened with constant rowing, and whose ears were
ringing with the never ceasing drone of the pipe to which they kept stroke in the voyage or
the onset of battle. It is not at all surprising then that Attic literature is full of direct
allusions to the structure of the ship of war and to all the incidents of sea-life. And in
point of fact nothing is more common than the occurrence of nautical metaphors. But
although this has been duly noticed, and though much has been written on the subject, there
are still some phrases in common use, which have not yet received an adequate explanation,
and consequently some passages, which still require to be illustrated by a more complete and
accurate investigation of the Athenian trireme. It is my intention, in the present paper, to
submit to you some of the conclusions at which I have arrived after a renewed survey of the
ancient authorities.

It is a well-known fact that ships of war in the most glorious days of the Athenian republic
were mainly, if not entirely, triremes, or galleys with three banks of oars. This convenient
form of the rowing-vessel, combining, as it seems, the maximum of speed and power, was
invented by Ameinocles the Corinthian about 700 b. c. The elementary form, of which it

Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 85

was an extension, and which kept its place by the side of the trireme, was the penteconter or
single-banked galley with fifty rowers. The short flat-bottomed barges of the earliest sea-
men were not adapted either for rapid navigation or for warfare. And as soon as the
Greek mariners put out to sea either to trade with or to plunder distant cities, they seem to
have adopted the long sharp-prowed vessel with its twenty-five rowers on each side. Herodotus
says expressly that the Phocaeans, who navigated the Archipelago, the Adriatic, and the
western Mediterranean as far as Tartessus, used for this purpose ov aTpoyyiiXricri vrjvai, aWd
■n-evTrjKovTopoKTi (i. 163)", and the mythical Argo, which represents the first of those voyages,
half piratical, half commercial, which the Thessalians made into the Black Sea, was undoubt-
edly regarded as a penteconter. The tradition generally reckons fifty Argonauts, and it was
not without a distinct reference to this, that Pindar describes the dragon killed by Jason as
"bigger in length and breadth than a penteconter, which blows of steel have perfected"
{Pyth. IV. 255). In these galleys it is presumed that all the rowers were armed men, and
Homer is careful to tell us this in speaking of the penteconters which Philoctetes took to
Troy {II. II. 227). Whether the ships of the Boeotians, to which Homer gives a complement
of 120 men (7^. ii. l6), were biremes, or large penteconters, with double crews, is a point
which can hardly be decided; Pliny mentions {H. iV. vii. 57), on the authority of Damastes, a
contemporary of Herodotus, that the Erythrseans were the first to introduce biremes, but we
do not know when this form was originally adopted, and it is clear that the galley with two
banks was never very common. And Thucydides seems to have understood that the pente-
conters only were rowed by the soldiers, who in that case were bowmen, so that the other
vessels would contain, beside the rowers, who served as archers, some seventy hoplites, who
only pulled on an emergency. There is a special reason for coming t& this conclusion.
Thucydides (i. 10) speaks of the TreoiVe^ or supernumeraries in the ships which went to Troy,
and limits them to the kings and their suite. But the Scholiast says that this term included
all the eir'ifiaTai or soldiers on board. Now in the nautical inscriptions published by Bockh,
we have a particular class of oars called by this name, a\ irepiveip Kwirai, and it is probable
that these were intended to be used by the synonymous eirifiaTcu whenever additional hands
were wanted, to make head against wind or tide.

All things considered, we may take the penteconter as the oldest and most permanent type
of the Greek war-ship. Both with regard to the number of the crew, and the vessel's length
and breadth of beam, it was the basis or starting-point of the trireme. The crew of the
trireme consisted of about 170 rowers and 30 supernumeraries. As the length of the vessel
over all from forecastle to poop was greater than that of its keel, there were more seats for
rowers in the upper tier than in the two lower tiers, and the Scholiast on Aristophanes
{Ran. 1074) tells us that at the stern the first thranite sat before the first zygite, and the first
zygite before the first thalamite. It seems indeed that there were 62 Opavtrai, or bench-
rowers, in the highest tier, 54 ^vylTai or cross-bit-rowers, on the second tier, and the same
number of OaXanlTat, or main-hold-rowers, on the lowest tier. Unless then some of the
thranites were employed to work the two great oars, or irti^aXia, at the stern, they must have
had four ports on each side more than the lower tiers. Supposing that the penteconter had
exactly 50 rowers, it must have been nearly as long as the trireme, for it had 25 ports or

86 Db DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME,

holes for the oars, whereas the corresponding or lower part of the trireme was pierced for
27 holes on each side. And as the interscalmium, or space between the ports, was two cubits
(Vitruv. I. 2), or 3 feet 6 inches, we should require a length of 105 feet above, and 91 feet
below, exclusively of the steerage and bow, or parexeiresia. That the trireme and the oldest
penteconter were exactly of the same breadth of beam, I will prove directly. And of course
the height was not increased more than was necessary for the accommodation of the additional
tiers of rowers.

Having regard then to that permanence of numerical arrangements which is so remarkable
among the ancient Greeks, we must see at once that the broad-side of the penteconter cor-
responded to the enomoty or triakad, a body of 25 to 30 men, sworn to act together, and
constituting the basis of the Greek military system. Consequently, the whole crew of the
penteconter corresponded to the pentekostys, and the crew of the trireme was a lochus, con-
sisting, with the epibatcB, of four pentekostyes, which was the Lacedaemonian arrangement at
the first battle of Mantineia (Thuc. v, 68), or it was two lochi of 100 men each, if we prefer
Xenophon's subdivision {Rep. Lac. ii, 4).

In regard to these general features all is plain enough. Our difficulty commences, when
we come to speak of the arrangements for seating the three tiers of rowers, and it is here that
I hope to clear up some obscurities, and throw a little new light on the subject, Dr Arnold
has called this " an indiscoverable" or " unconquerable problem " (Rom. Hist. in. 572 on
Thucyd, iv. 32), and Mr James Smith, in his elaborate and interesting Essay On the Voyage
and Shipwreck of St Paul, has proposed a solution quite at variance with the meaning of the
Greek words which distinguish the classes of rowers*. Even Bockh, in his Archives of the
Athenian Navy, can give us no definite information, and inclines to the erroneous belief that

The following is Mr Smith's tiansveise section of a trireme. (Voyage and Shipwreck of St Paul, p. 194.)

a. Oar of thalamite seated on deck,

b. Oar of zygite seated on stool on deck.

c. Oar of thranite seated on stool on gangway.

Besides the objection stated in the text, that this arrangement will not explain the Greek names of the three tiers of
rowers, it is impossible to conceiTe that the best rowers should have been placed on a platform within reach of the enemies'
shot.

Db DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME, 87

the rowers of all three tiers were furnished with seats of the same kind attached to the ribs
of the vessels. I shall now endeavour to show, I believe for the first time, that the names of
the three tiers of rowers accurately describe the manner in which they worked in the ships.

I. The Zygitce.

There is a very primitive description of the structure of a Greek ship in the Odyssey v.
243 sqq., but we can infer from it that the ribs were always bound together with cross-beams
before they were covered with planks. These cross-beams or cross-bits are called "iKpia in the
passage to which I refer, a name elsewhere limited to the planks of the partial deck fore and
aft, which till a late period was the only KardaTpw/jia of a war-ship. As the main-yard is
termed the eiriKpiov in this passage, and as the Christian cross was designated as an "iKpiov, we
may conclude that the word implied a transverse or cross direction of these timbers; the root
is probably that of i/co-ja»ji/, and therefore, as we shall see, the word is synonymous with aeXna.
These cross-bits are called KXtfi^es in Homer, because, like the collar-bone, they locked
together the two sides of the ship. The poets call them creXfiara, a word containing the old
root set or sal, " to go" {New Crat. ^ 269), and implying that they furnished the means of
walking from one end to the other of the undecked vessel. The common name, retained to
the last in the Athenian navy, was ^vyd, "the yokes" or bridges which joined the opposite
sides of the ship. There is a reason for these changes of designation. In a mere pinnace,
like that constructed by Ulysses, there would be no occasion for a hold, and the cross-planks
might be placed close together, like the foot-boards of a boat. In this case, ^Kpia would be

TRANSVERSE SECTION OF AN ATHENIAN TRIREME.

a. Thranus, or long stool, placed on the alternate zygon, or supported by the seUs, and extending 7 feet amidships.

b. Zygon, or cross-plank, running athwart the vessel at intervals.

c. Thalamitic seat, 4ft. 6in.

d. Thalamos, or hold leading to antlos.

e. Platform for Epibatse running along the traphex, and 6 feet wide, with bulwarks of 3 feet.
/. Selis, or gangway, fore-and-aft, 4 feet wide.

.4— B, (Breadth of beam) =18 feet. C—D, (Depth) = 12 feet.

88 Db DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME.

an appropriate designation. In larger vessels, however, these "iKpta would be remanded to the
decks fore and aft, the cross-pieces would be separate KktiiSe^ or ^1170, to furnish a ready
access to the hold; and, in the case of a trireme, both to allow ventilation for the lowest tier
of rowers who worked there, and also to permit the officers, who gave them the stroke, to hear
the whistle or word of command, to say nothing of the fact that there was no room for a
complete deck between them and the second tier of rowers. Still, however, these i^vyd would
be a-eXmara, or means of walking from stem to stern; for, by the nature of the case, there was
no other footing. As then we know that there were ^vya in a Greek trireme, as the middle
tier of rowers were called ^yylrat, because they sat there (Jul. Poll. i. 87 : ra fieaa t^s vem
^vyd, ov o'l ^vy7Tai KaOrji/Tai), and as it was necessary that room should be economized, and
the length of the upper oars kept at a minimum, we conclude that these middle rowers
actually sat upon the transfra or cross-planks of the vessel. Bockh is led to the opposite
conclusion by the phrase eSpa^ Kwirrji ^vyias in one of his Inscriptions (11. 40, p. 286). But
this merely means that the trireme in question had one of the tvyd broken close to the oar-
hole, just as the same vessel is stated to have been defective in its Tpdcprj^ or bulwark. And
in a subsequent part of the same inscription (p. 291) we have the phrase twv ^vywu Keird-
TTtjvTai vevre, " only five of the cross-bits are supplied with oars," which implies that the
^vyd were the proper place for one class of the rowers.

II. The Thalamitce.

That the OoKan'iTai got their name from having their seats in the QdXanos (Jul. Poll. i.
87: ddXaiJLOs ov o'l OaXdnioi eperToviri), and that this meant the hold of the vessel, is quite
obvious, and it would generally be supposed that the hold was so called, because, like the
women's apartments, the nursery, the store-room, &c. in a house, it was the inner part, the
least accessible quarter of the ship. It may however be doubted, whether, in its proper
meaning, OdXa/mos, like BoXos, did not imply specifically a vaulted chamber. If so, the
hold, sloping inwards to the keel, would represent an inverted OdXa/ioi, just as the bees' cells
were called by this name (Anth. Pal. ix. 404, 2):

airXacTTOL yeipwv avToirayet^ 6aXa/j.ai,
i. e. " chambers not formed by the hands, but all of a piece." We have a similar inversion in
the laquear or lacunar of the cieling, which was an inverted pit, bin, tray or trough, and in
the word 066a, which properly meant a drinking-vessel with a shaip point at the bottom,
but was also used to designate a cap, with a sharp point at the top. In fact the words "cap"
and "cup" might be taken as different forms of the same word denoting inverted uses of
the same object. Be this as it may, it is clear that the OaXaixlrai sat in the hold, with
their feet upon the water-line; and as there was no lower range of cross-bits, they must have
had benches projecting from the side of the ship. It is just possible that these benches were
technically called OdXafioi. At least, in the curious story told by Timseus (ap. Athen. p. 37)
of the young men at Agrigentuni who fancied that their house was a trireme at sea, one of
them says viro tov Seovs KaTafiaXwv efiavrov vtto to()s OaXafjiovi, ws evi fxaXiara Karwrarw

Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 89

eKeiixrjv, "having flung myself, in my fear, under the 9a\aiuot, I lay as low down as possible."
The bottom of the hold, however, was also called the ai/rXos, a name given afterwards to
the bilge-water which settled there, and to the pump, by which it was bailed out.

III. The Thranitce.

An examination of the name of the dpavlrai or "benchmen" of the highest tier, leads
to some very interesting results. The whole of this tier was called the Opavov, because the
rowers were seated on benches, which did not reach across the vessel, but rested by means of
short legs on the i[vyd beneath, so as to resemble a Qprjw's or foot-stool. It has been supposed
that Optjvvs and Opavo^ are other forms of Op6vo%, but this seems very unlikely. It would be
more reasonable to connect 6p6vo% with the root arop-, and to understand an original form
(TTpoi'o^, but to recognize in Gpavos or Oprjvvs the root of Opavw ; for the idea conveyed by
the latter is that of a fragment or separate piece, the dpovos being the seat with its cushion,
and the Oprjvvs the detached vTroiroSiov. And this view is not affected by the consideration that
the Qprjvvi in a trireme was really a seat and not a foot-stool. It could only have been high
enough to enable the QpaviTrj's to use the "(vyou immediately before him as a stretcher, and to
carry the handle of his oar clear of the '(vy'iTri's below and behind him ; and, by a proper arrange-
ment of the seats, less than one foot six inches would suffice for this. Now we know that
the Opijvvs was seven feet long, even in Homer's time. It was therefore just like a low foot-
stool placed on the (^vyoi'. Why it was so constructed may easily be shown. If the 6prjvvi
had run quite across the ship, the ^vylrai and OaXafUTOi could not have got to their places
without passing over the upper benches, and there would have been no passage fore and aft
for the officers of the vessel. It must always be recollected that the trireme was not a three-
decker, but a mere galley with three tiers of benches, and till a comparatively late period only
partially decked over all. When the deck was introduced, it was carried from the poop to the
forecastle, either so raised in the middle that there was room for a man to walk upright along
the ^vya, or else carried to the same height above the bulwarks on each side, in which case
the sides of the bulwark were an open grating for the whole length of the vessel. Originally,
however, the "iKpia were confined to the two ends of the vessel, and in going amidship it was
necessary to step down, first to a Qptjvvi and then to the "(iiya. In Homer''s account of the
attack on the Greek ships, which were drawn ashore, with their heads to the sea, it is stated
that Ajax, who was their chief defender, passed along the line of quarter-decks, jumping from
ship to ship, like a horse-vaulter, and driving off" the enemy with a punting pole 22 cubits long;
until at last he was obliged to yield to superior numbers, and retired a little way {ave\aXeTo
tvtQov) i. e. so as merely to get out of immediate danger, to a bench seven feet long (Opijwv
e(p' tiTTairo^riv), and "he left the deck of the equal ship" (XiVe c' 'iKpia vrjo^ e'iat]^); in this
lower position he stood watching, and repulsing with his long pole any Trojan who en-
deavoured to set fire to the vessels (Z?. xv. 674 — 731). That the Oprjvvs was always seven feet
long, in other words, that the war-ship had always the same breadth of beam, appears from
the following considerations. In order to give the full advantage of the leverage for the
longest oar, it is manifest that the rowers of the upper tier would sit as far as they could
Vol. X. Paet I. 12

90 Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME.

from the side of the vessel. Consequently the passage for the officers, &c. along the ^vyd
would be as narrow as possible. Now the minimum breadth for the free and rapid passage of
a man up to his knees is two feet. With seven feet then for each of the benches, and two
feet at least for the passage between them, we require sixteen feet for the minimum breadth of
the trireme, and I am informed by travellers, who have just returned from Athens, and who
have measured the slips in the docks of the Piraeus, that this was precisely the breadth allowed
for a Greek war galley under the water-line. Adding two feet for the breadth between
the tops of the ribs, we shall get the means of passing the mast, and the whole beam will be
eighteen feet, or, including the projecting gangways for the epibatcB, twenty-four feet over all.
For the height of the trireme's sides and its draught, we have no authority. I conjecture that
it drew about six feet, and that there was about the same depth from the platform of the
Epibatae to the water-line. Considering that the trireme was a sea-boat, and that the ports for
the oars were large enough to admit of a man's head being thrust through them (Herod, v. 33),
and to expose the rowers to missiles from boats rowing along-side (Thucyd. vii. 40), it is
extremely unlikely that the lower ports would be less than two feet above the water. And as
the oars were not too long to be carried by a single man on a march across the Isthmus
(Thucyd. n. 93) even those of the thranitcB must have been less than twenty feet long. The
inscriptions mention the length of the supplementary oars only, and these seem to have
varied from nine to nine and a half cubits. I have no doubt that the thranitic oars were
longer than this, and the epithet IdKi-^rtpeTnoi which Pindar applies to ^gina {01. viii. 20),
indicates that the length of the working oars in a trireme was as considerable as that of
the long spear which was similarly designated (Hom. //. xxi. 155 : ^oXiyeyKyK, iii. 346, &c. :
^oKiyocTKiov eyKo$). And this must have been the case if they were pulled with a good lever-
age. The best result that I can obtain by conjectural measurements gives about fifteen feet
for the thranitic oars, of which five feet were within and ten without the ship; twelve feet
for the zygitic oars, and nine or ten for the thalamitic. That there was a great difference
between the length of the thranitic oars and those of the lower tiers is implied by what Thu-
cydides says (vi. 31), as illustrated by the Scholiast: o\ ^e dpainrai ixerd fxaKpoTepwv kwttwv
epeTTovTei TrXeiova kottov e'^ovai tuov oKKuiv' oia tovto tovtok ij-ovoi^ eTricocreK enoiovvTo oi
Tpirjpdp^ai ov-)(t ^e irdai toI^ eperai^. It appears that all the oars were longest at the middle
of the ship. For though the oar-blades touched the water in the same line, the trireme was
broader in the middle, the thranus was longer there, and the rower sat farther from the side.
This is clear from what Galen says, when he compares the oars to the fingers of the human
hand when clenched (de usu partium corporis humani, I. 24, Vol. iii. p. 85, Kuhn) : KaOdirep
oi/xai Kav Tats Tpitjpecri Ta ireipara twv kcottoov eis tcrov e^iKveiTai kui toi y ovv ovk tcrwv
diraaoov ovawv, Kal yap ovv kuksi tos jueffas (leyieTTas- Aristotle makes a similar comparison
(de partibus animalium, iv. 10, ^ 27 : 6 /xeaoi [5a/cTuXoj] fiuKpos, uxnrep Kiiirt] fieaovewi) ; and
he enters more fully into the subject in his Mechanica, c. 4, where he answers the question :
Ota Ti o\ fxeaoveoi (xakiaTa xjyi/ vavv Kivovatv ; by referring to the principle of the lever —
though he takes the water as the weight and the rowlock as the fulcrum — and having asserted
the principle, he says : kv neari oe Trj vrfi irXelcTTov rfj^ Konrri's evros ea-riv' kui yap r; i/avs
TavTri evpvTUTrj effxtV, wcrre TrXeiov eir a/n(poTepa ei/oe^ea^at /mepo^ rijs koSttt)^ eKarepov

Dr DONALDSON, ON THE STRUCTUftE OF THE ATHENIAN TRIREME. 91

To«'^oi; ecTo? elvai Trji vews, — and at the end he adds : 6id tovto oi fxeaoveoi fiaXiaTa
KivovGiV [jieyiffTov yap ev fxecrr] vr)i to airo tow (TKaXfxou t^s Kwjnr)^ to ci'tos' ecmv. To a
strange misunderstanding of these statements respecting the oars at the middle of the trireme
combined with the remark of the Scholiast on Aristophanes (above, p. 4) that each zygite
sat between the thranite and thalamite immediately next to him, and the words of Pollux
(above, p. 7) that the ^vya were to. fxecra t^s vew^ {i. e. considering the three tiers as hori-
zontal lines), we owe the perplexing theory, first started, I believe, by Schneider in his Lexicon,
s. V. ixeaoveoi, that the ssygites, as a body, sat in the middle of the ship, and that their oars
were the longest ! The inferior position of the thalamites as compared with the other rowers
is coarsely intimated by Aristophanes (EancB 1074), and implied in the fact that they were left
on board when the rest of the crew disembarked to serve on shore (Thucyd. iv. 32). And
from what Aristophanes says, in his description of the bustle in the dockyard which attended
a sudden preparation for sea, I am disposed to infer that the first step in the equipment of a
trireme was to provide it with oars for the thalamites, who navigated the vessel provisionally,
and until it got its full complement or fighting crew ; for, in immediate connexion with making
the spars into oars (Kwrrewv irXaTovfxevwv), he speaks of fitting the lowest oars with thongs
{Iddkanitov TpoirovfjLevcov, Acharn. 552, 553). The interval between two oar-ports on the same
tier was two cubits (Vitruv. i. 2), or three feet six inches, and as the thranite sat before (i. e.
nearer to the stern than) the zygite, and he than the thalamite, it is not difficult to conceive
an arrangement by which the bodies of the lower rowers would have free play as they
bent forward to their work. The measurement, which I have proposed (p. 6), leaves ample
room for the thalamites to pull under the platform for the epibatae. It is not impossible
that the thranus rested on the selis, so that there were xyga or cross planks only where the
zygites sat. This seems to be suggested by the explanation in Julius Pollux (i. 87): to ^e
irepi TO KaTaaTpwua Opavos, oil o'l Opavtrat, for the only KaraaTpwixa was the gangway.

I will now apply these considerations to the removal of some difficulties which have
been very troublesome to editors.

(o) The conjecture that the interval between the ends of the upper benches or thranos was
intended to leave a passage along the cyeXiuara or i[vyd is supported by the fact that the
special name for this passage was aeXk, a name also given to the spaces between the benches
in the theatre. Hesychius defines the aeXiSas as ret fxera^u ^laCppaytiaTa rwv oiacrrrjiJLdTcov
T^s i/e<os, " the middle partitions of the passages in the ship." And that this was the primary
meaning is clear from the glosses in Eustathius and Julius Pollux, which connect aeX/s with
aeXfxa. In later times aeX'i's was commonly used to denote the blank space between two
columns in a written page. When Phrynichus says (Bekk. Anecd. 62, 27) : aeXh (iifiXiov
XeyeTai ce Kal creXh QeaTpov, like a grammarian, he confuses between the primary and the
secondary meaning. The application of this term to the intercolumnal space in a manuscript,
and hence to the page of a book in general, is due to the resemblance between the KepKiSes of
the theatre, which were divided by the creX/^es, and the lines of writing divided by the inter-
vening space of blank paper; and the corridors of the theatre again were called aeXiSei,
because they were flanked on each side by seated spectators, just as the creXi^es in the trireme

12—2

92 Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME.

passed between rowers seated below one another. And hence we derive the explanation of the
passage in Aristophanes (Equites 546), which has been found unintelligible :

cupecrO' avTw iroXv to poOiov, ■jrapairefi'^aT' e(f> evoexa KWTrajs
Qopvfiov ■^prjaTOV XrjvdiTTjv —
" raise for him a plash of applause in good measure, and waft him a noble Lenaean cheer with
eleven oars." It seems that there were eleven tiers of seats between each diazoma of the
Theatre at Athens, the diazoma itself being counted as the twelfth row. Accordingly, each
wedge would suggest the idea of eleven benches of rowers, and the applause, which the chorus
demands, would come like the plash of eleven oars striking the water at once.

(6) As the o-eXis was the only uninterrupted thoroughfare by which the officers could
pass to and fro to give their orders and keep the men to their work, we get at last the long
sought explanation of a passage in jEschylus, which all the commentators have failed to eluci-
date. In the course of the altercations between JEgisthus and the chorus at the end of the
Agamemnon, the usurper is made to address the senators as follows (v. 1588):

av TavTU <pwi/6is veprepa yr pocnj fuevo^
KWTrri, KpuTovvTwv Tcov em i^oyifi oopoSi
" These words from thee, that sittest at the oar
Below, while rulers on the cross-bits walk ?"

Here the editors are quite at sea. They cannot understand why the ^vy'irai should be
described as the Kparovvres instead of the dpav'irai. Dr Blorafield went so far, in his struggle
to get out of the difficulty, as to suppose that the old men of the Chorus were the daXa/ixioi,
iEgisthus and Cly toemnestra the i^vy'iTai, and the murdered Agamemnon the dpavirrj^ ! Paley
is satisfied with saying, that the third tier was as inferior to the second, as the second was to
the first, " quare satis recte se habet comparatio." And Klausen fancies he has unravelled the
perplexity by supposing that J^schylus is speaking of a bireme, being quite ignorant of the
fact, that if biremes had been used at Athens, the upper tier of rowers would still have been
QpaviTaiW The fact is that all these commentators have overlooked a refinement of Greek
Syntax, ^schylus, who was as well acquainted with sea-life as any of the men that pulled
at Salamis, has been careful to introduce the participle irpoarinevo^ in speaking of the rower,
while by writing eTrJ ^"'y^ instead of eirl X^vyuiv, he expressly tells us that the KpaTovvres
were not seated on the ^i'7a'> but had their feet upon them. Every Greek scholar is aware
that when we wish to say that a man is seated with his legs hanging from his seat, whether it
be on a chair, a rowing-bench, or on horse-back, we use kirl with the genitive; but 'eir\ with
the dative, when we wish to say that the whole man is upon that which serves as his footing.
If the officers had seats they were placed upon the Xvyd, and were much higher than the
stools of the 9paviTai, so that even when seated, the Kparovvre^, or officers, might speak of
the rowers of the highest tier as veprepa ■irpoar]fievov^ Kwirri. Their seats then being placed
on the ^vya, they might be said either KaOrjcrOat or earmevai eiri ^uyoii, because their feet
rested on them; but the l^vy'irai could only be said KaGfjaQai eiri (yywv. Hence we have
in Eurip. Phmniss. 74: evel ^ eirl ^vyoii KaOe^er ap)(fis, and Eustathius tells us that the

Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 93

Homeric epithet v\l/i([iryos is derived from the high seat of the pilot in a ship (p. 131,
18): Kal TovTo 06 oLTTo KvfiepvrjTiKtj's fj.eT)jveKTai KaTaaTaa-ews. For the same reason JEschy-
lus speaks of the Gods as aeX/ua aefipov ^fievoov {Agam. 176).

(c) Another difficult passage in the same play furnishes an illustration of the fact that
the middle part of the aeXjuara or ^vyd, in an old Greek vessel, belonged to the officers and
supernumeraries. In v. 1413 it is said of Cassandra, who came with Agamemnon from Troy,
that she was vavTiKwv areXfiarwu taTorpi/Sj/s, where some read iaoTpijirys. The allusion to
Chryseis a line or two before makes it probable that ^Eschylus had in his recollection the lines
in the Iliad, where Agamemnon says that old age shall find her : 'igtov eiroiyofxevr}v Kal emov
Xej^os di/Tiocoaav. Here it is implied that the atXnara were her only gynceceum, just as
Persius says (v. 146): "tun' mare transsilies .? tibi torta cannabe fulto, coena sit in transtroT''
Or if IffTos has its nautical meaning, it will imply that the captain's quarters were amidships
near the mast. But to this it may be objected with reason that, at all events in later times,
the captain or admiral occupied a pavilion or round-house on the poop ; Jul. Poll. i. 87 : efcel
irov Kal aKrjVT) ovofxaXeTai to Trriyvvfievov aTparrjytp rj Toit^papyw. And ^Eschylus himself
describes the sovereign of a state as a pilot or captain who keeps sleepless watch at the helm
on the quarter-deck of the city {Sept c. Theb. 2, 3 : ootjs (pvXdaaei irpdyos ev Trpv/xvri iroXew;
oiaKa vwfLwv, fiXeipapa /xrj Koifxwv vTrvw)-

(d) To the practice of moving fore and aft along these cross-planks with frequent
intervals, at least where the rowers sat, even if the selis was planked, I also refer the proverbial
expression of warning, that " we must take care not to step into the bilge-water, or put our
foot into the hold" (eh avrXov e/mfi^cTai rro^a, Eurip. Hercul. l68). It is clear, from this mode
of describing it, that the caution referred to some risk of common occurrence. Mr Haliburton
connects the corresponding American phrase of "putting your foot into it" with an incident in the
backwoods, where a bear grapples with a saw-mill, and is bisected accordingly. Some risk not
much less formidable is implied in the Greek expression. When ^schylus says {Choeph. 695) :
e^w KOfxii^oov dXeOpiov -rrtjXov tto^o, he refers to an escape from serious danger, and not to the
mere avoidance of dirt. So this phrase cannot apply to the fear of getting one's feet wet with
bilge-water, or with dirty water in general, but must mean that there was a constant risk of
tumbling between the ^vyd, to the very bottom of the ship, if those who walked across the
planks did not attend to their feet; and that this often happened with serious consequences to
the sailors, officers, and passengers in a trireme.

I submit these observations in the hope that they will tend to clear up some obscurities in
Greek history and antiquities, and, at all events, reconcile the language of the best authorities
with a probable theory respecting the structure and management of the swift war-boat whicli
dashed through the water and wheeled round at the command of some sea-captain like
Phormio, or, as the Greek poet says, sped across the main, keeping pace with the hundred
feet of the Nereids (Soph. (Ed. Col. 720 sqq.).

v. Of the Platonic Theory of Ideas. By W. Whewell, D.D. Master of Trinity

College, Cambridge.

^Read November 10, 1856.]

Though Plato has, in recent times, had many readers and admirers among our English
scholars, there has been an air of unreality and inconsistency about the commendation which
most of these professed adherents have given to his doctrines. This appears to be no captious
criticism, for instance, when those who speak of him as immeasurably superior in argument to
his opponents, do not venture to produce his arguments in a definite form as able to bear the
tug of modern controversy; — when they use his own Greek phrases as essential to the expo-
sition of his doctrines, and speak as if these phrases could not be adequately rendered in
English; — and when they assent to those among the systems of philosophy of modern times
which are the most clearly opposed to the system of Plato. It seems not unreasonable to
require, on the contrary, that if Plato is to supply a philosophy for us, it must be a phi-
losophy which can be expressed in our own language; — that his system, if we hold it to be
well founded, shall compel us to deny the opposite systems, modern as well as ancient; — and
that, so far as we hold Plato''s doctrines to be satisfactorily established, we should be able to
produce the arguments for them, and to refute the arguments against them. These seem
reasonable requirements of the adherents of any philosophy, and therefore, of Plato's.

I regard it as a fortunate circumstance, that we have recently had presented to us an
exposition of Plato's philosophy which does conform to those reasonable conditions; and we
may discuss this exposition with the less reserve, since its accomplished author, though
belonging to this generation, is no longer alive. I refer to the Lectures on the History of
Ancient Philosophy, by the late Professor Butler of Dublin. In these Lectures, we find an
account of the Platonic Philosophy which shews that the writer had considered it as, what it
is, an attempt to solve large problems, which in all ages force themselves upon the notice of
thoughtful men. In Lectures VIII. and X., of the Second Series, especially, we have a
statement of the Platonic Theory of Ideas, which may be made a convenient starting point
for such remarks as I wish at present to make. I will transcribe this account; omitting,
as I do so, the expressions which Professor Butler uses, in order to present the theory, not as
a dogmatical assertion, but as a view, at leas>t not extravagant. For this purpose, he says, of
the successive portions of the theory, that one is "not too absurd to be maintained;" that
another is " not very extravagant either;" that a third is " surely allowable;" that a fourth

Dr WHEWELL, on the PLATONIC THEORY OF IDEAS. 95

presents "no incredible account" of the subject; that a fifth is "no preposterous notion in
substance, and no unwarrantable form of phrase." Divested of these modest formulae, his
account is as follows: [Vol. ii. p. 117.]

*' Man's soul is made to contain not merely a consistent scheme of its own notions, but a
direct apprehension of real and eternal laws beyond it. These real and eternal laws are
things intelligible, and not things sensible.

" These laws impressed upon creation by its Creator, and apprehended by man, are some-
thing distinct equally from the Creator and from man, and the whole mass of them may
fairly be termed the World of Things Intelligible.

" Further, there are qualities in the supreme and ultimate Cause of all, which are mani-
fested in His creation, and not merely manifested, but, in a manner — after being brought out
of his superessential nature into the stage of being [which is] below him, but next to him —
are then by the causative act of creation deposited in things, differencing them one from the
other, so that the things partake of them (/meTe'x^ovai), communicate with them {KOivcovovai)'

" The intelligence of man, excited to reflection by the impressions of these objects thus
(though themselves transitory) participant of a divine quality, may rise to higher conceptions of
the perfections thus faintly exhibited; and inasmuch as these perfections are unquestionably
real existences, and known to be such in the very act of contemplation, — this may be
regarded as a direct intellectual apperception of them, — a Union of the Reason with the
Ideas in that sphere of being which is common to both.

" Finally, the Reason, in proportion as it learns to contemplate the Perfect and Eternal,
desires the enjoyment of such contemplations in a more consummate degree, and cannot be
fully satisfied, except in the actual fruition of the Perfect itself.

" These suppositions, taken together, constitute the Theory of Ideas."

In remarking upon the theory thus presented, I shall abstain from any discussion of the
theological part of it, as a subject which would probably be considered as unsuited to the
meetings of this Society, even in its most purely philosophical form. But I conceive that it
will not be inconvenient, if it be not wearisome, to discuss the Theory of Ideas as an attempt
to explain the existence of real knowledge ; which Prof. Butler very rightly considers as the
necessary aim of this and cognate systems of philosophy*.

I conceive, then, that one of the primary objects of Plato's Theory of Ideas is, to
explain the existence of real knowledge, that is, of demonstrated knowledge, such as the
propositions of geometry offer to us. In this view, the Theory of Ideas is one attempt to
solve a problem, much discussed in our times, What is the ground of geometrical truth ? I
do not mean that this is the whole object of the Theory, or the highest of its claims. As I
have said, I omit its theological bearings; and I am aware that there are passages in the
Platonic Dialogues, in which the Ideas which enter into the apprehension and demonstration of
geometrical truths are spoken of as subordinate to Ideas which have a theological aspect.
But I have no doubt that one of the main motives to the construction of the Theory of Ideas

* P. 116. "No amount of human knowledge can be adequate which does not solve the phenomena of these absolute certainties."

96 dk whewell, on the Platonic theory of ideas.

was, the desire of solving the Problem, " How is it possible that man should apprehend
necessary and eternal truths?" That the truths are necessary, makes them eternal, for
they do not depend on time; and that they are eternal, gives them at once a theological
bearing.

That Plato, in attempting to explain the nature and possibility of real knowledge, had in
his mind geometrical truths, as examples of such knowledge, is, I think, evident from the
general purport of his discourses on such subjects. The advance of Greek geometry into a
conspicuous position, at the time when the Heraclitean sect were proving that nothing could
be proved and nothing could be known, naturally suggested mathematical truth as the refu-
tation of the skepticism of mere sensation. On the one side it was said, we can know nothing
except by our sensations; and that which we observe with our senses is constantly changing;
or at any rate, may change at any moment. On the other hand it was said, we do know
geometrical truths, and as truly as we know them, we know that they cannot change. Plato
was quite alive to the lesson, and to the importance of this kind of truths. In the Meno
and in the Phcedo he refers to them, as illustrating the nature of the human mind : in the
Republic and the Timceus he again speaks of truths which far transcend anything which the
senses can teach, or even adequately exemplify. The senses, he argues in the ThecBtetus,
cannot give us the knowledge which we have; the source of it must therefore be in the mind
itself; in the Ideas which it possesses. The impressions of sense are constantly varying, and
incapable of giving any certainty: but the Ideas on which real truth depends are constant and
invariable, and the certainty which arises from these is firm and indestructible. Ideas are the
permanent, perfect objects, with which the mind deals when it contemplates necessary and
eternal truths. They belong to a region superior to the material world, the world of sense.
They are the objects which make up the furniture of the Intelligible World: with which the
Reason deals, as the Senses deal each with its appropriate Sensation.

But, it will naturally be asked, what is the Relation of Ideas to the Objects of Sense ?
Some connexion, or relation, it is plain, there must be. The objects of sense can suggest,
and can illustrate real truths. Though these truths of geometry cannot be proved, cannot
even be exactly exemplified, by drawing diagrams, yet diagrams are of use in helping ordinary
minds to see the proof; and to all minds, may represent and illustrate it. And though our
conclusions with regard to objects of sense may be insecure and imperfect, they have some
shew of truth, and therefore some resemblance to truth. What does this arise from.!" How
is it explained, if there is no truth except concerning Ideas?

To this the Platonist replied, that the phenomena which present themselves to the senses
partake, in a certain manner, of Ideas, and thus include so much of the nature of Ideas,
that they include also an element of Truth. The geometrical diagram of Triangles and
Squares which is drawn in the sand of the floor of the Gymnasium, partakes of the nature of
the true Ideal Triangles and Squares, so that it presents an imitation and suggestion of the
truths which are true of them. The real triangles and squares are in the mind: they are, as
we have said, objects, not in the Visible, but in the Intelligible World. But the Visible
Triangles and Squares make us call to mind the Intelligible; and thus the objects of sense
suggest, and, in a way, exemplify the eternal truths.

Dr WHEWELL, on the PLATONIC THEORY OF IDEAS. 97

This I conceive to be the simplest and directest ground of two primary parts of the
Theory of Ideas; — The Eternal Ideas constituting an Intelligible World; and the Partici-
pation in these Ideas ascribed to the objects of the world of sense. And it is plain that so far,
the Theory meets what, I conceive, was its primary purpose; it answers the questions, How
can we have certain knowledge, though we cannot get it from Sense ? and, How can we have
knowledge, at least apparent, though imperfect, about the world of sense ?

But is this the ground on which Plato himself rests the truth of his Theory of Ideas ?
As I have said, I have no doubt that these were the questions which suggested the Theory;
and it is perpetually applied in such a manner as to shew that it was held by Plato in this
sense. But his applications of the Theory refer very often to another part of it; — to the
Ideas, not of Triangles and Squares, of space and its affections; but to the Ideas of Relations —
as the Relations of Like and Unlike, Greater and Less; or to things quite different from
the things of which geometry treats, for instance, to Tables and Chairs, and other matters,
with regard to which no demonstration is possible, and no general truth (still less necessary
and eternal truth) capable of being asserted.

I conceive that the Theory of Ideas, thus asserted and thus supported, stands upon very
much weaker ground than it does, when it is asserted concerning the objects of thought,
about which necessary and demonstrable truths are attainable. And in order to devise argu-
ments against this part of the Theory, and to trace the contradictions to which it leads, we
have no occasion to task our own ingenuity. We find it done to our hands, not only in
Aristotle, the open opponent of the Theory of Ideas, but in works which stand among the
Platonic Dialogues themselves. And I wish especially to point out some of the arguments
against the Ideal Theory, which are given in one of the most noted of the Platonic Dialogues,
the Parmenides.

The Parmenides contains a narrative of a Dialogue held between Parmenides and Zeno,
the Eleatic Philosophers, on the one side, and Socrates, along with several other persons, on
the other. It may be regarded as divided into two main portions ; the first, in which the
Theory of Ideas is attacked by Parmenides, and defended by Socrates ; the second, in which
Parmenides discusses, at length, the Eleatic doctrine that All things are One. It is the
former part, the discussion of the Theory of Ideas, to which I especially wish to direct
attention at present : and in the first place, to that extension of the Theory of Ideas, to
things of which no general truth is possible ; such as I have mentioned, tables and chairs.
Plato often speaks of a Table, by way of example, as a thing of which there must be an
Idea, not taken from any special Table or assemblage of Tables ; but an Ideal Table, such that
all Tables are Tables by participating in the nature of this Idea. Now the question is,
whether there is any force, or indeed any sense, in this assumption ; and this question is
discussed in the Parmenides. Socrates is there represented as very confident in the existence
of Ideas of the highest and largest kind, the Just, the Fair, the Good, and the like.
Parmenides asks him how far he follows his theory. Is there, he asks, an Idea of Man, which
is distinct from us men ? an Idea of Fire.'' of Water ? " In truth," replies Socrates, " I have
often hesitated, Parmenides, about these, whether we are to allow such Ideas." When
Plato had proceeded to teach that there is an Idea of a Table, of course he could not reject
Vol. X. Part I. 13

98 dk whewell, on the platonic theory of ideas.

such Ideas as Man, and Fire, and Water. Parmenides, proceeding in the same line, pushes
him further still. " Do you doubt," says he, " whether there are Ideas of things apparently
worthless and vile? Is there an Idea of a Hair? of Mud? of Filth?" Socrates has not the
courage to accept such an extension of the theory. He says, " By no means. These are
not Ideas. These are nothing more than just what we see them. I have often been perplexed
what to think on this subject. But after standing to this a while, I have fled the thought, for
fear of falling into an unfathomable abyss of absurdities." On this, Parmenides rebukes him
for his want of consistency. " Ah Socrates," he says, " you are yet young; and philosophy
has not yet taken possession of you as I think she will one day do — when you will have
learned to find nothing despicable in any of these things. But now your youth inclines you
to regard the opinions of men." It is indeed plain, that if we are to assume an Idea of a
Chair or a Table, we can find no boundary line which will exclude Ideas of everything for
which we have a name, however worthless or offensive. And this is an argument against the
assumption of such Ideas, which will convince most persons of the groundlessness of the
assumption : — the more so, as for the assumption of such Ideas, it does not appear that Plato
offers any argument whatever; nor does this assumption solve any problem, or remove any
difficulty*. Parmenides, then, had reason to say that consistency required Socrates, if he
assumed any such Ideas, to assume all. And I conceive his reply to be to this effect; and to
be thus a reductio ad absurdum of the Theory of Ideas in this sense. According to the
opinions of those who see in the Parmenides an exposition of Platonic doctrines, I believe that
Parmenides is conceived in this passage, to suggest to Socrates what is necessary for the com-
pletion of the Theory of Ideas. But upon either supposition, I wish especially to draw the
attention of my readers to the position of superiority in the Dialogue in which Parmenides is
here placed with regard to Socrates.

Parmenides then proceeds to propound to Socrates difficulties with regard to the Ideal
Theory, in another of its aspects ; — namely, when it assumes Ideas of Relations of things ;
and here also, I wish especially to have it considered how far the answers of Socrates to these
objections are really satisfactory and conclusive.

" Tell me," says he (§ 10, Bekker), "You conceive that there are certain Ideas, and that
things partaking of these Ideas, are called by the corresponding names ;■ — an Idea of Likeness,
things partaking of which are called Like; — o{ Greatness, whence they are Great: of Beauty,
whence they are Beautiful?" Socrates assents, naturally: this being the simple and universal
statement of the Theory, in this case. But then comes one of the real difficulties of the
Theory. Since the special things participate of the General Idea, has each got the whole of
the Idea, which is, of course, One; or has each a part of the Idea? " For," says Parmenides,
" can there be any other way of participation than these two .''" Socrates replies by a simili-
tude : " The Idea, though One, may be wholly in each object, as the Day, one and the same,
is wholly in each place." The physical illustration, Parmenides damages by making it more
physical still. "You are ingenious, Socrates," he says, (§ 11) "in making the same thing be in

• Prof. Butler, Lect. ix. Second Series, p. 136, appears to I for the assumption of such Ideas ; but I see no trace of
think that Plato had sufficient grounds (of a theological kind) | them.

Db WHEWELL, on the PLATONIC THEORY OF IDEAS. 99

many places at the same time. If you had a number of persons wrapped up in a sail or web,
would you say that each of them had the whole of it.'' Is not the case similar.?" Socrates
cannot deny that it is. "But in this case, each person has only a part of the whole; and
thus your Ideas are partible." To this, Socrates is represented as assenting in the briefest
possible phrase; and thus, here again, as I conceive, Parmenides retains his superiority over
Socrates in the Dialogue.

There are many other arguments urged against the Ideal Theory of Parmenides. The
next is a consequence of this partibility of Ideas, thus supposed to be proved, and is ingenious
enough. It is this:

" If the Idea of Greatness be distributed among things that are Great, so that each has
a part of it, each separate thing will be Great in virtue of a part of Greatness which is less
than Greatness itself. Is not this absurd ?" Socrates submissively allows that it is.

And the same argument is applied in the case of the Idea of Equality.

" If each of several things have a part of the Idea of Equality, it will be Equal to some-
thing, in virtue of something which is less than Equality."

And in the same way with regard to the Idea of Smallness.

" If each thing be small by having a part of the Idea of Smallness, Smallness itself
will be greater than the small thing, since that is a part of itself."

These ingenious results of the partibility of Ideas remind us of the ingenuity shewn in the
Greek geometry, especially the Fifth Book of Euclid. They are represented as not resisted
by Socrates {§ 12) : "In what way, Socrates, can things participate in Ideas, if they cannot do
so either integrally or partibly .!*" '' By my troth," says Socrates, "it does not seem easy to
tell." Parmenides, who completely takes the conduct of the Dialogue, then turns to another
part of the subject and propounds other arguments. " What do you say to this?" he asks.

" There is an Ideal Greatness, and there are many things, separate from it, and Great by
virtue of it. But now if you look at Greatness and the Great things together, since they
are all Great, they must be Great in virtue of some higher Idea of Greatness which includes
both. And thus you have a Second Idea of Greatness ; and in like manner you will have a
third, and so on indefinitely."

This also, as an argument against the separate existence of Ideas, Socrates is represented
as unable to answer. He replies interrogatively :

" Why, Parmenides, is not each of these Ideas a Thought, which, by its nature, cannot
exist in anything except in the Mind ? In that case your consequences would not follow."

This is an answer which changes the course of the reasoning : but still, not much to
the advantage of the Ideal Theory. Parmenides is still ready with very perplexing argu-
ments. (§ 13.)

" The Idea, then," he says, " are Thoughts. They must be Thoughts of something.
They are Thoughts of something, then, which exists in all the special things; some one
thing which the Thought perceives in all the special things; and this one Thought thus
involved in all, is the Idea. But then, if the special things, as you say, participate in the
Idea, they participate in the Thought ; and thus, all objects are made up of Thoughts, and
all things think ; or else, there are thoughts in things which do not think."

13—2

100 Dr WHEWELL, on the PLATONIC THEORY OF IDEAS.

This argument drives Socrates from the position that Ideas are Thoughts, and he moves
to another, that they are Paradigms, Exemplars of the qualities of things, to which the things
themselves are like, and their being thus like, is their participating in the Idea. But here too,
he has no better success. Parmenides argues thus :

" If the Object be like the Idea, the Idea must be like the Object. And since the Object
and the Idea are like, they must, according to your doctrine, participate in the Idea of Like-
ness. And thus you have one Idea participating in another Idea, and so on in infinitum."
Socrates is obliged to allow that this demolishes the notion of objects partaking in their Ideas
by likeness: and that he must seek some other way. "You see then, O Socrates," says
Parmenides, " what difficulties follow, if any one asserts the independent existence of Ideas ! "
Socrates allows that this is true. " And yet," says Parmenides, " you do not half perceive the
difficulties which follow from this doctrine of Ideas." Socrates expresses a wish to know to
what Parmenides refers ; and the aged sage replies by explaining that if Ideas exist inde-
pendently of us, we can never know anything about them : and that even the Gods could not
know anything about man. This argument, though somewhat obscure, is evidently stated
with perfect earnestness, and Socrates is represented as giving his assent to it. " And yet,"
says Parmenides, (end of & 18) "if any one gives up entirely the doctrine of Ideas, how is
any reasoning possible ?"

All the way through this discussion, Parmenides appears as vastly superior to Socrates ; as
seeing completely the tendency of every line of reasoning, while Socrates is driven blindly
from one position to another ; and as kindly and graciously advising a young man respecting
the proper aims of his philosophical career ; as well as clearly pointing out the consequences
of his assumptions. Nothing can be more complete than the higher position assigned to Par-
menides in the Dialogue.

This has not been overlooked by the Editors and Commentators of Plato. To take for
example one of the latest ; in Steinhart's Introduction to Hieronymus Miiller's translation of
Parmenides (Leipzig, 1852), p. 26l, he says: "It strikes us, at first, as strange, that Plato
here seems to come forward as the assailant of his own doctrine of Ideas. For the difficulties
which he makes Parmenides propound against that doctrine are by no means sophistical or
superficial, but substantial and to the point. Moreover there is among all these objections, which
are partly derived from the Megarics, scarce one which does not appear again in the penetrat-
ing and comprehensive argumentations of Aristotle against the Platonic Doctrine of Ideas."

Of course, both this writer and other commentators on Plato offer something as a solution
of this difficulty. But though these explanations are subtle and ingenious, they appear to leave
no satisfactory or permanent impression on the mind. I must avow that, to me, they appear
insufficient and empty ; and I cannot help believing that the solution is of a more simple and
direct kind. It may seem bold to maintain an opinion different from that of so many eminent
scholars; but I think that the solution which I offer, will derive confirmation from a consi-
deration of the whole Dialogue ; and therefore I shall venture to propound it in a distinct
and positive form. It is this:

I conceive that the Parmenides is not a Platonic Dialogue at all ; but Antiplatonic, or
more properly, Eleatic: written, not by Plato, in order to explain and prove his Theory of

Dr WHEWELL, on the PLATONIC THEORY OF IDEAS. 101

Ideas, but by some one, probably an admirer of Parmenides and Zeno, in order to shew how
strong were his master's arguments against the Platonists, and how weak their objections to the
Eleatic doctrine.

I conceive that this view throws an especial light on every part of the Dialogue, as a
brief survey of it will shew. Parmenides and Zeno come to Athens to the Panathenaic festi-
val : Parmenides already an old man, with a silver head, dignified and benevolent in his appear-
ance, looking five and sixty years old: Zeno about forty, tall and handsome. They are the
guests of Pythodorus, outside the Wall, in the Ceramicus ; and there they are visited by
Socrates, then young, and others who wish to hear the written discourses of Zeno. These
discourses are explanations of the philosophy of Parmenides, which he had delivered in
verse.

Socrates is represented as shewing, from the first, a disposition to criticize Zeno's disser-
tation very closely ; and without any prelude or preparation, he applies the Doctrine of Ideas
to refute the Eleatic Doctrine that All Things are One. (^ 3.) When he had heard to the end,
he begged to have the first Proposition of the First Book read again. And then : " How is it,
O Zeno, that you say, That if the Things which exist are Many, and not One, they must
be at the same time like and unlike ? Is this your argument ? Or do I misunderstand you .''■'
" No," says Zeno, " you understand quite rightly.' Socrates then turns to Parmenides, and
says, somewhat rudely, as it seems, " Zeno is a great friend of yours, Parftienides : he shews
his friendship not only in other ways, but also in what he writes. For he says the same
things which you say, though he pretends that he does not. You say, in your poems, that
All Things are One, and give striking proofs : he says that existences are not many, and he
gives many and good proofs. You seem to soar above us, but you do not really differ."
Zeno takes this sally good-humouredly, and tells him that he pursues the scent with the keen-
ness of a Laconian hound. "But," says he (§6), "there really is less of ostentation in my
writing than you think. My Essay was merely written as a defence of Parmenides long ago,
when I was young ; and is not a piece of display composed now that I am older. And it was
stolen from me by some one ; so that I had no choice about publishing it."

Here we have, as I conceive, Socrates already represented as placed in a disadvantageous
position, by his abruptness, rude allusions, and readiness to put bad interpretations on what
is done. For this, Zeno's gentle pleasantry is a rebuke. Socrates, however, forthwith rushes
into the argument ; arguing, as I have said, for his own Theory.

" Tell me," he says, " do you not think there is an Idea of Likeness, and an Idea of
Unlikeness ? And that everything partakes of these Ideas .'' The things which partake of
Unlikeness are unlike. If all things partake of both Ideas, they are both like and unlike; and
where is the wonder.? {& 7-) If you could shew that Likeness itself was Unlikeness, it would
be a prodigy ; but if things which partake of these opposites, have both the opposite qualities,
it appears to me, Zeno, to involve no absurdity."

" So if Oneness itself were to be shewn to be Maniness" (I hope I may use this word,
rather than multiplicity) "I should be surprized; but if any one say that /am at the same
time one and many, where is the wonder ? For I partake of maniness : my right side is
different from my left side, my upper from my under parts. But I also partake of Oneness,

102 Dr WHEWELL, on THE PLATONIC THEORY OF IDEAS.

for I am here One of us seven. So that both are true. And so if any one say that stocks and
stones, and the like, are both one and many,' — not saying that Oneness is Maniness, nor Mani-
ness Oneness, he says nothing wonderful : he says what all will allow. (^ 8.) If then, as I
said before, any one should take separately the Ideas or Essence of Things, as Likeness and
Unlikeness, Maniness and Oneness, Rest and Motion, and the like, and then should shew that
these can mix and separate again, I should be wonderfully surprised, O Zeno : for I reckon
that I have tolerably well made myself master of these subjects*. I should be much more
surprised if any one could shew me this contradiction involved in the Ideas themselves ; in
the object of the Reason, as well as in Visible objects."

It may be remarked that Socrates delivers all this argumentation with the repetitions
which it involves, and the vehemence of its manner, without waiting for a reply to any of his
interrogations; instead of making every step the result of a concession of his opponent, as
is the case in the Dialogues where he is represented as triumphant. Every reader of Plato will
recollect also that in those Dialogues, the triumph of temper on the part of Socrates is represented
as still more remarkable than the triumph of argument. No vehemence or rudeness on the
part of his adversaries prevents his calmly following his reasoning ; and he parries coarse-
ness by compliment. Now in this Dialogue, it is remarkable that this kind cff triumph is given
to the adversaries of Socrates. " When Socrates had thus delivered himself," says Pythodorus,
the narrator of the conversation, " we thought that Parmenides and Zeno would both be
angry. But it was not so. They bestowed entire attention upon him, and often looked at each
other, and smiled, as in admiration of Socrates. And when he had ended, Parmenides said : ' O
Socrates, what an admirable person you are, for the earnestness with which you reason ! Tell
me then, Do you then believe the doctrine to which you have been referring ; — that there
are certain Ideas, existing independent of Things; and that there are, separate from the Ideas,
Things which partake of them ? And do you think that there is an Idea of Likeness besides
the likeness which we have ; and a Oneness and a Maniness, and the like ? And an Idea of
the Right, and the Good, and the Fair, and of other such qualities?'" Socrates says that he
does hold this ; Parmenides then asks him, how far he carries this doctrine of Ideas, and
propounds to him the difficulties which I have already stated ; and when Socrates is unable to
answer him, lets him off in the kind but patronizing way which I have already described.

To me, comparing this with the intellectual and moral attitude of Socrates in the most
dramatic of the other Platonic Dialogues, it is inconceivable, that this representation of Socrates
should be Plato's. It is just what Zeno would have written, if he had wished to bestow
upon his master Parmenides the calm dignity and irresistible argument which Plato assigns
to Socrates. And this character is kept up to the end of the Dialogue. When Socrates
(^ 19) has acknowledged that he is at a loss which way to turn for his philosophy, Parmenides
undertakes, though with kind words, to explain to him by what fundamental error in the course
of his speculative habits he has been misled. He says ; " You try to make a complete

• I am aware that this translation is different from the | of my view ; but I do not conceive that the argument would
common translation. It appears to me to be consistent with I be perceptibly weaker, if the common interpretation were
the habit of the Greek language. It slightly leans in favour I adopted.

Dr WHEWELL, on the PLATONIC THEORY OF IDEAS. 103

Theory of Ideas, before you have gone through a proper intellectual discipline. The impulse
which urges you to such speculations is admirable — is divine. But you must exercise yourself
in reasoning which many think trifling, while you are yet young ; if you do not, the truth will
elude your grasp." Socrates asks submissively what is the course of such discipline : Parmenides
replies, " The course pointed out by Zeno, as you have heard." And then, gives him some
instructions in what manner he is to test any proposed Theory. Socrates is frightened at
the laboriousness and obscurity of the process. He says, " You tell me, Parmenides, of an
overwhelming course of study ; and I do not well comprehend it. Give me an example of
such an examination of a Theory." " It is too great a labour," says he, " for one so old as I
am." " Well then, you, Zeno," says Socrates, " will you not give us such an example.?" Zeno
answers, smiling, that they had better get it from Parmenides himself; and joins in the peti-
tion of Socrates to him, that he will instruct them. AH the company unite in the request.
Parmenides compares himself to an aged racehorse, brought to the course after long disuse, and
trembling at the risk; but finally consents. And as an example of a Theory to be examined,
takes his own Doctrine, that All Things are One, carrying on the Dialogue thenceforth, not
with Socrates, but with Aristoteles (not the Stagirite, but afterwards one of the Thirty), whom
he chooses as a younger and more manageable respondent.

The discussion of this Doctrine is of a very subtle kind, and it would be difficult to
make it intelligible to a modern reader. Nor is it necessary for my purpose to attempt to do
so. It is plain that the discussion is intended seriously, as an example of true philosophy;
and each step of the process is represented as irresistible. The Respondent has nothing to say
but Yes; or No; How so? Certainly; It does appear; It does not appear. The discussion is
carried to a much greater length than all the rest of the Dialogue; and the result of the rea-
soning is summed up by Parmenides thus : " If One exist, it is Nothing. Whether One exist
or do not exist, both It and Other Things both with regard to Themselves and to Each other.
All and Everyway are and are not, appear and appear not." And this also is fully assented
to; and so the Dialogue ends.

I shall not pretend to explain the Doctrines there examined that One exists, or One does
not exist, nor to trace their consequences. But these were Formulae, as familiar in the Eleatic
school, as Ideas in the Platonic ; and were undoubtedly regarded by the Megaric contempo-
raries of Plato as quite worthy of being discussed, after the Theory of Ideas had been over-
thrown. This, accordingly, appears to be the purport of the Dialogue; and it is pur-
sued, as we see, without any bitterness towards Socrates or his disciples ; but with a persuasion
that they were poor philosophers, conceited talkers, and weak disputants.

The external circumstances of the Dialogue tend, I conceive, to confirm this opinion, that
it is not Plato's. The Dialogue begins, as the Republic begins, with the mention of a
Cephalus, and two brothers, Glaucon and Adimantus. But this Cephalus is not the old man
of the Piraeus, of whom we have so charming a picture in the opening of the Republic. He
is from Clazomense, and tells us that his fellow-citizens are great lovers of philosophy ; a
trait of their character which does not appear elsewhere. Even the brothers Glaucon and
Adimantus are not the two brothers of Plato who conduct the Dialogue in the later books
of the Republic: so at least Ast argues, who holds the genuineness of the Dialogue. This

104 Db WHEWELL, on THE PLATONIC THEORY OF IDEAS.

Glaucon and Adimantus are most wantonly introduced; for the sole office they have, is to say
that they have a half-brother Antiphon, by a second marriage of their mother. No such
half-brother of Plato, and no such marriage of his mother, are noticed in other remains of
antiquity. Antiphon is represented as having been the friend of Pythodorus, who was the
host of Parmenides and Zeno, as we have seen. And Antiphon, having often heard from
Pythodorus the account of the conversation of his guests with Socrates, retained it in his
memory, or in his tablets, so as to be able to give the full report of it which we have in the
Dialogue Parmenides*. To me, all this looks like a clumsy imitation of the Introductions
to the Platonic Dialogues.

I say nothing of the chronological difficulties which arise from bringing Parmenides and
Socrates together, though they are considerable ; for they have been explained more or less
satisfactorily ; and certainly in the Thecetetus, Socrates is represented as saying that he
when very young had seen Parmenides who was very oldj". Atheuseus, however |, reckons
this among Plato's fictions. Schleiermacher gives up the identification and relation of the
persons mentioned in the Introduction as an unmanageable story.

I may add that I believe Cicero, who refers to so many of Plato's Dialogues, nowhere
refers to the Parmenides. Athenaeus does refer to it ; and in doing so blames Plato for his
coarse imputations on Zeno and Parmenides. According to our view, these are hostile
attempts to ascribe rudeness to Socrates or to Plato. Stallbaum acknowledges that Aristotle
nowhere refers to this Dialogue.

* In the First Alcibiades, Pythodorus is mentioned as having paid 100 minse to Zeno for his instructions (119 a).
t p. 183 e. + Deipn. xi. c. 15, p. 106.

VI. On the Discontinuity of Arbitrary Constants which appear in Divergent
Developments. By G. G. Stokes, M.A., D.CX., Sec. R.S., Fellow of Pembroke
College, and Lucasian Professor of Mathematics in the University of Cam-
bridge.

[Read May 11, 1857.]

In a paper " On the Numerical Calculation of a class of Definite Integrals and Infinite
Series," printed in the ninth volume of the Transactions of this Society, I succeeded in

developing the integral / cos — {vf - mw) dw in a form which admits of extremely easy

•'o 2

numerical calculation when m is large, whether positive or negative, or even moderately large.
The method there followed is of very general application to a class of functions which
frequently occur in physical problems. Some other examples of its use are given in the
same paper ; and I was enabled by the application of it to solve the problem of the motion
of the fluid surrounding a pendulum of the form of a long cylinder, when the internal friction
of the fluid is taken into account *.

These functions admit of expansion, according to ascending powers of the variables, in
series which are always convergent, and which may be regarded as defining the functions for
all values of the variable real or imaginary, though the actual numerical calculation would
involve a labour increasing indefinitely with the magnitude of the variable. They satisfy
certain linear differential equations, which indeed frequently are what present themselves in
the first instance, the series, multiplied by arbitrary constants, being merely their integrals.
In my former paper, to which the present may be regarded as a supplement, I have employed
these equations to obtain integrals in the form of descending series multiplied by exponentials.
These integrals, when once the arbitrary constants are determined, are exceedingly convenient
for numerical calculation when the variable is large, notwithstanding that the series involved
in them, though at first rapidly convergent, became ultimately rapidly divergent.

The determination of the arbitrary constants may be effected in two ways, numerically or
analytically. In the former, it will be sufficient to calculate the function for one or more
values of the variable from the ascending and descending series separately, and equate the
results. This method has the advantage of being generally applicable, but is wholly devoid
of elegance. It is better, when possible, to determine analytically the relations between the

• Comb. Phil. Trans. Vol. IX. Part U.

Vol. X. Part I. 14

106 PROFESSOR STOKES, ON THE DISCONTINUITY

arbitrary constants in the ascending and descending series. In the examples to which I have
applied the method, with one exception, this was effected, so far as was necessary for the
physical problem, by means of a definite integral, which either was what presented itself in
the first instance, or was employed as one form of the integral of the differential equation, and
in either case formed a link of connexion between the ascending and the descending series.
The exception occurs in the case of Mr Airy's integral for m negative. I succeeded in
determining the arbitrary constants in the divergent series for m positive ; but though I was
able to obtain the correct result for m negative, I had to profess myself (p. 177) unable
to give a satisfactory demonstration of it.

But though the arbitrary constants which occur as coefficients of the divergent series may
be completely determined for real values of the variable, or even for imaginary values with
their amplitudes lying between restricted limits, something yet remains to be done in order to
render the expression by means of divergent series analytically perfect. I have already
remarked in the former paper (p. 176) that inasmuch as the descending series contain radicals
which do not appear in the ascending series, we may see, a priori, that the arbitrary con-
stants must be discontinuous. But it is not enough to know that they must be discontinuous ;
we must also know where the discontinuity takes place, and to what the constants change.
Then, and not till then, will the expressions by descending series be complete, inasmuch as
we shall be able to use them for all values of the amplitude of the variable.

I have lately resumed this subject, and I have now succeeded in ascertaining the character
by which the liability to discontinuity in these arbitrary constants may be ascertained. I may
mention at once that it consists in this ; that an associated divergent series comes to have
all its terms regularly positive. The expression becomes thereby to a certain extent illusory ;
and thus it is that analysis gets over the apparent paradox of furnishing a discontinuous
expression for a continuous function. It will be found that the expressions by divergent
series will thus acquire all the requisite generality, and that though applied without any
restriction as to the amplitude of the variable they will contain only as many unknown con-
stants as correspond to the degree of the differential equation. The determination, among
other things, of the constants in the development of Mr Airy's integral will thus be rendered
complete.

1. Before proceeding to more difficult examples, it will be well to consider a com-
paratively simple function, which has been already much discussed. As my object in treating
this function is to facilitate the comprehension of methods applicable to functions of much
greater complexity, I shall not take the shortest course, but that which seems best adapted to
serve as an introduction to what is to follow.

Consider the integral

u = 2 f

Jo

e'"' sin2ax(Lv = H (l)

1 2.3 3.4.5 '

OF ARBITRARY CONSTANTS, &c. 107

The integral and the series are both convergent for all values of a, and either of them
completely defines u for all values real or imaginary of a. We easily find from either the
integral or the series

du

-r- + 2aM = 2 (2)

da

This equation gives, if we observe that m = <then a = 0,

« = 2e-«^ e«da=2e-«Ja + — +^-^+^-^^ + ...} (3)

This integral or series like the former gives a determinate and unique value to u for
any assigned value of a real or imaginary. Both series, however, though ultimately conver-
gent, begin by diverging rapidly when the modulus of a is large. For the sake of brevity I
shall hereafter speak of an imaginary quantity simply as large or small when it is meant that
its modulus is large or small.

2. In order to obtain m in a form convenient for calculation when a is large, let us seek
to express u by means of a descending series. We see from (2) that when the real part of a*
is positive, the most important terms of the equation are Saw and 2, and the leading term of
the development is a~'. Assuming a series with arbitrary indices and coefiicients, and deter-
mining them so as to satisfy the equation, we readily find

1 ] 1.3

M = - + — - +

a 2a' To/

This series can be only a particular integral of (2), since it wants an arbitrary constant.
To complete the integral we must add the complete integral of

du

-— + 2aM = 0,

da

whence we get for the complete integral of (2)

1 1 1.3 1.3.5

- + — + 1-

a 2a' 2V a^a"

M= Ce-"" + - + — - + -f- + -V^ + (4)

This expression might have been got at once from (3) by integration by parts. It remains
to determine the arbitrary constant C.

3. The expression (l) or (3) shews that « is an odd function of a, changing sign with a.
But according to (4) u is expressed as the sum of two functions, the first even, the second odd,
unless C = 0, in which case the even function disappears. But since, as we shall presently see,
the value of C is not zero, it must change sign with a. Let

a = p (cos Q + v/— 1 sin 6).

Since in the application of the series (4) it is supposed that p is large, we must suppose a
to change sign by a variation of 9, which must be increased or diminished (suppose increased)
by IT. Hence, if we knew what C was for a range tt of 0, suppose from Q = a to = o + 7r,
we should know at once what it was from = a -^ v to & = o+ 27r, which would be sufficient

14—2

108 PROFESSOR STOKES, ON THE DISCONTINUITY

for our purpose, since we may always suppose the amplitude of a included in the range a to '
a + 2;r, by adding, if need be, a positive or negative multiple of 27r, which as appears from
(1) or (3) makes no difference in the value of m.

4. When p is large the series (4) is at first rapidly convergent, but be p ever so great it
ends by diverging with increasing rapidity. Nevertheless it may be employed in calculation
provided we do not push the series too far,* but stop before the terms get large again. To
shew in a general way the legitimacy of this, we may observe that if we stop with the term

1 .3.5...(2i- l)

2V*+» '

the value of u so obtained will satisfy exactly, not (2), but the differential equation

du 1.3.,.(2i + ])

— - + 2o« = 2 -T-j

da 2*0'^'

• + 2a« = 2- • ■'. (5)

Let «o be the true value of u for a large value a^ of a, and suppose that we pass from a^ to
another large value of a keeping the modulus of a large all the while. Since u ought to satisfy
(2), we ought to have

whereas since our approximate expression for u actually satisfies (5) we actually have, putting
Af for the last term,

u

. + e-'' f\2-Ji)e^'da (6*)

•'Oo

If a be very large, and in using the series (4) we stop about where the moduli of the terms
are smallest, the modulus of Jf will be very small. Hence in general A^ may be neglected in
comparison with (2), and we may use the expression (4), though we stop after i + 1 terms of
the series, as a near approximation to u.

5. But to this there is an important restriction, to understand which more readily it will
be convenient to suppose the integration from Oq to (>■ performed, first by putting

da = (cos 9 + s/ - 1 sia 6) dp,
and integrating from p„ to p, 9 remaining equal to 9^, and then

da = p {- sin 9 + \/ - 1 cos 9) d9,

and integrating from 9^ to 9, p remaining unchanged. This is allowable, since « is a finite, con-
tinuous, and determinate function of a, and therefore the mode in which p and 9 vary when a
passes from its initial value a,, to its final value o is a matter of indifference. The modulus
of c"' will depend on the real part p^ cos 29 of the index. Now should cos 20 become a
maximum within the limits of integration, we can no longer neglect Af in the integration. For
however great may be the value previously assigned to i, the quantity p~^'~^e''^ ™'''* will become,
for values of 9 comprised within the limits of integration, infinitely great, when p is infinitely
increased, compared with the value of e''"'^"'^* at either limit. And though the modulus of the
quantity 2e"' under the integral sign will become far greater still, inasmuch as it does not con-

OF ARBITRARY CONSTANTS, &c. 109

tain the factor p~^'^, yet as the mutual destruction of positive and negative parts may take
place quite differently in the two integrals jie^'da and fAfC^'da, we can conclude nothing as
to their relative importance.

6. Now cos 20 will continually increase or decrease from one limit to the other, or else
will become a maximum, according as the two limits 0^ and lie in the same interval to tt
or TT to 27r, or else lie one in one of the two intervals and the other in the other. Hence we may
employ the expression (4), with an invariable value of C yet to be determined, so long as
< < TT, and we may employ the expression obtained by writing C' for C so long as
IT <6 < Stt, but we must not pass from one interval to the other, retaining the same expression.
Now we have seen (Art. 3) that the constant changes sign when is increased by tt, and there-
fore C' = - C And since u is unchanged when is increased by any multiple of 27r, we readily
see that in order to make the expression (4) generally applicable, it will be sufficient to change
the sign of the constant whenever d passes through zero or a multiple of ir.

7. We may arrive at the same conclusion in another way, which will be of more general
or at least easier application, as not involving the integration of the differential equation.

The modulus of the general term (Art. 4) of the series (4), expressed by means of the
function F, is

r(i)p^*+'-

Suppose i very large. Employing the formula

r (a? + l) = v 27r« (- I » nearly, when a? is large,
observing that r(^) = tt^, and calling the modulus yUj, we find

which, since (i + cf = i'e", nearly, becomes

Me = 2*i«e->-^'-i (7)

We easily get, either from this expression or from the general term,

,'^=^, nearly, (8)

Mi P

Hence when p is large the ratio of consecutive moduli becomes very nearly equal to unity
for a great number of terms together, about where the modulus is a minimum. To find
approximately the minimum modulus n, we must put i = p^ in (7), which gives

M-gV'e-"' (9)

If we knew precisely at what term it would be best to stop, the expression for n would
be a measure of the uncertainty to which we were liable in using the series (4) directly, that
is, without any transformation. For although it is clear that we must stop somewhere about
the term with a minimum modulus, in order that the differential equation (5) which our
function really satisfies may be as good an approximation as can be had to the true differential

110 PROFESSOR STOKES, ON THE DISCONTINUITY

equation (2), the number of terms comprised in this about will increase with i, the order of the
term of minimum modulus. If we suppose that we are uncertain to the extent of n terms, the
sum of the moduli of these n nearly equal terms will be

nearly. It seems as if « must increase with i, but not so fast as i. If we suppose that it is of
the form ki or kp, the sum of the n terms will be a quantity of the order e~^\ But even
if n increased as any power p of i, however great, still the sura of the n terms would be a quan-
tity of the order p'"'~^e'''', which when p was infinitely increased would become infinitely small
in comparison with the modulus e-p"«o*28 ^f ^^^ ^^^.^ multiplied by C in (4), provided 6 had
any given value differing from zero or a multiple of tt. Hence if B have any value lying
between a and v — a, or else between tt + a and Stt — a, where a is a small positive quantity
•which in the end may be made as small as we please, the quantity C in (4) cannot pass from
one of its values to another without rendering the function u discontinuous, which it is not.
But when = or = tt, the term Ce""' becomes merged in the vagueness with which, in this
case, the divergent series defines the function. Hence we arrive in a way quite different from
that of Art. 5 at the conclusions enunciated in Art. 6,

8. Nor is this all. When the terms of a regular series are alternately positive and
negative, the series may be converted by the formulae of finite differences into others which
converge rapidly. In the present case the terms are not simply positive and negative alternately,

except when 6 is an odd multiple of — , but the same methods will apply with the proper

modification. Suppose that we sum the series (4) directly as far as terms of the order i - 1
inclusive. Omitting the common factor e-<"+'>* -i, which may be restored in the end, we have
for the rest of the series

If we denote by X) or 1 + A the operation of passing from Hi to /x^^^, and separate symbols
of operation, this becomes

(1 + e-'^~'D + e-">^~'I)' + ...)^i,

or {l-(l + A)e-^''^^|-Vi.

Now 1 -e-'*^-'= 1 - COS 20 + \/- 1 sin20 = 2sin 0e ^ ,
which reduces the expression to

(2 sin0)-^e('"^^''^{l - (2 sin0)-»e"^^^*)^^A}-V„
or, putting q for (2 sin 0)"S to

Now if p be very large, and m belong to the part of the series where the moduli of con-
secutive terms are nearly equal, the successive differences Afji<, A^^l^,... will decrease with great
rapidity. Hence if 6 have any given value different from zero or a multiple of tt, by taking

OF ARBITRARY CONSTANTS, &c. Ill

p sufficiently great, we may transform the series about where it ceases to converge into one
which is at first rapidly convergent, and thus a quantity which may be taken as a measure
of the remaining uncertainty will become incomparably smaller even than ^, much more,
incomparably smaller than the modulus of e'"^. But if 6 = or = tt, the above transform-
ation fails, since q becomes infinite. In this case if we want to calculate u closer than to
admit of the uncertainty to which we are liable, knowing only that we must stop somewhere
about the place where the series begins to diverge after having been convergent, we must
have recourse to the ascending series (1) or (3), or to some perfectly distinct method. The
usual method by which Sm^ is made to depend on Ju^dx would evidently fail, in consequence
of the divergence of the integral.

9. In applying practically the transformation of the last article to the summation of
the series (4), it would not usually, when p was very large, be necessary to go as far as the
part of the series where the moduli of consecutive terms are nearly equal. It would be
sufficient to deduct /,' 2Z... from the logarithms of ^£^.„ Atj+j..,, where I is nearly equal to
the mean increment of the logarithms at that part of the series, to associate the factor/ whose
logarithm is I with the symbol D, and take the differences of the numbers,

However, my object leads me to consider, not the actual summation of the series, but the
theoretical possibility of summation, and consequent interpretation of the equation (4).

10. The mode of discontinuity of the constant C having been now ascertained, nothing
more remains except to determine that constant, which is done at once. Writing v — la for
a in (4) after having put for u its first expression in (3), we have

Se"' fe-'^'da = -y/ - xCe"'-- + — - ...
> a 2a3

whenee, putting a=<X) , we have C= v — 1 ir^. Hence we get for the general expression
for C in (4),

C = \/^ TT^, when < fl < tt, I , ^

. I (10)

C = - V - 1 TT^ when tt < 0<27r; I

and therefore from (3) and (4)

1 1 1.3 1.3.5

2e

-' f e'-da = W- 1 TT^e-V 1 + -^ + il^ + ^_^+ (11)

the sign being + or — according as 6, the amplitude of a, is comprised within the limits
and TT, or tt and 27r. •

Writing a\/— 1 for a in (11), which comes to altering the origin of by — , we find

, r 1 , 1^1 1 1-3 1-3.5

Jo a ^a^ 9.'a^ 2° a'

112 PROFESSOR STOKES, ON THE DISCONTINUITY

the sign being + or - according as the amplitude of a lies within the limits and - , or

— and — . It is worthy of remark that in this expression the transcendental quantity tt^
appears as a true radical, admitting of the double sign.

Two cases of the integral j e'''da occur in actual investigations, namely when = — ,

when the integral leads to / e~*'dt, which occurs in the theory of probabilities, and when

•'o

0=—, when it leads to Fresnel's integrals T' cos (-«') ds and / sin (-«'] ds. In the

latter case the expression (11) is equivalent to the development of these integrals which has
been given by M, Cauchy.

11. If in equation (11) we put a = p (cos ± v - 1 sin 9), where is a small positive
quantity, and after equating the real parts of both sides of the equation make 9 vanish, we
find, whichever sign be taken,

1 1.31.3.5 , rf

- + h +

= 2e-'''r^'dp (13)

p 2/3' ay 2'p'

The expression which appears on the second side of this equation may be regarded as a
singular value of the sum of the series

1 1 1.31.3.5

a "^ 2^' ■*" 2^ "•■ ~2W" "^ ^^*^

a series which when 9 vanishes takes the form of the first member of the equation. The
equivalent of the series for general values of the variable is given, not by (13), but by (11).
It may be remarked that the singular value is the mean of the general values for two infinitely
small values of 9, one positive and the other negative.

These results, to which we are led by analysis, may be compared with the known theory
of periodic series. If /"(a?) be a finite function of a?, the value of which changes abruptly
from a to 6 as a? increases through the value c, a quantity lying between and ir, and /{x)
be expanded between the limits and tt in a series of sines of multiples of a?, and if (p {n, x)
be the sum of n terms of the series, the value of (p (n, x) for an infinitely large value of n and
a value of a> infinitely near to c is indeterminate, like that of the fraction

(tv + yy + x - y
{x -yy + 0! + y'

which takes the form - when a? and y vanish, but of which the limiting value is wholly

indeterminate if a? and y are independent. We may enquire, if we please, what is the limit
of the fraction when <t? first vanishes and then y, or the limit when y first vanishes and then x,
for each of these has a perfectly clear and determinate signification. In the former case
we have, calling the fraction >|/ (tv, y).

OF ARBITRARY CONSTANTS, &c. 113

■.2

y — y
lim.^ol'in-x=o^('».y) = ""'-^=0 -7— = - 1 ;

in the latter

So in the case of the periodic series if we denote by ^ a small positive quantity

lim.f^lim.„^„ 0(n, c + ^ = lim-f=o/(<' + = * 5
but we know that

lim.„^„ lim.f^o ^ («. c ± ^) = lim.„,„ («, c) = 1 (a + 6).

Similarly in the case of the series (14) if we denote its sum by ■^{a) = sr (p, 9), and use
the term limit in an extended sense, so as to understand by lim.p^„jP(p) a function of p to
which F(p) may be regarded as equal when p is large enough, and if we suppose to be a
small positive quantity, we have from (11)

lim.e=olim.,=„ ■ar(p, 6) = lim.^^o |2c-»' fef^'da -y/^l -n-h-"'}

°

Hm.^„lim W(p,-e) = lim.^^o {Se-"' TV da + n/^ tt^c""'}

''0

''0
whereas equation (13) may be expressed by

lim.^=„ lim.j^o w (p, ± 0) = lim.^„;^(^) = Sc""' fef'dp.

*'o

There is however this difference between the two cases, that in the case of the periodic
series the series whose general term is A0 (w, c) is convergent, and may be actually summed
to any assigned degree of accuracy, whereas the series (13), though at first convergent, is
ultimately divergent ; and though we know that we must stop somewhere about the least
term, that alone does not enable us to find the sum, except subject to an uncertainty com-
parable with e"*"'. Unless therefore it be possible to apply to the series (13) some transfor-
mation rendering it capable of summation to a degree of accuracy incomparably superior to
this, the equation (13) must be regarded as a mere symbolical result. We might indeed
define the sum of the ultimately divergent series (13) to mean the sura taken to as many
terms as should make the equation (13) true, and express that condition in a manner which
would not require the quantity taken to denote the number of terms to be integral ; but
Vol. X. Part I. 15

114 PROFESSOR STOKES, ON THE DISCONTINUITY

then equation (13) would become a mere truism. However I shall not pursue this subject
further, as these singular values of divergent series appear to be merely matters of
curiosity.

12. In order still further to illustrate the subject, before going on to the actual

application of the principles here establisWd, let us consider the function defined by the

equation

i.l , 1.1.3,

M = l + la? - — ,v^ + -ai'- (15)

^ 2.4 2.4.6 ^ '

Suppose that we have to deal with such values only of the imaginary variable a; as have
their moduli less than unity. For such values the series (15) is convergent, and the equation

(15) assigns a determinate and unique value to u. Now we happen to know that the series
is the development of (l + a?)'. But this function admits of one or other of the following
developments according to descending powers of x : —

,,,1.1.1.1.3,

^ 2.4 2.4.0 ^ '

u = — X^ — %X 2 + UO * X~^ ■>r (17)

^ 2.4 2.4.6

Let X = p (cos Q Jrs/ -\ sin 0), and let x^ denote that square root of x which has \Q for
its amplitude. Although the series (l6), (17) are divergent when o < 1, they may in general,
for a given value of Q, be employed in actual numerical calculation, by subjecting them to
the transformation of Art. 8, provided p do not differ too much from 1. The greater be the
accuracy required, Q being given, the less must p differ from 1 if we would employ the series

(16) or (17) in place of (15). It remains to be found which of these series must be taken.

If Q lie between (2i - l) tt + a and (2i + 1) tt - a, where i is any positive or negative
integer or zero, and a a small positive quantity which in the end may be made as small as
we please, either series (l6) or (17) may by the method of Art. 8 be converted into another,
which is at first sufficiently convergent to give u with a sufficient degree of accuracy by
employing a finite number ohly 6f terms. If m terms be summed directly, and in
the formula of Art. 8 the n**" difference be the last which yields significant figures, the
number of terms actually employed in some way or other in the summation will be wi + « + l .
And in this case we cannot pass from one to the other of the two series (l6), (17) without
rendering u discontinuous. But when Q passes through an odd multiple of tt we may have
to pass from one of the two smes to the other. Now when Q is increased by 27r the Series
(l6) or (17) changes sign, whereas (15) remains unchanged. Therefore in calculating w for tWo
values of 9 differing by Stt we must employ the two series (16) and (17), one in each case.
Hence We must employ one of the series from = — tt to = tt, the other from = tt to
B a= 3rr, and so on ; and therefore if we knew which series to take for some one value
of 8r everything Would be determined.

Now when '|0 it 1 the Series '(15) becomes identical Vith (1 6) when Q has the particular
Value 0. Hencfe (T6) ^U 'not (17) gives the true value of « wheal - tt < Q <-k.

OF ARBITRARY CONSTANTS, &c. 115

13. Let p, 6 be the polar co-ordinates of a point in a plane, O the origin, C a circle
described round O with radius unity, S the point determined by a; = - 1, that is, hy p= 1,
= IT. To each value of x corresponds a point in the plane ; and the restriction laid down
as to the moduli of a> confines our attention to points within the circle, to each of which
corresponds a determinate value of u. If Pg be any point in the plane, either within the
circle or not, and a moveable point P start from Pg, and after making any circuit, without
passing through S, return to P, again, the functipn (1 + my will regain its primitive value m^,,
or else become equal to - u^, according as the circuit excludes or includes the point S,
which for the present purpose may be called a singular point. Suppose that we wished to
tabulate u, using when possible the divergent series (l6) in place of the convergent series (15).
For a given value of 6, in commencing with small values of p we should have to begin with
the series (15), and when p became large enough we might have recourse to (l6). Let OP be
the smallest value of p for which the series (l6) may be employed ; for which, suppose, it will
give u correctly to a certain number of decimal places. The length OP will depend upon Q,
and the locus of P will be some curve, symmetrical with respect to the diameter through S.
As 9 increases the curve will gradually approach the circle C, which it will run into at the
point 8. For points lying between the curve and the circle we may employ the series (l6),
but we cannot, keeping within this space, make 9 pass through the value tt. The series
(16), (17) are convergent, and their sums vary continuously with x, when p> \; and if \ve
employed the same series (l6) for the calculation of u for values of w having amplitudes
TT - /3, IT + fi, corresponding to points P, P', we should get for the value of m at P' that
into which the value of m at P passes continuously when we travel from P to P outside the
point S, which as we have seen is minus the true value, the latter being defined to be that
into which the value of u at P passes continuously when we travel from P to P' inside
the point S.

In the case of the simple function at present under consideration, it would be an arbitrary
restriction to confine our attention to values of x having moduli less than unity, nor
would there be any advantage in using the divergent series (l6) rather than the convergent
series (15). But in the example first considered we have to deal with a function which
has a perfectly determinate and unique value for all values of the variable a, and there is
the greatest possible advantage in employing the descending series for large values of p,
though it is ultimately divergent. In the case of this function there are (to use the same
geometrical illustration as before) as it were two singular points at infinity, corresponding
respectively to = and 9 = n.

14. The principles which are to guide us having been now laid down, there will be no
difficulty in applying them to other cases, in which their real utility will be perceived. I will
now take Mr Airy's integral, or rather the differential equation to which it leads, the treatment
of which will exemplify the subject still better. This equation, which is No. 11 of my paper
" On the Numerical Calculation, &c,," becomes on writing u for U, — 3^ for n

— -9a7M = (18)

15—2

116

PROFESSOR STOKES, ON THE DISCONTINUITY

The complete integral of this equation in ascending series, obtained in the usual way, is

«

-1

X

9^

+

9V

+

9'

1-9

2

3 .5

.6

2

.3

.5

6

8

.9

9«'

3.4

+

9V

+

9'

.r'"

3

4.6

7

3

4

6

7.

9.

10

...)J

(19)

These series are always convergent, and for any value of x real or imaginary assign a
determinate and unique value to u.

The integral in a form adapted for calculation when x is large, obtained by the method of
my former paper, is

u

.( 1.5 1.5.7.11 1.5.7.11.13.17 U

[ 1.144a;* 1.2.144a^ 1.2.3.144V j[

^ f 1.5 1 .5.7.11 1 .5 .7.11 .13.17 1

+ Dx-ie'^ {l+ i+ + r— + —\

I 1 . 1440?* 1.2. 144'a' 1.2.3. 144V J

(20)

The constants C, D must however be discontinuous, since otherwise the value of u deter-
mined by this equation would not recur, as it ought, when the amplitude of x is increased by
2n-. We have now first to ascertain the mode of discontinuity of these constants, secondly,
to find the two linear relations which connect A, B with C, D.

Let the equation (20) be denoted for shortness by

«= Gr-i/,(<p) +Z>a?-i/2(.i7) ; (21)

and let f(jv), when we care only to express its dependance on the amplitude of x, be denoted
by F{d). We may notice that

-F'i(0 + |t) = /'.(0); F,{e + %^)^F,{e) (22)

16. In equation (21), let that term in which the real part of the index of the exponential
is positive be called the superior, and the other the inferior *''^' ^'

term. In order to represent to the eye the existence and
progress of the functions fi(x), /j,(*) for different values of
d, draw a circle with any radius, and along a radius vector
inclined to the prime radius at the variable angle take two
distances, measured respectively outwards and inwards from
the circumference of the circle, proportional to the real part
of the index of the exponential in the superior and inferior
terms, 6 alone being supposed to vary, or in other words
proportional to cos ^9. For greater convenience suppose
these distances moderately small compared with the radius.
Consider first the function Fi(9) alone. The curve will evidently have the form represented
in the figure, cutting the circle at intervals of 120", and running into itself after two complete
revolutions. The equations (22) shew that the curve corresponding to F„{9) is already

OF ARBITRARY CONSTANTS, &c. 117

traced, since F^ (0) = Fi(9 + Stt). If now we conceive the curve marked with the proper
values of the constants G, D, it will serve to represent the complete integral of equation (18).

In marking the curve we may either assume the amplitude 6 of a? to lie in the interval
to 27r, and determine the values of C, D accordingly, or else we may retain the same value of
C or D throughout as great a range as possible of the curve, and for that purpose permit 9 to
go beyond the above limits. The latter course will be found the more convenient.

16. We must now ascertain in what, cases it is possible for the constant G or D Xo
alter discontinuously as 9 alters continuously. The tests already given will enable us to
decide.

The general term of either series in (20), taken without regard to sign, is

1.5... (6i-5)(6t-l)
1 .2... i(144.r*)* '

and the modulus of this term, expressed by means of the function F, is

r (e + j) r (i + 1)
r(^)r(f)r(i + i)(vt)''

which when i is very large becomes by the transformations employed in Art. 7, very nearly,

\/y(;)Vr(i)r(i)(4/,i)*.

Denoting this expression by /Ujj and putting for r(^) T{^) its value Trcosec- or 27r,

we have

"^-^^""'^''Wef'' ^''^

whence for very large values of i

—-A (24)

IXi 4p«

For large values of p the moduli of several consecutive terms are nearly equal at the part
of the series where the modulus is a minimum, and for the minimum modulus n we have very
nearly from (24), (23)

i = 4pi, fi = (27ri)-^e-* = (2ni)~ie~*''*.

If the exponential in the expression for j^i be multiplied by the modulus of the exponential
in the superior term, the result will be

g-(4T2c08| 9)p*

the sign — or + being taken according as cos § ^ is positive or negative. Hence even if the
terms of the divergent series were all positive, the superior term would be defined by means of
its series within a quantity incomparably smaller, when p is indefinitely increased, than the
inferior term, except only when =tcos50= 1, and in this case too and this alone are the
terms of the divergent series in the superior term regularly positive. In no other case then

lis

PROFESSOR STOKES, ON THE DISCONTINUITY

Fig. 2.

can the coefficient of the inferior term alter discontinuously, and the coefficient of the other
term cannot change so long as that term remains the superior term. Referring for conve-
nience to the figure (Fig. l), we see that it is only at the points a, b, c, at the middle of the
portions of the curve which lie within the circle, that the coefficient belonging to the curve can
change.

It might appear at first sight that we could have three distinct coefficients, corresponding
respectively to the portions aAb, hBc, cCa of the curve, which would make three distinct
constants occurring in the integral of a differential equation of the second order only. This
however is not the case ; and if we were to assign in the first instance three distinct con-
stants to those three portions of the curve, they would be connected by an equation of
condition.

To shew this assume the coefficient belonging to the part of the curve about B to be equal
to zero. We shall thus get an integral of our equation with only one arbitrary constant.

Since there is no superior term from = - — to 6 = + — , the coefficient of the other term

cannot change discontinuously at a {i.e. when Q passes through the value zero); and by what

has been already shewn the coefficient must remain unchanged

throughout the portion hBc of the curve, and therefore be equal

to zero ; and again the coefficient must remain unchanged

throughout the portion cCaAb, and therefore have the same

value as at a ; but these two portions between them take in the

whole curve. The integral at present under consideration is

represented by Fig. 2, the coefficient having the same value

throughout the portion of the curve there drawn, and being

equal to zero for the remaindOT of the course *.

The second line on the right-hand side of (20) is what the
first becomes when the origin of 9 is altered by =•=§■"-, and
the arbitrary constant changed. Hence if we take the term
corresponding to the curve represented in Fig. 3, and having
a constant coefficient throughout the portion there repre-
sented, we shall get another particular integral with one
arbitrary constant, and the sum of these two particular
integrals will be the complete integral.

In Fig. 3 the uninterrupted interior branch of the curve

Fig. 3.

is made to lie in the interval — to ir.

3

It would have done

equally well to make it lie in the interval to — tt ; we should thus in fact obtain th^

3

same complete integral merely somewhat differently expressed.

• A numerical verification of the discontinuity here represented is given as an Appendix to this paper.

OF ARBITRARY CONSTANTS, &c. 119

The integral (20) may now be cftnveniently expressed in the following form, in which the
discontinuity of the constants is exhibited :

/ 47r ^.ttN^ , ,4/ 1.5 1.5.7.11 h

\ 3 3 J I 1 . 1440?* 1.2- 144V j

/ 27r \ ^ 1 ,* { 1-5 1.5. 7.11 1

V 3 y 1 1.144.r* 1.2.144V j

(25)

to + — j denotes that the function written after it

is to be taken whenever an angle in the indefinite series

...0 - 47r, 0-27r, 0, + Ztt, + tv,...
falls within tbe specified limits, which will be either once or twice accoi'ding to the value of 0.

17. If we put 2) = in (25), the resulting value of u will be equal to Mr Airy's integral,

— 1 — . When = we have the

integral belonging to the dark side of the caustic, when = ir that belonging to the bright
side. We easily see from (25), or by referring to Fig. 2, in what way to pass from one of
these integrals to the other, the integrals being supposed to be expressed by means of the
divergent series. If we have got the analytical expression belonging to the dark side we
must add + tt, - tt in succession to the amplitude of x, and take the sum of the results. If
we have got the analytical expression belonging to the bright side, we must alter the ampli-
tude of 00 by TT, and reject the superior function in the resulting expression. It is shewn in
Art. 9 of my paper " On the Numerical Calculation, &c." that the latter process leads to a
correct result, but I was unable then to give a demonstration. This desideratum is now
supplied.

18. It now only remains to connect the constants A, B with C, D in the two different
fdrms (19) and (25) of the integral of (18). This may be done by means of the complete
integral of (18) expressed in the form of definite integrals.

Let V = /" e-'^'-'^dX,

Jo

then ^ = - r"e-^^-'=^U3\= + cw) - ca>\ d\
da? 3 Jo * *

whence

mx) ;

3 3

im PROFESSOR STOKES, ON THE DISCONTINUITY

In order to make the left-hand member of this equsrtion agree with (18), we must have
c' =— 27, and therefore

c = - 3, or 3a, or 3/3,

a, )8 being the imaginary cube roots of - 1, of which o will be supposed equal to

TT / . IT

COS — + v — 1 sin — . •

3 3

Whichever value of c be taken, the right-hand member of equation (26) will be equal to — 9i
and therefore will disappear on taking the difference of any two functions cv corresponding
to two different values of c. This difference multiplied by an arbitrary constant will be an
integral of (18), and accordingly we shall have for the complete integral

u = E ["e-'-'i^ + ae-''^)dX + F f e-^'(e'^+ /3e-^^) dX (27)

Jo Jo

That this expression is in fact equivalent to (19) might be verified by expanding the
exponentials within parentheses, and integrating term by term.

To find the relations between £1, F and A, B, it will be sufficient to expand as far as
the first power of x, and equate the results. We thus get

A + Bx= f "c-^'{(l +a)E + +li)F+S [(l -a')E + - (^)F1 '^M d\

which gives, since

a»=-i8, /3^ = -«, /V^'d\ = |r(i). /■V^'xrf\ = ir(f),

*/q Jo
A = ir(i){(l+a)E+{l+fi)F},\
S= rQ){0+fi)E+(l+a)F\.f ^ ^

19. We have now to find the relations between E, F and C, D, for which purpose we
must compare the expressions (25), (27), supposing w indefinitely large.

In order that the exponentials in (25), may be as large as possible, we must have 9 = —

in the term multiplied by C, and = in the term multiplied by D. We have therefore
for the leading term of «

Cfe"i^-ie**, when 0c= -;

Dp-i^, when = 0.

Let us now seek the leading term of u from the expression (27), taking first the case in
which = 0. It is evident that this must arise from the part of the integral which involves
e^ or in this case e"^, which is

{E + F)f''e-'-'->-^d\.

OF ARBITRARY CONSTANTS, &c. 121

Now SpX - X^s a maximum for X = pk Let \ = p^ + ^; then

3pX-X^=2pi-3pi^'-^',
and our integral becomes

e'>'*f''^e-'<'*^-^d^.
Put r= 3'ip~i^ ; then the integral becomes

Let now p become infinite ; then the last integral becomes / e'^^d^ or 7ri For though

the index - P- S"^^"^^-' becomes positive for a sufficiently large negative value of ^, that
value lies far beyond the limits of integration, within which in fact the index continually
decreases with ^, having at the inferior limit the value — 2pi. Hence then for 6 = 0, and for
very large values of p, we have ultimately

u = 3-i',r-'{E + F)p-ie^''*.

Next let = — . In this case ax = - p, and we get for the leading part of «

3

aETe-'-'+'i'^dX,
Jo

I (29)

which when p is very large becomes, as before,

S-^7riaEp-ie'i'\

27r
Comparing the leading terms of u both for = — and for 6=0, we find, observing

that a = e^ '

C=\/^3-iniE,
D=S-iTri{E + F)
Eliminating E, F between (28) and (29) we have finally

^ = ,r-r(l){ C4-e-^^2)f,l _ _ ^^^^

5= 37r-^r(f){-C + e?^^'Z>^.| ,

20. As a last example of the principles of this paper, let us take the diflFerential equation

<f M 1 du

-— + - — -M = (31)

dx x dx

The complete integral of this equation in series according to ascending powers of a? involves a
logarithm. If the arbitrary constant multiplying the logarithm be equated to zero we shall
obtain an integral with only one arbitrary constant. This integral, or rather what it becomes
Vol. X. Part I. 16

122

PROFESSOR STOKES, ON THE DISCONTINUITY

when \/ — 1 OB is written for a?, occurs in many physical investigations, for example the problem
of annular waves in shallow water, and that of diffraction in the case of a circular disk. I had
occasion to employ the integral with a logarithm in determining the motion of a fluid about a
long cylindrical rod oscillating as a pendulum, the internal friction of the fluid itself being
taken into account*. In that paper the integral of (31) both in ascending and in descending
series was employed, but the discussion of the equation was not quite completed, one of the
arbitrary constants being left undetermined. A knowledge of the value of this constant was
not required for determining the resultant force of the fluid on the pendulum, which was the
great object of the investigation, but would have been required for determining the motion of
the fluid at a great distance from the pendulum.

21. The three forms of the integral of (31) which we shall require are given in Arts. 28
and 29 of my paper on pendulums. The complete integral according to ascending series is

M = (.4 + Z/ log a?) 1 1 + —

J?'

a'-'.*^

;^2 +

where

&= l-' + 2-' +3-

2' . 4- .

2^4^6'

+ 1'

■■)1

(32)

The series contained in this equation are convergent for all real or imaginary values of a?,
but the value of u determined by the equation is not unique, inasmuch as log as has an infinite
number of values. To pass from one of these to another comes to the same thing as changing
the constant A by some multiple of 27rJ5\/ — 1. If p, 9, the modulus and amplitude of w, be
supposed to be polar co-ordinates, and the expression (32) be made to vary continuously by
giving continuous variations to p and 9 without allowing the former to vanish, the value of
log a? will increase by 27rv — 1 in passing from any point in the positive direction once round
the origin so as to arrive at the starting point again. In order to render everything definite
we must specify the value of the logarithm which is supposed to be taken.

The complete integral of (31) expressed by means of descending series is

1^ .3^ l^ 3' . 5'' Y

.4.t? 2 . 4 (4a7)' ~ 2 . 4. . 6 (4ia?)' "*" "'/
1'

1

2.4.1?

+ Dx-ie' |l

1 +

2 . 4 (4a7)-
1=.3«

2.4.6 (4a?)'

l^ 3^ 5^

+ ■

...}

(33)

2 . 4a? 2.4 {'ixf ' 2.4.6 {4.xf

These series are ultimately divergent, and the constants C, D are discontinuous. It may
be shewn precisely as before that the values of 9 for which the constants are discontinuous are

... - 47r, - 27r, 0, Ztt, 47r ... for C,

... — Stt, — 7r, TT, Stt, ... for D.

• Comb. Phil. Tram. Vol. IX. Part II. p. [38.]

OF ARBITRARY CONSTANTS, &c. 123

Hence the equation (^33) may be written, according to the notation employed in Art. l6,
as follows :

M = (0 to Stt) Cw-ie'Ul + ...) + (-TT to +7r)Z)a?-^e'(l + + ...) (34)

2. 4a? 2.4.J?

22. It remains to connect A, B with C, D. For this purpose we shall require the third
form of the integral of (31), namely

u

= f^{E+ F\og{a>sm''w)\(g'"'"'+e-'''°"')dw (35)

(36)

As to the value of logx to be taken, it will suffice for the present to assume that whatever
value is employed in (32), the same shall be employed also in (35).

To connect A, B with E, F, it will be sufficient to compare (32) and (35), expanding the
exponentials, and rejecting all powers of x. We have

A + B logx = 2 j [E + F log (a> sin'^w) \ dw

= 7r(£+i^log.r) +27rlog(i).F;
whence

^ = TT^ - 27r log 2 . F,l

B = 7rF. J

To connect C, D with E, F, we must seek the ultimate value of u when p is infinitely
increased. It will be convenient to assume in succession 6=0 and 9 = tr. We have ulti-
mately from (34)

u = Dp-ie^ when 6 = 0; u =- \/ - 1 Cp'hi' when 6 = ir (37)

It will be necessary now to specify what value of log a; we suppose taken in {S5). Let it be
log p + \/ — l6, 6 being supposed reduced within the limits and 2ir by adding or subtracting
if need be 2i7r, where i is an integer.

The limiting value of u for 6 = from (35) may be found as in Art. 29 of my paper on
Pendulums, above referred to. In fact, the reasoning of that Article will apply if the
imaginary quantity there denoted by m be replaced by unity. The constants

C, D, C, D', C", D",
of the former paper correspond to

A, B, C, D, E, F,
of the present. Hence we have for the ultimate value of u for =

M=[^]V{i;+(7r-ir'(J)+log2)/'| (38)

For = TT, i^5) becomes

IT

u= f''{E+ TrFy/- 1 + /'log (p sm^ w)l (e-'"=<''"+ e^~'") dw ;

Jo

and to find the ultimate value of u we have merely to write E + irF y/— 1 for E in the above,
which gives ultimately for = tt

^ = (^y^l^ + '^^y/^^ + {-^'^^'(D + ^^s^l ^ (39)

16—2

(40)

124 PROFESSOR STOKES, ON THE DISCONTINUITY

Comparing the equations (38), (39) with (37), we get

C= ^J] [eV-1-^F+ {,r-^r'(i)+log2| x/IT/^,

z»= g)\£ + {,r-ir'(i)+iog2}r].

Eliminating E, F between (36) and (40), we get finally

C = (27r)-^[\/3T^ + {(^-.n^'(l) +log8) V'::! -7r( B],
Z> = (2,r)-i[^ + {tt-^T (^) + log8| 5].

Concltision.

23. It has been shewn in the foregoing paper,

First, That when functions expressible in convergent series according to ascending powers
of the variable are transformed so as to be expressed by exponentials multiplied by series
according to descending powers, applicable to the calculation of the functions for large values
of the variable, and ultimately divergent, though at first rapidly convergent, the series contain
in general discontinuous constants, which change abruptly as the amplitude of the imaginary
variable passes through certain values.

Secondly, That the liability to discontinuity in one of the constants is pointed out by the
circumstance, that for a particular value of the amplitude of the variable, all the terms of an
associated divergent series become regularly positive.

Thirdly, That a divergent series with all its terms regularly positive is in many cases a
sort of indeterminate form, in passing through which a discontinuity takes place.

Fourthly, That when the function may be expressed by means of a definite integral, the
constants in the ascending and descending series may usually be connected by one uniform
process. The comparison of the leading terms of the ascending series with the integral
presents no difficulty. The comparison of the leading terms of the descending series with
the integral may usually be effected by assigning to the amplitude of the variable such a
value, or such values in succession, as shall render the real part of the index of the expo-
nential a maximum, and then seeking what the integral becomes when the modulus of the
variable increases indefinitely. The leading term obtained from the integral will be found
within a range of integration comprising the maximum value of the real part of the index
of the exponential under the integral sign, and extending between limits which may be supposed
to become indefinitely close after the modulus of the original variable has been made in-
definitely great, whereby the integral will be reduced to one of a simpler form. Should a
definite integral capable of expressing the function not be discovered, the relations between
the constants in the ascending and descending series may still be obtained numerically by
calculating from the ascending and descending series separately and equating the results.

G. G. STOKES.

OF ARBITRARY CONSTANTS, &c.

125

APPENDIX.

QAdded since the reading of the Paper.]

On account of the strange appearance of figures 2 and 3, the reader may be pleased to
see a numerical verification of the discontinuity which has been shewn to exist in the values
of the arbitrary constants. I subjoin therefore the numerical calculation of the integral to
which fig. 2 relates, for two values of x, from the ascending and descending series separately.
For this integral D = 0, and I will take C = 1, which gives, (equations 30,)

^ = 7r-^r(l); B= -S7r-5r(f);
and log J = 0-1793878 ; log (- B) = 0-3602028.

The two values of x chosen for calculation have 2 for their common modulus, and 90«,
1.60°, respectively, for their amplitudes, so that the corresponding radii in fig. 2 are situated
at 30" on each side of the radius passing through the point of discontinuity c. The terms of
the descending series are calculated to 7 places of decimals. As the modulus of the result
has afterwards to be multiplied by a number exceeding 40, it is needless to retain more than
6 decimal places in the ascending series. In the multiplications required after summation,
7-figure logarithms were employed. The results are given to 7 significant figures, that is, to
5 places of decimals.

The following is the calculation by ascending series for the amplitude 90" of a?. By
the first and second series are meant respectively those which have A, B for their coefficients
in equation (19).

First Series.

Second Series.

Order of

Coefficient

Coefficient

tenn.

Real pan.

ofV=T.

Real part.

of V^T.

+ 1 -000000

+

2-000000

1

-

12-000000

+ 12-000000

2

- 28-800000

-

20-571429

3

+ 28-800000

- 16-457143

4

+ 15-709091

+

7-595605

.5

-

5-385974

+ 2-278681

6

- 1-267288

-

0-479722

7

+

0-217249

- 0-074762

8

+ 0-0i8337

+

0-008971

9

-

0002906

+ 0-000855

10

- 0-000?40

-

000066

11

+

0000016

- 0-000004

12

+ 0-000001

Sum

- 13-330099 +

1 1-628385 v/- 1

- 2-252373 -

11-446641 <v/-l

Sum multiplied by A,

- 20-14750 +

17-57548 \/-l;

hyB,

+ 5-16230 +

26-23499 \/-l-

126 PROFESSOR STOKES, ON THE DISCONTINUITY

When the amplitude of x becomes 150° in place of 90°, the amplitude of ar^ is increased
by 180", Hence in the first series it will be sufficient to change the sign of the imaginary
part. To see what the second series becomes, imagine for a moment the factor uc put outside
as a coefficient. In the reduced series it would be sufficient to change the sign of the imagi-
nary part ; and to correct for the change in the factor oo it would be sufficient to multiply by
cos eo" +■%/- 1 sin 60°. But since the amplitude of w was at first 90", the real and imagi-
nary parts of the series calculated correspond respectively to the imaginary and real parts of
the reduced series. Hence it will be sufficient to change the sign of the real part in the

product of the sum of the second series by B, and multiply by - (l + \/3 \/- 1), which

gives the result

- 25-30132 + 8-64681 \/- 1.

Hence we have for the result obtained from the ascending series :

for amp. a? = 90*, for amp. a? = 150",

From first series - 20-14750 + 17-57548 \/- 1 - 20-14750 - 17-57548 \/- 1

From second series + 5-16230 -|- 26-23499 \/- 1 -25-30132+ 8-64681 -s/- 1

Total - 14-98520 + 43-81047 v/- 1 - 45-44882 - 8-92867 \/- 1

On account of the particular values of amp. x chosen for calculation, the terms in

the ascending series were either wholly real or wholly imaginary. In the case of the

descending series this is only true of every second term, and therefore the values of the

moduli are subjoined in order to exhibit their progress. The following is the calculation

for amp. x = 90", in which case there is no inferior term.

Coefficient
Real part. ofV— 1.

+ 1-0000000

+ 0-0086806 +0-0086806

+ 0-0011604

- 0-0001484 + 0-0001484

- 0-0000563

- 0-0000142 - 0-0000142
- 0-0000089

+ 0-0000033 - 0-0000033
+ 0-0000029

+ 0-0000015 +0-0000015
+ 0-0000017

- 0-0000010 + 0-0000010

- 0-0000007 - 0000017

+ 1-0084677 + 0-0099655 v/- 1.
The modulus of the term of the order 12 is 14 in the seventh place, and is the least of
the moduli. Those of the succeeding terms are got by multiplying the above by the factors

Order,

Modulus.

1-0000000

1

0-0122762

2

0-0011604

3

00002099

4

0-0000563

5

00000200

6

0-0000089

7

0-0000047

8

0-0000029

9

0-0000021

10

0-0000017

11

0000015

Remainder

Sum

OF ARBITRARY CONSTANTS, &c 127

VO616, 1*2208, 1-5116, 2-0053, &c., and the successive differences of the series of factors headed
by unity are

A'= + 0-0616, A'= + 0-0976, A'= + 00340, A*= +0-0373, &c.
These differences when multiplied by 14 are so small that in the application of the
transformation of Art. 8, for which in the present case q = 1, the differences may be neglected,
and the series there given reduced to its first term. It is thus that the remainder given above
was calculated.

The sum of the series is now to be reduced to the forp p (cos 9 + \/- l sin 6), and
thus multiplied by e~''"^ and by x~i. We have

for series log. mod. = 0-0036832 amp. = + o* 33' 58". 21

for exponential log. mod. = 1-7371779 amp. = + ISO* 49' o". 78

for x'i log. mod. = 1-9247425 amp. = - 22* 30'

1-6656036 + ]08« 52' 58". 99

When the amplitude of x is 150", there are both superior and inferior terras in the ex-
pression of the function by means of descending series. It will be most convenient, as has
been explained, to put in succession, in the function multiplied by C in equation (20),
amp. x = 150" and amp. a? = - 210°, and to take the sum of the results. The first will give
the superior, the second the inferior term.

For the amplitudes 90*, 150* of *, or more generally for any two amplitudes equidistant
from 120*, the amplitudes of xi will be equidistant from 180°, so that for any rational and
real function of a?^ we may pass from the result in the one case to the result in the other by
simply changing the sign of v — 1, or, which comes to the same, changing the sign of the
amplitude of the result. The series and the exponential are both such functions, and for the
factor x-i we have simply to replace the amplitude - 22* 30' by - 37° 30'. Hence we have
for the superior term

log. mod. = 1-6656036 ; amp. = - 168* 52' 58". 99.

When amp. x is changed from 150° to - 210°, amp. x^ is altered by 3 x 180°, and there-
fore the sign of x^ is changed. Hence the log. mod. of the exponential is less than it was by
2 X 1-737... or by more than 3. Hence 4 decimal places will be sufficient in calculating the
series, and 4-figure logarithms may be employed in the multiplications. The terms of the
series will be obtained from those already calculated by changing first the signs of the imagi-
nary parts, and secondly the sign of every second term, or, which comes to the same, by
changing the signs of the real parts in the terms of the orders 1, 3, 5..., and of the imagi-
nary parts in the terms of the orders 0, 2, 4... Hence we have

Real part.

Coefficient of V-1.

+ 1-0000

- 0-0087

+ 0-0087

- 0-0012

+ 0-0001

+ 00001

+ 0-9914 + 0-0076 \/- 1
log. mod. = 1-9963 ; amp. = + 26'-5.

128 PROFESSOR STOKES, ON THE DISCONTINUITY OF ARBITRARY CONSTANTS.

Hence we have altogether for the inferior term,

log. mod. = 2-1838 ; amp. = + 183" 45'. 5

Hence reducing each imaginary result from the form p (cos 9 + v - 1 sin 9) to the form
a +v/— 1 6, we have for the final result, obtained from the descending series:

For amp. a? = gO". For amp. x = 150«.

From superior term - 14-98520 + 43-81046\/- 1 ; - 45-43S60 - 892767 \/- 1
From inferior term - 0-01524 - 0-00100 \/- 1

- 45*44884 - 8-92867 \/- 1
Had the asserted discontinuity in the value of the arbitrary constant not existed, either
the inferior term would have been present for amp. x = 90^, or it would have been absent
for amp. x = 150", and we see that one or other of the two results would have been wrong in
the second place of decimals.

In considering the relative difficulty of the calculation by the ascending and descending
series, it must be remembered that the blanks only occur in consequence of the special values
of the amplitude of x chosen for calculation: for general values they would have been all
filled up by figures. Hence even for so low a value of the modulus of a? as 2 the descending
series have a decided advantage over the ascending.

VII. On the Beats of Imperfect Consonances. By Augustus De Morgan, F.R.A.S.
of Trinity College, Professor of Mathematics in University College, London.

[Read Nov. 9, 1857.]

The subject of this paper was treated in full, for the first and only time, by Dr Robert
Smith, in the two editions of his Harmonics (Cambridge, 1749, 8vo. ; London*, 1759, 8vo.).
The results are the same in both editions, but the improvements of the second edition add
considerably to the learned obscurity in which the subject is involved. Dr Smith presents, so
far as I know, the strongest union of the scholar, mathematician, physical philosopher, and
practical musician, who ever treated of mathematical harmonics: and his book is not only the
most obscure and repulsive in its own subject, but it would be difficult to match it in any sub-
ject. The consequence has been that the point in which Robert Smith made an important
addition to acoustics has been little more than a resultf in the hands of some of the organ-
tuners. Dr Young certainly did not understand Smith's theory. He was also a remarkable
union of the scholar, mathematician (a character in which he deserves to stand much higher
than he is usually placed), and physical philosopher: and was a successful student in music;
but he wanted a musical ear (Peacock's Life, pp. 59, 79, 81). I have my doubts whether
Robison had read more of Smith's theory than its results. For myself, I made out what
ought to have been the theory from the formulce, and then was successful in mastering Smith's
explanations.

Before proceeding to the subject, I make some remarks upon the method of dividing the
octave. Should this paper fall into the hands of any mathematician unused to musical mea-
surement, he must be informed that proximity and longinquity are measured by ratio, not by
difference. Thus notes of p and q vibrations per second are at the same interval as notes of
hp and kq vibrations per second, be k what it may. Consequently, an interval remains con-
stant, not with p - q, but with logp - log g. The octave of any note, which has with that
note a sort of identity of effect which no words can describe, makes two vibrations while
the note makes one vibration. Any note makes p vibrations while its upper octave makes
Zp vibrations : hence log 2p - log^, or log 2, is the measure of every interval of an octave.

• It is worthy of note that at this period the book bears the
name of the place where it is printed, not of the place where
the publisher sells it. Both these editions are printed for
Cambridge publishers (the Merrills).

■f- So long as unequal temperament was in use, and even

now when it is adopted, the beats were and sometimes are used
in tuning: but when equal temperament is required (and this
system has gained ground rapidly) the tuners have nothing to
do with beats, except to get perfect octaves by destroying them.
I speak of the organ, and of this country.

Vol. X. Pakt I. 17

130

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

Many writers, from Sauveur downwards, have seen the convenience of using the figures of
'3010300, the common* logarithm of 2. Thus Sauveur, for one method, divides the octave
into 301 parts, so that if the higher of two notes make m vibrations while the lower makes
n, the integer in 1000 (log wi — logw) is the number of subdivisions contained in the interval,
quam proxime. The tuner of the pianoforte is required to estimate half a subdivision: for the
fifth of equal temperament is 175'60 subdivisions, and the perfect fifth is 176"09 subdivisions.
Even in practice, then, a smaller subdivision is required: and theory will hardly be content
without the representation of the 50th part of the smallest interval in common practical use.
I should propose to divide the octave into 30103 equal parts, 2508"6 to a mean semitone.
Each part may be called an atom; and we have the following easy rules, which suppose the
use of a table of five-figure logarithms.

To find the number of atoms in the interval from m io n vibrations per second, neglect
the decimal point in log m — log n, or in log n — log m, whichever is positive. To find the
ratio of the numbers of vibrations in an interval of k atoms, divide by 100,000, and find the
primitive to the result as a logarithm.

To find the number of mean semitones in a number of atoms, divide the number of atoms

by — logs X 100000, which may be done thus. Multiply by four; deduct the 300th part of this

product and its 10,000th part, adding one-ninth of this 10,000th part; make four decimal
places, and rely on three. Thus a perfect fifth has 100,000 (log 3 — log 2) atoms, or 17609,
which multiplied by 4 is 70436. The 300th part of this is 235, and the 10,000th part is 7, of
which one-ninth may be called 1. And 70436 — 241 is 70195, whence 70195, say 7"020, is the
number of mean semitones in a perfect fifth.

To find the atoms in a number of mean semitones, multiply by 10,000 ; add to the result
its 300th part and its 10,000th part, and divide by 4. Thus 12 mean semitones gives 120,000
increased by 400 + 12, or 120412, which divided by 4 gives 30103. This rule is as accurate
as the value of log 2; the one which precedes is a near approximation. Both are consequences
of the equation

1 1 / 1 1 \

— X -30103 = — 1 -) + .

12 40 \ 300 10,000/

Dr Smith found that the D of his organ, the first space below the lines of the treble, gave 254,
262, 268, double vibrationst in the common temperatures of November. September, and August.

* Euler, and after him Lambert, suggested the use of
acoustical logarithms; and proposed systems, of which the
bases are 2 and '^2. Prony gave both tables in his Inslruc-
tions Elimentaires sur les mot/ens de calculer les intervalles
musicaux, Paris, 1832, 4to. The second table shows at once,
in log m — log n, the number of mean semitones in the interval
whose ratio of vibrations is m : n. Prony has also calculated,
but I cannot give the reference, a table of logarithms to the base

81

— , which gives the number of commas in m iti, by logm— logn.

The atom which I have proposed, which is the 540th part of a
comma, gives the commas by division by 60 and 9. 1 have my
doubts whether any tables will be so convenient as those of
common logarithms, used in the way I propose. Special tables.

for purposes which do not often occur, are of value only when
tliey save complicated operations. Such tables are not in
the way when wanted ; and when they are found, their struc-
ture and rationale have to be remembered.

It is a sufficient proof of the state of knowledge of the theory
of beats that a work which goes so deeply into the formulae
connected with musical vibrations as Prony 's makes no allusion
to beats. Previously to the use of logarithms, the arithmetical
calculations of the scale were very laborious. Mersenne makes
58| commas in the octave, the true number being 55J. Nicolas
Mercator corrected this in a manuscript seen by Dr Holder,
and then proposed an artificial comma of 53 to the octave,
which gave all the intervals very nearly integer.

+ Writers are very obscure in their use of the word vibra-

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

131

Here we have intervals of -54 and -39, altogether '93, of a mean semitone. Mr Woolhouse's
experiment gives 254 double vibrations to the C immediately below; and other experiments
give nearly the same, for our day. The common tradition is that concert-pitch has risen
about a note in the last century. The change can be traced in its progress. Robison, at
the end of the last century, found the ordinary tuning-forks gave 240 vibrations for C, that is,
270 vibrations for D, a little higher than Dr Smith's organ at its warmest. Possibly some of
this effect may have arisen as follows. The organs being tuned in the cold to the usual pitch
of the day, the orchestras, on tuning with them after the air had been warmed by a crowd,
would find it necessary to raise their pitch. This would have a tendency to cause a perma-
nent rise, which the organ-tuners would of course follow, and then the same effect would be
repeated. The convenience of representing the Cs by powers of 2 has led many writers to
choose 256 as the number of double vibrations in the first C below the lines of the treble: I
trust this power of 2 will be enough to prevent the pitch from making any further ascent.

The subject to which I now come has been perplexed from the beginning by a confusion
of different things under one word. By a beat, I mean any acoustical cycle derived from
composition of ordinary vibrations ; whether the returns can be distinguished by the ear as
separate occurrences, or whether they are rapid enough to cause a sound. The first kind* of
beats were used by Sauveur: but as there is a confused discussion about them in which his
name occurs, it will be more convenient to call them Tartini's heats, because, when they
become rapid enough to give a note, that note is the grave harmonic detected by Tartini in or

Hon; they make it difficult to know whether they mean the
single wave, be it of condensation or of rarefaction, or the
double wave made up of one condensation and one rarefaction.
Much confusion might have been saved in many subjects if
terms of contempt, or of slang, had been seriously adopted :
for such terms are very often more expressive than the solemn
words which they are directed at. The " previous examina-
tion" is very feeble compared with the "little-go." For the
present case, when the pendulum was brought into use, it was
called in derision a swing-swang. If this word had been
adopted by writers on acoustics, all the confusion I speak of
would have been prevented; for no writer would have left it in
doubt whether he reckoned in swings, or in swing-swangs, as
I shall do. There is the same difficulty in medical descrip-
tions, occasionally : some have counted inspiration and respira-
tion as one, most as two.

• The organ tuners must in all time have known the beats
which disappear when the concord becomes perfect. The first
writer who is cited as having mentioned them is Mersenne
( Harmonie Universelle, Paris, 1636, folio, book on instruments,
p. 362). But Mersenne does not attempt any explanation. He
observes that two pipes which are nearly unisons tremble,
and make the hand which holds them tremble. But the trem-
bling goes off when the unison is made perfect ; which, says
Mersenne, is the exact opposite of what takes place in strings.
That is, he imagined the beats were to be compared with the
sympathetic vibrations. Dr Smith, with that habit of indis-
tinctive citation which is one of the manias of much learning,
cites Mersenne and Sauveur together as his predecessors in
the subject.

There is another writer who is better qualified to be classed

as the immediate predecessor of Sauveur, because he distinctly
opposes the sympathy of consonant vibrations, and its effects, to
the clashing of dissonant vibrations. I mean Dr Wm. Holder,
r.R.S., who died in January 1696-7, and was the opponent of
Wallis on a question of priority in the method of teaching
the deaf and dumb. In his Natural Grounds and Principles of
Harmony, published in 1694, he describes beats in a manner
which is worth quoting, were it only as an instance of the
poetry of explanation which science has driven out (pp. 34, 35,
ed. of 1731):—

" It hath been a common Practice to imitate a Tabour and
Pipe upon an Organ. Sound together two discording Keys
(the base Keys will shew it best, because their Vibrations are
slower), let them, for Example, be Gamut with Gamut sharp,
or F Faut sharp, or all three together. Though these of them-
selves should be exceeding smooth and well voyced Pipes, yet,
when struck together, there will be such a Battel in the Air
between their disprbportioned Motions, such a Clatter and
Thumping, that it will be like the beating of a Drum, while a
Jigg is played to it with the other hand. If you cease this, and
sound a full Close of Concords, it will appear surprizingly

smooth and sweet Being in an Arched sounding Room

near a shrill Bell of a House Clock, when the Alarm struck,
I whistled to it, which I did with ease in the same Tune with
the Bell, but, endeavouring to whistle a Note higher or lower,
the Sound of the Bell and its cross Motions were so predomi-
nant, that my Breath and Lips were check'd, that I could not
whistle at all, nor make any sound of it in that discording
Tune. After, I sounded a shrill whistling Pipe, which was
out of Tune to the Bell, and their Motions so clashed, that
they seemed to sound like switching one another in the Air."

17—2

132

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

about 1714. And even when they give a sound, it will still be convenient to call them Tar-
tini's beats. These beats are in their perfect theoretical existence when a consonance is quite
true, and they owe their usual existence to its approximate truth. Tartini* used to tell his
pupils that their thirds could not be in tune when they played or sang together, unless they
heard the low note: assuming, doubtless, that their perceptions were as acute as his own.

The second kind of beats I shall call SmitJCs beats, because Dr Smith first made use of
them, and gave their theory. They are entirely the consequence of the imperfection of a con-
sonance, and become more rapid and more disagreeable as the imperfection increases, vanishing
entirely when the consonance is perfectly true.

I cannot find the means of affirming that Smith was acquainted with Tartini's grave har-
monic. In the place in which one would have expected him to mention it, namely, when he
mentions the Jlutterings, as he calls them, which I name Tartinis beats, he does not make
the slightest reference to those flutterings becoming rapid enough to yield a note, though he
complains that he could hardly count them.

Smith accuses Sauveur of confounding the beats of an imperfect consonance with the
flutteringsf of a perfect one. It is true that Sauveur makes the same use of Tartini's beat

• Tartini published his treatise on harmony at Padua in
1754. D'Alembert's account of this work ia so precisely what
he might have written of Smith, that I quote it. " Son livre
est e'crit d'une maniere si obscure, qu'il nous est impossible
d'en porter aucun jugement : et nous apprenons que des Savans
illustres en ont pense' de menie. II seroit k souhaiter que
I'Auteur engageat quelque homme de lettres verse' dans la
Musique et dans I'art d'&rire, ^ developper des idees qu'il
n'a pas rendues assez nettement, et dont Part tireroit peut-etre
un grand fruit, si elles e'toient mises dans le jour convenable."
M. Romieu, of Montpellier, published a memoir in 1751, in
which he described Tartini's grave harmonic : and hence some
have made him the first discoverer. But Tartini had been
teaching the violin, on which instrument he was the head of a
celebrated school, a great many years : that he should not have
published t'he grave harmonic to every pupil whom he taught
to tune by fifths, is incredible. He himself affirms in his
work that he always did so from 1728, when he established
his school : and further, that he made the discovery on his
violin, at Ancona, in 1714; this was the year after he dreamed
the Devil's Sonata. As it is stated that he told how the devil
played to him in his sleep, many years after, to Lalande, who
could make astronomical gossip of any thing, I should not be
at all surprised if a certain four-volume work contained evidence
of the date of the grave harmonic.

Kameau, not Romieu, is the natural counterpart of Tartini.
In 1750 he published his celebrated treatise on harmony, the
completion of a system which he had sketched in previous
works : and he and Tartini are thus related. Tartini makes
his grave note the natural and necessary bass to the consonance
which produces it : Rameau makes the harmonics of any given
note the natural and necessary treble of the given note as a bass.
These contemporary counter-systems are now exploded : they
have an uncertain coiuiexion with the truth, no doubt ; but the
are demands and obtains a great number of combinations which
neither system will allow.

It is due, however, to Rameau to observe that his discovery,

which appears independent of Tartini's, is that of a physical
philosopher, and is developed in a masterly manner. He gave
the theory, and detected the beats which occur when the grave
harmonic becomes inaudible by lowness. His memoir was pub-
lished by the Royal Society of Montpellier in 1751, in a coUec
tion headed Assemblie Publique &c. I have never seen this
memoir. There ia a long extract from it in a curious and ex-
cellent work, which I never see quoted, the Essai sur la musique
ancienne et moderne^ Paris, 1780, 4 vols. 4to, attributed by
Brunet to Jean Benjamin de la Borde.

Chladni {Acotistique, p. 253) says that the first mention of
the grave harmonic which he knew of is by G. A. Sorge (An-
weisung zur Stimmung der Orgelwerke, Hamburg, 1744), who
asks why fifths always give a third sound, the lower octave of
the lower note, and concludes that nature will put 1 before 2, 3,
that the order may be perfect. If Tartini's evidence in his own
favour be disallowed, then Sorge becomes the first observer.
But to me the uncontradicted assertion of a teacher whose
pupils were scattered through Europe, and included men so
well known on the violin as Nardini, Pugnani, Lahoussaye,
&c. &c., that he had pointed out the third sound to all his
school from 1728 to 1750, is real evidence. Chladni's mention
of Tartini is as uncandid as possible: — ' Tartini, auquel on
a voulu attribuer cette de'couverte, en fait mention dans son
Trattato . . .' Mentions it .' No one knew better than Chladni
himself (as he proceeds to show, the moment the paragraph
about priority is finished) that Tartini's whole book is a system
founded upon it. D'Alembert, La Borde, Rousseau, &c. do not
dispute Tartini's claim ; and the common voice of Europe
gives no other name to the discovery.

On this subject in general see the Article Fondamental in
the Encyclopedia, by D'Alembert ; Rousseau's Musical Dic-
tionary, Ilarmonie and Systime ; Matthew Young's Enquiry,
&c.

f Smith does not, so far as 1 can find, attempt to explain
these Jlutterings; though I think it may be collected that he
knew their cause.

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

133

which Smith shows how to make of his own beat; namely, the deduction of the number of
vibrations in a note. It is also true that Sauveur applies the term hattemens to both, and
quite correctly; for both are hattemens, though arising from different sorts of cycles. But it
is not true that Sauveur confounds the phenomena by imagining them to be the same, by put-
ting one in the place of the other, or by giving to either the reason of the other. His object is
(Mem. Acad. So. 1701, Paris, 1719, p. 359) to find the son fixe, as he calls it, which makes 100
vibrations in a second. He directs us to take organ-pipes, at least two feet long, and to tune
diatonic intervals so perfect that not the smallest battement shall be perceived. Here he
speaks of what I call Smith''s beats, of which he clearly knows the negative use, namely, the
acquisition of perfect concords by avoiding them. Having thus procured a perfect major and
minor third to one note, he sounds them together, the interval being 25 : 24 in ratio of vibra-
tions, and thus procures a battement (but this is Tartini's beat) at each 25th vibration of the
upper note. By taking nearer* consonances, though certainly not harmonic ones, he procures
beats which can be easily counted. Dr Smith {Harmonics, 2nd Ed. p. 96) complains that he
cannot count Sauveur's beats: but, though he used low notes, he took the prominent concords
or discords of the scale, which are not near enough.

Dr Young pronounced Smith's work " a large and obscure volume, which for every pur-
pose except the use of an irapracticablef instrument leaves the whole subject precisely where
it found it." If Dr Young had said that the work was largely obscure, he would have
been correct: had the volume been larger, it had probably been less difficult; it is a small
volume for the quantity of subject-matter. It leaves the subject where it found the subject
only in the minds of those who do not master it; in which number we must place Young
(Peacock, Life of Young, pp. 128, 129; Works, Vol. i. pp. 83, 84, yS, 134 — 139; Robison,
Mech. Phil., Brewster's edition, Vol. iv. pp. 408, 411, 412). One sentence from Young will
make it clear that he confounded Tartini's beat with Smith's, though Smith had distinctly
stated (p. 97) that "a judicious ear can often hear, at the same time, both the flutterings and
the beats of a tempered consonance, sufficiently distinct from each other." But Young says
(i. 84), " The greater the difference in the pitch of two sounds the more rapid the beats, till
at last, like the distinct puffs of air in the experiments already related, they communicate the
idea of a continued sound; and this is the fundamental harmonic described by Tartini."

• He inserts between the two, 24 and 25, the pipe 24J, and
making the three sound together, gets a three-pipe beat of 48,
49, 50 vibrations. He then inserts 48.J and 49J, and gets a five-
pipe beat of 96, 97, 98, 99, 100 vibrations. These are the beats
which he proposes to count ; so that, though he sets out with
Tartini's beat, his experiment is as far removed as can be, even
from the mere use of this, and has nothing to do with Smith's
theory. Strange that Young, who actually refers to Sauveur,
should call Smith's theory nothing but an extension of this
multipipe clatter : strange also that Robison should imply the
same thing. It is said that Sauveur's musical ear was very
bad. That he sounded these pipes together is clear ; for of the
three first mentioned he says, that the beat of the first and
third is faintly audible through the beat of the three. When

his five pipes sounded together, each of the consecutive inter-
vals was something less than the fifth part of a mean semitone.
Any one whose ear was thus guillotined might well have ex-
claimed. Oh ! Mtisique ! que de crimes on commei en ton
nom !

f This entirely relates to the second edition. No doubt
some readers of Dr Young have searched their copies of Smith's
first edition for this instrument, without finding it. It is the
account of an enharmonic harpsichord, which is described in
the work, and with improvements in a postscript to the second
edition, with a separate title page, in 1762, three years and a half
after the publication of the work. The enharmonic piano-forte
would not be impracticable, if people cared enough about the
accession to pay for it.

184

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

Never was anything* more inaccurate: it would make the whole passage from unison to
the minor third a preparation for the grave harmonic of that concord. When the unison or
other simple concord is gradually mistuned, the " beating becomes more and more rapid,
changes to a violent rattling flutter, and then degenerates into a most disagreeable jar."
These phenomena are reversed as continued increase of the interval brings us towards another
simple concord. The description is due to Robison, who (iv. 414 — 421) goes through the
whole phenomena of the octave. It is clear that Young confounded the two kinds of beat:
and even Robison, while animadverting on Young's opinion of Smith, gives strong reason to
think that he does not make the distinction. He informs us (iv. 410) that Sauveur had applied
beats, and that his method is operose and delicate, " even as simplified and improved by Dr
Smith." In common with a great number of other writers, he ventures on no explanation of
any beats except those which occur in imperfect unisons, in which Tartini's beat is no other
than the vibration of the note itself. When he comes to mention the beats of badly-tuned
fifths, he declines explanation, and (iv, 409) states what " Dr Smith demonstrates." He calls
the method of beats, and to my mind very justly, the greatest discovery (iv. 411) made in the
subject since the time of Galileo: but he goes on to depreciate the value of his own opinion by
asserting that the theory of Tartini's harmonic is included in Smith's theory of the beats of
imperfect consonances. The great defect of Smith's theory is its exclusion of Tartini's har-
monic. Young, in replying, writes as follows (i. 136): "Why then are we obliged to call it
Dr Smith's discovery, or indeed any discovery at all .'' Sauveur had already given directions
for tuning an organ-pipe by means of the rapidity of the beating with others, Mem. de
I' Acad. 1701, 475, ed. Amst. Dr Smith ingeniously enough extended the method; but it appears
to me that the extension was perfectly obvious, and wholly undeserving the name either of a
discovery or of a theory." This amply proves that neither Robison nor Young had read
Smith's theory; and I have very strong doubts that any pei'son who has written on the subject
ever did read it, Chladni makes precisely the same mistake as Young. He tells us {Acous-

* Except, perhaps, Young's reiteration of his own mistake,
several years after, in the Course of Lectures ( London, 2 vols.
4to, 1807, Vol. I. p. 390). Young here begins by describing the
Smith's beat of imperfect unisons, clearly and correctly. He
then takes, as his second instance, the Tartini's beat of a well-
tuned diatonic semitone, and then repeats the account of the
Smith's beats giving the grave harmonic. The terms in which
he has spoken of Ur Smith's labours are sucli as can only be
met by convicting him of clear and palpable mistake. Those
who may be inclined to wonder that Young should have so sig-
nally failed in a matter connected with the distinct conception
of a complex undulation, may be reminded that many an inves-
tigator has fallen into some singular error in the subject which
he had, of all others, made completely his own.

t 1 think this is a mistake. 1 find nothing in Sauveur's
memoir of 1701 about beats, except what I have described.
Lagrange, in his celebrated Turin memoir on sound, refers
(p. 75) to Sauveur's memoir of 1700 (not 1701) in so vague
a manner that he might be supposed to have Smith's beats in
view. On looking at the volume for 1700, I find, not a memoir
by Sauveur, but the description of one, forming part of the
abstracts called Histoire. Here we find that Sauveur did
actually commence with imperfect unisons, which give Smith's

beats, that he had a notion of the rationale of such beats, that
he had made some experiments, and that a commission of tbe
Academy was appointed to inspect their repetition. The ac-
count of this experiment is a part of the history of the subject.
" M. Sauveur en rendit conte luy-meme et avoUa que pour
cette fois elle n'avoit pas bien r^ussi, car d'autres fois, et en
presence des plus habiles Musiciens de Paris, elle avoit paru
trds juste et tr^s prdcise. La difficult^ de la recommencer,
I'appareil qu'il faut pour cela, d'autres occupations plus
pressantes de M. Sauveur, et meme d'autres recherches
d'Acoustique, ou il a it6 oblige de s'engager par la liaison
qu'elles avoient avec le Son fixe, ont e'te cause qu'on en est
demeurf? la, mais on sjait qu'en fait d'experiences il ne faut
pas se d^courager aistiment, et qu'elles ont pour ainsi dire, leur
caprices que I'on surmonte avec le temps " (p. 139). All this
means that Sauveur commenced with the beats of imperfect
unisons; that he made experiments which satisfied the musi-
cians, bat broke down — by caprice — before the academicians;
that he had in the mean time commenced his acquaintance with
Tartini's beats, and was pursuing the researches which led to
thepaper of 1701, in which Smith's beats are wholly abandoned.
It is singular that Smith himself did not see this.

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

135

tique, p. 252), that when the vibrations of two sounds come together very rarely, we perceive
the coincidences like beats (comme des battemens, tres-desagreables...) very disagreeable to the
ear in a badly-tuned instrument. The more nearly, he goes on to say, the consonance is
made perfect, the more insensible the beats become, until at last they are lost in the sensation
of a feeble resonance with a grave sound. And he ends by telling us that an instrument is not
in tune if any one of its intervals allow beats to be heard. According to Chladni, then, uni-
sons ought to give a grave sound when perfectly tuned; to say nothing of his appearing to
believe in some system of tuning a whole instrument in which there are no beats.

All the modern writers with whom I am acquainted content themselves, at the utmost,
with describing the phenomenon, and giving some account of the beats of imperfect unisons,
except as I proceed to mention. Some time ago, after detecting the explanation from the
formulas, and then unravelling the demonstration of the same formulae with very great diffi-
culty, I searched far and wide to see if any writer had appreciated and acknowledged the skill
with which Dr Smith had concealed his truth at the bottom of a well of learning. The only
writers in whom I found a solution of the problem were as follows. William Emerson, a
sound and once well-known, but now nearly obsolete, writer, gave a true solution (p. 484) of the
problem in his Algebra, published in 1763. His method is very obscure just at the pinch of
the demonstration; we see that certain recurrences are established, but are left wholly in the
dark as to why those recurrences should explain the beats; it is quite as likely that two of
them should go to a beat as one. Mr Woolhouse, in his Essay on Musical Intervals (Lon-
don, 1835, 8vo. p. 84), the best modern manual of mathematical harmonics which I know of,
has treated the problem in the same manner, arriving at another variety* of the formula.
Both these methods want the introduction of Tartini's beat in its connexion with Smith's;
and this the following treatment of the subject will supply.

Let m and n be two numbers prime to one another, m > n, and let the higher note make m
vibrations while the lower note makes n. In the diagram I shall suppose m = 5, n = 3, or the
interval a major sixth. I shall also suppose each whole wave to be one of condensation, for
simplicity. And first, let two zeros of condensation, one in each wave, be synchronous.
The following diagram represents the whole of one wave of Tartini's beat, whether it be the

• Emerson arrives at the formula which I presently mark as
( 1 — j;) Mn -=- X ; Mr Woolhouse arrives at (.1 — x) Nm. Look-
ing at all probabilities, as derived from Emerson's life, habits,
and access to books, I very much doubt his method being
derived from Dr Smith. He was a musician, and an amateur
tuner of instruments; and he was mechanic enough to enrich
his own virginal with additional semitones. He was nearly
fifty before the first edition of Smith appeared, he lived in the
county of Durham on a very small fixed income (about i;60
a-year), his writings show very little reading, and the library
which he sold before his death, the collection of nearly forty
years, was valued by himself under £50. If I could only
establish a high probability of acquaintance between Emerson
and Thomas Wright, now known as the speculator on the milky
way, who lived within twelve miles of Emerson, I should con-
sider the united chances of Wright having possessed the book

and having lent it to Emerson as giving a liigher probability to
Emerson having seen it than anything I can create from com-
parison of the two methods. It is very likely, then, that he
had not seen Smith's Harmonics. The amusing biography of
Emerson, which is prefixed to his collected works, and which
appears to have been written by some one who had ample in-
formation, states that he was a very desultory student till after
thirty years of age. Having been treated with contempt by his
wife's uncle, he determined to gain a name, that he might
prove himself the better man of the two. This he has done :
if the name of his relative were now worth inserting, it would
only be in connexion with the statement, true or false, that,
though possessed of two livings and a stall, he made a large
income by the practice of surgery. Emerson died in 1782, in
his 81st year.

136

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

pulse of a grave harmonic, or only one of Smith's flutters: namely, five waves of the upper
note, three of the lower, and the resultant wave.

The abscissa represents 15 equal portions of time, of which the component waves take suc-
cessive threes and fives; the ordinates represent the condensations at the end of the times
represented by the abscissas. The thick line, whose ordinate is always the sum of the other
two, represents the wave of Tartini's beat, which is repeated in the next fifteen portions of
time.

The united effect of the two waves is one particular phase of a major sixth: a pulse of
the grave harmonic in which gradations of loudness and faintness are distributed in a certain
manner through 15 portions of time, to be strictly repeated in the next 15 portions, and so on.
An unlimited number of other phases exist, one for every mode in which the zero of conden-
sation of the shorter wave can be laid down in the longer wave, so as to produce a law of
loudness and faintness which is not found in any other mode. Thus the following is the dia-
gram in which the maximum condensation of the shorter wave synchronises with the zero of
condensation of the longer wave.

We have now Tartini's beat under a different type, in which the loudness and faintness
are distributed in another way: the consonance of a major sixth, as before, with a different
kind of pulse for the grave harmonic, if there be one. Whether the ear would acknowledge
any difference between two major sixths of these different types, cannot be settled; for it is not
in our power to start the pulses as we please. But the ear does acknowledge the gradual
progression through all the types, by recognizing what I have called Smith''s beat. If the
consonance be a very little mistuned, Tartini's cycle is not sensibly altered in character, but
its recommencement undergoes a very small change. If the higher note be tuned a little too
sharp, for example, so that the shorter wave is a very little less than three-fifths of the longer
wave, Tartini's cycle, or something excessively like it, begins a little sooner the second time
than it should do; and the zero of condensation of the shorter wave is thrown back a little.
This effect is doubled at the next commencement, trebled at the next one, and so on: accord-
ingly, in a consonance slightly mistuned, the approximate compound pulse goes through all the
phases which variations in the mode of setting off can give to the true one. This is the
most marked geometrical effect upon the pulses; and Smith's beat is the most marked acousti-
cal effect upon the ear. The connexion of the two is then of the highest probability : and this
becomes certainty so soon as, and not until, the study of the beats, and their application to

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

137

questions of temperament, shows that the theory agrees with other theories, and with practice.
Smith's beat* is a kind of disturbed orbit, of which Tartini's beat is the instantaneous orbit.

The phenomenon itself is different to different ears. To some it consists in alternations of
louder and softer: and undoubtedly there are changes from condensation reinforcing conden-
sation, and rarefaction rarefaction, to condensation balanced by rarefaction, and rarefaction by
condensation. To others it consists in alternate perception of the two sounds of the conso-
nance; and this also is intelligible, as the stronger parts of the two waves alternate. For
myself, though I can perceive both the effects above mentioned when I look out for them, the
phenomenon which forces itself on my ear is an alternation of vowel-soundsf, as in u-a u-a
u-a, &c. pronounced in the Italian way.

The time of a beat depends upon a circumstance which I suppose, by the manner in
which many writers have confined themselves to the case of imperfect unisons, has not been
clearly apprehended. The diagrams are only detached portions of a succession unlimited in
both directions. If the times of vibration be 3a and 5a, (so that a represents the greatest
common measure of the times of vibration, which is repeated 15 times in Tartini's beat,) and
if one of the shorter waves begin at zero with one of the longer ones, the first, third, and
fifth of the shorter waves are advanced 0, a, 2a, upon the several longer waves. If the first
of the shorter waves be advanced ,v (<a) upon the longer one with which it began, then the
advances just spoken of become x, a + w, 2a + ai. Accordingly, looking at the full succes-
sions, while w progresses from to a the consonance passes through all its phases. Every
possible variety of Tartini's cycle is exhibited during the motion of the beginning of one wave,
not through the whole of the other, but through that portion of the other which is performed
in the time which is the greatest common measure of the times of the two waves. In disen-
tangling Dr Smith's explanation, I thought it looked much as if he had first counted on the
whole of the other wave, had found that his results would not agree with experiment, had
detected tiie true submultiple of the other wave by comparison of his theory with experiment,
and had then corrected his former theory. This may be fancy: but, should any one ever read
Dr Smith's book again, I should recommend his attention to this point. And should he find

" 1 have looked in many places to see If 1 could discover the
two beats in any other connexion than that of confusion between
tile two. I once thought I had succeeded; for there is a paper
by Lord Stanhope (Tilloch's Phil. Mag. Vol. xxviii. 1807,
June — September, p. 150) which is sometimes cited in a manner
which would make one suppose he had the distinction. But on
looking at this paper, I find that his distinction between a beat
and a beating is this, that the first is merely the vibration of tlie
not«, the second is Smith*s beat. There is a rather obscure
paragraph at the end, in which he speaks of the beating of a
beating occasioned by two badly turned fifths DA, DA, in
difTerent octaves, sounding together.

f And so it struck Emerson, who says (I. c.) — " Its noise
is such as this, waw, aw, aw, aw, or ya, ya, ya, ya, ya. Our
business is to find out in how many vibrations this perturbation
happens, or how many yaws in a second of time." In the
organ-pipe, in which the efl'ect is much coarser than in the
string, the alternation of vowels is not very self- asserting.
Since the text was written, I have tried tlie comparison of pipes

Vol. X. Part I.

again, which I had never heard, with any reference to beats, for
many years. Mr Davison (of the firm of Gray and Davison)
instructed one of his tuners to prepare an octave of equal
temperament, by ear in the usual way. On trying one of the
fifths, the first thing which struck me was that the beating
seemed to be about double what it ought to be. Without say-
ing anything, I asked tlie tuner to count the beats in a minute
in his own way ; his counting gave the half of mine, and
agreed with the theory, nearly. So little did the alternation of
vowels present itself, that is, so like was each half beat to the
other, that I did not even remember the phenomenon. It was
only on a subsequent day, after more practice, that I caught
the two vowels. The equal temperament, tuned by a practised
tuner, brings the beats near enough to the theory to prove that
the complete cycle of Tartini's beat, and not any multiple or
sub-multiple of it, is the cause of the phenomenon called
Smith's beat, namely, the sound of pulsation which is heard in
two halves, with different vowels in the two.

18

1S8

Mb DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

the reading very easy, I would desire him to put himself, if possible, into the position of a
reader who had had no one but Dr Smith to help him. For of all the difficulties I have ever
encountered with any success, I have no hesitation in calling the theory of beats, as presented
by its author, the very greatest. I would compromise such another job, if such another there
be, by choosing rather to explain to a pupil of reasonable preparation, any fifty pages of the
Principia, and of the TMorie des Prohabilitis, and of the Disquisitiones Arithmeticce.

I now proceed to the formulse which the subject requires. Let the higher note (perfect)
jnake m while the lower note makes n vibrations; m : n being > l and in its lowest terms. Let
k be what we may call the adjusting factor, that is, let nk and mk be the actual numbers of
vibrations in one second of the lower and higher notes. Let ma and na be the actual times
of vibration, in seconds, of the lower and higher notes. Then mnka = 1. Let na +9 he the
time of vibration of the upper note in the imperfect consonance which gives the beats. When
9 is positive, the consonance is tuned flat, the commencements of the more rapid vibrations
advance upon those of the less rapid, and the beats may be said to move forwards. The con-
trary when 9 is negative. It is the same thing to the ear whether the beats move forwards or
backwards. Let as be the ratio of the consonance of the perfect and imperfect upper note;
that is, let on = na : na + 6. Thus d?< 1 when the upper note is too flat. And let N and M
be the actual numbers of vibrations per second in the lower and higher notes of the imperfect
consonance. Hence

m M ^^ ,, . -^ na

— .v = — , Nma = 1, M {na + 0) = i , x =

n N na + 9

1 - a;
9 = na, kmna = I, kn = N.

w

Let /3 be the number of beats in one second. A beat, as shown, lasts through as many

„ , , ., . , . . a . . . . {na + 0) a

of the shorter vibrations as there are units m - : its time is then ; so that we

9 9

have

)8 =

e

{na + 9)a a

1 — <2? 1 ■" ^

= (l - a?) kmn = (l - !c) mN = nM = mN - nM.

Dr Smith does not elicit * any of these formulae, the last of which is remarkably simple-
Thus if a fifth be tuned imperfectly to 200 and 301^ vibrations per second, we have

200 X 3 - soil X 2 = - 3,

or the consonance is tuned sharp to 3 beats per second. The number of beats per second
depends only on the number of vibrations by which the upper note is wrongly tuned, and
the smaller of the two lowest terms of the perfect consonance. Let M' be the proper num-
ber of vibrations for the upper note, so that M' : N = m : n, then /3 = {M' - M) n. Or

• Since this paper was written the article ' Beats ' in the
Edinburgh Encycloptedia, attributed to Mr John Farey, has
been pointed out to me. This article contains Smith's formula,

with two varieties arising out of difl'erent modes of expressing
the division of the octave, Emerson's method, and tlie formula
mN-nM. But no explanution is given.

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

139

thus : — In every consonance of which the lower number is n, every wrong vibration per second
in the upper note is n beats per second*. With this theorem as a key, a rationale can be
obtained without difficulty; but it does not connect the two beats, and would, I think, be
subject to the doubt I have cast on Emerson's method.

The formula given by Dr Smith are obtained as follows. The comma, or difference of a
major and minor tone, being 81 : 80, let 9 correspond to the fraction q : p oi a. comma. Then

Now (1 - aY = 1 -

r8o\?

2wa

2 + (« - 1) a

/■80\J 27

whence a> = — \p= \ nearly.

\81/ l6lp + q- •'

Zq

nearly; a being small:

And /3 = (1 - x) mN =
= nM'

\6\p + q
29

mN
nM

X l6lp - q

which are Smith's formulae (2nd ed. p. 82).

When the upper note is too sharp, q must be made negative, the negative sign of /3
being neglected.

If /J. be the fraction of a mean semitone by which the upper note is flat, we have, for
the number of beats in a minute,

60 (1 - 2~'*) mN, or ■ mmN, or I04u ( —]mN

^ ' 30 \30 1000/

nearly, and more nearly. If the octave be composed of 30103 atoms, of which the upper
note is tuned flat by a atoms, the number of beats in a minute will be

•001381551a (1 - "OOOOl 15129a) miV very nearly,
4x8x13

or

301000

amN nearly.

These formulae are not accurate enough to give the beats in a minute within three or four,
unless both terms be used: and, the vibrations being given, mN - nM is much more easy.

• The passage over the greatest common measure being
fairly arrived at, as the time of a beat, the transition to the
formula mN—nM may be very briefly made. We know that,
m and n being prime to one another, there is, before we arrive
at mn, one way and one only in which ym — 971 = 1 ; and one
way and one only in which qn—pm = 1. The ratio N : M o(
the numbers of vibrations in the erroneous consonance, and also
of the lengths of the waves, is not n : m, but

M

mN - nM
M

consequently the commencement of the shorter wave gains the
of a common measure in every vibration of

fraction

mN — nM
Af "

the higher note, than is mN — nM common measures in one
second, or in M higher vibrations; and each gain of a com-
mon measure is a beat. This demonstration, a little more
developed, will be, 1 should think, the best that can be given.

18—2

140

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

Let the notes of the imperfect consonance be P, Q, and let P' be the octave above P.
If the interval PQ be tuned too flat, then QP^ is two sharp, and vice versa. All remaining
as above for PQ, in passing from PQ to QP^ we must change N, M into M, ZN. If m be
an odd number, we must change n, m, into m, 2w; but if m be even, n, m, must change
into im, n; since the fundamental ratio must be in its lowest terms. And we must also
change the sign of q, neglecting the negative sign of the value of /3, when it occurs. Conse-
quently, jS' being the number of beats of QP^ in a second, we have

(m odd) /3 =

(m even) /3 =

2q

l6lp + q

Nm, /3'=

2?

I6lp - q

M.Zn = 2/3;

9.q

\6\p + q

Nm, /3' =

^q

l6lp - q

M.n = fi.

That is, when the fundamental number (in the ratio m : n) of the mistuned note is odd, the
interval complemental to the octave beats twice as fast as the lower interval first given. But
when this fundamental number is even, the interval and its octave complement have the
same rate of beating. This is one of Smith's* experimental verifications, and is a very easy
one. He is of opinion that an octave might probably be tuned with more perfection by the
isochronous beats of a minor and major concord composing it, than by the judgment of the
most critical ear.

What precedes is a particular case of the following theorem: — Let N, M, L, be three ascend-
ing notes represented by their numbers of vibrations per second. Let N make n vibrations
while M makes m: let M make m vibrations while L makes I: the fractions m : n and I : m
being in their lowest terms. Let the imperfect consonances NM, ML, NL, beat severally /3,
j8', B, times per second: )3 being positive when the higher note is flat, and negative when it is

• There must needs be some way of explaining the excessive
difficulty of this one work of Dr Smith's. His Optics, if not
a model of perspicuity, is by no means notable for obscurity;
on the contrary, I find it abounding insufficiently good descrip-
tions of machinery, a point in which an obscure writer is
generally most perplexed and perplexing. I take the cause
of Dr Smiih's failure of clearness in the Harmonics to be that
he was a practical musician, well versed in the practical writers.
I suppose others have agreed with myself in noting tliat the
worst explainers are those who have to describe the purely con-
ventional, without having had it distinguished from the natural
or the essential in their education. First come the writers
on games of chance, who all, or with the rarest exception,
proceed to explain whist or hazard by commencing at the point
at which they imagine a priori knowledge of the arrangements
ceases. Next come the musicians, with whom a five-line stave,
&c. are In the nature of things. Now Dr Smith had got into
the way of interchanging the practical and theoretical, the
accidental and the essential, &c. The manner in which he
treats the theorem on which this note is written is perhaps
the easiest instance to produce. He gets into the theorem in a
way which leads him to the table of ratios of vibrations, and
he arrives at this result, that wlien the minor consonance is
above the major, the higher consonance beats twice as quick as

the lower, but when the minor consonance is below the major,
the beats are the same. And not until he has pointed this out,
does he proceed to note that the greater term of the ratio of
a minor consonance is even, &c. And his final theorem is
stated in terms of major and minor consonance, it being merely
accidental, so far as our knowledge is concerned, that the nume-
rators of minor consonances happen to be even, in the cases in
which they are useful. The usual minor intervals are the

tone (-g); the third \t) ; the sixth lA; the seventh
(— -). The usual major intervals are the tone (5); the third
(- j ; the fourth (-1, in which there is a failure; the fifth

(- j ; the sixth (- 1 ; the seventh (-0-). In the minor and

lib 16\ ^ , . .
major semitone I — , jr I the rule is inverted ; and also in the

minor fifth ( 55 )• Keeping, however, to common intervals

used in tuning, and calling ihe fourth a minor to ihe fifth, it is
a pretty practical rule that the duplication of beats takes place
when the minor interval is above the major.

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 141

sharp; and the same of the rest. Let g be the greatest common measure of ml and mn.
Then

m + w/3' = gB.

From this we may obtain such theorems as the following. The beats of a minor third

exceed those of the following major third by twice the beats of the whole fifth which they

make up. Twice the beats of a minor third exceed three times the beats of the major third

which it follows by five times the beats of the fifth they make up.

a
Smith's beats themselves have a long inequality whenever - is not an integer; of which I

u

suppose (though I am by no means sure) the ear could hardly be made sensible. The theory

of the beats of a consonance of more than two notes would offer no difficulty, if there be any

thing presented to the ear which it would be of any interest to explain.

A. DE MORGAN.

Univbrsitt College, London,
August n, 1857.

POSTSCRIPT.

A FEW observations on tuning and on temperament will not be out of place. The method
of tuning employed in this country at present is simply adjustive. In equal temperament,
for example, the tuner gets one octave into tune, with its adjacent parts so far as successions
of fifths up and octaves down require him to go out of it; and the notes thus tuned are
called the bearings: all the rest is then tuned by octaves from the bearings. The method
of tuning the bearings, after taking a standard note from the tuning-fork, consists merely
in tuning the successive fifths a little flat, by the estimation of the ear, making corrections
from time to time, as complete chords come into the part which is supposed to be in tune, by
the judgment of the ear upon those chords. Proceeding thus, if the twelfth fifth appear to
the ear about as flat as the rest, the bearings are finished : if not, the tuner must try back.
The system generally used is the equal temperament: when any other is adopted, beats are
sometimes, but not always, employed, that is, counting the beats. For the ordinary tuner,
even in equal temperament, learns to help himself by a perception of the rapidity of the
beating : but without numerical trial.

Now it appears to me that there is in this a loss of time and a loss of accuracy. Difl'erent
tuners, however excellent their ears, do not agree in their results. Two men, tuning different

142 Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

compartments of the same organ, produce two systems which do not agree : they take care that
their tuning-forks shall give them the same standard-note ; but this is all they can get. Many
years ago I had two dulcimers, as 1 suppose they must be called, of a couple of octaves each :
the notes were given by single strings, and the sound was produced by a hammer held in the
hand; they stood exceedingly well in tune, and the sound was as pure as that of a tuning-
fork. When I tuned one to equal temperament, as I thought, and tlien the other, I never
found agreement, though each was satisfactory by itself. I soon left off, setting down the
discordance to my own inexperience. But an old professional tuner, to whom I mentioned the
subject, assured me that he did not believe either that any tuner gained equal temperament, or
that any one tuner agreed with himself or with any other. He summed up by saying that
"equal temperament was equal nonsense."

An octave of tuning-forks might easily be prepared, adjusted with exactness to any tempe-
rament by beats. These beats can be heard in a consonance of tuning-forks as well as in one of
strings or of pipes. The preparation of a standard set, for the manufacturer's own use, would
cost time and trouble : but the standards once at hand, copies might be taken off by unisons
with comparative ease. The labour of obtaining the bearings from the tuning-forks would be
small compared with that of adjustment, as now practised. In tuning the organ, I feel certain
that the ear of the tuner must be much injured, for the moment, by the hideous squalling slides
which the pipe sounds while the tuning-instrument is inserted and turned about at the top.
He might still be a judge of a perfect unison ; but I should no more imagine him able to ,
know the fiftieth part of a mean semitone from the twenty -fifth, when his ear is just out of this
abominable clamour, than I should rely on the tenth part of a second from the wire of an
astronomer who had the instant before been tossed in a blanket. The sensibility to false
intonation languishes and almost dies during a powerful crash of the whole orchestra ; but it
is fostered and nourished by soft passages performed on a few instruments.

When beats are employed at the instrument itself, a watch is in several respects a difficult
standard. The counting should begin when the ear is well in gear with the beats, which will
not happen just at the five seconds or the quarter minute. And the employment of the eye
at the very commencement of counting is confusing to the ear. A regulated metronome might
be used, but I suspect it would be a troublesome instrument. A half-minute sand-glass (emery
powder should be used) would probably be found the best time-piece ; this could be turned
over when the ear is in repose on the beats ; and the counting would begin from the tuner's
own perception of his own act, with that composure which would arise from the act being in
his own power.

The system of equal temperament is to my ear the worst I know of. I believe that the
tuners obtain something like it. A newly-tuned pianoforte is to me insipid and uninteresting,
compared with the same instrument when some way in its progress towards being out of tune.
Now as every bearable change must be called temperament, and not maltonation, I suppose
that, in passing from key to key by modulation, the variety which the temperament of wear
and accident produces is more pleasing than the dead flat of equal temperament. I give the
results of four systems, which I shall now describe.

P is equal temperament, on which I need say no more.

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 143

Q is a system in which the change of temperament of the fifth, in passing from a key to
that of its dominant, is always of the same amount, one way or the other. That is, the tem-
peraments of the fifths in the keys of C, G, D, A, E, B, F|^, are m, 2m, 3m, 4-m,, 5m, 6m, Im ;
while those in the keys of C^, G||, D|^, A|f, F are 6m, 5m, 4m, Sm, 2m. Here 4m must be
the temperament of the fifth in the equal system. I have described this system in the article
Tuning in the Penny Cyclopcedia.

R is a system in which all the major thirds are equally tempered : and the variety of the
fifths in passing from key to key is made as great as, consistently with this condition, it can be.

5* is a system in which all the minor thirds are equally tempered, the varieties of the fifths
being made as great as they can then be.

In the article cited above, I have exhibited all the relations of the temperaments in tlie
form of three theorems, including 25 equations, as follows. The temperament of fifths and
minor thirds is considered positive when they are tuned flat : that of major thirds is positive
when they are tuned sharp.

1. The sum of the temperaments of the fifths in all the 12 keys must be •2346 of a mean
semitone.

2. The keys being arranged dominantly, that is, in the order C, G, D, A, E, B, F|f, Cijif,
^% 1^1^' -^tt' 1^' C' ^' I>,...the temperament of the major third in any key together with the
temperament of the fifth in that key and the three succeeding keys will always amount to
a comma, or "SlSl of a mean semitone.

3. The temperament of the minor third in any key, together with the temperaments of
the fifths in the three preceding keys, will always amount to a comma.

Thus in all systems, the temperament of AC^, together with those of AE, EB, BFj:f,
F^C|f, will make a comma. And the temperaments of AC, together with those of CG, GD,
DA, will make a comma.

If then the temperaments of the fifths go in cycles of four, that is, if the twelve keys,
dominantly arranged, have the temperaments p, q, r, s, p, q, r, s, p, q, r, s, in their fifths, the
temperament of every major third will be p+q+r+s less than a comma, or -0782 of a mean
semitone less than a comma. In the system R, I have taken p=0, q=-0391, r=0, s=-039i :
that is, the dominantly consecutive fifths are alternately perfect and tempered as much again
as in equal temperament. This is the way of satisfying the condition 3 (p+q+r+s) = -2346,
which gives most variety of key. The temperaments of the minor thirds in dominantly con-
secutive keys are alternately '1369 and •136.9 + -0391, equal temperament giving -1564 to all.

If the temperaments of the fifths run in cycles of three, as in p, q, r, p, q, r, p, q, r, p, q, r,
it follows that the temperament of every minor third is p + q+r less than a comma. And
p+q+r must be '0587. In system S I have made p=0, 5'=-01955 as in equal temperament,
r=2q; which satisfies 4(p+g' + r) = '2346. The temperaments of the major thirds in dominantly
successive keys are -1564, •1564-5', •1564-2g': that is, the major third is never more tempered
than the minor third in equal temperament.

144 Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES.

The tables here seen are described in the following paragraphs : —

Intervals

in Mean

Semitones.

P

Q

R

S

C

0-00000

0-00000

0-00000

c

c^

1

1-00000

1-01955

1-01955

c#

D

2

9-02444

2-00000

2-01955

D

D#

3

2-98534

3-01955

3-00000

I^tt

K

4

4-02932

4-OOOOC

4-01955

E

F

5

4-99022

5-01955

5-01955

F

F#

6

6-01466

6-OOOOC

6-00000

F^

G

7

7-01466

7-01955

7-01955

G

Gjt

8

7-99022

8-OOOOC

8-01955

Gtt

A

9

9-02932

9-01955

9-00000

A

A^

10

9-98534

10-00000

10-01955

A#

B

11

11-02444

11-01955

11 -01955

B

Vibrations in

one Minute.

Beats in one

Minute.

P Q

R

S

P Q

R

S

c

14400-0 14400-0

14400-0

14400-0

48-8 12-S

00-0

00-0

c

c#

15256-3 15256-3

15273-5

15273-5

51-7 77-5

103-4

51-7

cfi

D

16163-5 16186-3

16163-5

I6I8I-7

54-7 41-1

00-0

109-5

D

D#

17124-6 17110-1

17143-9

17124-6

58-0 58-0

116-0

00-0

Dft

E

18142-9 18173-6

18142-9

18163-4

61-4 76-c

1 000

61-5

E

F

19221-7 19210-8

19243-4

19243-4

65-1 32-5

130-2

130-2

Ftt

F#

20364-7 20381-9

20364-7

20364-7

69-0 120-7

00

000

F

G

21575-6 21593-9

21600-0

21600-0

73-0 36-5

146-2

73-2

G

G#

22858-6 22845-7

22858-6

22884-4

77-4 96-(}

00-0

154-9

Gtt

A

24217-8 24258-9

24245-2

24217-8

820 82-]

164-1

00-0

A

Aft

25657*9 25636-2

25657-9

25686-9

86-8 65-1

00-0

870

A*|

B

27183-6 272

22-0

27214-3

27214-3

92-0 138-S

! 184-2

184-2

B

The vibrations are calculated from the formula

log itf =log iV+-J^ log 2 X iP,

where M and N are the vibrations in the higher and lower notes, and x the number of mean
semitones in the interval. The beats are calculated from the formula mN—nM', for the fifths
SN—ZM. The beats are those of each note with the fifth above it : thus A^ F' (the octave
above F) l)eats 86-8 times in a minute in equal temperament (P).

The vibrations are taken as in the pitch frequently used for organs, when not wanted to
combine with the orchestra, that is, a diatonic semitone (15 : 16) below the ordinary concert-
pitch of our day, in which C (on the first line below the treble) gives 256 double vibrations
per second. In tuning to the concert-pitch, each number in the lower table, be it of vibra-
tions or of beats, must be increased by its 15th part. For an octave above, the number of
beats must be doubled : for an octave below, it must be halved. Thus, CG beating 48*8
times in a minute, C,Gi beats 24-4 times, and C'G' beats 97-6 times in a minute.

Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 145

I feel sure that the results of this principle of variety in the keys would, if fairly tried, be
found more satisfactory than those of equal temperament. Nor do I at all apprehend that
the principle is carried too far : on the contrary, I should predict that the system R, in which
the difference between dominantly successive keys is greater than in the others, would be the
best of all. But by making p, q, r, s, in R, and p, q, r, in S, more nearly equal than in the
instance given, any less amount of adherence to the distinctive feature might be secured.

It is useless to speculate on systems with any view of materially diminishing the number
of beats/ in the thirds and sixths. In equal temperament, the consonance G A^ beats more
than 1150 times in a minute, while GD' (D' the octave above D) beats only 73 times. Nor
can the beats be reduced, in the different consonances of a chord, either to equality, or to near
commensurability, throughout any considerable portion of the scale. It is the irregularity of
the beating which is its chief disadvantage : regularity would give merely the effect of a faint
drum-accompaniment ; but such change as that from C F C, in which C F and F C beat
equally, to C G C, in which G C beats twice as fast as C G, is the real annoyance. A
further disadvantage is that the multitudinous beats are thrown on the consonances which
are least suited to take them. The fourths and fifths should be called martial consonances,
the thirds and sixths pastoral : but the bray of the beats is thrown on the thirds and sixths,
and is never so distressing in the fourths and fifths.

The subject will never be fairly entered upon, as to true comparison of systems of tem-
perament, until the bearings are tuned from a system of forks, one to each semitone. I think
it probable that nothing but the general ignorance of the theory of beats, arising out of the
obscurity under which the subject has been presented, has hitherto prevented the construction
of such standard bearings.

A. DeM.

January 18, 1858.

Vol. X. Paet I. 19

VIII. On the Genuineness of the Sophista of Plato, and on some of its philosophical
bearings. By W. H. Thompson, M.A., Fellow of Trinity College, and Regius
Professor of Greek.

[Read Nm. 23, 1 857-3

In selecting the Sophista of Plato for the subject of this paper, I have been influenced by
certain passages in an interesting contribution to our knowledge of some parts of the Platonic
system which was read by the Master of Trinity at a former Meeting'. I have principally
in view to assert what was then called in question, the genuineness of this dialogue, and the
consequent genuineness of the Politicus, which must stand or fall with it ; but I am not without
the hope of throwing some new light upon the scope and purpose of the Sophista in particular,
and upon the philosophical position of Platonism in reference to two or three now forgotten,
but in their day important schools of speculation. Such an inquiry cannot fail, I think, to be
interesting to those members of the Society whose range of studies has embraced the fragmen-
tary remains of the early thinkers of Greece, as well as the more polished and mature compo-
sitions of Plato and Aristotle : for such persons must be well aware that it is as impossible to
account for the peculiarities of these later systems without a clear view of their relation to
those which went before them, as it would be to explain the characteristics of Gothic archi-
tecture in its highest development without a previous study of those ruder Byzantine forms
out of which it sprang ; or to account for the peculiar form of an Attic tragedy without
a recognition of the lyrical and epic elements of which it is the combination. Nor is this all.
The writings both of Plato and Aristotle abound with critical notices of contemporary systems,
with the authors of which they were engaged in life-long controversy : and whoever refuses to
take this into account will miss the point and purpose not only of particular passages, but,
in the case of Plato, of entire dialogues. In the search for these allusions to the writings or
sayings of contemporaries, we have need rather of the microscope of the critic than of the sky-
sweeping tube of the philosopher : and a task so minute and laborious is not to be required of
any man whose literary life has loftier aims than the mere elucidation of the masterpieces of
classical antiquity.

I say then at the outset of this inquiry, that I not only hold the Sophista to be a genuine
work of Plato, but that it seems to me to contain his deliberate judgment of the logical doc-
trines of three important schools, one of which preceded him by nearly a century, while the
remaining two flourished in Greece side by side with his own, and lasted for some time after
his decease. I hold the Sophista to be, in its main scope and drift, a critique more or less

Cambridge Philosophical Transactions, Vol. ix. Part iv.

PROFESSOR THOMPSON, ON THE GENUINENESS OF THE SOPHISTA, &c. 147

friendly, but always a rigorous and searching critique of the doctrines of these schools, the
relation of which to each other is traced with as firm a hand, as that of each one to the scheme
which Plato proposes as their substitute. These positions I shall endeavour to substantiate
hereafter, but I shall first produce positive external evidence of the authenticity of the dialogue
under review.

1. The most unexceptionable witness to the genuineness of a Platonic dialogue is, I pre-
sume, his pupil and not over-friendly critic Aristotle. Allusions to the writings of Plato abound
in the works of this philosopher, of which the industry of commentators has revealed manv, and
has probably some left to reveal. These allusions are frequently open and acknowledged; the
author is often, the dialogue occasionally named': but in the greater number of instances no
mention occurs either of author or dialogue, and the (pacri Tives of the philosopher has to be
interpreted by the sagacity of his readers or commentators. I shall begin with an instance of
the last kind, where however the identity of phraseology enables us to identify the quotation.
In the treatise De Anima, iii. 3. 9, we read thus: (pavepov on ovSe So^a fxer alaBtjaeoos
ovce 01 a'lcrOijcrewi, OV06 cvixTrXoKtj oo^r/^ Kai a'laOtjcreMs (pavTaaia ai> e'ltj. A "combi-
nation of judgment and sensation" is evidently the same thing as "judgment with sensation;"
why then this tautology ? It is explained by a reference to Plato's Sophista, ^ 107, p. 264 b,
where we are told that the mental state denoted in a previous sentence by the verb (paiverai,
is "a mixture of sensation and judgment," au/jL/uLi^is aiaOijcrewi kuI ^o^jys; and just before, that
when a judgment is formed, one of the terms of which is an object present at the time to the
senses, we may properly denote such judgment as a (pavTacria- ''Orav ixrj Kaff avrrjv aWd
oi aicrOrjaews iraprj Ttvt to toiovtov au Traaoy, ap oiou tc opOws e'cTreii' erepov ri
7rX»?r (pavTacr'iav. A (pavraaia is, it will be seen, according to Plato a variety of So^a.
The distinction was perhaps not worth making, but it is perfectly intelligible ; and in restrict-
ing a popular term to a scientific sense, Plato is taking no unusual liberty. Aristotle, how-
ever, needs the word for another purpose, and accordingly pushes Plato's distinction out of
the way.

The only word used by Aristotle which Plato does not use is av/jiTrXoKri : he wrote av/x-
fjLi^ii, but it is remarkable that the word av/ixTrXoKij does occur two or three times over in this
part of the dialogue ; hence Aristotle, writing from memoiy, substitutes it for the a-un/jn^i^ of
the original. One of the most learned and trustworthy of his commentators, Simplicius, has
the gloss : tov YlXarajvos ev tb Tip "^oCpiaTtj Kat ev Tip OjXr;/3o> ti/i; (pavTaaiai' ev fx'i^ei
oo^t]^ Te Kai a'ladrjcrew^ TiOefievou, eviaTaaOai ttoos Trji' Oeaiv oia tovtwv ooKei. Now in
the Philebus the definition in question does not occur, though the mental act which Plato calls
(pavraaia is graphically described, and the cognate participle (pavrai^ofjievov is used in the
description (p. 38 c). The passages quoted from the Sophista are therefore here alluded to,
for there are none such in any other dialogue, and the restricted use of the term is peculiar
to the author of the Sophista.

' Sometimes without Plato's name, as en tw 'lirwia, iv tw
iaiSmvi. It is remarkable that these are the only two dia-
logues quoted by name in the Metaphysics : though Plato's

entire system comes under review in that work, of which one
book is appropriated to the theory of ideas alone. The Par-
menides, which is largely drawn from, is not once named.

19—2

148

PROFESSOR THOMPSON, ON THE GENUINENESS OF

2. The next passage 1 shall quote refers not to the Sophista, but to the Politicus,
which is a continuation of it. It is familiar to readers of the Politics, in the first
chapter of which Aristotle writes thus : ''Oc70£ fxev ovv o'lovrai ttoXitikov kuI fiacriXiKov
KOI o'lKovoixiKov Kut Seo-TTOTiKov eTvui Tov uvTov ov KoXws Xeyovaiv' TrXriOcL yap Kat o\i-
yoTtjTi i'oixii^ov(Ti ^ia(pfpeiv dX\' ovk ei^ei tovtcov €KaarTov...UK ovotv dta(pepova-av ixeyaXrjv
oiKiav tj atiiKpav ttoXiv. "Those persons are mistaken who pretend that the words
statesman, king, housemaster and lord mean all the same thing, differing not specifically, but
only in respect of the number of persons under their controul ; for, say they, a large house-
hold is but a small state." With this compare Plato's Politicus, 258 e: irorep' ovv tov ttoXc-
TiKov Koi (iaaiXea Kai oe(nroTt]v Kat er o'lKovofxov drjaoixev w^ kv iravra ravra Trpoaayopev-
ovTe<;, ri Toaavra^ rej^i/a? awras elvai (puifiev, oaairep ovofxaTa eppijdtj. " Are we then to
identify the statesman with the king, the lord, or the master of a family ; or are we to say
that there are as many separate arts as we have mentioned names .'"' The young Socrates is
not prepared with an answer, whereupon he is further asked: "What.-' can there be any
difference, as regards government, between a household of large and a town of small dimen-
sions ?" (rt Se; fxeydXrii c^W" o'lKtiaews, tj afxiKpa^ av TroXecos o'yifos uwv ti irpoi ap-
Ynv ^toiaerov). " There can be none," says the facile respondent. " Is it not then clear,"
rejoins the other, " that there is but one science applicable to all four, and that it is a mere
question of words whether we choose to call such science Kingcraft or Politic or (Economic ?"
{e'lTe ^aaiXiK^v e'lxe -TroXiTiKrjv e'lre o'ikovo/ixiktjv t<s ovofxa^ei /xr)d€v auTw oia<pepwfi.e6a.)

3. There is a passage in Aristotle's treatise De Partibus Animalium (i. c. 2), too long for
quotation, in which he describes and criticizes that method of division or classification of which
the author of this dialogue gives us specimens, styling it /ueaoToiuia or ^t-^oTomia, the method
of mesotomy or dichotomy. " Some persons," says Aristotle, " get at particulars by dividing
the genus into two differentiae : but this method is in one point of view difficult, in another
impracticable." " It is difficult in this process," he observes, " to avoid discerption or lacera-
tion of the genus {^lacnrav to yevos), for example, to avoid classing birds under two distinct
heads, an error is committed in the 'written divisions' (jyeypa/i/nevat Siaipeaei^), in which some
birds -come under the genus Terrestrial, and some under that of Aquatic Animals (e/ce? yap
Tovs fJ-ev (xera twv evvcpwv (7Ufij3a'cvei otrjpfjo'Oai tov^ o ev aXXw yevet), so that birds and
fishes are both classed under the term Aquatic Animals." In a zoological treatise, nothing
could have been worse than such a classification ; which occurs both in this dialogue and
in the Politicus^. Again, in the Politicus, 264 a, animals are divided into tame and wild,
^iriprjTo ^ufXTrav to ^(fov tm Ti6da(p nal ayp'tw. This distinction does not escape
Aristotle, who in the treatise referred to, proceeds to observe that a classification of this
popular kind mixes up creatures widely diverse in structure {waff otiovv Xwov ev TavTai^
{tcu^ ^laipea-ecTiv) inrdp-)^eiv), and not only so, but the distinction itself is a conventional one :
for nearly all tame animals exist also in a wild state; for instance, man, the horse, the ox,

^ Soph, 220 A : TO juey ire^ou yevov^ t& 6' eTepou vcwtlkov
^wov. Politic, 264 c : tj/? fjikv d-yeXaioiv TpoipTJv etrn fxev
evuSpov, ej-Ti oe ^tjpofiaTLKov, The words 'written divisions'
are supposed to refer to a work now lost, a collection of Pla-

tonic 'Divisions' similar perhaps to that of the 'Definitions'
attributed by some to Speusippus, and compiled partly from
the Dialogues and partly from Plato's oral teaching.

THE SOPHISTA OF PLATO, &c.

149

Kviies ev rfi '[v^tKrj, ves, aiye^, -n-pofiaTa. In the Aristotelian treatise itself I am not aware
that any system of classification is proposed which would obtain the approbation of modern
zoologists. The Politicals and the Sophista are not zoological works, and Aristotle's censure
is therefore irrelevant. But the coincidences seem too special to have been accidental.

4. In a work similar in its scope to the Sophista, the curious treatise irepl '2.o<pi(yTiKwv
eXey-^^wv, occurs a definition of " Sophistic," which to my ear is an echo of the Platonic
Dialogue. I allude to the often repeated definition, eariv t] aotpiaTtKrj (paivo/xevri (jotpia u\X' ovk
ovaa, Koi o aoCpiaTTj^ ~^prjiJiaTiaTri<i airo (paivo/uevrjs aocpias aW ovk o'varj^ (S. E. I. 6).
" Sophistic is a wisdom seeming but not real, and the Sophist is a tradesman, whose capital
consists of such unreal wisdom." What is this but an abridgment of the ^taipsTiKos X0709 of
the Sophista, a definition identical with the vewv kuI irXovalwv efx/uaOoi OtjpevTtj^ — " the
hireling hunter of the rich and young," with the very addition which Plato proceeds, with an
affectation of logical accuracy, to graft upon it ?

5. In the same treatise, c. 5, S 1, we read as follows: "Other paralogisms depend on an
ambiguity in the terms employed: — whether they are used absolutely or only in a certain
sense: for instance, if you say that "that which ' is not ' may be a term in a judgment," they
infer the contradiction, ' That which is not, is :' but this is a fallacy, for ' to be this or
that' and 'to be' in the abstract are not the same thing. Or conversely, they argue
that that which is, is not, if you tell them that any entity is not so and so — say that A is not
a man. For not to be this or that is not the same as absolute non-existence'."

This is but an Aristotelic translation of the following in the Sophista : " Let no one
object that we mean by the /u.i^ ov the contrary of the ov, when we dare to affirm that the /u>) ov
is : the truth being, that we altogether decline to say anything about the contrary of the oc,
whether any such contrary is or is not conceivable by the reason." j;fie?s fxev yap trepl
evavTiov tivos avrip (sc, tw ovri) ■^aipeiv iroXXa Xeyo/mev, e'lT eaxiu e'lre fir/ Xoyov e-^ov rj Kat
iravTairacnv aXoyov. p. 258 E. *

To this same passage I suppose Aristotle to allude in the Metaphysica (vi. 4. 13, Bekk,
Oxon.) a'XX' w<nrep eK tov uri ovto<s XoyiKws (pacri rtces eivat to fxr) ov oy^ wkXw^ aXXa fxri
ov, K. T. X. (Where XoyiKok = ' sensu dialectico,'' as distinguished from (pvaiKw^-)

6. I shall have more to say on these passages hereafter : for the present they are mentioned
for the sake of the coincidence. The ^aai rivei, as already observed, is Aristotle's frequent
formula of acknowledgment. If any one doubt that the rives are in this instance a ti's, or if he
doubt who the r/y may be, let him hear Aristotle in another part of the same work ; Sio
YlXaTwv TpoTTOV TWO. OV KaKWi TJyi; aoCpicTTiKtjv irepl to /nt] ov eTa^ev , Met. v. 2, ^ 3,
and then turn to the Sophista, pp. 235 a, 237 a, 258 b, 264 d, passages which it would be
tedious to quote, but the upshot of which is the very distinction to which Aristotle alludes.
Add p. 254. A of the same dialogue, where the Sophist is described as " running to hide himself
in the darkness of the Non Ens," {dTroSiSpcujKwv el? tjji/ tov fit] ovtos (TKoreivoTrira), taking

* aTrXaJs To^e 17 tt^ \6ye<Tdai Kai fitj Kvpiut^y orav t6 ev
fJiepcL Xeyofievov ws d-TrXws eiprj/ieirov \ij<l>6y, olov el to /jlij otf
icTt ^o^affTOl/, oTi TO fxi] ov €(TTiy' oil yap TavTdv elvai re Tl
Kat elvai a-TrXeo?. ij iraXii/ oxt to ov oiiK earnv ov el twv ''"'^

Tt [ITJ e(TTty, oiov ei /itj dv6pwTros,

' " Plato was right to a certain extent, when he represented
the Non-ens as the province of the Sophist."

150

PROFESSOR THOMPSON, ON THE GENUINENESS OF

into account that the description occurs in no other part of Plato's writings, and nothing will
be wanting to the proof that Aristotle had not only read with attention two dialogues answering
to those which bear the titles of the Sophista and the Politicus^, but that he knew or believed
them to have been written by his Master.

The recognition of a dialogue by Aristotle is at least strong evidence of its genuineness :
and it would require stronger internal evidence on the other side to justify us in setting such
recognition at defiance". Of the dialogues generally condemned as spurious, .some owe tiieir
condemnation to the voice of antiquity ; others betray by their style another hand ; while those
of a third class have fallen into discredit on account of the comparative triviality of their
matter or the supposed un-Platonic cast of the sentiments they contain. To objections
founded on the matter of a suspected dialogue I confess that I attach comparatively little
weight, except when they are supported by considerations purely philological. We need have
little scruple in rejecting a dialogue so poor in matter and dry in treatment as the Second
Alcibiades, when we find the evidence of its spuriousness strengthened by the occurrence of
grammatical forms which no writer of the best times would have used^. But it would be
rash criticism to condemn the Second Hippias, in which no such irregularities occur, merely
because it contains paradoxes apparently inconsistent with other parts of Plato's writings.
Tried by this test, the Lysis and the Laches, and perhaps the Charmides, would fare but
ill. Yet in them, those who have eyes to see have not failed to recognize the touches
of the Master's hand, and the perfection of the form has outweighed the doubtfulness of
the matter.

Now I am not aware that any philological objections have been urged against the
Sophista. So far as the mere style is concerned, there is no dialogue in the whole series
more tiioroughly Platonic. In their structure the periods are those of Plato, and they
are unlike those of any other writer. Throughout, as it seems to me, the author is writing
his very best. His subject is a dry*one ; and he strives to make it palatable by a more than
ordinary neatness of phrase, and by a sustained tone of pleasantry. His style is terse or
fluent, as terseness or fluency is required : but the fluency never degenerates into laxity, nor
the terseness into harshness. The most arid dialectical wastes are refreshed by his humour :
and bloom in more places than one with imagery of rare brilliancy and felicity. Few besides
Plato would have thought of describing the endless wrangling of two sects who had no

> I cannot but think that had the Master of Trinity exa-
mined the Polilicus with the same care which he has bestowed
on the Sophista, he would have formed a different opinion of
the genuineness of the two dialogues. The Politicus contains
passages full not only of Platonic doctrine, but of Platonic
idiosyncrasy. I may mention, as a few out of many, the
grotesque definition of Man as a "featherless biped" {Pol.
p. 2(i(;E. 99) which exposed the philosopher to a well-known
practical jest : the somewhat wild but highly imaginative
mythus, redolent of the Timceus, (p. 269 foil.) : and, finally,
the fierce onslaught on the Athenian Democracy, (p. 299),
breathing vengeance against the unforgiven murderers of
Socrates. On reading these and similar passages, it would be
difficult for the most sceptical to repress the exclamation,
" Aut Plato aut Diabolus!"

° The Sophista is also recognized, as we have seen, by the
vigilant and profoundly learned Simplicius, also by Porphyry
(ap. Simp, ad Phi/s. p. 335, Brandis). Clemens Alexandrinus
and Eusebius quote it as Plato's. If it is not named by Cicero,
neither are the Philebus and ThecBtetus. The omission of any
mention of this latter dialogue by the Author of the Academic
Questions is really remarkable.

^ e.g. dirOKptdilval for aTroKpiuacrQaL, iyK€TrT€tjdat for <TKO-
■jreicrdai. The latter barbarism, I presume, would be defended
from Laches, p. 185b. ti wot etn-t Trepi ou (iovXevo/^eda kul
aKevTo/jLeSa, but to me it seems clear that o-KeTrxo'|ne6a is an
interpretamentum of /SouXeuo/xfea, which is used in a sense
not strictly its own, as in the same passage, paulo supra;
el eo-Tt Tis TexviKd^ irepl ou (iovXevoneda.

THE SOPHISTA OF PLATO, &c.

151

principle in common, under the image of a battle between gods and giants ; and fewer still,
had they conceived the design, would have executed it with a touch at once so firm and so
fine. What inferior master could have kept up so well, and with so little effort, the fiction of
a hunt after a fierce and wily beast, by which the Eleatic Stranger sustains the ardent
ThejBtetus amid the toil and weariness of a prolonged logical exercitation ? Or who could so
skilfully have interwoven that exercitation itself with matter so grave and various as that
of which the dialogue in its central portion is made up ? If vivacity in the conversations,
easy and natural transitions from one subject to another, pungency of satire^ delicate persi-
flage, and idiomatic raciness of phrase are elements of dramatic power, I know no dialogue
more dramatic than the Sophista. The absence of any elaborate exhibition of character or
display of passion is, under the circumstances, an excellence and not a defect : as such
elements would have disturbed the harmony of the composition, and have been as much out
of place as in the Timceus, or in some of the later books of the Republic — to say nothing of
the Cratylus and Parmenides, which resemble this dialogue in so many particulars that those
who condemn it, logically give up the other two also.

The Sophista, it is well known, is professedly a continuation of the Thecetetus. The
same interlocutors meet, with an addition in the person of an Eleatic Stranger, and they meet
by appointment : for at the conclusion of the Thecetetus Socrates bespeaks an interview for
the following day, of which he is reminded by Theodorus in the opening sentence of the
Sophista. The Politicus or Statesman is, in like manner, a professed continuation of the
Sophist. The connexion, however, between these two is on the surface much closer than
that between the Thecetetus and the Sophista. In the Thecetetus we are not informed what is
to be the subject of the next day's talk, but in the Sophista^ three subjects are proposed for
consideration — the Sophist, the Philosopher, and the Statesman; and the choice is left to the
new-comer, who selects the Sophist as the theme of that day's conversation. The third day is
devoted to the Statesman, who is made the subject of an investigation similar to that pursued
in the case of the Sophist. In both dialogues the professed object of the persons engaged is
to obtain a definition, and the method pursued is that called by the ancient Logicians, and by
the Schoolmen after them, the method of Division. We are left to infer that the Philosopher
was to be handled on the fourth day in like fashion. Instead of this projected Tetralogy, we
have only a Trilogy. No dialogue exists under the title of (^CKoaoCpo^, and the ingenuity
of commentators has been taxed to account for the deficiency^. It is tolerably certain
that Plato never wrote a dialogue under this title, and it seems idle to speculate on the
causes or motives of this omission. It is more to the purpose to observe, that there is no
connexion apparent on the surface between the subject-matter of the Tlwoetetus and that of

' As a specimen of this, take the argument with the ytj-
yeveis, 246 D, seq., and the mock solemnity with which the
' Ens' of the eio<oj» <^i\oi is described, 249 a.

- P. 217 A.

' Schleiermacher, for instance, conceives that the omission is
intentional, and that we must look for the missing portrait in
the Symposium and Phaedo ; of which the first teaches us how
a philosopher should live, the latter how he should die. This is

one of those "Schleiermachersche Grillen" which contribute to
the amusement even of his admirers. Stallbaum seems to think
that the title of the Parmenides may originally have been
4>i/\do-o</)o9, a conjecture which does not seem to me probable,
and which I should not have noticed, had it not found
favour in the eyes of a gentleman of this University, for whose
critical acumen I entertain the greatest respect.

152

PROFESSOR THOMPSON, ON THE GENUINENESS OF

the two succeeding dialogues : and no resemblance between the method of investigation pursued
in the Sophista and in the Thecetetus. A definition, it is true, is the professed object of both:
the question proposed in the one being, "What is knowledge?" in the other, "What is a
Sophist?" Each dialogue is, therefore, a hunt after a definition; but the instruments of the
chase are not the same in both instances.

I propose the following as a plausible, though I do not put it up for a certain explanation
of the connexion intended by Plato to subsist between the two dialogues.

The art of Definition, it is well known, was an important constituent part of the Platonic
Dialectic. It held its ground in the Dialectic of Aristotle, who, however, devotes a larger
share of attention to the Syllogism ; a branch of Dialectic for which Plato had omitted to give
rules. Both are elaborately investigated by the Schoolmen, as by Abelard in his Dialectice ;
nor was it, I believe, until the commencement of this century, or the end of the last, that
Definition dropt out of our logic books', and the art of Syllogism reigned alone, or nearly
alone. Now, in the Phcedrus of Plato, a dialogue written for the purpose of magnifying
the art of dialectic at the expense of its rival, Rhetoric, occurs a passage in which two methods
are marked out for the dialectician to pursue in searching for definitions'. Either, it is
said, he may start from particulars, and from these rise to generals : or he may assume a
general, and descend by successive stages to the subordinate species (the species specialissima)
which contains the thing or idea which he seeks to define. He may begin, to take the
example given in the dialogue, with examining the different manifestations of the passion
of Love, and after ascertaining what element or elements they possess in common, and
rejecting all those in which they differ, he may frame a definition or general conception
of Love, sufficiently comprehensive to include its subordinate kinds, and sufficiently restricted
to exclude every other passion. Or he may reverse the process, and divide some higher
genus into successive pairs of sub-genera or species, until he " comes down " upon the
particular kind of Love which he seeks to distinguish. The first of these processes is styled
by Plato avva'yw'yrj. Collection: by Aristotle etra-yoDyr), Induction: the second is called
by both Plato and Aristotle ^laipsarK, or the ^laipeTiKt) ixeOo^o^, Division, or the Divisive
method. Whoso is master of both methods is styled by Plato a Dialectician, and his art, the
Art of Dialectic^ We have, therefore, in this passage of the Phcedrus a Platonic organon
in miniature.

Now it so happens, that the Thecetetus and the Sophista pretend, each of them, to be
an exemplification of one of these two dialectical methods : the Thecetetus of a avvayarytj, the
Sophista of a SiaipeaK*. It is this fiction which gives life and unity of purpose to the Thece-

' It was first re-instated, so far as I know, by Mr Mill.

^ See Appendix I. Phcedr. 265 D, foil.

' Those who are unskilled in the application of these pro-
cesses are termed IpicrTiKol in the PMlebus, 16 E. ol ik vuv
Twv dvQpunritiV <TO<pot ev fxeit^ ottws dif TU)(a)<rt, Kai daTTOv Koi
PpaovTcpov nrowvai tov Scovto^ p-erd Be rd ej/ aireipa ev6v9'
Ta Sk fieaa axiToii^ cKtpeuyeL' oTs BiaKexiopiarTai to Te iiaXcK-
TtKui TrdXiv Kai TO cptffTtKtos Tj^ot nToteTadaL irpoi
dWnXovt Tout Xoyous. It is needless to enlarge on the im-
portance of this quotation towards the illustration of the

Sophista, as well as of the passage from the Phadrus now
under review. In the received text we read xal voWd
flaTToi/, K.T.X. The sense manifestly requires the omission of
nroWd. The Eristics admit a One and an Infinite: the Pla-
tonists divide the One into Many, and define the number of
the Many (Phileb. paulo supra). In other words, they employ
the method of Division or Classification, as well as that of Col-
lection or Induction.

♦ Compare Theat. 145 D — 148, with Sophista, init, and
253, §§ 82, 83, Bekk.

THE SOPHISTA OF PLATO, &c 153

tetus, a dialogue which is in reality a critical history of Greek psychology as it existed down
to the fourth century, just as the Sophista is virtually a critique of the logic or dialectic of
the same and previous eras. The one dialogue exposes the unsoundness or incompleteness of
the mental theories of Protagoras, of the Cyrenaics, whose founder Aristippus was Plato's con-
temporary and rival, and perhaps of certain other schools whose history is less known to us'.
The Sophista, in like manner, passes under review the logical schemes of the Eleatics, of their
admirers, the semi-Platonic Megarians, and finally of Antisthenes and the Cynics. Both dia-
logues, as I have said, profess to be at the same time exemplifications of the processes which
the true dialectician, or, as he is styled in the Sophista, 216e, 253 d, the true philosopher must
adopt in his search for scientific truth. The one is a hunt after the true conception of 67ri-
aTijuti or science, the other an investigation of the genus and differentiae of the conception
implied in the term Sophist; and this fiction^ serves in both cases to bind together the critical
and polemical investigations which make up the main body of either dialogue. It lends to
each the unity of an organic whole'; and infuses into a critical treatise on an abstruse branch
of philosophy the vivacity and interest of a drama. Add to this, that the Sophista helps
materially towards a solution of the question, What is Science .-' which is the professed aim of
the dialogue which precedes it. It attains this object in two ways. First, by enlarging the
conception of that which is not Science, treating the subject on its logical or dialectical, as
the ThecEtetus regarded it chiefly on its real or psychological side : and, secondly, by giving
rules, illustrated by example, for what Plato considered, as we have seen, one of the main
elements of scientific method. And the same analogy holds in respect of the critical or con-
troversial portion of either dialogue. As in the Thecetetus it is shewn that the Protagorean
dictum, that Truth exists only relatively to its percipient {travTaiv (lerpov uvOowtto^), and the
kindred, tliough not identical Cyrenaic dogma, that sense is knowledge, and the sensations the
sole criteria of truth (KpiTt'jpia ra irdOri), so far from furnishing tenable definitions of
Science, in effect render Science impossible : so in the Sophista the Logic of the Cynics and
Eleatics is proved to be more properly an Anti-logic, annihilating all Discourse of Reason, and
rendering not only Inference but Judgment, or the power of framing the simplest propositions,
a sheer impossibility.

I have said that the Sophista is first a dialectical exercitation, and secondly a critique more
or less hostile of three rival systems of dialectic ; two of which, it may be added, evidently
sprang out of the third, and presuppose, if they do not assert, the false assumptions on which
that third is founded. It may conduce to greater clearness if I take this critical portion of
the dialogue first in order. In defending my position, I shall make no assertions at second
hand ; an indulgence to which there is the less temptation, as Plato himself tells us pretty
plainly what he means, and where he fails us, Aristotle and the ancient historians of Philosophy
supply all that is wanting.

' The theory that " Science is right Opinion combined with
Sensation " is given by Zeller to Antisthenes on grounds which
seem highly probable.

' 1 would not be understood to mean that the pursuit of the
Definition is a mere feint in either case, but only that it serves
as a Trpotpacris — a natural and probable occasion for the intro-
duction of important controversial discussions. It constitutes

the framework or "plot" of the drama. At the same time I
conjecture that the end Plato had most at heart in these two
dialogues was the confutation of opponents. In the Politicus,
on the other hand, a didactic or constructive intention appears
to predominate.

' Comp. Phcedr. 2G4 C : tel irdirra \6yo¥ uatrtp {oioK avre-
aTtivaif ic.T.X.

Vol. X. Part I. 20

154

PROFESSOR THOMPSON, ON THE GENUINENESS OF

The oldest, and in the history of Speculation the most important, of these three schools was
the Eleatic, founded, as the Stranger from Elea tells us in this dialogue, by Xenophanes^,
though its doctrines underwent some modification, and received extensive development in the
hands of Parmenides and Zeno, his successors. When Plato wrote this dialogue, there is every
reason to suppose that the Eleatic school had ceased to exist. The latest known successor of
Parmenides, Melissus, flourished, as the phrase is, about the year b.c. 440, and Zeno is placed
a few years earlier. The earliest date which it is possible to assign to the Thecetetus, and a
fortiori to the Sophista, is about 393.^ There can therefore be no question of an Eleatic
author of this dialogue, an " opponent of Plato," resident in Athens, and writing in the Attic
dialect. Socrates may have had such opponents, though we read of none ; but the hypothesis
is inadmissible in the case of his disciple.

The Eleatic Stranger however leaves us in no doubt of his intentions. In the course of his
investigation of the attributes of the Sophist, he is on the point of obtaining from Thesetetus
an admission that his, the Sophist's, art is a fantastic and unreal one : but he affects to
hesitate on the threshold of this conclusion, because, as he says, " The Phantastic Genus," to
which they are about to refer the Sophist, is one difficult to conceive ; and the fellow has
very cunningly taken refuge in a Species the investigation of which is beset with perplexity \
Thesetetus assents to this mechanically, but the Stranger, doubting the sincerity of his
assent, explains his meaning more fully. The word (pavraa-TiKW implies that a thing
may be not that which it seems, and it is a question with certain schools whether there
is any meaning in the phrase, to say or think that which is false, in other words, that
which is not: for, say they, you imply by the phrase that that which is not, is — that
there exists such a thing as non-existence : and thus you involve yourself in a con-
tradiction*. But if we assert that 'Not-being is' (quod Non Ens est,) then, says the
speaker, "we fly in the face of my Master, the great Parmenides, who both in oral
prose and wi-itten metre adjured his disciples to beware of committing themselves to this
contradiction ^ To extricate ourselves then from the diropia in which the Sophist has con-
trived to plant us, it is necessary," proceeds the Stranger, " to put this dictum of our
Father Parmenides to the torture, and to extort from it the confession that the contra-

^ Soph. 242 D ; to Sk irap' tj/iui/ 'EXeaTiKOv £0j/os diro
^evotpdvov^ . . .dp^dfievov .

^ Apuleius, de Dogm. Plat. 569, says that Plato took up the
study of Parmenides and Zeno (inventa Parmenidis et Zenonis
studiosius executus) after his second visit to the Pythagoreans
in Italy : having been compelled to give up his intention of
visiting Persia and India by the wars which broke out in Asia
at the time. Does this imply that he visited Elea instead ?
If so, and if he composed the Sophista and its sister-dialogues
on his return, we obtain a clue to the fiction of an Eleatic
Stranger. He was Plato, on his return from a sojourn at Elea,
laden, it may be, with Eleatic lore.

The circumstance that the conduct of the dialogue devolves
upon this Stranger is pointed to as one proof that the Sophista
was not written by Plato, whose custom is to make Socrates
his Protagonist. The secondary part which Socrates plays in

the Timceus and his entire absence from the colloquy in the
Laws seem fatal to the major premiss in this reasoning. It
should also be observed, that the author of the Sophista, if not
Plato, took pains to pass himself off as Plato : else why did he
tack on the Sophist to the Thecetetus? But if the author of
the Sophista wished to pass for Plato, why did he deviate from
Plato's ordinary practice, by putting a foreigner from Elea into
the place usually occupied by Socrates ?

^ *E7rei Ka\ vvv /ua\' eu /cat Ko/juj/tii}^ eh diropov eldoi Siepev-
vt'iaaadai KaTa'jretpevyev. 236 D.

* TeToXfjLrjKei/ 6 Xoyo^ outos uiro6eV6at to fxij on eli/at*
^eu£oi ydp ouk dv aWws hylyveTo ov. 237 A.

^ 'AirefiapTupaTO ire^^ Te caSe eKaVxcTe Xeytov Kai fieTa
fieTpwii'

ov ydp p\]-iroTe touto ^aps, eJi/at ptj eovTa,
dXXd crv Tijard' d(p' odou dt^tjaiov eipye voijpa. lb.

THE SOPHISTA' OF PLATO, &c.

155

diction is in fact no contradiction, but that there is a sense in which the fxtj ov is, and in
which the ov is noi'." In this passage the Eleatic, who is Plato's mouthpiece, formally declares
war against the logical system of his master Parmenides, in one of its most vital parts. His
words, I conceive, admit of no other explanation. A question here suggests itself as to the mean-
ing of this Eleatic denial of the conceivableness of non-entia. " You can never learn," says
Parmenides, " that things which are not are''." Does he mean to forbid the use of negative
propositions ? His words will bear, I think, no other sense, and so, as we shall see, Plato
understands them. In fact two misconceptions, both arising from the ambiguity of language,
seem to lie at the root of the Eleatic Logic. Parmenides first confounds the verb-substantive,
as a copula, with the verb-substantive denoting Existence or the Summum Genus of the
Schoolmen. He secondly assumes that in any simple proposition the copula implies the
identity of subject and predicate, instead of denoting an act of the mind by which the one
is conceived as included in the other, in the relation of individual or species to genus. It
may seem strange that so great a man should have thus stumbled in limine. But enough
is left of his writings to enable us to perceive that he was notwithstanding a profound, or if
that be questioned, certainly a consistent thinicer. In the first place he altogether repudiates the
distinction of ' subjective ' and ' objective.' " Thought," he says, " and that for which thought
exists are one and the same thing';" and more distinctly still, "Thought and being are the
same," to yo-p avTo voeiv ecTTiv re Kai elvai : and, -^pri to Xeyeiv Te voetv t eov enfievai*,
" Speech and thought constitute reality." A man who thus thought must therefore have
repudiated the antithesis between Logic and Physics, between Formal and Real Science, a
distinction which appears to us elementary and self-evident. Logic wa^ to Parmenides
Metaphysic, and Metaphysic Logic. That which is conceivable alone is, and that is which is
conceivable. The abstraction "To Be" is the same as Absolute Existence. The "Ens
logicum " and the " Ens reale " are the same thing. The only certain proposition is the
identical one " Being is," for " not-Being is Nothing*." Hence the Formula which served
as the Eleatic watchword: ev Tci iravTa, " unum omnia."

If it be asked, what did Parmenides make of the outward universe? we are at no loss
for an answer. He denied its claim to reality, or any participation of reality, in toto". And on
the principles of his Logic he was bound so to do. For every sensible object, or group of
sensible objects, being distinct from every other object or group of objects, is at once an Ens
and a Non-ens, it is this and it is not that, e. g. If Socrates is a man, Socrates is not a beast :
for the genus "man " excludes the genus "beast." {avOpwrro^ eVrt ntj 9tjpiov, as Parmenides
would have expressed it.) But a /uj) Orjpiov is, according to his logic, a (dj ov; therefore all so-
called ovra are at the same time fi^ ovtu: non-existent, and therefore inconceivable, and so
altogether out of the domain of Science.

* Tow Tov -Trarpos HapfieviSov \6yov dvayKaTov tlfxlif
dfX.vvofi€ifot9 etTTat ^affavt^eiv, Kal l3iaX^€adtti to xe fiij ov
(lis eo-Ti KaTti T(, Kai to 01/ av Trdkiv a!s evn irrt. p. 241 D,
Comp. Arist. Soph. El. c. 5, § 1, quoted above.

* oil yap fit'tiroTe tovto day^, elvai fi,} eovra,

* Tai/T&v S' €<TTt voeiv Te Ka\ ovveKev etrTt v6r]fia. Frag,

V. 94, MuUach.

* Frag. v. 43, ed. Mullach.

' SCTTi yap elvai, piriSev i' ouk elvau

ovdev yap ij effriv »/ eoTai

'AWo 'TrapcK tov eovToi. Ibid.
« Ibid. V. no.

20 — 2

156

PROFESSOR THOMPSON, ON THE GENUINENESS OF

From the dicta of Parmenides which I have been endeavouring to explain, the Eleatic
Stranger in the dialogue proceeds to deduce various conclusions: the most startling of which
is, that Being is, on Eleatic principles, identical with Not-being, — that the worshipt ov is after
all a pitiful ntj ov^ I He is enabled to effect this reductio ad absurdum by the incautious
proceeding of Parmenides, who instead of entrenching himself in the safe ground of an identical
proposition, and thence defying the world to eject him, must needs invest his Ens with a
variety of attributes calculated to exalt it in dignity and importance. It is " unbegotten," it
is " solitary," it is " immoveable," it is " a whole," it is even " like unto a massive orbed
sphere^." {Soph. 246 e.) In one of these unguarded outworks the Stranger effects a lodgment,
and by a series of well-concerted dialectical operations, succeeds, as we have seen, in carrying
the citadel.

Having shewn the Nothingness of the Eleatic Ontology, the Stranger proceeds to pass
in review two other systems of speculative philosophy. *' We have now," he says, " discussed —
■ not thoroughly it is true, but sufficiently for our present purpose, the tenets of those who pre-
tend to define strictly the oi' and the fir] ov : we must now take a view of those who talk
differently on this subject. When we have done with all these, we shall see the justice of our
conclusion that the conception of Being is involved in quite as much perplexity as that of Not-
being'." Of one of the two sects who " talk differently," I venture to hold an opinion
varying from that generally received — an opinion formed many years ago in opposition to that
advanced by Schleiermacher and adopted without sufficient consideration by Brandis, Heindorf
and others. Careful students of Plato are aware that his dialogues abound with matter
evidently polemical, to the drift of which his text seems on the surface to offer no clue. I
mean that, like Aristotle, he frequently omits to name the philosophers whose tenets he
combats : characterising them, at the same time, in a manner which to a living contemporary,
versed in the disputes of the schools and personally acquainted with their professors, would at
once suggest the true object of his attack*. Such well-informed persons constituted doubtless
the bulk of Plato's readers and formed the public for whom he principally wrote. It was they
who applauded or writhed under his sarcasms, as they happened to hold with him or his
adversaries. It is to place himself in the position of this small but educated public that the
patient student of Plato should aspire: neglecting no study of contemporary monuments, and
no research among the scarcely less valuable notices which the learned Greeks of later times
have left scattered in their writings. Of these notices, emanating originally from authorities

* Soph. 245 c, 964 Bekk. : /ixij oi'tos &e ye to irapdirau

TOV o\oVt TaifTU T€ Tui/TU VTTitp^fl T(p OVTt, Kui TTpOS TlS fit}

tlvat fiiiS' uu yei/etrQat Trore ov.

' irdvTo^eu euKUKXou (ripaipij^ evaXiyKtov oyKta. I'arm.
V. 103.

** 'Iv €K irdvTiav XStufjiev OTt Td 5lt tov fxij ovtov ovdiv
eviroptvTepov'eiTTeil' o tc ttot* effTlj/. p. 245 E.

* This reticence, of which it is rot difficult to divine the
motives, is most carefully practised in the case of the living
celebrities who claimed like himself to be disciples of Socrates,
8uch as Euclides, Aristippus and Antisthenes. A cursory
reader of Plato has no conception that such men existed as the
heads of rival sects with which the Platonists of the Academy

were engaged in perpetual controversy. On the other hand,
Plato never scruples to name the dead, nor perhaps tliose living
personages with whom he stood in no relation of common pur-
suits or common friendships, e.g. Lysias, Gorgias, &c. The
Pyihagoreans, though remote in place, were his friends and
correspondents, and in speaking of them he observes the same
rule as in the case of his living Athenian contemporaries, in-
dicating without expressly naming them. Thus, in the Poli-
licus, p. 285, they are merely denoted as (so/ui//ol, " ingenious
persons." This, by the way, is a passage of great importance,
as indicating the limits within which Plato '"pythagorized,"
and the particulars in which he dissented from his Italic
friends.

THE SOPHISTA OF PLATO, &c.

157

contemporary or nearly contemporary with the philosopher himself, many have been embalmed
in the writings of Eusebius and Sextus Empiricus, the Aristotelian Commentators, Cicero,
and others : not to mention the vast store of undigested learning amassed by Diogenes
Laertius.

Now of the two sects who here come under revision, and who enact the part of Gods and of
Giants in the famed Gigantomachy*, which is familiar to most readers of Plato, the occupants
of the celestial regions are rightly, as I think, judged to mean the contemporary sect of the
Megarics. They are idealists in a sense, but their idealism is not that of Plato. They so
far relax the rigid Eleatic formula " unum omnia" as to admit a plurality of forms (e'l^rj or
ovra or ovaia). They are complimented in the dialogue as ^/xepwTepoi, "more civilized" or
" more humane," than their rude materialistic antagonists : but they are at the same time
taken sliarply to task by the Eleatic Stranger : and for what ? For the absence, from their
scheme of Idealism, of tiiat very element which constitutes the differentia of the Platonic
Idealism. " They refuse to admit," says the Stranger, " what we have asserted concerning sub-
stance, in our late controversy with their opponents :" ov avy^^wpovaiv rif^^v to vvv S>} prjOev
•n-pos Toy's yrjyeve^s ovala% irepi, 248 B ; the thing they refuse to admit being neither more
nor less than that Koivwvia or fxeOe^K tw;' ei^wn'', which Aristotle cannot or will not under-
stand in his critique of the Platonic Doctrine of Ideas. Like Plato, they distinguish the two
worlds of sense and pure ideas, the ytveat^ from the ovaia {yevecriv ttjv oe ova'iav ^(o^is nov
^leXo/uevoi Xeyere, 248 a), but, unlike him, they deny that the one acts or is acted upon by
the other : they even deny that Being (e'iS^ or ovaiit) can be said to act or suffer at all ; nay,
when pressed, they seem to admit that it is impossible to predicate of it either knowledge or
the capacity of being known^. The arguments by which the "Friends of Forms" {el^wv
<pl\oi, 248 a) are pushed to this admission may not ring sound to a modern ear; but my
business is not with the soundness of Plato's opinions, but with their history: and it would
be easy to produce overwhelming evidence both from his own writings and those of Aristotle
to the truth of the statement, that however the phrase is to be interpreted, there is, according
to Plato, a fellowship, Koti'wvia, between the world of sensibles and the world of intelligibles,
and that the conception of this fellowship or intercommunion distinguishes his Ideal Scheme
from that of the Eleatics*, and, as appears from this passage, from that of the semi-Platonic school

' Soph. 246 A, I 65 Bekk.

" Aristotle objects to the term fiede^i^'on the ground that it
is metaphorical. Now as a logical term, the Platonic /nf6cgi5
is but the counterpart of ii-n-apjis, the Aristotelian word denot-
ing the relation of subject to predicate. The one term is
as metaphorical as the other, and not more so. " A belongs
{iirdpxfi) to B" and " B partakes of A " (/nexe'xf) ^'^ '""^
in a sense metaphorical phrases, and the metaphor employed is
the same in both cases. The Platonic term marks the relation
between subject and predicate as not one of identity, and thus
serves to distinguish the Dialectic of Plato from that of the
Eristics, who denied that the "One" includes a "Many."
The same purpose is equally well, but not better answered by
the iirdpx,ei of Aristotle.

' Ti)V otiaiav iij Ka-ra roll \6yov toutoii yiyviaaKOiiein)u
VTTO Tijs yi/too-eois, KaO* otrov ytyvwtTKfTat Kara to(tovtov
KivfTaSai itd to ■7ra'<rX"»', " ^'1 <t><'h^i' <>>"< «" 7"'^''^^'" '^^p'^
t6 I'lpe/iou '. p. 248 E.

* Compare 249 D, § 76 : tw Stj (piXotrotpui xal -rad-ra fidXi-
trra tiixSivti iraaa mi eoiKCv dvdyKn Sid Tavra ^lire tmi; er
li Kal Ta troWd eUv XeyovTmu to irav 60-ti|K(Js diroSexet'^at,
Tuii/ t' ail TravTaxV ''■" ^■' KiuoivTmv fiitle t6 irapdirav dKoieiir,
dXXd Kara t^v twv vaiSuii/ eliX'i'', o<ra (lus?) aM'i/ijTa icol
Ke».-ii/i)n«'vo, tA ov Te Kal to ttuv, ^vvafKpoTepa Xe'^eii'. This
passage, as 1 understand it, expresses Plato's dissent alike from
the Eleatics and Megarics, and from those Ephesian followers
of Heraclitus whom he had discussed in the Theatetus. This
is not the only echo of that dialogue heard in the Sophiita.

158

PROFESSOR THOMPSON, ON THE GENUINENESS OF

of Megara also'. I will only add, that the passage on which I have been commenting deserves,
in my opinion, a more careful study and closer analysis than it has yet received, and I shall
be very thankful for any remarks in elucidation of it which may be contributed either by those
who agree with my notions of its general import, or by those who take a totally opposite view".

We pass now from the heavenly to the earthly ; from the serene repose of the transcenden-
talists, fxdXa evXafiws avwQev e^ aopuTov iroOev dfxuvo/uieviov, to the violence and fury of the
giant brood below, who seek to eject these divinities from their august abodes, " actually
hugging rocks and trees in their embrace," toTs x^P^'" "'''eX'"^* -n-erpas Kal Spv^ irepiXaix-
fiavovTe^, 24.6" a.

Of these materialists — for such in the coarsest sense of the word they are — I remark,
first, that they are evidently the same set of people as those described in terms almost
identical by Plato in the Thecetetus, p. 155 e. At this point of the last-named dialogue
Socrates is about to expound the tenets of the Ephesian followers of Heraclitus ; whose
sensational theory, as he afterwards shews, agrees with that of the Cyrenaics in essentials,
though it was combined with cosmical or metaphysical speculations in which it may be doubted
whether they were followed by the Socratic sect. Before, however, he enters upon these
highflown subtleties, he humorously exhorts Theatetus to look round and see that they were
not overheard by " the uninitiated :" " those," he says, " who think nothing real, but that
which they can take hold of with both hands'; those who ignore the existence of such
things as ' actions,' and ' productions,' — in a word, of anything that is not an object of sight,"
(irdv TO doparov ovk uTro^e-xpiJ-evot w<f ev ovcxias iiepei). These persons are garnished with
the epithets "hard," "stubborn," "thoroughly illiterate," aKXrjpol — dvTiTvirov — /xdX' ev
afxovcroc.

Now the only contemporary philosopher to whom these epithets of Plato are applicable is
the founder of the Cynic school, Antisthenes, a man whose nature corresponded with his
name, and to whose name, as well as to his nature, the dvTLTviroi of the Thecetetus would be
felt to convey an allusion "intelligible to the intelligent." The fidX' eJ a/uovaoi finds its
echo in the synonymous epithet uTral^evToi, which Aristotle in the Metaphysica bestows on
Antisthenes and his followers*. Every one, however, must see, without further argument, that
the description in the Thecetetus tallies in all points with that in the Sophista, and that both
are in perfect agreement with what we know from Diog. Laertius and a host of others, of the
moral characteristics of the Cynic school ^. The materials of the comparison may be found in

» This epithet I conceive to be justified by Cicero's notice,
'■Hi quoque (sc. Megarici) multa a Platone," Acad. Qu. ii.
42, and also by the brief statement of the Megaric dogmas
which Cicero gives us in the context of this passage.

' In the Philebus — a dialogue which treats of therelation of
t>b<rla to yeveci! in its moral and physical, that is to say its
real, in distinction from the purely logical or formal aspect
under which it is presented in the Sophista — Plato postulates
a Tetrad, composed of the principles he there denotes as Limit,
the Unlimited, the Mixed or Concrete, and Cause. The third
principle he denominates yci/eert? eh ovaiav, the possibility of
which process is precisely what the clouiv <^i'\oi — the pure

idealists of this dialogue— deny. Phileb. p. 24, foil. The dis-
tinctness of the Causal Principle from the Ideas is clearly laid
down in the Philebus, and is recognized in the Sophista also,
p. 265, §§ 109, 110.

^ Compare Soph. 247 c : ciareii/otvr' av wav o fiti Suua-rol
Tals x^P*^' ^v/xirie^etf eltrii/ w9 dpa tovto ovSev to
'jrapdwav eiTTiv.

VII. 3. 97 '. oi 'AuTtrrdeveLoi Kai oi outws aTraiSevTO^.

* I have shewn in Appendix II. that the only other schools
who can in fairness be called "materialists," are out of the
question here.

THE SOPHISTA OF PLATO, &c. 159

any manual of the history of philosophy. For our present purpose it were to be wished that
some portion of the voluminous writings of Antisthenes had been preserved, in addition to the
meagre declamations, if they are really his, which are commonly printed with the Oratores
Attici. The notices, however, which Aristotle and his commentators have preserved to us,
countenance the assumption just made, that the Earth-born are the Cynics. Hatred of Plato
and the Idealists seems to have been the ruling passion of Antisthenes, and this passion drove
him into the anti-Platonic extremes of Materialism in Physics, and an exaggerated Nominalism
in Dialectic. " He could not see Humanity, but he could see a Man," is one of his recorded
sarcasms upon the doctrine of ideas'. "Your body has eyes, your soul has none," was the
curt reply of Plato. Many other stinging pleasantries were interchanged by the leaders of the
two schools: and Antisthenes, less guarded than his antagonist, wrote a dialogue "in three
parts," entitled So^coi/, which was avowedly directed against Plato in revenge for a bitinw
reply (Diog. Laert. iii. § 35 ; vi. ^ l6). The subject of this dialogue has been recorded, and
it is not a little curious that it was written to disprove the very position which Plato devotes a
large proportion of the Sophista to establishing; viz. that there is a sense in which " the Non-ens
is," in other words, that negative propositions are conceivable. Antisthenes maintained in this
book, oTi ovK eariv avriXeyew. If we add, that he also wrote four books on Opinion and
Science {-irepi So^Tji kuI etnar^fxri^), we shall hardly think the conjecture extravagant, that
the remainder of this dialogue is, in the main, a critique of the Cynical Logic. Another
paradox of this school, closely connected with the last, is recorded by Aristotle^ and sarcasti-
cally noticed at page 251 b of the Sophista, in terms which leave little doubt as to the object
of Plato's satire. If Antisthenes really pushed this paradox to its legitimate results — and
from the character of the man it is not unlikely he did — he must be understood as maintaining
that identical propositions are the only propositions which do not involve a contradiction : a
theory which, as Plato shews, renders language itself impossible^ as well as that inward
" discourse of reason*," of which language is the antitype.

The resemblance of the Cynical Logic to the Eleatic is usually accounted for by the cir-
cumstance that Antisthenes had been a hearer of Gorgias, who wrote a treatise, preserved or

' Tzetzes, Chil.vii. 605; Schol. im Arist. Categ. ed. Brandis,
p. 666, 45 and 68 6, 26 ; Zeller, G. P. ii. p. 116, note 1.

^ Metaph. v, 29 : 'AvTitrdevit^ wcto euijOws jutjBci/ d^iuiv
\ey€(Fdat irXufV tw olKeiay Xoyw eu e(p* evov e£ caf rjvve^aive
fiij eivai dvTiXeyetv^ ff-)^eSdl/ 5' oiide \t/€ude<T6ai. Plat. Soph.
1. 1.: OVK eoii'Te? dyadou Xeyetv dvQpwjrov, dWd to fjLCv dyaOdu
dyaQov t^v 51 dvdpanrov dvQpwirov. The latter passage explains
the olKeiu) \6ym of Aristotle, and the allusion is further deter-
mined by the dfiovorov Tii/09 Kal drj>t\oa-6<pov applied to the
upholder of the similar sophisms noted at p. 259 D. In the
latter passage occur the following words : ou -re tis eXeyxot
0VT09 aA-tjCiyos, dpTL Te T(jav ovtojv tiv6<i ktpa'WTop.evov 5^\os
wfoyei/ijs mi/. " This is no genuine or legitimate confutation:
but theinfant progeny of a brain new to philosophical discussion."

This hangs together with the yepoi/Twu toIs 6tlfip.a9e(Ti " the

old gentlemen who have gone to school late in life," p.
251 B, and both passages are illustrated by a notice in Diog.
Laert. vi. 1, init. that Antisthenes, having been originally a
hearer of Gorgias, became at a later period a disciple of Socrates,

and brought with him as many of his pupils as he could induce
to follow his example. A similar sarcasm is hurled at Diony-
sodorus and Euthydemus, in the Euthyd. p. 272 c, which not
improbably was designed to glance oft' from them upon some
contemporary Eristic. Antisthenes, we know, was present at
the battle of Tanagra, in B.C. 426. He may therefore have
been Plato's senior by some 20 years.

^ Kal yap tjo 'yade, to ye irdv dir& iraj'TOS iTTi\eipeXv
airo^wpi^eti/, aWws -re ovk eju/xeXe? Kal 61] Kal ■jravTairatrnr
dfioixTov Tivoi Kal d(piXotT6fpov. 6. xi Sij ; S. TeXeio-raTq
nrdvTUjv Xoywv earTiv d<pdv ivi-i td StaXveiv ^/caffTOf dird
'Trdirrwv' Std yap xi|V a\X»|\(Oi/ Tton eldtjou arvpLTrXoKtjt/ 6 Xdyos
yeyovev r\pXv, Soph. 259 D.

* didXoyo^ dvev (pwvjjs yiyv6p.evoi gtt oivofxaoQi) Stdvota,
Soph. 263 E. Van Heusde first pointed out the infamous ety-
mology lurking in this passage {Stdvota—didXoyo^ ditev)
The sentiment, without the etymology, occurs in Theat. 189 E :
{to 3e ^tavoeXa^ai KaXm) Xoyov 3v aiiTii irpov auTiji/ jj ^vX'i
die^epxsTal Trepl wv dv ctkott^.

160

PROFESSOR THOMPSON, ON THE GENUINENESS OF

epitomized by Aristotle, in whicli the paradoxes of Parmenides and Zeno are put forward in
their most paradoxical form, and pushed to their consequences with unflinching consistency.
Gorgias was also a speculator in physics, and so was Antisthenes*; in whom, moreover, we may
observe other characteristics of those accomplished men of letters of the fifth century, who are
usually called " the Sophists." His ethical opinions on the other hand were borrowed from
Socrates ; but in passing through his mind they took the tinge of the soil, and seem to the
common sense of mankind as startling as any of his dialectical paradoxes. It is remarkable,
however, that when Plato handles the Cynical Ethics, he treats their author with far more
leniency than in this dialogue. In comparing it with the Pleasure Theory of Aristippus, he
speaks of the Cynical system with qualified approbation. ISva-^epri^', "austere or morose,"
is the hardest epithet he flings at Antisthenes in the Philebus : he even attributes to him
a certain nobleness of character ((puaiv ouk dyeui/rj), which had led him, as Plato thought,
to err on the side of virtue. The Philebus is a work of wider range and profounder bearings
than the Sophista, but the dialogues have this in common, that in both the broad daylight of
reason is shed on regions which had been darkened by the one- sided speculations or the wilful
logomachy of earlier or inferior thinkers. The way in which Antisthenes is dragged from his
hiding-place among the intricacies of the Non-existent into the light of common sense, at the
close of the present dialogue, appears to me an admirable specimen of controversial ability ; and
the broad and simple principles on which Plato founds the twin sciences of Logic and Grammar'
stand in favourable contrast to the sophistical subtlety of his predecessors and contemporaries.
At this point of the discussion I would gladly stop : but I feel bound to say a few words
on what I have ventured to call the " logical exercise," which is the pretext under which
Plato takes occasion to dispose of the doctrines of certain formidable antagonists. That the
^laperiKol \6yot, the " amphiblestric organa*," in which he endeavours to catch and land
first the Sophist and then the Statesman, were regarded by Plato himself in this light, we
learn from his own testimony in the Politicus, 285 d, ^ 26 Bekk. " Is it," asks the Eleatic
Stranger, " for the Statesman's sake alone, that this long quest has been instituted, or is it not
rather for our own sake, that we may strengthen our powers of dialectical enquiry upon
subjects in general ? S. J. It was doubtless for this general purpose. E. S. How much
less then would a man of sense have submitted to a tedious enquiry into the definition of the
art of weaving, if he had no higher object than that !" He then proceeds to apologize
for the prolixity of this method of classification : but adds, " The method which enables us to
distinguish according to species, is in itself worthy of all honour ; nay, the very prolixity of an
investigation of this kind becomes respectable, if it render the hearer more inventive. In that

' Hence the explanation of Philebus, 44 b : xal nd\a Sil-
vui/i Xeyofxevou-i Ta irepi tpvaiv,

* /'Ai/. 44 c : /lavTevo/xeuoli ov Tt'^vp dWd th/i Sutrx^-
/u«la <f)V<rfws uuk dyei/i/ov^, Kiav fxefxtatjKOTwv TtjV Tijt i;-

ooi/j/s Suvafntff Kui vevofJiiKoTfav ouSev u-yte? <rK€i//a/iev»s Irt

Kal TaWa ai/Tcov dv<r-)(^epdar fiwra. lb, D : kutu to tt/s
tuaxepeia^ avTuiti ixyoi. The accomplished and unfortu-
nate Sydenham first pointed out the reference in these epithets
to the Cynics and their master. The oi rix'V "f Plato tallies

witli the oVaWeuTot of Aristotle, and with his own anovaoi,
&c.

' P. 262 D. Simple as the analysis of the Proposition into
omfia Kal prjun (subject and predicate in logic, noun and verb
in grammar) may seem to a modern reader, it appears to have
been a novelty to Plato's contemporaries. Plutarch expressly
attributes the discovery to Plato (Plat. Qu. v. I. 108,
Wytlenb.), Apuleius, Doctr. Plat. iii. p. 203. Comp. Plat.
Crat. 431 B. * Soph. 235 B.

THE SOPHISTA OF PLATO, &c.

161

case we ought not to be impatient, be the enquiry short or long." If we say it is too long,
" we are bound to shew that a shorter discussion would have been more effectual in improving
the dialectical powers of the student, and helping him to the discovery and explanation of the
essential properties of things'." "Praise or blame, founded on any other consideration, we
may dismiss with contempt."

This passage, the importance of which for the appreciation of these two dialogues it
is superfluous to point out, derives unexpected illustration from an amusing fragment of a
contemporary comic poet, preserved by Athenseus^. In this passage we are introduced into
the interior of the Academic halls, and the curtain rises upon a group of vouths who are
"improving their dialectical powers" by a lesson in botanical classification. The subject
proposed is not a Sophist, but a pumpkin, and the problem they have to solve is, to what
genus that natural production is to be referred. Is a pumpkin a herb.-* Is it a grass.''
Is it a tree ? The young gentlemen are divided in opinion — each genus having its sup-
porters. Their enquiries, however, are rudely interrupted by a " physician from Sicily,"
who happened to be present, and who displays his contempt for their proceedings in a
manner more expressive than delicate. " They must have been furious at this," says the
second speaker. "Oh!" says the other, "the lads thought nothing of it: and Plato, who was
looking on, quite unruffled, mildly bade them resume their task of defining the pumpkin and
its genus. So they set to work dividing."

In this transaction it is possible that the Sicilian physician may have been in the right,
and the philosopher and his pupils in the wrong. And probably the result of their researches,
could it be recovered, would add little or nothing to our knowledge of pumpkins. But one
thing the passage proves; and that one thing is enough for my purpose. The SiaiperiKol
\6yoi of the Sophista and Politicus represent what really occurred within the walls of the

* «s ^pa\(iTepa av yevofieva xoi? cui/ovTas aTreipya^ero
d laXcKT tK(jOTepoV9 Kai Tr/s Toil/ ovTojv XoyM 0TlXu)<r6W9
evpeTtKWTepov^. Polit. 286 E.

' II. p. 59. As this fragment has not yet received the
attention it deserves, it is printed in full.

A. Ti nXaVlov

Kai STreufftTTTTOs Kal Mevedljfio^y

•Jrpos Ti(n vvi/i ^iaTpt^ovaiv ;

iroia (^poyrt's, ttoTos ^e A-oyo?

&tepevvaTai irapd ToTtrtv ;

TaSe fiot TTivuTws, et xl KaTCi^m^

tjKet«, \e^ov, irpdi -yas * •
S. aW olSa Xeyeii/ xepi TwvSe traipwi'

Havadijvaiot^ yap Idmv dyeXrjv

/xeipaKtuyv * *

iif yvfivatxioL^ ^AKadrjptai

ilKova-a Xoyoyv dtfyaTiov dToirtav

irepi yap {putrecoi d(popiX^6fievot

iie\wpi*^ov ^w'u)v Te ^iov

iei/6pa]v Te tpOtriv Xa^dvuiv Te yettrj,

KOt' kv TOi5tOIS XtJI/ KoXoKVVT1\V

€^»jTa^ov Tii/os eirri yivovi.

Vol. X. Part I.

KoX TL ttot' ap' wptffavTO Kai tIvo^ yevov^

elvai TO <Pvt6v\ o-i)\(i}(Tov^ el KctTOKydd Tt.
S, irptoTitTTa fxeu ovv Trai/res aVau^ets
TOT enrea-TTitTav, Kal Ku\J/avTe9
Xpovov ovK oXiyov 6i€(pp6vTi^ov,
kSt^ e^atcfu/tjs cti kvittovtwv
Kal "^riTovvToiv tcov fietpaKiwv
\a.)(av6v xis etpr] crTpoyyvKov tlvaiy
iroiav d' d\\o^, deuSpov 5' €Te/>os.
TavTa 6' aKovoiV laTpOi tic
^iKeXa^ diro yas KaTcirapS' auTwu

ftls XlJpOUl/TWVt

A, i] irov Seivuj^ topyiGQr\orav
xXeua'^eo-flai t €(36ij(Tav'

TO yap €V Xdtry^ats TaTorde TOtavTi
"TToteTv dTrpeire^.

B. oi}?*' ip.4Xij(Tev ToTs fieipaKtOK*

HXaTwit Se irapcai/ Kal fidXa irpauiv,
ovSev dptvdehf CTreVa^' auroTs
irdXiu [e^ dpxu^ Trii* KoXoKvvTiiv]
d^opi^eadat xtvos etrri yevovv

01 Se di^povv.

Com. Gr(sc. Fragm. v. iii. p. 370, ed. Meineke.

21

162

PROFESSOR THOMPSON, ON THE GENUINENESS OF

Academy: and we can have no doubt that Plato regarded such long-drawn chains of dis-
tinctions in the light of a useful exercise for his pupils. They became " more inventive" and
" more dialectical" — may we not say, clearer-headed — by the process.

I may add that the Invention of the Divisive Method is traditionally attributed to Plato
by the Greek historians of philosophy. Aristotle devotes several chapters of his Posterior
Analytics to the discussion of this method: he points out its uses and abuses, and defends it
against the cavils of Plato's successor Speusippus, who abandoned the method because, as he
alleged, it supposed universal knowledge on the part of the person employing it. The
method discuss.ed is that which we have been considering, for Aristotle describes it as
Division by contradictory Differentiae^ He also replies to the objection that this process
is not demonstrative — that it proves nothing — by the remark that the same objection
applies to the counter process of collection or induction. Tiiis defence, I presume, would
not in the present day be accepted as satisfactory ; for, as the able translator of the
Analytics observes, "This is the chief flaw in Aristotle's Logic: for some more vigorous
method than the Dialectical, the method of Opinion, ought to be employed in establishing
scientific principles." To shew the superiority of modern over ancient methods of
arriving at truth, is a gratifying, if it is not the most profitable employment of the Historian
of Ancient Philosophy. At the same time, I must confess my inability to discover the
flaw in the principle of dichotomy, as a principle of classification, in cases where the
properties of the objects to be classified are supposed to have been ascertained. A Class
can exist as such only by exclusion of alien particulars. The Linnean Class Mammalia
for instance, implies a dichotomy of Animals into Mammal and Non-Mammal — into
those which give suck and those which do not. The distinction may or may not be a
natural or convenient one, but in any other which may be substituted, some " differentia,"
some property or combination of properties must be fixed upon, which one set of species
or individuals possesses, and which all others want. And this is all that is essential in " dicho-
tomy," or the " method of Division by contraries^." The application of the method will,

' Anal. Post. ii. c. xm. § 6, and Schol. in loc. So
Abelard (Ouvrages Inedits. Op. 5fi!), ed. Cousin : coll. pp. 451,
461), distinguishes between those divisions which imply di-
chotomy and those which do not : e.g.

animal. animal.

man. horse. ox, &c. man. not man.

Porphyry attributes the latter or dichotomous method to Plato.
It could not be " Eleatic," tor each of the contraries would be
in that scheme a "non-ens." It is remarkable that a similar
Divisio Divisicnum occurs in the Potiticus, p. 287, § 27, where
in lieu of the regular dichotomy a rougher form of classi-
fication is for once adopted. This Plato, keeping up the
original metaphor in the Phtzdrus, describes as a fieXoroixia.
Kaxa fieXt) Toivvv auTa^ oiou lepe'iov Siaipufxeda, eireiSri
iiX^ (iSuvaTOVfieUt ^** ywp cis tow eyyuraTa OTt fid\i(TTa
Tc'/ui/eiK dpiBfjiiv dei. The division he proceeds to make, is a
distribution of "accessory arts" avvairioi reX""') into seven
co-ordinate groups. A similar relaxation is permitted in the
Philebut, p. 16 D: Aei ouv tj^as dfi /ui'av ISeav irepl

traVTOi eKaV-roT* deflevov^ X.iiTelv...edv ouv [neTa]\d^fi€v,
juera pi'iav Suo, et ttws elcrij iTKOireXv, el Si /.it;, t/>c7s
»? Til/' dWov dpid fJLOV, Kai twu ev enelvmif SKacnov
irdXw ciiTauTa}^ /te')(pnr6p dv to »c«t' dpxf'^^ ^^ M^i ^'Ti ei/
Kai uTretpd tcTt fiuvov iSri -ris, dWd Ktil o-Kuva. 1 understand
this passage as conveying Plato's di>tinction between his own
method and that of the Eleatics and tlieir Eristic successors,
who acknowledged only a ev and an direipov,

' For the length of the process will evidently depend on the
distance, so to apeak, between the Species generalissima and
the Species specialissima, between the remote and the proxi-
mate class in the tabulation of species. The very brief dicho-
tomy in the Gorgias, p. 464, is evidently the same in principle
as the long drawn divisions in the Sophisla, as will be seen
fiom the following scheme :

Qepaireia^

11 TOV <TtOfiaT0^.

rill ^vx^^'

yvfivaiTTtKti,

laTptKtJ. UOflodeTlKIJ*

SlKaVTLKlj,

THE SOPHISTA OF PLATO, &c.

163

as Plato acknowledges, be more or less successful in proportion to the insight and knowledge of
the person employing it. The specimens with which he favours us in these dialogues may be
arbitrary, injudicious, or even grotesque: but as logical exercises they are regular — and
logic looks to regularity of form rather than to truth of matter, which must be ascertained by
other faculties than the discursive. And even in judging of these particular divisions, we must
bear in mind the object in view. In the Sophista it is Plato's professed intention to dis-
tinguish the Sophist from the Philosopher, the trader in knowledge from its disinterested
seeker : surely no unimportant distinction, nor one without its counterpart in reality, either in
Plato's day or in our own. The ludicrous minuteness with which the successive genera
and sub-genera of the " acquisitive class " are made out in detail, would not sound so strange
to ears accustomed to the exercises of the Schools ; while it subserves a purpose which
the philosophic satirist takes no pains to conceal, that, namely, of lowering in the estimation of
his readers classes or sects for which he harboured a not wholly unjust or unfounded dislike
and contempt. It serves, at the same time, to heighten by contrast the dignity and importance
of the philosophic vocation, and in either point of view must be regarded as a legitimate
artifice of controversy in a dialogue unmistakeably polemical.

APPENDIX I.

In the foregoing discussions it is assumed that the method of Division sketched in the
Phcedrus is the same with the Dichotomy or Mesotomy of which examples are furnished in the
Sophista and Politicus. This I had never doubted, until the Master of Trinity gave me
the opportunity of reading his remarks on the subject, in which a contrary opinion is
expressed. I have therefore arranged in parallel columns the description of the process of
Division, as given in the Phcedrus, and in the two disputed dialogues ; from which it will
appear that the onus probandi, at any rate, lies with those who deny that the processes meant
are the same. I must premise that the Master of Trinity's question, " If this," viz. the
method in the Sophista, "be Plato's Dialectic, how came he to omit to say so there .^" has
been already answered by anticipation in p. l6, note 1, but more fully in Soph. 235, quoted
presently.

Where it is implied that all ♦tendance" is either corporal or
mental; that all tendance of the body is comprised in the
" antistrophic arts" of the gymnast and tlie physician, and all
tendance of the soul in those ot the legislator and the judge.
There is, therefore, no room under either for the four pretended
arts of the sophist, the rhetorician, the decorator of the person,
and the cuisinier. In Politicus, 302 E, the dichotomy is com-
prised in a single step: iu Taurais S,] tA wapdtio/iou sui
evvofjLOi/ knaaTtiv StxoTo/iel Toy-rwi/.

I ttust I shall not be understood as consciously advancing

opinions contrary to those of the Master of Trinity on the
subject of Classification. But so far as I comprehend his views
they do not seem necessarily inconsistent with my own. The
tyjiical principle of Classification seems, in its spirit at least,
strikingly Platonic; but it surely involves physical or meta-
physical ideas which transcend the limits of formal Dialectic,
lie this as it may, 1 should be sorry to have it supposed that
1 conceive my opinion on such a subject to be of any value in
comparison with that of the historian of Inductive Science.
This would be to " lecture Hannibal on the Art of War."

21—2

164

PROFESSOR THOMPSON, ON THE GENUINENESS OF

Phwdrus, 265 e, § 110.

OAI. To S' e-repov Zti eiSo? t'i Xfyeti la "EioxpaTe^ ;

2i2. To irdXiv Kar el^r) hvva<r0at Te/jiveiv, kot
apdpa, p 7re0uK6, kui fitj iirf^itpeTv Ka-rayvivai ixepo<s
ixritev, KOKov piayeipov rpoira y^pta/jkevov' dw' wairep
apTi Tin \oyu> TO fxev atppov Ttji Ztavo'ia<s ev ti KOii/p
EiCof «\a/36T»7i', uairep 8e (rionnTOi e^ era? SittAo koi
Ofxaivvfia irecpvKe, <TKaia, to Se Se^ia KXrjdevTa^ ovtio
Koi TO Tij<! irapavoiav o)« ev tJiMV ire<pvKoi etiot ijyt]-
aafievia Tie \oyu>, o /xev to €7r a p i <r t e pa TCfi-
V o pi € V o ^ piepo^y iraXtv touto Tepvwv ovk 67ray^K6,
irpiv ev auToT? c(pevptav ovopa^ofxevov ff Ka ton tiv
epu)Ta eXoiioptjae pd\' ev Sikj;. o ei? to ev Ze^ia
Trj<! pav'ia^ dyayiov ripdi, dpiavvpov pev eKe'wia deTov
S' au Ttv epioTa i<pevpiav koi tt p ot eivd pe vo<s eir-
tjveffev w? peyKTTov atTiov ijpiv ayavwv.

•I'Al. ' A\r]de<TTaTa XeyCK.

2n. TouTMi/ ir) eyiaye oi/toc t6 epa(TTij<;, m
<Pdiipe, Tiav Ziaipeveiav icai avvayioytov, "v oio? t u>
\eyeiv tc koi (ppoveTv eav Te tiv dWov >jyt]<Ta>pat
oiivaTOV eU ev koi eVi troWd ire (pvKod' dpdv, tov-

SlCdl

TOV CttUKtO KaTOWKTVe pCT I^I/IOI/ w<TTe oeoio.

deoTo

Ka\

pevTot KOI Tov<s tvvapevovi avro ipdv tl pev dpdm^
t) prj irpoaayopevia Oedt die, Ka\(o B' ovv peypi TOvZe
iiaXeKTiKoui,

Sopkista, 264 e.

HE. TlaKiv Toivvv eiri^eipiSpev, <T'^i j^ovTei ?'y5,
TO irpoTcBev yevo<;, iropeveaQai KOTa Tovir\ Ze^ta de\
pepo<i TOV T fxt} &evT ot e'y^opevoi t^<; tou ito^kttou
Kotvwvtatj etat av avTov Ta Kotva iravTa trepteAovTe';^
Ttjv oiKeiav XiTTovTet (pvtriv eir toe i^ta tAev fia\t(TTa
pev tjp^v auToZ^:^ eireiTa de Kai Tot? eyyvTUTta yevei t^?
ToiavTrjt peOooov iretpvKouiv,

lb. 253 D, § 82. To koto yevri ItaipeTa-dat, koi
pr]Te TavTov etho<! CTepov tjyij<Taa6ai prid' CTepov ov
TavTov pcov ou Tfji; iiaXeKT iKrj t (ptja-onev emaTtj-
fttji eivai; Q. [Nai,J (pt]<ropev . , , S. dwd ptjv to ye
i ia\eKT I K 6v ouK a\Ku) Boxreic, a>? eyiapai, irKtjv
T<S Kudapai Te koi BiKaioic (pi\o(TO(p(o.

lb. 229 B, § 31. T»)i/ dyvotav iBo'i/Tec ei Trp k oto
pe cr o V avTrji t o p t] v eyei Tiva. o i ir \rj yap avTf/
ytyvofxevtj Zri\ov oTi Kat Ttjv BiSa<rKa\iK»;i/ ivo avay-
Ka^ei pdpia evf'i', ev e(p' ev) yevet Tav avTtj<s eKaTepio,

Politicus, 263 B. E i B o <; pev orai/ rj tow, koi
pepo'; avTO avayKaTov etvai tou irpaypaTot otov irep
dv elhoi XeyrjTui' pepo<s Be elioi ouZepia dvdyKrj.
(Tliis explains the kot' dpdpa r) irecpvxe of the
Pkwdrus.)

lb. 265 A. Kai ptjv e<p' o ye pepov upptjKev
6 \dyo<s eir CKetvo Zvo Ttvt naOopav o'Bm TCTopeva
(paiveTUi, Triv pev daTTia, frpoi peyci pepoi (rptxpov
htaipovpevov, Trjv B' oirep ev tw trpoirdev eXeyopev, oti
Be? p e (T o T o peTv oti paXiara^ tout eyovaav juaA-
\ov, paxpoTepav ye pr)v.

lb. 262 D, occurs ai specimen of the " unskilful
carving" (KaKov payelpou Tpdirov) of the Phcedrus.
Et TK Tdv6 pwirivov e'TTii/eiprja'ai otya BieXetrdat yevov
itaipoirj Kadairep oi ito\\oi...to pev EWf/i/iKov (to
Be) ^dp/3apov...fi tov apiOpov T15 av vopi^oi kot' elir]
hvo iiaipeTv pvpiada aTroTefUvopevoi awo iravTwv,
a>5 ev eiBos a'/roj^upij^iDv, k.t.K,

In allusion to Xen. Mem. iv. ^ 11, a passage noticed by the Master of Trinity, p. 595 of
his paper, I may observe that the etymology of Dialectic, a-rro tov SiaXeyeiv, is undoubtedly
vicious, and is nowhere countenanced by Plato. On the contrary, Dialectic is described in the
Philebus, 58 e, as >; tov SiaXeyeaOai Suva/nis. He could not have adopted Xenophon's
etymology, for as we have seen, the Platonic Dialectic includes dwaywy^ as well as Siaipeais.
The etymology was tempting, and Xenophon, who writes very nrtich at random upon
philosophical subjects, was unable to resist the temptation. A similar error is that of Hegel,
who in his History of Philosophy, derives aocpiaTtjs from ao(p'i^6iv instead of aoCpi^eaOai,
an error in which he has been followed by English scholars who ought to have known better.

THE SOPHISTA OF PLATO, &c. 165

APPENDIX II.

On the Earth-born (yriyevels) of Sophista, 246.

Of the three contemporary sects professing some form of Materialism, I have singled out
the Cynic as that which alone answers the conditions of Plato's description. The following
extracts from the fragments of Democritus, and from Aristotle's notices of his opinions, seem
conclusive against his claim to a share in the Gigantomachy.

1. The sect in question held that, tovto fxovov
eaTtv, o TTiipe^ei wpoajioXriv Kai firatprjv Tiva,

2. -rau-roii (TiSixa koi ova'iav <ofji(^ovTO' they defined
" substance " to mean corporeal substance only.

3. They despised tow? (pa<rKovra<; ftri (TWfxa ei^ov
eivai.

1. Democritus, on the contrary, says, i/oV«)
Trai/TO TO uiirdrjra, £ t e j; aTOfxa Koi kcvov. — Frag.
ed. Mullach. p. 204.

2. Democritus denies that the sense of touch
conveys any true knowledge. 'Hiuc?? tw /ueV
eovTi ovcev aroeKe? PvvUfxeVf jjieTairTirTop Sc KaTCt
T6 trwfxaTO^ oiaotyriv Kai twv eirttirtovTiav koi twv
ai/TiO'TrjfJtQoi/TOJv,

3. Democritus held " on oidev iiaWov to on tow
fxrj oyTO? e<TTiVj on ovc€ to kcvov tov o"to/jtaTO?. —
Arist. Met. I. 4. In other words, that vacuum (his
nrj 6v) was in every respect as real as corporeal sub-
stance.

The Cyrenaics are not the -yrf^evel^, for they admit nothing to be real except the affection
(tto^os), of which we are conscious in the act of sensation, an affection produced by some cause
unknown. The objects of sense are to them as unreal as they were to Berkeley. Sext. Empir.
adv. Matth. vii. igi : ^acrlv ol KvprndiKoi Kpirripia elvai ra TraOt], koi (xova KUTaXa/xfiavfa-Oai
Kat dSia'd/evaTa Tvyyaveiv' twv ce TreTroirjKOTWV ra Tradrj nrjhkv elvai KaTaXrjtrTov /xijce
aota\l/€V(TTov.

The case of the Ephesian peovres is not worth considering, for they acknowledged no
ovaia, as the Earth-born know nothing of 'yevean, which they properly class with the aoparov.

The view I have adopted, that the passages in the Thecetetus and Sophista both refer
to Antisthenes, and that the latter dialogue is in the main a hostile critique of his opinions,
occurred to me in the course of my lectures on the Thecetetus in 1839, as I find from MS. notes
in an interleaved copy. I mention this, because Winckelmann in his Fragments of Antisthenes,
published in 1842, observes in a note: " Omnino in multis dialogis ut in Philebo, Sophista,
Euthydemo, Platonem adversus Antisthenem celato tamen nomine certare, res est nondura satis
animadversa." Some of the allusions to this philosopher which Winckelmann detects in the
Thecetetus appear to me doubtful, but I observe with pleasure that he acknowledges the
double bearing of the epithet di'TjTi/Tro?, the perception of which first put me on the enquiry
of which I have given some of the results in the foregoing paper.

IX. On the Substitution of Methods founded on Ordinary Geometry for Methods
based on the General Doctrine of Proportions, in the Treatment of some Geo-
metrical Problems. By G. B. Airy, Esq. Astronomer Royal.

[Read Bee. 7, 1857-]

The doctrine of Proportions, laid down in the Fifth Book of Euclid's Elements, is, so far
as I know, the only one which is applicable to every case without exception. It is subject
only to the condition, that the quantities compared, in each ratio, shall be of the same kind
(without requiring generally that the quantities in the different ratios shall be of the same
kind); a condition which appears essential to the idea of ratio.

This generality, however, as in other instances, is not without its inconvenience. The
methods of demonstration which are applied by Euclid are very cumbrous and exceedingly
difficult to retain in the tnemory, and I know but one instance (that of the proposition ex cequali
in ordine perturbata, as amended by Professor De Morgan) in which it has been found prac-
ticable to simplify them. It is therefore natural that attempts should be made, in special
applications of the doctrine of proportions, to introduce the facilities which are special to each
case.

In the special application in which numbers are the subject of proportion, methods have
long since been introduced, departing widely in form from Euclid's, yet demonstrably leading
to the same results, and possessing all desirable facility of application.

No attempt, I think, has been made to avoid the necessity for employing Euclid's gene-
ralities, when geometrical lines alone are the subject of consideration. Yet there are cases in
which these generalities have always been openly or tacitly employed, but in which the nature
of the investigation seems to indicate that there is no need to introduce proportions at all.
I was led to this train of thought by considering the well-known theorem, " If pairs of tan-
gents be drawn externally to each couple of three unequal circles, the three intersections of the
tangents of each pair will be in one straight line." This, I believe, has always been proved
by the use of certain propositions of proportion. Yet the theorem starts from data without
proportions, and leads to a conclusion without proportions; and it seems wrong that it should
be conducted by intermediate steps of proportions, the theorems of which have been proved by
methods based fundamentally on considerations of arbitrary equimultiples.

It appeared to me, on examination, that this and similar investigations, of which lines only
are the subject, might be put in a simple and satisfactory form, referring to nothing more
advanced than the geometry of Euclid's Second Book, by a new treatment of a theorem equi-
valent to Euclid's simple ex cequali, and of the doctrine of similar triangles. I beg leave to

G. B. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS, &c.

167

place before the Society the series of propositions which I suggest as sufficient for these pur-
poses, and (as an example) their application to the particular Theorem to which I have
alluded. I have omitted several merely formal steps in the demonstrations. It will be seen
that the demonstrations which I offer, though applying to the properties of lines only, require
the use of areas; but in this respect they are simpler than Euclid's, which, though applying
to lines only, require the use both of areas and of the process of equimultiples.

Proposition (A). If the rectangle contained under the sides a, B, be equal to the rect-
angle contained under the sides 6, A ; and if these rectangles be so applied together that the
sides a and b shall be in a straight line and that the side B shall meet the side A ; the two
rectangles will be the complements of the rectangles on the diameter of a rectangle.

7?

X ^

D

y^ K

A

L
T

Because the opposite vertical angles of the two rectangles are equal at the point of meeting,
A and B will be in the same straight line. Produce the external sides of the rectangles till
they meet in D, join DE ; and, as the sum of the angles GFD, EDF, is less than two right
angles, produce the lines FG, DE, till they meet in H; and draw ^/parallel to FD or GE.
If the rectangle under h and A is not terminated in the line HI, let it be terminated by the
line KL. Since KL is parallel to 6 or GE and therefore parallel to HI, it will be entirely
above or below HI. Now by Euclid, the complements FE, EI, are equal ; but, by hypothesis,
FE, EL, are equal ; therefore EL is equal to EI, which is impossible if the line KL is above
or below ///; therefore KL coincides with HI, and the rectangle 6, A, coincides with the
complement EI, and the two given rectangles therefore are the complements, &c. a.E.D.

Peoposition (B). If the rectangle contained under the lines a, B, is equal to the rect-
angle contained under the lines A, h ; and if the rectangle contained under the lines 6, C, is
equal to the rectangle contained under the lines B, v ; then will the rectangle contained under
the lines a, C, be equal to the rectangle contained under the lines A, c.

[This is equivalent to the ordinary ex cequali theorem.

If a : b :: A : B,

and b : c :: B : C,

Then will a : c :: A : C]

168

G. B. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS

w

L M

A

P

8
T

F /r^^^''^

a

n

B B

^ B

a

I

^^

b

N

c

^^"^"^

/:;

Construct the similar and equal rectangles DE, FG, with sides 6 and B ; and apply them
with their angles meeting at H, in such a manner that the side DH or B of one shall be in
the same straight line with HG or B of the other ; then will the side FH or b of one be in
the same straight line with HE or b of the other. In the right-angled triangles IDH, HFK,
the sides including the right angles are equal, therefore the angle FHKis equal to the angle
DIH, and is the complement of the angle DHI; therefore IH and HK are in the same
straight line.

To DH apply the rectangle DM whose side DL or HM is equal to a ; the sides DL and
HM will be in the same straight lines with DI and HE. To HE apply the rectangle HO,
whose side HN or EO is equal to A ; the sides HN and EO will be in the same straight
lines with HD and EL Produce LM, ON, to intersect in P, and join KP.

Then, because the rectangle LH, which is the rectangle contained under a and B, is equal
to HO, which is the rectangle contained under b and A ; by Proposition (A), LH and HO
are the complements of the parallelograms about the diameter of the rectangle LO ; therefore
/ZT and consequently IHK (which are in the same straight line) are in the diameter; therefore
LHKP is a straight line.

In like manner, to HG apply the rectangle HQ whose side GQ or HR is equal to c;
and to HF a.^^\y the rectangle HS whose side FS or HT is equal to C; and produce ST
and QR to meet in V; and join LV. Then, proceeding from the hypothesis that the rect-
angle contained under c and B is equal to the rectangle contained under 6 and C, it will be
shewn in like manner that KHIV is a straight line.

Therefore PKHIV is one straight line.

Complete the rectangle WX. Then WH, HX, are complements of the parallelograms
about the diameter of WX, and are therefore equal. But WH is the rectangle contained
under a, G, and HX is the rectangle contained under c, A ; therefore the rectangle contained
under the lines a, C, is equal to the rectangle contained under the lines A, c. a.E.D.

Corollary. By repeating the operation, the theorem may be extended to four or any
number of terms of comparison of rectangles, following in a similar order.

Proposition (C). If two right-angled triangles are equiangular, and if a, A, are their
hypothenuses, and b, B, homonymous sides ; the rectangle contained under the lines a, B, is
equal to the rectangle contained under the lines A, b. -

FOUNDED ON ORDINARY GEOMETRY, &c.

169

[The equivalent theorem in proportions is

a : b :: A : B.^

>^^

Apply one triangle upon the other as in the right-hand diagram, so that the side h meets
the hypothenuse A at right angles, and the vertex of the angle opposite 6 meets the vertex of
the angle included by A and B. Since the angle GFH is equal to the angle FDE, it is the
complement of the angle DFE ; and GFE is therefore a right angle ; and GF is parallel to
DE. Now the rectangle under a and B is the double of the triangle GFE ; and the rect-
angle under 6 and A is the double of the triangle GFD. But because GF is parallel to DE,
the triangle GFE is equal to the triangle GFD. Therefore the rectangle under a and B is
equal to the rectangle under A and b. q.e.d.

Proposition (D). If a, c, and A, C, are homonymous sides of equiangular triangles, the
rectangle contained under a, C, will be equal to the rectangle contained under c, A.

From the angles included by the sides A, C, and a, c, let fall the perpendiculars B, b,
upon the third side. The corresponding right-angled triangles thus formed are easily shewn
to be equiangular. Hence, by Proposition (C),

Rectangle under a, B, is equal to rectangle under A, b.

Again, Rectangle under b, C, is equal to rectangle under B, c.

Therefore by Proposition (B),

Rectangle under a, C, is equal to rectangle under A, c. q.e.d.

Proposition (E). If b, c, and B, C, are homonymous sides including the right angles
of two equiangular right-angled triangles, the rectangle contained under 6, C, will be equal
to the rectangle contained under c, B.

This may be considered a case of the last proposition, or it may be treated independently
thus.

Vol. X. Part I.

22

170

G. B. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS

Apply the two triangles together, so that their right angles coincide, and their homony-
mous sides are in the same straight lines. In consequence of the equality of the remaining
angles, the hypothenuses EG, FH, will be parallel. Therefore the triangle FEG is equal
to the triangle HEG. To each add the triangle EDG, then the triangle FDG is equal to
the triangle EDH. But the rectangle under 6, C, is double of the triangle EDH ; and the
rectangle under c, B, is double of the triangle FDG. Therefore the rectangle under 6, C>
is equal to the rectangle under c, B. a.E.D.

Pkoposition (F). If the rectangle contained under the lines a, B, is equal to the rect-
angle contained under the lines J, b ; the parallelogram contained under the lines a, B, will
be equal to the equiangular parallelogram contained under the lines J, b.

[This is equivalent to the proposition,

U a : b :: J : B,
Then a : b :: A.cosa : B.cosa.^

In the figure, produce the upper sides of the parallelograms to cut the vertical sides of
the rectangles in D and H. The rectangles DG, HL, are equal to the given parallelograms,
therefore it is to be proved that the rectangle DC? is equal to the rectangle HL, or that the
rectangle under a, EG^ is equal to the rectangle under b, IL.

Since the parallelograms are equiangular, the right-angled triangles EGF, ILK, are equi-
angular ; and therefore by Proposition (C), the rectangle under EG, A, is equal to the
rectangle under IL, B. But by hypothesis, the rectangle under B, a, is equal to the rectangle
under A, b ; therefore by Proposition (B), the rectangle under EG, a, is equal to the rectangle
under IL, b. Or, the parallelogram under the lines B, a, is equal to the equiangular paral-
lelogram under the lines A, b. a.E.D.

FOUNDED ON ORDINARY GEOMETRY, &c. 171

These Propositions, I believe, will suffice for treatment of the first thirteen Propositions
of Euclid's Sixth Book (Prop. I. excepted), and for all the Theorems and Problems appa-
rently involving proportions of straight lines (not of areas, &c.) which usually present them-
selves. As an instance of their application, I will take the theorem to which I alluded at
the beginning of this paper.

Theorem. If pairs of tangents are drawn externally to each couple of three unequal
circles, the three intersections of the tangents of each pair will be in one straight line.

I shall omit the demonstration that, for each couple of circles, the pair of tangents and
the line passing through the two centers all intersect at the same point ; and I shall use only
the intersection of one tangent with the line passing through the center. Also I shall omit
the construction and its demonstration, for inserting between the greatest and least of the
three circles a circle equal to the remaining circle, having its center upon the line joining
their centers, and being touched by their tangent.

M

Let A, B, C, be the centers of the given circles. Let N be the center of the circle
whose radius NO is equal to the radius BK, and which is touched at O by the tangent
DE. Join NB, MF, FI, MN, NI, FB.

First we shall prove that MF is parallel to NB.

The triangles NOF, CEF, have each one right angle, and they have another angle
common; hence they are equiangular; and by Proposition (C), the rectangle under CF, NO,
is equal to the rectangle under NF, EC; or, the rectangle under CF, BK, is equal to the
rectangle under NF, CL. Again, the triangles BMK, CML, are equiangular, for each has
one right angle, and they have another angle common; therefore the rectangle under CL, MB,
is equal to the rectangle under BK, MC. Consequently, by Proposition (B), the rectangle
under CF, MB, is equal to the rectangle under NF, MG. Therefore, by Proposition (F),
the parallelogram under CF, MB, which has one angle equal to MCF, is equal to the paral-
lelogram under NF, MC, which has one angle equal to MCF. But the former of these

22—2

178

G. B. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS, &c.

parallelograms is double of the triangle BMF, and the latter is double of the triangle
MNF. Therefore the triangle BMF is equal to the triangle MNF, and therefore MF is
parallel to NB.

Secondly. To prove that FI is parallel to NB.

It will be shewn in exactly the same way that the parallelogram under AF, BI, with the
angle FAI, is equal to the parallelogram under AI, NF, with the angle FAI. But the
parallelogram under AF, BI, with the angle FAI, is the excess of the parallelogram under
AF, AI, with the angle FAI, above the parallelogram under AF, AB, with the same
angle; or is the excess of double the triangle AFI above double the triangle AFB, or is
double the triangle BFI. Similarly the parallelogram under AI, NF, with the angle FAI,
is double the triangle NFI. Therefore the triangles BFI, NFI, are equal; therefore FI is
parallel to NB.

And as MF and FI are both parallel to NB, MF and FI are in the same straight line.

a. E. D.

' ADDENDUM.

I AM permitted by Professor De Morgan to transcribe the simple process for demon-
strating the theorem of ex cequali in ordine perturhata, to which allusion is made above.

li a : b ■.: B ; C,

and b : c :: A : B,

Then will a : c :: A : C.

i

then a

A : C.
:: c : d.

To exhibit the process more clearly to the eye, use the connecting mark /- — v for one
ratio and ^^i^ for the other; then the theorem stands thus,

If a ^-N 6 ;

and A^:=:::B^—^c,

To prove it, take a fourth quantity d, such that a : b

Then b ^p^:::: c ^— >. d.
But A^:=:::By-~>.C.

Therefore, ex eequali, b : d :: A : C.
But, because a : b :: c : d, therefore alternando, a
fore the ratio a : c for 6 : d in the analogy just found,

a : c :: A : C. q. k. d.

c '.: b : d. Substituting there-

RoTAL Observatoby, Greenwich.
September 2, 1867.

G. B. AIRY.

X. On the Syllogism, No. Ill, and on Logic in general. By Augustus De Morgan,
F.R.A.S., of Trinity College, Professor of Mathematics in University College,
London.

QRead Feb. 8, 1858.]

I PUT this paper under the title here given, for the sake of continuity of reference: in
scope, however, it is more extensive than those which precede (Vol. viii. Part S; Vol. ix.
Part 1). It will best be disposed under two heads. I shall first put together remarks on
the object of logic ; on its present state ; on the opinion of the world with respect to it ; on
the views which I take of it, in opposition to the world at large as to its advantages, and to
the writers upon it as to its details. I shall incidentally answer some objections to my former
paper ; objections, not objectors : and I would gladly do something, be it ever so little, to
hasten the time when logic shall again be a part of education in the University of Cambridge.
I am satisfied that there is no study, however useful, no exercise of the intellect, however
essential, but has its own short-comings which can only be made good by the study of mind
as mind, psychology ; and induces its own bad habits which can only be eradicated by the
study and practice of thought as thought, logic. But psychology and logic, in their turn,
require other studies even more than other studies require them.

In the second part, I shall present the elementary points of the system which I advocate.
Which of the two parts should be taken first is a question which each reader must decide for
himself.

Section I. General Considerations.

I. Eleven years ago, when I began to put together details on which I had been thinking
during several previous years, I had not the encouragement which would have arisen from
a knowledge of what was then going on in the logical world. In my own mind I was facing
Kant's* assertion that logic neither has improved since the time of Aristotle, nor of its own
nature can improve, exceptf in perspicuity, accuracy of expression, and the like. I did not
know that very high authority was then teaching its alumni to assert that logic had always

♦ There is an intell