Abe ΤΟΝ eS. os Seite = rise ee ARRON τς ' Pat iaetnnc σιν το ὑσὶ See een one ᾿ a wre \ hy ‘7 M A Set Anna aa Ἴ ΩΝ ARC an nA \ . MAN ἣ Ni ce Ν Soe Ἀλλ AMNG REG AMR BEE AERA AMAR Ι Pott xe fhe AMARA ΝᾺ ; ὶ Hn AMAIA NS ee “atl! Naan EREANON suet : ΤᾺΣ Se SNE αὶ A ANWAR AS Rm en ~ a Sean nena cut! — aS Δλλαλὰ ἀξ Ae eae SEIERRAR ον AAT Annas NON AN Ma — d : : de . Ἶ ς q (sin usin ν) τα — w= sin FU.sin UK. sin FUK é€ . = - Ὁ --- Β1ὴ “.COS yp δος 4 or after reduction du Gri εν tania. . 4: — -, τ --:}.---ττς + + = 0...... . dt * e τ =) tanya @) The equations A and (2) together ne tle two equations (B) serve to determine (after € . F ὅ . eliminating «4 and ») — τ and oo, when "4 —~» ~~ are given, that is, when the intensity and > dt VELOCITIES, ἄς. WITH RESPECT TO AXES MOVEABLE IN SPACE. 5 variation in direction of the acceleration of τ are given for every instant. And we have also from the triangle FU’F” ultimately dy ᾿ a . —. 810 μι ΞΞ ----- 8581 ee a to determine πὶ and therefore we have equations to determine the intensity and variation in direction of τὸ itself. Hence we have obtained a solution of the problem, ‘‘ Given the path of F and the variable intensity of f, to determine the path of U and the intensity of w,” 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 w,, Uy, τύ, denote the resolved parts of w along the moveable rectangular axes Ow, Oy Ox, and let Q,, Q, Q, and f,, f,, f, denote in like manner the resolved parts of Q and f. Now by reason of the acceleration f, τό, receives in the time dé the increment f,d¢: also Ox changes its position by reason of the rotations Q,, Q,, the first of which shifts it in the plane of xa through the angle Q,dt from O,, and the latter in the plane of wy through the angle Q,dt towards O,; and from the first of these causes w, receives the increment τ; COS (= + Q, 4) + u, cos (Q,dt) -- τι,» or — u,Q,dt ultimately, while from the second it receives the increment y U, COS G - 0,4t) + Uz cos (Q,dt) -- uz, or τὸ, ἀξ ultimately. Hence the total increment of w,, being the sum of these partial increments, we obtain the equation : or =f, + uy, Q, - u, Ay Similarly for w,, τὸ, we should obtain το κὸν ΩΝ 0,0, Γ st du $F] =f, + Uz Qy — Uy Qu, 1} 8. Τὸ 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 τι,» uy, u, may then be replaced by the co-ordinates, w, y, x: also Ff, denoting the acceleration of u, will in this case denote the absolute velocity of P, and /,, fy, Ff. the absolute velocities resolved in the directions of the axes, which we will denote by »,, Vy v,. Then by the equations above we have three equations, of which the type is 6 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING ἀν 1 Ox - τῶ and which determine the relative velocities mp Ὥς = of the point with respect to the co-or- dinate axes. If the point be fixed relatively to the axes, and a, Y, % be its co-ordinates, the above equation becomes Oy = Qy. πο = Ὡς. Yos 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 w, so that y, x both vanish, —=v, O0=0,-aQ,, 0=4,+ ay In these, if δ᾽, y, x 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, 0, @ denote radius vector, altitude and azimuth, then ἀφ dé ὥ τεῦ, Qy = τις cos 8; Q,= 7 whence dr αθ d τ ΣΉΝ soa haere υ, τ τοοθ 2, the common expressions for the components, relatively to polar co-ordinates, of the velocity of a point. ὃ. Accelerations, radial, transversal in the vertical plane, and perpendicular to that plane. In our general formule wu will now denote a velocity, and f an acceleration strictly so called. And in this case dr dé dp τι wore uy = τος 0s ἀφ. ἀφ dé Q, = ποτὶ and, Oy = — τ; “956, Q,=7 wherefore, by equations (1) 2, 2 radial acceleration =f, = a - (, : 4 Foca ὃ | ‘ transversal acceleration in the vertical plane = Ἢ -5(+5) - (-rsin6. cox of" - τ “| dt \ dt dt dt dt 1d/,d\ υοἱ de]? ΞΞ eel PP len ὁ — ’ = (¢ a) Ὁ 7 5159 con 066} : : d azimuthal. acceleration rat Pad πὲ [τος θ a ὧς ( = τ σῶς θ -- Υ sin wi τ VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 7 Wit ps νὰ Op = ab a (, cos 9.2). c. Let the axes of a, y, x be always parallel to the tangent, principal normal and normal to the osculating plane of any curve. Then ds Ue Uy = 0, u, = 0, dr de Q,= 755° Q, = 0, Q,= 7? where de, dr denote respectively the angle between consecutive tangents, and that between consecutive osculating planes. Hence a. tangential acceleration = f, = F ; oe Ce seca 1 ds de _ (a) de _ 1 (5): acceleration in principal normal =f, = ἜΚ eh ere, : sal 3 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. 10. 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 ὦ in the previous section, the Calculus there developed is applicable to them. 8 Mr R. Β. HAYWARD, ON A DIRECT METHOD OF ESTIMATING 11. Consider a material system at any instant of its motion. Tach 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 Ov, Oy, Ox be denoted by u,, uy, w,, and the corresponding angular momenta by h,, h,, h, respectively ; then it may easily be shewn that the linear momentum relatively to the axis, whose direction-cosines are /, m, m, is lu, + mu, + NU,, and that the angular momentum relatively to the same axis is th, + mh, + nh, The first expression will be a maximum, and equal to {u,” + u,’ + u7}3, when Lim: Mi Up? Uy : πὸ} and if this be denoted by τι, it is plain that the linear momentum along any line inclined to the direction of w at an angle @ will be wcos@. Hence we may regard the whole linear momentum of the system as equivalent to the single linear momentum wu determinate in intensity and direction. In like manner we may conclude that the whole angular momentum is reducible to a single angular momentum A 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 w is independent of the origin O both as regards direction and intensity, but the angular momentum ἢ 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 ἢ remains constant, while O moves along the direction of the linear momentum u, but changes, as Ὁ 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 Rdé in the direction of the resultant force R of the forces (P), and an angular momentum Gd¢ 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, ἃς. WITH RESPECT TO AXES MOVEABLE IN SPACE, 9 summation over the whole system; we may therefore regard R and G as the resultant force and resultant couple of the ewternal forces. Then the linear momentum w along the line OU must be compounded with the linear momentum #d¢ in the line OR in order to obtain its value at the time ¢ + dt: and in like manner the angular momentum hf relatively to the axis OH must be compounded with the angular momentum Gd¢ relatively to the axis OG. 15. Hence the method of the previous section applies to momenta of both kinds, replacing f in one case by R and in the other case by G. Thus the equations (B) give us du d ε qi = Roo RU; uP = Rsin RU, where ἀφ is the are through which U moves towards # in the time dt: and dh dy ἡ αἰ F008 GH, hoa, = Gain GH, where dy is the are through which H moves towards G in the time dé. 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 diy yyy = ty deck we. cat dt . dh, " dh, _ dh, ee — eo. which are really the six fundamental equations of motion of our works on Dynamics. For rectangular axes moveable about O, the equations (Z) of the last section furnish two sets of three equations, of which the types are du, dt dh, Leh dt be] a νῶ, = "Ὧν. 16. If the system be acted on by no external forces, it follows that both w and h 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 O 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 Vor. X. Part I. 2 = Χ σοι, -- τ,» 10 Mr R. Β. 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 «w and h, when the forces are given or vice versa. It is to be observed that the equations involving A, 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,a,ya, denote the linear velocities of the origin in the directions of the axes, the equation for h, becomes dh, ; Ξ 7 hyQ, -- hQy + τἰγας -- τὐ,αν» The equations involving uw, 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 a in 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 ὦ 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 w. 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, ἄς. 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,,v, denote the resolved velocities along Oz, Oy, Ox of the point O, to which the translation is referred, and let w,, ων» w, denote the resolved angular velocities about the same lines; then the velocity of any particle m, whose co-ordinates are ὦ, Y, ὧν ἰδ, + ωγῷ — wy in the direction of Ox, with similar expressions for the directions Oy, Ox. Hence summing the linear and angular momenta of the several particles of the system, we find τ; = =(m) οὖ, + w,. =(mz) — w,=(my), * T avoid the use of the term 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 a 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 itis 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 αὐ 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 co-ordinate 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 of 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 a 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 of 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 eaternal forces, or those whose opposite Reactions are 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 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING and h, = Σρι(ψ οὖ, + WY το Wt το 2. Vy + WH — W,8) = (my) .v, — E(msz).v, + T(m.y? + 2°). ὦ, — X(may)wy — U(mxx) .w, with similar expressions for u,, 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 Ὁ, each depends in general on both the linear and the angular momentum. If however O 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 =(mwx), =(my), =(mz) all vanish, and the equations become those, of which the types are Uz = X(M) . Ves hy, = =(my’ + 2)w, — =(may) . ὦν — =(mzxa)w,. 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,, h,, 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 A, B, C denote the moments of inertia about these axes, the equations become (substituting 1, 2, 3 as subscripts for a, y, # respectively) h, = Aw, hz = Bor, hz = σὰν hence the axis of angular momentum OH, whose equation is @o ey 8 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, ἄς. WITH RESPECT TO AXES MOVEABLE IN SPACE. 13 Aa? + By? + Cx* = 1, at the point, where the axis of angular velocity OJ, whose equation is eo oY 8 @, We ως ? meets it. Also reciprocally OJ is parallel to the normal to the ellipsoid, whose equation is y” 2? 4*B'C at the point where OH meets it. Thus a simple geometrical construction enables us to determine OJ, when OF is given, and vice versa. If now ὦ be the angular velocity about OJ, and J the moment of inertia about the same line, the angular momentum about it must be Jw, since w is the ¢otal angular velocity, and therefore the angular velocity about a line perpendicular to OI is zero; hence Iw =h.cos HI, an equation connecting h and , the quantities J and HJ 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 OJ will not in general remain fixed, nor ὦ constant, for, by the motion of the system about OJ, the position of OH in the system is altered, and to this new position of OH a new position of OJ will correspond, and then w will change by reason of the variation of cos HI There is an exception however in the case where OH and OJ coincide, for then the rotation does not change the position of OH in the system: this can only be the case when the radius OJ 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 OJ 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 ΠΟ], if OJ 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. 14 Mr R. Β. 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 ὧι» We, ὡς for Qy, Q,, Q,, and hy hy, hg, or Aw, Bw, Cw; for ἢ,» hy, ἢ,» _and then we obtain three equations, of which the type is, either dh, ἀν (sd dé =[D+ (ς - 5) hls ἃ or A— = ἢ +(B- C). 0,0 dt 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 OJ, 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 dé 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 OI is OH: thus the angular acceleration « 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 corresponding 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 w about OJ; and apply equation (C) of section I., putting ἢ for wu; therefore we = —hw.sin JH.sin HP.sin JHP, which determines the acceleration of momentum for any line OP. This acceleration will be zero, if OP bein the plane ΠΟ, and a maximum, if OP be perpendicular to HOJ, when its value is hw sin HI: we may therefore regard the total acceleration* (f) due to the motion of the body as being about the line OF, perpendicular to HOI, and equal to + hw sin HJ, when OF is taken on that side of HOI on which a positive rotation about OF would move OH towards OI. Now to this acceleration of momentum (f) about OF will correspond an acceleration of angular velocity (A) about a line OL which is related to OF, just as OL is to OH. 29. Tosum up our results, we have shewn that, if OH be the axis of angular momentum (h) and OJ that radius of the central ellipsoid at whose extremity the normal is parallel to OH, OF is the axis of angular velocity (w): if OG be the axis of the impressed couple (6); and OK the radius for which the normal is parallel to OG, OX is the axis of angular accele- * This result is that which M. Poinsot states thus: “‘The | sion.’’”—M. Poinsot’s “couple d’impulsion” is our angular axis of the couple due to the centrifugal forces is perpendicular | momentum. at once to the axis of rotation and to that of the ‘ couple d’impul- VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 15 ration due to the forces («): lastly, if OF be perpendicular to the plane HOJ, 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 (A). Also we have the three equations for w, x, d, Iw =h cos HI, Kr = GeosGk, ΤᾺ =f cos FL, where f = hw sin HJ, I, K, L denoting the moments of inertia about OI, OK, OL respectively. It will be observed that OJ is the direction, to which the plane through O perpendicular to OH is diametral, and that OL is the direction to which the plane ΠΟ] 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* OJ, OL, OM are 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 OZ. In this case OH remains fixed in space, and the motion of OJ is conveniently referred to its motion in the plane HOJ and the motion of that plane about OH. LT = ἤ 2 Let the conjugate radii ΟἹ, OL, OM be denoted by r, γ΄, γ΄, then the moments of inertia about them are “: > = ‘ aa by the property of the central ellipsoid : also let the angles HOJ, FOL be denoted by 6, 6’: then our last equations become (1) w=hr'cos@, (2) «=Gr'cosO, (8) A= (hwsin θ). τ΄" cos. Resolve w, x, along the axes OH, OM, OF; the component velocities are then w cos @ along OH, wsin@ along OM, and zero along OF, while the component accelerations are «cos@ along OH, «sin @ + sin @ along OM, and ἃ cos @ along OF ; whence, by applying either the equation (C) or the equations (£), ἕω 6080) =.«.008.0 = Gr* COS? O...005.0crercccrcveccecse cee (4) * Hence if no forces act, the instantaneous motion of the axis of rotation OJ will be towards OZ, the radius with respect to which the plane ΠΟΙ is diametral. . 16 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING w sin @.Q =) cos θ΄ = (hw sin 0) . 7 cos? 0',... 002000 νον κεν σον ..(6) where © is the angular velocity of OM (i. e. of the plane HOTZ) about OH. Also we have bent WR he innpaieec vas vos sec ccecccccctessesteeuernehieee Let p, p’ denote the perpendiculars from O on the tangent planes to the central ellipsoid at I, L respectively, then p = r cos 0, p’ = γ΄ cos 6’. Equation (4) becomes by (1) τ (hp*) = Gp, whence by (7), p is constant. This shews that the tangent plane at J to the central ellipsoid is fixed, and that the central ellipsoid therefore rolls on it as a fixed, plane. Also by (4) and (5) d(tan6) d/wsin@\ Asin’ , ,, ; i -5(*= 5) = gt hp =n θ.. τη 9 sicgeysuas (8) and from (6) Se ee eee eee ΔΝ μεν λυσάνον ἐς, τ 31. Now 7, τ΄, γ΄ being conjugate radii of the central ellipsoid, there exist three relations between them and the conjugate axes; these are, (putting psec 0, p’sec @ for 7, στ΄ respectively and denoting the angle JOL by x) 1 1 1 p’ sec? 0 + p® sec? θ΄ + 7? = 5." E, suppose, 1 1 1 2/2 ΩΣ 3.,3 coc? Ἐς Obed © ar pr”? + pr” + pp” sec’ θ sec’ θ΄, sin’ χ Bet ὉΑ 48 F,, suppose, »» Fag (- aH = G, suppose, and by reason of the rectangularity of the planes JOM, LOM, we have cos x = sin @ sin 6’. Eliminating r” and x, we obtain RUG pr sec! O + p'* sect + = Β, G = + δ + p’p*(sec? @ + sec’ θ΄ -- 1) =F. From these eliminating sec® θ΄, we obtain Look ὦ 2? ly 4 (1- τς ὅς - G)cot* a, fo { pp Pp which, (remembering what E, F, G denote, and putting a, β, Ὑ for the three quantities 1 - 1 - 1 : respectively) ag ce is equivalent to p® = p°(1 + aBy cot? @); VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 17 also, since ρ΄, θ΄ are involved in precisely the same manner as p, 6, it follows that p? =p(1 + a'p'ry cot? 6’) ; where a’, β΄, γ΄ are what a, B, Ὕ become, when p’ is put for p. From these equations we obtain wit apy cot?@ αβΎ 1+ ay τοῦθ᾽ 7 1 1 1 1 + cot? θ but a el-—=1- ——. -Ξα By το 4" Ap? τ-ὸοβγοοῦῦθ 1 + αβγ cot’ é cot® whence, with the corresponding expressions for β΄, +’, (1 + aBry cot? 6)? (1 + By cot’ 0)(1 + γα cot® @)(1 + αβ cot* 6)’ hence ρ΄, 6’ are known in terms of p, 0. cot? 0’ = -- cot? @. 32. Substituting now for μ΄, θ΄ in terms of p, 0, we obtain from equation (8) d(cot @) ,, cot 8 soi eta dt P cot = + hp*S - (1 + By cot® 6)(1 + γα cot? O)(1 + a cot® θ)}},.....6.6 010) and from equation (9) Q = hp? (1 + aBy cot? 6). If h be known by means of (7), these two equations determine completely the motion of OI the axis of angular velocity in altitude and azimuth, since p, and therefore a, B, Ὑ, are constants. If @ denote the azimuth at any instant, τ =Q, and dividing the last equation by the preceding, we obtain a relation involving @ and @ only, which will therefore be the differential equation to the conical path of OJ in space; and it is worth notice that, this relation being independent of ἡ, the path of OJ 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 (1) gives w = hp? sec θ, from which ὦ is known when 6 is known by means of equation (10), and thus the velocity about OJ is known completely as well as its position at any time, 33. If there be no forces acting, i. 6, if G= 0, ἢ is constant, as is also ὦ 605, the re- solved angular velocity of the body about OH. Also the vis viva of the body w h’p! =],?=—= r 2 cos@ and is therefore constant; and hence ~ is constant, or ὦ « 7; both well known results. It may r Vou. A. Pant ck, 3 18 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING vis viva also be well to note that p® = iainc wieunegemay" even if G do not vanish, and therefore g somentum ἢ that the vis viva « (angular momentum)*, when the angular momentum has a fixed direction. It is needless to carry the solution farther by investigating the path of OJ in the body, the position of the principal axes relatively to OH, ΟἹ at any time, &c., since all these questions are discussed with the utmost completeness and elegance in M. Poinsot’s Théorie 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. σ ᾿8ὅ. When the polar axis of the central spheroid always lies in the plane of the meridian, let 9 denote the north polar distance of its extremity 4. 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 about its axis, the motions of OA, OB, OC will be due to the velocities Q cos θ, Q sin θ, = about them respectively: also the actual velocities of the body about the same axes are d respectively w, Q sin 0, =, and the consequent angular momenta 4w, BQsin 0B, where w, τᾷ are reckoned positive when the motion about their axes is in the same direction as the earth’s about its axis. VELOCITIES, ἄο. 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 d do. ae a 4”) “5: . 5'ηθ -- BQ sin 8. = d μ dé dé qi BO sin θ)- α + do. πο “ἂν cos 0, d;_,d0o P ; (83) -- ΒΩ sin θ. οο5θ -- 4ω. Ὡ sin 6; from the first equation, w is constant, and from the last ἂν -- (Fe - Q cos 6) asin θ: 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, @0 dt? whence the motion of the axis OA is precisely similar to that of the circular pendulum, whose =— < sin 0, A : length is J, where © = Roe and therefore /= Ξ 3 the direction of the earth’s axis taking the ω place of the direction of the force of gravity. 2, Also since 75 = 0, when sin 9 = 0, there are two positions of equilibrium of the axis OA, namely, when θ = 0, and@ = 7: 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 @ denote its azimuth from the south towards the east. Taking OB and OC as before, the latter will now coincide with the vertical. If ¢ denote the co-latitude, Q may be resolved into Ω cose vertical and Q sinc horizontal in the north direction: hence the angular velocities by which the axes move, are relatively to OA, OB, OC respectively dp -Qsinecosd, —Qsincsin gd, ag + cose, and the corresponding angular momenta are 4w, -- BQsin ὁ sin ᾧ: a(t + Ω cose), 40 Mr R. B. HAYWARD, ON A DIRECT METHOD, &c. whence as before, d(4w) - -- 0, dt ale ΒΩ sine sin Φ) = G+ Aw (t+ cos ο + B(Z + 2cose) .Qsin ecos p, αἀ{. αἱ : (22 + Qeos¢) = BQ sine sind .Qsin 6 cos ᾧ + Aw. Qsine sin d, dt\ dt and therefore w is constant, and i "δ. = ΞΖ εἰ esin φ + QO? 5ἰπ" 6. sin ᾧ cos φ, or approximately α' ΤῸΝ < oO sin e. sin gs 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 ———-——. 8 A@Q sin Ἢ Duruam, Feb. 19, 1856. R. B. H. Il. On the question, What is the Solution of a Differential Equation? A Supple- ment to the third section of a paper, On some points of the Integral Calculus, printed in Vol. IX. Part Il. By Avaustus De Morean, of Trinity College, Vice-President of the Royal Astronomical Society, and Professor of Ma- thematics in University College, London. [Read April 28, 1856.] Trustine 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* — ada’ = 0, to which Euler assigns the integral form (y-aw+b) (y+av+c)=0, where ὃ and ¢ are independent constants. Most other writers insist on the condition ὦ = ὁ. 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 ( -- αὐ +b) ( -- αὐ -Ὁ δ) for (y—aw+b) (Ψ Ὁ αὦ Ὁ 6)» makes two remarks. The first,— chacun de ses facteurs doit étre considéré isolément; the second, alluding to the form with two constants, is—on n’en tire pas d’autres lignes que celles qui résulteraient de l’intégrale renferm- ant une seule constante. M. Cauchy (Moigno, ii. 456) says—On ne restreindra pas la généralité de cette intégrale en désignant toutes les constantes arbitraires par la méme 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 problem being relation between y and x, the equation (y — x) (y — x*) = 0 implies power of choice between the relations y = a”, y=, The equation (y — a’) (ὦ — 1) =0 implies the relation y = αὐ with a dispensation from all relation in the case of # = 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 Mr 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 f(x, y, y’) = 0 involve any number of relations between «, y,y’: and let (a, y, a,b) = 0 be a relation between ὦ and ὃ, or any number of relations, Consequently, selecting one relation by which to satisfy ᾧ = 0, values of a and b can be found to satisfy both p(w, y, a,b) =0, and also p(a + h,y + k, a, Ὁ) =0, for any values of x, y,h,k. Hence, for any values of @# and y, y’ may have any value whatever: and this is incompatible with f(a, y,y’)=0. But this is no argument against any form of $(,y, a, b,) = 0, in which the constants are not in relation ; as Wa, y, 4) - χίω, ψ, δ) = 0. For we cannot pretend to satisfy Ve, Y; a). χίω, ψ, δ) =0, Ve +h,y +k, a) -x(@ +h,y +k, b) =0, for any values of w,y,h,k, except by W(#,y,a) = 0, and y(w~+h,y+k,b) =0, or else by W(a +h, y + k, a) = 0, x(#, y,b) =0. And from neither set can we deducey’. If W(a, y, a) = 0 be a primitive of f(x,y, ν΄) = 0, there appears nothing ἃ priori to prevent our saying that V(a, y, 4). ψίω, y, b) = 0 isa 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'=/(«,y) be satisfied by different relations (a, y, a) =0; V/(a, y, b) = 0, then, taking a and ὃ so as to satisfy both at a given point (v,y), we find, generally, two values of y’ at (wy). But y'=f(#,y) may give these two values; irreducibly connected, as in ψ' = 1 + ./y, or reducibly, as in ψ' =14,/y*. The great point of algebraical interest, namely, that when the two values of y’ are irreducibly connected = 0 and ψ =0 are the alternatives of an equation which can be rationalised or otherwise inverted into χ =0, where χ 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 = #, which is certainly one relation for determination of y from w, is to be considered as giving one or two relations for determination of # 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 letter 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 a f sinav.v-'de from +4 to -- ἐπ, as w passes through 0, might be discovered by the 0 WHAT IS THE SOLUTION OF A DIFFERENTIAL EQUATION? 23 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= which ends at w =0 joins the branch of y= +e-* which begins at w =0 with acontact of the order co, as order of contact is usually defined. We may even propound the question whether (— #)* and (+ «)* 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 - οὐ to τῷ c 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 wnity into a number, and its factor into a multiplier, be extended to contain the usual passage through infinity. Let 2-ordinal continuity be when and so long as y, Μ΄, y”,...y are of non-ordinal continuity. These definitions being premised, we have in the passage from the positive to the negative value of w} an interminable continuity, and a change of form answering to, and indeed derived from, the change of form seen in (+ )* and (— @)*. 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 (a + a)} is non-ordinally continuous, and not so much as primordinally, when ὦ = 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 ( -- x)’ to(+a)*. 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.y' we announce three solutions. To ψ' = 0, any parallel to the axis of # To ψ' =2 x the positive value of ,/y, the right hand branch, from # =a onwards, as figures are usually drawn, of any parabola y=(v-a)*. To oy =2x the negative value of 4/y, the left hand branch of the same up to #=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 « 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. Ina 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 = (« — δ)", which we please, but only one, for the solution of y? =2,/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=0 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 «, and of the other factor for others. The isolation of the factors is the postulation of a certain permanence of form. 24 Mr DE MORGAN, ON THE QUESTION, 3. In asolution 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 Φ 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 ψ' =24/y any number of parabolas with as much of the singular solution y = 0 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 are 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 = x(#, y). Aw, with a very small value of Aw, which may be as small as we please in the reasoning, a solution of y= x(a, y) is shewn to exist: but it may be one of the kind just alluded to. The draughtsman employed to construct such a solution, when his are 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 are 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 ἃ ‘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’ = 2\/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 = (A)?. It may now be affirmed that ( -- αὦ -- Ὁ) (y+aa+c) =0, ὁ and e being perfectly independent constants, is a solution of y’*—a*=0; nothing in the general theory of the primordinal differential redation in any way withstanding. It remains to examine the assertion that the generality of this solution is not restricted by the supposition ὃ = ὁ. 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 ὃ = belongs to one mode of grouping a solution of y' = α with a solution of ψ' = —a: but there is an infinite number of modes in ὃ = de. If ordinal continuity be held sufficient, and if φίω, y, b) = 0, ψίω, y, 6) = 0 be independent relations satisfying f(a, y, y’) = 0, and if P =0 be the most complete singular solution, then P. (a, y, by) « Pla, Y, bz) 0-10 W(@, Ys 61). ψίω, Y, 62)... = 0 is the most general solution, where 6,, b,,...c;, 625... are in any number, and of any values, This however is but equivalent to P. f(a, y, 6). ψίω, y, 6) = 0 with the usual addition ‘for any values whatever of 6 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, ὃ and ὁ being in relation in φίῳ, y, ὃ; 6) = 0, we may designate all the curves contained in (a, y, fc, 6) = 0 as the group (fe,c). Generally speaking, the curves of the group (fc, 6) are different from those of (Fc, c). The unlimited number of cases of (fe, 6) 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 (2, y, ὃ). ψίω, y, ὁ) = 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 @ =0 and Ψ = 0 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 (Ρ -- δ) (0 -- ὁ) - R=0, P,Q, R, being each a function of w and y: and let P’ represent ἢ, + P,.y', &e. When b = fc, the primordinal equation of the group (fe, 6) is Ro +/(R? +4P’'QVR) R-,/(R? +4P QR) Gt 2P’ ae \P τ 40’ | ; Let R=,V, where V is a finite function, and » a constant. When μ diminishes without limit, and finally vanishes, each primordinal equation becomes either P’=0 or Q’=0, for otherwise we have only Q = /P, the algebraic result of eliminating ¢ between (P — δ) (Q -- c) =0, and (P — fe) Q' + (Ω -- ὁ) P’ =0, And the biordinal equation is determined by differentiating R+f(R?+4P'VR) 2P’ i Do this fully, clear the result of fractions, and write »V for R: it will then appear that y’’ is seen only in terms multiplied by positive powers of 4; and so that μ =0 gives P’Q’ = 0 in place b=Q+ of a biordinal equation. The correction which the common theory requires is as follows ;—An equation in which n constants are in relation with w and y, cannot have any differential equation clear of those constants under the mth order; and an equation of single and irreducible relation between X,Y, Yi .ey™ must have a primitive containing 2 constants in relation to ὦ and y. But a primitive equation in which m constants are contained in alternative relations, m, in one relation, m in a second, &c. does not require a differential equation of the mth order ; but has an equation of alternative relations, one of the mth order, one of the nth order, &c. From a primitive having m constants, in relation with a 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(w, y,a) =0 give a = (#,y,), and therefore ᾧ, + ®,.y’ = 0 for a differential equation, in which ὦ has disappeared and y’ is introduced, it is easy to give this differential Vou. X. Parr I, 4 40 Mr DE MORGAN, ON THE QUESTION, WHAT IS THE SOLUTION &c. equation a primitive containing any number of separate and independent constants. For A, + A, O(a, y) + 4. {P(x, y)}* + ... = 0 cannot give any relation in which one of these constants disappears in favour of y’ except ©,+,.y' = 0, in which they all disappear. But this is merely formal; for 4, +A, ®(#,y) + ... = 0 is but a transformation of some case of O(a, y) = SF (Ao, Ay...) or of Pha, y, f(Ay .4.»...)} =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 ἢ - ψ' can only be a factor in the differential of one class of forms. If {F(a,y)}’ give M(P+y), nothing but ᾧψ ζω, y)}’ can give N(P + y’): and F(#,y)=const. and ψ ζω," ) = const. are the same equations. : But it is otherwise with P + γ΄, P being a function of 2,y,y’. This occurs, as previously shewn, in the differentiations of two distinct classes of forms. Thus 0+y” is a factor in § f(ay’ —y)}’ and in { Fy'}'. The equation f(y -y) = 4, + 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 vy’ — 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. University Cotitece, Lonpon, March 29, 1856. Ill. On Faraday’s Lines of Force. By J. CuerK Maxwett, B.A. Fellow of Trinity College, Cambridge. (Read Dec. 10, 1855, and Feb. 11, 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 formule, 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 formule 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 the 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 in form between the laws of light and those of vibrations. By stripping it of its physical dress and reducing it to ” 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 a theory of “transverse alternations, 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 formule. These formule have a distinct mathematical significance, 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 force 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 formule 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 formulz, there would be nothing to distinguish between the one set of phenomena and the other, Mr. MAXWELL, ON FARADAY’S LINES OF 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 especially Series XX XVIII. of the Experimental Researches, and Phil, Mag. 1852. 80 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. I. 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 Mr 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 of fluid 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 fluid 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. (8) 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 wnit tube of fluid 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 92 Mr MAXWELL, ΟΝ 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) Τὸ 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 different 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. Tn 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. 11. 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. Vous ΣΧ, .Parr I, 5 34 Mr 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 difference 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 ina direction perpendicular to two of its faces, Then the resistance will be kv, and therefore the difference 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, 3 &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 difference 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 we find by the last equation 8 = 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 wnit cells. In each of them unity of volume of fluid passes from a pressure p to a pressure (p—1) 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. ΜΕ MAXWELL, ΟΝ 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 systems 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 p, 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 shouldbe the 5—2 36 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 (16) 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 4πτ΄ : k , The rate of decrease of pressure is therefore kv or i=’ and since the pressure = 0 7 when ¢ is infinite, the actual pressure at any point will be Ss Tr 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 4πτ΄ If we have a source formed by the coalition of S unit sources, then the resulting F ΠῚ : 5 . pressure will be p = Poa! that the pressure at a given distance varies as the resistance 7? and number of sources conjointly. (19) Ifa 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, ΟΝ 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 &, 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 & 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 &, and that of the interior. by Κ΄. 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 Xk’, 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 &, 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 & and the sink by ζ΄, so that the result is for every external unit source on the surface, a source = (k — 1). By means of these three systems of sources the original system of pressures may be produced in a medium for which k = 1, 38 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. (24) Let there be no resistance in the medium within the closed surface, that is, let Μ΄ = 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 withia 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 zero 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 , τν and wherever a tube leaves it we must conceive a source producing fluid at a rate , 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 different directions. In such cases the lines of motion will not in general be perpendicular to the surfaces of equal pressure. If a, ὃ, ὁ be the components of the velocity at any point, and a, /3, Ὑ 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=Pa+QB+ Ry, = PB + Qy+ R,a, Pry + Qa + RB. In these equations there are nine independent coefficients of conductivity. In order to simplify the equations, let us put Q+h, Ξ 95.) Q-R,=2lT, δον θενος δῦ; φουτου τονε ΟΣ where 47" = (9, -- R,)* + (Q,- Δ)" + (Q, - κ᾿), and 7, m, m are direction cosines of a certain fixed line in space. o I The equations then become a=Pat+s8,8+S,y +(nB- my)T, b= PB + δι + δια + (ly -- na)T, c= ΡΟ + S,a + 8,8 — (ma -- 1B)T. By the ordinary transformation of coordinates we may get rid of the coefficients marked S. The equations then become 40 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. a= Ρία- (nB-m'y)T, b= Ρ( β + (ly - n’a)T, c = Py +(m'a - VB)T, where 7’, m’, n’ are the direction cosines of the fixed line with reference to the new axes. If we make dp dp dp =o = -- d so a da’ dy’ ὮΝ the equation of continuity da db if de ΕἸ da” dy .ds ’ becomes d’p , ap ,d’p Piss Pay OF faa and if we make aar/Plt, y= VPin #= VPS op 2 ae d‘p =0 then Pig 5 +a > the ordinary equation of conduction. It appears therefore that the distribution of pressures is not altered by ‘the existence of the coefficient 1. 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 P,, P,, P, 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. X XI. Part I.) It is evident that all that has been proved in (14), (15), (16), (17), with respect to the superposition of different 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 (p -- 1) 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 ρ΄, then the number of cells in it will be p — p’. Now if the tube had extended from the Mr MAXWELL, ΟΝ 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 ,ϑ' from which S tubes proceed to be p, Sp is the number of cells due to the source δ: but if S’ of the tubes terminate in a sink at a pressure p’, then we must cut off Sp’ cells from the number previously obtained. Now if we denote the source of S' tubes by S, 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 = (Sp). (30) The same conclusion may be arrived at by observing that unity of work is done on each cell, Now in each source 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, ,5΄ units of fluid sink into nothing under pressure ρ΄; 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 W = Sp -- 3S"p’, or more concisely, considering sinks as negative sources, W = X(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 mp, 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 R is a coefficient 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, oe R k PS ry = 4πΆ ° Aa 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) Ina uniform medium the pressure due to a source S ew Pt" At another source §” at a distance 7 we shall have SPR ar ge TOP if p’ be the pressure at § due to S’. If therefore there be two systems of sources Σ(8) and =(S’), and if the pressures due to the first be p and to the second ρ΄, then =(S"p) = =(Sp’). For every term S’p has a term Sp’ equal to it. Vou. X. Past I, 6 42 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. (38) 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 & 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 A be the distance between consecutive surfaces of equal pressure and s the section of the unit tube, we have by (13) s = kh. But s is the product of the breadth and depth; but the depth is &, therefore the breadth is ἃ 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 Jines 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 § at a distance x varies inversely as γ΄. Let us see what will be the effect 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 * See Cambridge and Dublin Mathematical Journal, Vol, 111. p. 286. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 43 By this assumption we find that if V be the potential, dV = Xdxv+ Ydy + Zdx = — dp, or since at an infinite distance V = 0 and p =0, V = —p. v= -Σ( 7. In the electrical problem we have Inthe Auid-pim Z (= =); 4nr 7 4 . S= = am. If & 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 k k = (pdm) = ---, =(pS)=— W. 4π 4π If =(dm), = (dm’) be two systems of electrical particles and pp’ the potentials due to them respectively, then by (32) k k , ’ Σ (pdm) =, Σ(»5γ- =, Σ( 8) -- Σ ἀπὸ, π 4π 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 : V=—-p, : d Χ-- = ku, k dm = — 8S, Amr whole potential of a system = — 2Vdm = -w, where W is the work done by the fluid in over- π 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, ΟΝ 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 effects, 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. ‘ 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 gst of Magnetism, Chapters III. & V. Phil. Trans. 1851. + Experimental Researches (3292). Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 4 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 κ΄, the resistance within the second medium, is greater than &, 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 ὦ, 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 ζ΄, 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 pinned 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 +. 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. Theory of Magnecrystallic Induction. The theory of Faraday + 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 a Experimental Researches (2797), (2798). See Thom- + Exp. Res, (2429), (3320). See Weber, Poggendorff, son, Cambridge and Dublin Mathematical Journal, May, Ixxxvii. p. 145. Prof. Tyndall, Phil. Trans. 1856, p. 237. 1847. ‘ ath + Ezp. 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 formule. 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. x11. 496, and of M. Quincke, xivir. 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 +. 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 Exp. Res. (371). + Exp. Res. Vol. 111. 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 an 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 difference 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, J 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=IK=p-p, where p is the electric tension in the fluid and p’ in the zine. 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 ρ΄. p—p’,, or F affords 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 J 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 JK, where K is the resistance of the whole circuit. The same idea of quantity and intensity may be applied to the case of magnetism t. 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 & be the resisting power of the material, and ,ϑ' the area of the section, and Z the number of lines of force which pass through it, then the whole intensity throughout the section k ἘΠ 7 ts 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 Pal fedo=K=p-y. * Exp. Res, Vol. 111. p. 519. + 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 #. 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 I = Sidydz, where dydz 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 ki =f, and the total intensity of magnetization in the circuit is measured by the sum of the local intensities all round the circuit, F = (fda), where dz is the element of length in the circuit, and the summation is extended round the entire circuit. In the same circuit we have always 1 = IK, where K 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 Ampére, 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 Ampére’s formule 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. Ampére 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 Xda+ Ydy+Zdz a complete differential. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 49 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, ceteris 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 + 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. + Art. (13). Vous.) Parr dl, ῆ δ0 Mr 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 by 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 acurrent in it. If the If a closed conductor move transversely to the lines of magnetic induction, then, if the number of lines conductor is closed, there is a continuous current, if open, tension is the result. which pass through it does not change during the motion, the electro-motive forces in the Hence the electro-motive forces If the motion be such that a greater number of lines pass through the circuit formed by the conductor circuit will be in equilibrium, and there will be no current. depend on the number of lines which are cut by the conductor during the motion, 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 f. * 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. Mr MAXWELL, ΟΝ 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 volta-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 +. Part 11. 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). tion of Electric, Magnetic and Galvanic Forces. Camb. and + See Prof. W. Thomson On a Mechanical Representa- | Dub. Math. Jour, Jan. 1847. 7—2 52 Mr 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, ΟΝ FARADAY’S LINES OF FORCE. 53 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. : Der. 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 (yz) estimated in the directions of the axes w, y, x respectively will be denoted by a, ὅ; cs. The quantity of electricity which flows in unit of time through the elementary area dS’ = dS (la, + mb, + ne), where Jmn 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 caemical composition or of temperature in contiguous parts of the conductor. Let p, represent the electric tension at any point, and X, 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 a, B, ry, be the effective electro-motive forces dp, a, = X,- “δε dp, phage (A) dp, | aw ier ἢ 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,, ας = KA, Be = Κ,}.» y= kCo, (B) but if the resistance be different in different directions, the law will be more complicated. These quantities a, (8, Ὑ5 may be considered as representing the intensity of the electric action in the directions of wyz. δ4 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. The intensity measured along an element do of a curve e=la+m3+ny, where Zmn are the direction-cosines of the tangent. The integral fedo 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 aB-y from equations (A) fedo = [(Χάω + 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 Jedo = {(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 +n¢, inn being the direction-cosines of the normal, fedS = ffadydz + [[bdzdx + f{edxdy, the integrations being effected over the given surface. When the surface is a closed one, then we may find by integration by parts fas = [2 + ἦν Ὁ 12) 4 dy de. If we make feds = 40 | [fedadyde, 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.”” Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 55 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, 6, 6, k, a, β, y; p, and p, are the same in form as those which we have just given, a, 6, ὁ are the symbols of magnetic induction with respect to quantity ; 4, denotes the resistance to magnetic induction, and may be different in different directions; a, β, ‘y, are the effective magnetizing forces, con- nected with a, 6, c, by equations (B); p, is the magnetic tension or potential which will be afterwards explained; p denotes the density of real magnetic matter and is connected with a, ὃ, ¢ 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. Ampére 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 Ampére 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 Jaws 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 dyds. Let the axis of # point towards the west, x towards the south, and y upwards. Let ayz be the position of a point in the middle of the area dydx, then the total intensity measured round the four sides of the element is τῇ (x oF) ax d, Total intensity = [- -- oS) ay dz. The quantity of electricity conducted through the elementary area dydz is a,dydz, 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 πος ὐμαν .} 2 ds ~ dy’ δ: τὼ dry, _ day yada? Sih eg da, dB, Cy ant nee tee dy da These equations enable us to deduce the distribution of the currents of electricity whenever we know the values of a, β, yy, the magnetic intensities. Ifa, β, Ὑ be exact differentials of a function of wyz with respect to #, y and s respectively, then the values of a, ὃς ὁ5 disappear ; * See Experimental Researches (3265) for the relations between the electrical and magnetic circuit, considered as mutually embracing curves. Mr MAXWELL, ΟΝ 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, de, da rf dy μὲ 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. Tueorem I. The equation oot eV as : 0, + ay? * Sagick: Sp = (where V and p are functions εν wys never she and vanishing for all points at an infinite distance,) can be satisfied by one, and only one, value of V. See Art. (17) above. Tueorem II. The value of V which will satisfy the above conditions is found by integrating the expression pdrdydx ie -a'|bty- ψ + z- z' |)! where the limits of zyx 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 Taylx’s Scientific Memoirs. Tueorem ITI, Let U and V be two functions of vyz, then ΟὟ at av CGY dUdV dUdV I e Se 1.5) ttyl = ΤῊΝ { a da da * dy dy * de dz ap) ἀπάγάς ἐμ .5 +f eg . where the integrations are supposed to at over all the space in which U and V have values differing from 0.—(Green, p. 10.) This theorem shews that if there be two attracting systems the actions between them are equal and opposite. And by making U = V 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 Vor. 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. 12, 13), and therefore the number of cells in a given space is directly as the square of the velocity. Turorem IV. Let a, 3, y, p be quantities finite through a certain space and vanishing in the space beyond, and let & be given for all parts of space as a continuous or discontinuous function of wyz, then the equation in p d i dp dil dp ac are (6-3 va fae τίν - SE) + 4x0 το, 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 aBy be the electro-motive forces, p the electric tension, and & the coefficient of resist- ance, then the above equation is identical with the equation of continuity day Poni de, da * dy 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 α β as magnetizing forces, p as magnetic tension, and p +47p = 0; 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 " d avy ἢ d dV\2 1 d dV a= (if, («- a τ} Ls (8 - ar ia) εχίν- Ἢ ε a ‘Ste ay de where V is got from the equation Vv ¢g av ie Bee ore dye * ae + 47p = 0, 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 & has various values did not affect the distribution of forces, then the : P dV : : dV “quantity” resolved in # would be simply nm and the intensity & ἼΣ But the actual quan- 1 d, d tity and intensity are τ(« - =) and a -- τῇ , and the parts due to the distribution of media alone are therefore Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 59 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 z 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. TuHeorEM V. If a, ὃ. e be three functions of w, y, z satisfying the equation i Backes spe de dy dz Ὁ it is always possible to find three functions a, 8, Ὑ which shall satisfy the equations Ge Py dz dy ᾿ dy ἀα. ἀν dz ᾿ da ἀβ ὩΣ Let A = fedy, where the integration is to be performed upon ὁ considered as a function of y, treating w and κ' as constants. -Let B = Jadz, C = [bdv, A’ = /bdz, B’ = fedx, C’ = (κὰν, integrated in the same way. Then ee Pa gh a dx B= B-B + oA, i 7 y=C-C } a ἃ: will satisfy the given equations; for d d d db τ -τῆς [τ ds -{¢ abi pi dv + [Fay and om (Bde + [dos [Fass a8 dy da da da i a τσ αν ἂν + fa dy + [π =a. 00 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. In the same way it may be shewn that the values of a, , Ὑ satisfy the other given equations. The function y, 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, 6, 6 are discontinuous functions of ἃ, y, 2, 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 PA @A ada dat * αν * ae 7 sens TB aB ὌΣΣΕ Phe a] dy? ae aa ad ro. PC @&C da * dy? * ae by the methods of Theorems I. and II., so that 4, B, C are never infinite, and vanish when a, y, or αὶ is infinite. b=0, +c=0, Also let ἀβ dC dy "a ee dC dA <& Bs dx dz * dy’ . 44 dB LW dy “dy da dz’ then ἀβ dy _ Ὁ (442, ὩΣ ἘΞ +54+ 54) ds dy daw\duw ἂν ἀξ da dy ds d (dA dB adc * da (= dy * ay If we find similar equations in y and x, and differentiate the first by , the second by y, and the third by x, remembering the equation between a, ὃ, c, we shall have es +.) (4 dB dC) _o. da? * dy’ * dz? (τ 2" ; and since A, B, C are always finite and vanish at an infinite distance, the only solution of this equation is and we have finally with two similar equations, shewing that a, 8, y have been rightly determined. Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE. 61 The function y, is to be determined from the condition da dB dy ee - αἵ ad La γα ore ἜΣ ΣῊ da * dy + oe if the left-hand side of this equation be always zero, ψ' must be zero also. Tueorem VI. Let a, ὃ, ὁ be any three functions of #, Y, 2, it is possible to find three functions a, β, Ὕ and a fourth V, so that du * dy * ds" ” dB dy dV and hace hag ἣν + ao? b dy da dV de dz dy’ ᾿ da dB dV dy ἄν dz Let da db de ‘ dx * dy dz BE? and let V be found from the equation ¢ BV ον dav da® + dy? * age =~ ΠΡ then ; dV a ee > δ ἢ αὶ | dy | ee dV a ee satisfy the condition da db dé ὃ ἀφ * dy Ἢ ἀπ᾿ Ὁ and therefore we can find three functions 4, B, C, and from these a, B, Ὑ, so as to satisfy the given equations. TueEorEm VII. The integral throughout infinity Q = fff(aa, + 6,8; + evy:)dadydz, 62 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. where a,b, 6)» a; 3; Ὑι are any functions whatsoever, is capable of transformation into Q = + [if f4mrppr — (acts + Bobo + γ.0.}} dadydz, in which the quantities are found from the equations da, μὲ db, δι ie ‘ ag a ee da, dp, Ν Δ] , ae Beat —— + 4 = 0; ἼΣ dy = a + 4701 a, Bo yo V are determined from a, ὃ, ¢, by the last theorem, so that ya eee πως OE dz dy ‘dx a, b, c, are found from a, 8; y, by the equations pd at &e ς ΞῚ Ἐπ . ἀξ dy and p is found from the equation For, if we put a, in the form dBy ἀγ. dV dz dy da’ and treat b, and 6) similarly, then we have by integration by parts thfugh infinity, remem- bering that all the functions vanish at the limits, 9- - Π γα τὰ +B) το φῇ) als - ἀ):» or Q=+ [ἀπ Κρ) = (a,d,+ Bobo + yote) } dadydz, and by Theorem ITI, da, dp, (> ky ia) [αυάγάς, [[[Vp'dadydz = [[/ppddydz, so that finally Q = [If ξ4πρρ — (αγας + Bibs = yrs) } dadyds. If a,b, οἱ represent the components of magnetic quantity, and a, βι γι those of magnetic intensity, then p will represent the real magnetic density, and p the magnetic potential or tension. ὧς ὃ; c, will be the components of quantity of electric currents, and ay By Ὑο will be three functions deduced from a, 6,¢,, 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 (,). Thus a,6,¢, are the components resolved in the directions of #, y, z of the Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 63 quantity of magnetic induction acting through a given point, and a,B,y; 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 When there are no electric currents, then a,dx + B,dy + yidz = dp,, a perfect differential of a function of a, 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 dp, dp, —m— , —-m—, and —m— dx’ 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 = S/ppidadyds 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 1 Q = + re [[faa + b,B, + yy: )dadydz, in which a, 8, γι are the differential coefficients of p, with respect to a, y, x respectively. If we now assume that this expression for Q is true whatever be the values of a, βι γι» 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. ‘Q= [ff\ pe -- = (aes + Bib. + γι) ἡμιάγαν. So that in the case of electric currents, the components of the currents have to be multiplied by the functions a,3,ry. 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 ay 8, yo 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. ΤῸ these I give the name of Electro-tonic functions, or components of the Electro-tonic intensity. 64 Mr 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, 6, ¢, and their intensity by a, B. ys Then the amount of work due to this cause in the time dé is dt {ff (α;ας ὙΠ b3. + Cxry2)dadydz in the form of resistance overcome, and Bed Sf if (α,ας + b, By + Cory))dudydz 4π dt 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 Ὶ dt d dt fj f f (aya. + ὃ.. + Cyty2)dadydz + oe I [f/f (asa + 6,3, + Cory )dadydz, where the integrals are taken through any arbitrary space. We must therefore have 1d Az + b.B. + ΡΝ Δ = ree a) + 8, a C20) for every point of space; and it must be remembered that the variation of Q is supposed due to variations of αὐ 3,7, and not of a,b,c,. We must therefore treat a,b,c, as constants, and the equation becomes 1 da, 1 15. 1 dry, ee) ab by | Py ictees + — =0. α[α,- a) + (βεε ας a) +6, (ve + 5 x) In order that this equation may be independent of the values of a, ὃ; ¢,, each of these co- efficients must =0; 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, : 25 Τα : It appears from experiment that the expression on 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. Ifa, be expressed as a function of a, y, x, and ¢, and if a, y, x be the co-ordinates of a moving article, then the electro-motive force measured in the direction of a is ἀν dt + dy dt” ds dict sae} Cg Ss σο 1 (= dx da, dy da dz day’ 4π * Translated in Taylor’s New Scientific Memoirs, Part 11. Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE. 65 The expressions for the electro-motive forces in y and x 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 formule, 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. Prov. I. Jf on any surface a closed curve be drawn, 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. Lawl. 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). Vor. X. Part I. 9 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. LawIV. 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 circutt. 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 the 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, 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 by 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 Eaperimental 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 Llectro-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. | tal and theoretical, renders the study of his theory necessary to Weber’s Memoir is translated in Taylor’s Scientific Memoirs, | every electrician. Vol. V. Art. x1v. The value of his researches, both experimen- Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 67 ee’ a dr 7 (1 ater es br =) : (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 Ampére’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 waysof 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-dynamiec 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 froin 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. IX. Part 1. 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-- 68 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. EXxamPLes. 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 = @) giving out 2 Fiat ie > : Ἶ a : ΠΡ ΒΥ) fluid at the rate of 47Pa? units in unit of time, is ΚΙ — outside the shell, and /Pa 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 47Pa*, and the other absorbing at the rate 4a P’a’, then the expression for the pressure will be, outside the shells, 2 "Ὁ a a p =4rP —-4rP’—, r r where 7 and γ΄ 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 s P 'q'2 Pa*® Σ 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 __ Pay CO = CC χης Path and its radius Pa’. Pa? Ὁ Ree a ae hae Pa‘? - Pa? a 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, ΟΝ 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 ὃ. ὃς &c. from the centre, and let their rates of production be 4aP,, 4aP, &c. : a a : Bilin Then if at distances ar &c. (measured in the same direction as b,b, &c. from the 1 % centre) we place negative sources whase rates are : a the pressure at the surface r= a will be reduced to zero. Now placing a source 4 ΜΗ 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= ὦ 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 4nP,, 42P, to be small electrified bodies, containing e, ¢ 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 be 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 2 a F : a . : os ξ and at that point we must place ἃ quantity=-e ᾿Ξ of imaginary electricity. 1 1 At the centre we must put a quantity EZ’ such that a E'=E+e,— +05 a +e, —+ &e.; ὃ, then if R be the distance from the centre, 7,7, &c. the distances from the electrified points, and 1’,r’, the distances from their images at any point outside the sphere, the potential at that point will be 70 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. This is the value of the potential outside the sphere. At the surface we have τ ὃ a ὃ a Rea and + = —, +=— &e. ee γι T] *1Ts so that at the surface θι 64 —+-— 4+ -Ξ ὧς, Pp ἀῶ vag 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 ool and 6, is infinitely distant along axis of «, 1 and ΕΞ. The value p outside the sphere becomes then ae pale(— 5), and inside p=0. 11. 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 w is d (m Φ -- (=) =-lm—. dx \r 7 The effect of the sphere in disturbing the lines of force may be swpposed 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=In. Let the sphere produce an additional potential, which for external points is p= we, and let the potential within the sphere be P= Bu. 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, | induction (not the intensity) within the sphere to that without. Phil. Mag. March, 1851. The inductive capacity of the sphere, It is therefore equal to ‘ Moll Eas ccording to our notation according to that paper, is the ratio of the guantity of magnetic ἜΡΙΣ a 1° R "eee ee : Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE. 7? surface, and that the induction through the surface should be the same whether deduced from the external or the internal potential. Putting «=r cos θ, we have for the external potential 8 p= (w+ 45) cos 0, and for the internal p= Br cos 6, and these must be identical when r = a, or Lf + A = iB: The induction through the surface in the external medium is 1 dp ki dr r=a = (I-24) c0s 6, and that through the interior surface is dp, 1 ya Je B cos@; and .". Η (I-24) = ; B. These equations give 1- κα 3k A= = cba k Be The effect outside the sphere is equal to that of a little magnet whose length is ἐ and moment ml, provided k—-k [= —_—__ a] . 2a+k εν Suppose this uniform field to be that due to terrestrial magnetism, then, if & is less than Κ΄ 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 zyx are represented by a, B,y;, 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 72 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 Δ᾽, y, and x, and their moments equal to 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 J, /,,, and strengths m, m, m, exist at the point 2yx with their axes parallel to the axes of wy x; then, resolving the forces on the three magnets in the direction of X, we have Substituting the values of the moments of the imaginary magnets , ! 43 The force impelling the sphere in the direction of w is therefore dependent on the variation of the square of the intensity or (αὐ + β᾽ + y*), as we move along the direction of #, and the same is true for y and x, 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 R24 8.2 @ 4.3 at R= k-¥)5 (Sy + waar e tee ate) a. ὃ : il a ἐξόν When τ small, the first term gives a sufficient approximation. The repulsion is then as 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. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 73 IV. Two Spheres in uniform field. Let two spheres of radius ὦ be connected together so that their centres are kept at a dis- tance b, 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 = Ircos0, and the potential due to the sphere is ; k-k a =e 4 - Κ' r οὐδ el hae: Par(i-2s id <, ) £08 6, ἀ ak+k τ 1 dp k-Kk a, dp - ----Ξ -- ---------.-- — -Ξ =0, τ αθ ΤῸ πεν =) ἴα θ, ἀφ dp|? 1 ah 1 A k-k αϑ k—-¥ |'a° & Ps eos pease pepe toes he oe -- 13}1 Rea oe, na 2 ἘΣ 2 : Te te ἘΠ dd as rsin’?@ dp εἰ + oe 7 (1 pack dey ary πα + eos] 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 πὶ os Vimy 1 Π aoe FT τ ss k—-K |* a8 k-k @ = 8. eal ma Poe soy, ΤΥ oe 9, 5 ΤΣ ΣῊ πὶ οὗ +k 5) anda: ᾽ This expression, which must be positive, since 6 is greater than a, gives the moment of ὴ when r = 6, 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 (%— Κ΄), 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 «. ἈΠ See Prof. Thomson in Phil. Mag. March, 1851. Vor, X. Paezr I. 10 74 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. The expression for the potential, the middle of the line joining the poles being the origin, is M 1 1 Ps (FS cos @cr 4 c?+ 7 +2 cos i)" From this we find as the value of 1", ΤΣ re - 18“ γ sin 26, and the moment to turn a pair of spheres (radius a, distance 2b) in the direction in which @ is increased is k-k Mab? ar ara co sin 20. 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 diamagnetic 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, x. . Let k,, ἴω» ks, be the coefficients of resistance in these three directions, and let ζ΄ be that of the external medium, and a the radius of the sphere. Let 7 be the undisturbed magnetic intensity of the field into which the sphere is introduced, and let its direction-cosines be J, m, n. Let us now take the case of a homogeneous sphere whose coefficient is k, placed in a uniform magnetic field whose intensity is 11 in the direction of 2 The resultant potential outside the sphere would be : k,-k αὐ = 1 —_—_—- — P u( toa a)” Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE. 75 and for internal points 3k, = 1] ——— a. Pi ah +k So that in the interior of the sphere the magnetization is entirely in the direction of «. 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 &; without disturbing this distribution of magnetism. We may therefore treat the sphere as homogeneous for each of the three components of J, but we must use a different coefficient for each. We find for external points πὰ ey Ke In-K Na ραν See nz) ; arnt ( = p ω my + Ns + 2k, + Ke a and for internal points oe ( 38k, a 3k, ss 3k, Pi= ΕΝ χ' 2k, + Kk Yy + 2hs+ i na). The external effect is the same as that which would have been produced if the small magnet whose moments are k,-k’ kp— κ ky— kK kes mg) πο Nee fates anak” re eae (Becdadets PET aa 3 2k, + Ie ia had been placed at the origin with their directions coinciding with the axes of a,y,x. The effect of the original force J in turning the sphere about the axis of # 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 s on the second magnet is iy hed 2k,+ he mni*a’*. 2 The whole couple about the axis of 2 is therefore 3k! (Key — ks) (le, + I’)(2k, +) tending to turn the sphere round from tlie axis of y towards that of =. Suppose the sphere to be suspended so that the axis of δ is vertical, and let J be horizontal, then if @ be the angle which the axis of y makes with the direction of 7, m= cos 0, n = — sin@, and the expression for the moment becomes 7": 8 K (k.- ks) (2h, +k’) (2h,+ k’) tending to increase 0. 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 & are nearly the same as in a vacuum, 10—2 I*a? sin 20 70 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. the coefficient of this quantity can be but little altered by changing the value of k’ tok, the value in space. The expression then becomes k,—k ne ἡ 3 733 sin 20, 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 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 were away, whether the substance of the shell be diamagnetic or paramagnetic. p= 81 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 3 »κ--- 1 5 cos 8, and the internal potential p,= — Ir cos 0. ’ Now let us investigate the effect of filling up the shell with some substance of which the resistance is ἔφ 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 Ja’, and that of the imaginary superficial distribution due to the medium k= Aa*, Then the potentials are 3 external p’= (I + A) = cos@, internal p, = (J + A) r cos θ. The distribution of real magnetism is the same before and after the introduction of the medium &, so that Gnd fo 1 4 τιν πετῶ εν(1:4), 5- ἢ Ὁ The external effect of the magnetized shell is increased or diminished according as & is greater or less than ζ΄. 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- in nature, we must admit that 7'=0 in all substances, with ferred to in Art. (28), we find, in the expression for the moment respect to magnetic induction. This argument does not hold of the magnetic forces, a constant term depending on 7’, besides those terms which depend on sines and cosines of 0. The result is, that in every complete revolution in the negative direction round the axis of 7', a certain positive amount of work is gained; but, since no inexhaustible source of work can exist 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. Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE. 77 VIII. Llectro-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 ¢, and let its external effect be that of a magnet whose moment is Ja*. 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, αϑ p' = 1-- οο5θ, p, = Ar cos 0, r : ἄρ’ ἃ and since there is no permanent magnetism, — = =, when r = a, Α - -- 4]. If we draw any closed curve cutting the shell at the equator, and at some other point for which @ is known, then the total magnetic intensity round this curve will be 3Ia cos @, 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 ¢d@ is — 87. sin θάθ, so that the quantity of the current referred to unit of area of section is a ; - 31: sin 8. 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 0, 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 7, be wound round a sphere of radius a 6 fe Ξ Fi . 24 so that the distance between successive coils measured along the axis of w is —, then n there will be m coils altogether, and the value of J, for the resulting electro-magnet will be n = Gal” The potentials, external and internal, will be ; n a? nr p= τς τε 088, Pp, =~ 21, = = cos 8. 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 3k α n 88 Ξ,,- = cos 8, =— 21,— ——. — ΤΕΥ, di The external effect is greater or less than before, according as # is greater or less than &, 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 & 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 & 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 a = 0, By = 8, Yo= - ὧν. where ὦ is some function of γ. Where there are no electric currents, we must have dy», ὃ» cs each = 0, and this implies d (s 73: me dec fey” the solution of which is Ὁ a= Ο᾽ + τ . Within the shell ὦ cannot become infinite; therefore w = C, is the solution, and outside α must vanish at an infinite distance, so that Cs o=— "3 is the solution outside. The magnetic quantity within the shell is found by last article to be n 8 dB, dry -21,— ——;=+a4= 26a ak+k dr ἀν therefore within the sphere H=- — Outside the sphere we must determine w so as to coincide at the surface with the internal value. The external value is therefore "ga Ske τῇ where the shell containing the currents is made up of m coils of wire, conducting a current of total quantity J,. Let another wire be coiled round the shell according to the same law, and let the total number of coils be m’; then the total electro-tonic intensity EZ, round the second coil is found by integrating 2π El,= f wa sin Ads, Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 79 along the whole length of the wire. The equation of the wire is φ cos 9 = ——, n'a where 7 is a large number; and therefore ds = a sin 6ddq, = — an'x sin*6d0, 4π 2π 1 “ς. Ε1,π---- wan =~ —ann [——. ae Wes 8 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 mn” for nm’. 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 1 and J’. Let NW stand for pono , then the electro-motive force of the first wire on the second is 8 (8k+k) dI Ud — Nnn ae That of the second on itself is ar’ - Nn” —., dt The equation of the current in the second wire is therefore ,al i) amma — Nnn a ae ace Ee Peeves ὙΠ ΞΟ ἢ The equation of the current in the first wire is dI al’ = 3 ΤΥ ΑΞ: = eeeeeeveseeoe ni Ἢ Nnn ae Fe RI (2) Eliminating the differential coefficients, we get pay gti ΤΣ ον n n n n> mn’ αἱ F n® dF dN (G4 E) ἘΞ Gt Go τ RR) at Rte a ἰὼ from which to find Zand I’. For this purpose we require to know the value of F in terms oft. Let us first take the case in which F is constant and 7 and J’ initially = 0. This is the case of an electro-magnetic coil-machine at the moment when the connexion is made with the galvanic trough. Ρ ἂ 80 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. n2 12 Putting 37 for N (7 +7) we find - F # T= (1-67) , n’ -Ξ ΤΑ," . 9 . . F The primary current increases very rapidly from O to R and the secondary commences at , F wes ~ and speedily vanishes, owing to the value of + 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 ἐν P Rat. 0 The total quantity of current is o f Tat. 0 For the secondary current we find τἄρα τ΄ Fr <2 PRdit= —— — PAB koa cet τὰ Rit=T > [ ἄ! τ τς The work done and the quantity of the current are therefore the same as-if a current 7, of quantity 7’ = Rs had passed through the wire for a time 7, where n? n'2 roan (E+) 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 J, and in the secondary =0. Then we shall have for the subsequent time at 1 at I=I,e *, a cs The equivalent currents are 1 J, and ἃ ἢ ἘΞ — , and their duration is τ᾿ 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 7+, 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, ΟΝ 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 [1., and references are there given to previous experiments. Let the axis of # be the axis of revolution, and let the angular velocity be w. Let the magnetism of the field be represented in quantity by J, inclined at an angle @ to the direction of x, in the plane of sa. Let R be the radius of the spherical shell, and 7 the thickness. Let the quantities αν» Bo Yn be the electro-tonic functions at any point of space; a@,, ὃ,» 6)» a, βι» γι symbols of magnetic quantity and intensity; a, be, 62» ag, Bo, ‘2 of electric quantity and intensity. Let p, be the electric tension at any point, dp. ) ag = a + kay d By = ay tHe Vics atone edd etl) dps =— +ke Ὕ2 dz 2 } da, db, de, nica ee de er τς (2)3 ty, Bs ty ‘de dy dz i The expressions for ap, 39, yo due to the magnetism of the field are ἢ ἡ ay = Ay + 5 y cos 0, yaaa στὸ By = By +5 (x sin 9 — x cos 8), tae As C, --ἰς ysin 0, A,, B,, C, being constants; and the velocities of the particles of the revolving sphere are da _ ἂν τι dz ᾿ Ἢ" ci aa ἀρ We have therefore for the electro-motive forces 1 day 1 err etry ΞΊΣ an Oe, 1 dB, 1 ae rie ree Pe | 1 dy, a a ae emir pe τς 7m Coe Vor. X. Parr J, 11 82 Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE. Returning to equations (1), we get ἐ{15- Ξὴ _ 4β,, ty _ dz dy ds dy ’ de, da,\ dy, da, 1 1 (3 - 2)- pon ae gee hey g (Sn) oe a, dy da} dy ds From which with equation (2) we find 1 a las eral sin Owz, bale Uy oa : ΑΝ θωω 2K 4 4 : Po= sia Iw { (x* + y*) cos @ — a sin 0}. 167 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 3 Ἵ. cute! ot τος ς of a small magnet whose moment is ἘΝῚ wI sin θ, with its direction along the axis of y, 7 so that the magnetism of the field would tend to turn it back to the axis of «*. 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 ay inclined to the axis of # at an angle @ and producing an external effect equal to that of a magnet whose moment is J’ R’. The magnetic inductive components within the shell are 10 sin -- 21’ cos in a, - 21 sind in y, I, cos @ . in x. 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, (ὦ, sin 6 -- 2I’ cos Φ) in # produces 2 =e (J, sin @ - 2I’ cos φ) in y, 7 Ἐπ : SO sss (- 27’ sin φ) in y produces 2 ak” (2I’ sin φ) ina; 1 cos θ in produces no currents, * The expression for pz 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, ΟΝ FARADAY’S LINES OF FORCE. 83 We must therefore have the following equations, since the state of the shell is the same at ΄ every instant, : ἢ ἢ Τ' : I, sin 6 -- 21΄ cos φ = ἢ 5ἰὰθ + κατ CF sing - af’ sing = = w(I, sin 6 — 2I' cos φ), π. whence Ἵ TR Q4are ; cot p= - w, =} = I, sin 0 24k T ᾿ γι + ——_w 24k ᾿ 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 J be the total intensity of the terrestrial magnetism, and let the dip be θ, then I cos @ 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 = (απ πὶ 24 currents is the same as that of a magnet whose moment is Tw = 5] cos 0. 1 2 / 24k P+ Tut The moment of the couple due to terrestrial magnetism tending to stop the rotation is 2Q4ark Tw ao to the south of magnetic west, and the external effect of these RF cos? 0 2 dark ἢ + Jo" al and the loss of work due to this in unit of time is 2 + ad sie 1515 cos? 0. 2 24ark | + T*o* 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 Μ, Foucault, (see Comptes Rendus, xu. p. 450). 11—2 IV. The Structure of the Athenian Trireme ; considered with reference to certain difficulties of interpretation. By J. W. Donaupson, D.D. late Fellow of Trinity College, Cambridge. [Read November 6, 1856.] Tuer 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 Pirseus, 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 s.c, The elementary form, of which it Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN ΤΕΙΒΕΜΕ. 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 Phoceans, who navigated the Archipelago, the Adriatic, and the western Mediterranean as far as ‘T'artessus, used for this purpose ov στρογγύλῃσι νηυσί, ἀλλὰ πεντηκοντόροισι (1. 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. tv. 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 (Jl. 11. 227). Whether the ships of the Beeotians, to which Homer gives a complement of 120 men (Ji. 11. 16), were biremes, or large penteconters, with double crews, is a point which can hardly be decided; Pliny mentions (H. N. vit. 57), on the authority of Damastes, a contemporary of Herodotus, that the Erythrzans 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 (1.10) speaks of the περίνεῳ 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 ἐπίβαται or soldiers on board. Now in the nautical inscriptions published by Béckh, we have a particular class of oars called by this name, αἱ περίνεῳ κῶπαι, and it is probable that these were intended to be used by the synonymous ἐπέβαται 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 θρανῖται; or bench- rowers, in the highest tier, 54 ζυγῖται or cross-bit-rowers, on the second tier, and the same number of @adayirat, or main-hold-rowers, on the lowest tier. Unless then some of the thranites were employed to work the two great oars, or πηδάλια, 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 860 Dr 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. 1. 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 epibate, of four pentekostyes, which was the Lacedemonian arrangement at the first battle of Mantineia (Thuc. v. 68), or it was two locht of 100 men each, if we prefer Xenophon’s subdivision (Rep. Lac. τι. 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. 111. 572 on Thucyd. rv. 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 Béckh, 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 transverse section of a trireme. (Voyage and Shipwreck of St Paul, p. 194.) a. Oar of thalamite seated on deck. ὃ. Oar of zygite seated on stool on deck. 6. 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 conceive that the best rowers should have been placed on a platform within reach of the enemies’ shot. Dr 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 Zygite. There is a very primitive description of the structure of a Greek ship in the Odyssey v. 248 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 ἴκρια 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 κατάστρωμα of a war-ship. As the main-yard is termed the ἐπίκριον in this passage, and as the Christian cross was designated as an ἴκριον, we may conclude that the word implied a transverse or cross direction of these timbers; the root is probably that of ἱκό-μην, and therefore, as we shall see, the word is synonymous with σέλμα. These cross-bits are called κληῖδες in Homer, because, like the collar-bone, they locked together the two sides of the ship. The poets call them σέλματα, a word containing the old root sel or sal, “to go” (New Crat. ᾧ 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 ζυγά, “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, ἴκρια would be TRANSVERSE SECTION OF AN ATHENIAN TRIREME. 3 Cc 6 A oe || |e ΤΙ 18 d ry 12 I | {D Thranus, or long stool, placed on the alternate zygon, or supported by the selis, and extending 7 feet amidships. Zygon, or cross-plank, running athwart the vessel at intervals. Thalamitic seat, 4ft. 6in. Thailamos, or hold leading to anilos. Platform for Epibate running along the ¢raphex, and 6 feet wide, with bulwarks of 3 feet. Selis, or gangway, fore-and-aft, 4 feet wide. —B, (Breadth of beam) =18 feet. C—D, (Depth) =12 feet. fe ee SRS SR 88 Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. an appropriate designation. In larger vessels, however, these ἴκρια would be remanded to the decks fore and aft, the cross-pieces would be separate «Aides or ζυγά, 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 ζυγά would be σέλματα, 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 ζυγά in a Greek trireme, as the middle tier of rowers were called ζυγῖται, because they sat there (Jul. Poll. τ. 87: τὰ μέσα τῆς νεὼς ζυγά, οὗ οἱ ζυγῖται κάθηνται), 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 transtra or cross-planks of the vessel. Béckh is led to the opposite conclusion by the phrase ἕδρας κώπης ζυγίας in one of his Inscriptions (11. 40, p. 286). But this merely means that the trireme in question had one of the ζυγά broken close to the oar- hole, just as the same vessel is stated to have been defective in its τράφηξ or bulwark. And in a subsequent part of the same inscription (p. 291) we have the phrase τῶν ζυγῶν κεπώ- anvra πέντε, “only five of the cross-bits are supplied with oars,” which implies that the ζυγά were the proper place for one class of the rowers. 11. The Thalamite. That the θαλαμῖται got their name from having their seats in the θάλαμος (Jul. Poll. 1. 87: θάλαμος ov οἱ θαλάμιοι ἐρέττουσι), 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, θάλαμος, like θόλος, did not imply specifically a vaulted chamber. If so, the hold, sloping inwards to the keel, would represent an inverted θάλαμος, just as the bees’ cells were called by this name (Anth. Pal. 1x. 404, 2): ἄπλαστοι χειρῶν αὐτοπαγεῖς θαλάμαι, 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 obba, which properly meant a drinking-vessel with a sharp 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 θαλαμῖται 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 θάλαμοι. At least, in the curious story told by Timeus (ap. Athen. p. 37) of the young men at Agrigentum who fancied that their house was a trireme at sea, one of .᾿ μ A Ὡς ἐϑ ‘ , «. ἢν ᾿ , them says ὑπὸ τοῦ δέους καταβαλὼν ἐμαυτὸν ὑπὸ τοὺς θαλάμους; ws ἔνι μάλιστα κατωτάτω Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 89 ἐκείμην, “having flung myself, in my fear, under the θάλαμοι, I lay as low down as possible.” The bottom of the hold, however, was also called the ἄντλος, a name given afterwards to the bilge-water which settled there, and to the pump, by which it was bailed out. III. The Thranite. An examination of the name of the θρανῖται or “benchmen” of the highest tier, leads to some very interesting results. The whole of this tier was called the @pavos, because the rowers were seated on benches, which did not reach across the vessel, but rested by means of short legs on the ζυγά beneath, so as to resemble a θρῆνυς or foot-stool. It has been supposed that θρῆνυς and Opavos are other forms of θρόνος, but this seems very unlikely. It would be more reasonable to connect θρόνος with the root orop-, and to understand an original form στρόνος, but to recognize in Opavos or θρῆνυς the root of Opavw; for the idea conveyed by the latter is that of a fragment or separate piece, the θρόνος being the seat with its cushion, and the θρῆνυς the detached ὑποπόδιον. And this view is not affected by the consideration that the θρῆνυς in a trireme was really a seat and not a foot-stool. It could only have been high enough to enable the θρανίτης to use the ζυγόν immediately before him as a stretcher, and to carry the handle of his oar clear of the ζυγέτης 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 θρῆνυς was seven feet long, even in Homer’s time. It was therefore just like a low foot- stool placed on the ζυγόν. Why it was so constructed may easily be shown. If the θρῆνυς had run quite across the ship, the ζυγῖται and θαλαμῖται 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 ζυγά, 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 ἴκρια were confined to the two ends of the vessel, and in going amidship it was necessary to step down, first to a θρῆνυς and then to the ζυγά. 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 (ἀνεχάζετο τυτθόν) i.e. so as merely to get out of immediate danger, to a bench seven feet long (θρῆνυν ἐφ᾽ ἑπταπόδην), and “he left the deck of the equal ship” (Aime δ᾽ ἴκρια νῆος ἐΐσης); in this lower position he stood watching, and repulsing with his long pole any Trojan who en- deavoured to set fire to the vessels (Jl. xv. 674-731). That the θρῆνυς 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 Vou. X. Parr 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 ζυγά 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 Pirzeus, 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 epibate, 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 Epibate 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. vu. 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. 11. 93) even those of the thranite 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. 1 have no doubt that the thranitic oars were longer than this, and the epithet δολιχήρετμος which Pindar applies to AUgina (Ol. vi11. 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. 11. xxi. 155: δολιχεγκής» 111. 346, &e.: δολιχόσκιον Ery«os). 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 ¢hranitic oars and those of the lower tiers is implied by what Thu- cydides says (vi. 31), as illustrated by the Scholiast: οἱ δὲ θρανῖται μετὰ μακροτέρων κωπῶν ἐρέττοντες πλείονα κόπον ἔχουσι τῶν ἄλλων" διὰ τοῦτο τούτοις μόνοις ἐπιδόσεις ἐποιοῦντο οἱ τριηράρχαι οὐχὶ δὲ πᾶσι τοῖς ἐρέταις. 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 ¢hranus 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 usw partiwm corporis humani, I. 24, Vol. 111. p.85, Kuhn): καθάπερ οἶμαι κἀν ταῖς τριήρεσι τὰ πείρατα τῶν κωπῶν εἰς ἴσον ἐξικνεῖται καί τοι γ᾽ οὖν οὐκ ἰσῶν ἁπασῶν οὐσῶν, καὶ yap οὖν κἀκεῖ τὰς μέσας μεγίστας. Aristotle makes.a similar comparison (de partibus animalium, tv, 10, ᾧ 27: ὁ μέσος [δάκτυλος] μακρός, ὥσπερ κώπη μεσόνεως) ; and he enters more fully into the subject in his Mechanica, c. 4, where he answers the question: διὰ τί οἱ μεσόνεοι μάλιστα τὴν ναῦν κινοῦσιν ; 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: ἐν μέσῃ δὲ τῇ νηὶ πλεῖστον τῆς κώπης ἐντός ἐστιν᾽ Kal γὰρ ἡ ναῦς, ταύτῃ εὐρυτάτη ἐστίν, ὥστε πλεῖον ἐπ᾿ ἀμφότερα ἐνδέχεσθαι μέρος τῆς κώπης ἑκατέρου Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 91 τοίχου ἐντὸς εἶναι τῆς vews,—and at the end he adds: διὰ τοῦτο οἱ μεσόνεοι μάλιστα κινοῦσιν" μέγιστον γὰρ ἐν μέσῃ vnt τὸ ἀπὸ τοῦ σκαλμοῦ τῆς κώπης τὸ ἐντός ἐστιν. To ἃ 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 xygite sat between the thranite and thalamite immediately next to him, and the words of Pollux (above, p. 7) that the ζυγά were τὰ μέσα τῆς νεώς (i.e. considering the three tiers as hori- zontal lines), we owe the perplexing theory, first started, I believe, by Schneider in his Lexicon, 5. V. μεσόνεοι; that the sygites, 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 (Rane 1074), and implied in the fact that they were left on board when the rest of the crew disembarked to serve on shore (Thucyd. rv. 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 (κωπέων πλατουμένων), he speaks of fitting the lowest oars with thongs (θαλαμιῶν τροπουμένων; Acharn, 552, 558). The interval between two oar-ports on the same tier was two cubits (Vitruv. 1. 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 epibate. 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 (1. 87): τὸ δὲ περὶ τὸ κατάστρωμα Opavos, ov ot θρανῖται, for the only κατάστρωμα was the gangway. I will now apply these considerations to the removal of some difficulties which have been very troublesome to editors. (a) The conjecture that the interval between the ends of the upper benches or thranos was intended to leave a passage along the σέλματα or ζυγά is supported by the fact that the special name for this passage was σελίς, a name also given to the spaces between the benches in the theatre. Hesychius defines the cedidas as τὰ μεταξὺ διαφράγματα τῶν διαστημάτων τῆς νεώς, * 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 σελίς with σέλμα. In later times σελίς was commonly used to denote the blank space between two columns in a written page. When Phrynichus says (Bekk. Anecd. 62, 27): σελὶς βιβλίου" λέγεται δὲ καὶ σελὶς θεάτρου, 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 κερκέδες of the theatre, which were divided by the σελίδες, and the lines of writing divided by the inter- vening space of blank paper; and the corridors of the theatre again were called σελέδες; because they were flanked on each side by seated spectators, just as the σελίδες in the trireme 122—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 (Eqwites 546), which has been found unintelligible: αἴρεσθ᾽ αὐτῷ πολὺ τὸ ῥόθιον, παραπέμψατ᾽ ἐφ᾽ ἕνδεκα κώπαις θόρυβον χρηστὸν ληναΐτην--- ‘¢ raise for him a plash of applause in good measure, and waft him a noble Lenzan 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. (Ὁ) As the σελὶς 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 Aischylus, which all the commentators have failed to eluci- date. In the course of the altercations between A.gisthus and the chorus at the end of the Agamemnon, the usurper is made to address the senators as follows (v. 1588): ov ταῦτα φωνεῖς νερτέρᾳ προσήμενος κώπη: κρατούντων τῶν ἐπὶ ζυγῷ δορός: ςς ΤΉρβο 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 ζυγῖται should be described as the κρατοῦντες instead of the θρανῖται. Dr Blomfield went so far, in his struggle to get out of the difficulty, as to suppose that the old men of the Chorus were the θαλάμιοι, Agisthus and Clyteemnestra the ζυγῖται, and the murdered Agamemnon the Opavirys! 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 Aischylus 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 Opavira!! The fact is that all these commentators have overlooked a refinement of Greek Syntax. Aischylus, who was as well acquainted with sea-life as any of the men that pulled at Salamis, has been careful to introduce the participle προσήμενος in speaking of the rower, while by writing ἐπὶ ζυγῷ instead of ἐπὶ ζυγῶν, he expressly tells us that the κρατοῦντες were not seated on the ζυγά, 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 ἐπὶ with the genitive; but ἐπὶ 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 ζυγά, and were much higher than the stools of the θρανῖται, so that even when seated, the κρατοῦντες; or officers, might speak of the rowers of the highest tier as veptépa προσημένους κώπῃ. Their seats then being placed on the ζυγά, they might be said either καθῆσθαι or ἑστηκέναι ἐπὶ ζυγοῖς, because their feet rested on them; but the (uyira: could only be said καθῆσθαι ἐπὶ ζυγῶν. Hence we have in Eurip. Phoniss. 74: ἐπεὶ δ᾽ ἐπὶ ζυγοῖς καθέζετ᾽ ἀρχῆς», and Eustathius tells us that the Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 99 Homeric epithet ὑψίζυγος is derived from the high seat of the pilot in a ship (p. 131, 18): καὶ τοῦτο δὲ ἀπὸ κυβερνητικῆς μετήνεκται καταστάσεως. For the same reason ZEschy- lus speaks of the Gods as σέλμα σεμνὸν ἡμένων (Agam. 176). (c) Another difficult passage in the same play furnishes an illustration of the fact that the middle part of the σέλματα or ζυγά, in an old Greek vessel, belonged to the officers and supernumeraries. In y. 1413 it is said of Cassandra, who came with Agamemnon from Troy, that she was ναυτίλων σελμάτων ἱστοτρίβης, where some read ἰσοτριβής. The allusion to Chryseis a line or two before makes it probable that Auschylus had in his recollection the lines in the Iliad, where Agamemnon says that old age shall find her: ἱστὸν ἐποιχομένῆν καὶ ἐμὸν λέχος ἀντιόωσαν. Here it is implied that the σέλματα were her only gynewceum, just as Persius says (v. 146): ‘tun’ mare transsilies? tibi torta cannabe fulto, coena sit im transtro 7” Or if ἱστός 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. 1. 87: ἐκεῖ που καὶ σκήνη ὀνομάζεται τὸ πηγνύμενον στρατηγῷ 7 τριηράρχῳ. And schylus 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 ὁ. Theb. 2, 3: ὅστις φυλάσσει πρᾶγος ἐν πρύμνῃ πόλεως οἴακα νωμῶν, βλέφαρα μὴ κοιμῶν ὕπνῳ). (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 sedis 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” (εἰς ἄντλον ἐμβῆσαι πόδα, Eurip. Hercul. 168). 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 Atschylus says (Choeph. 695) : ἔξω κομίζων ὀλεθρίου πηλοῦ πόδα, 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 ζυγά, 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 which 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. Gd. Col. 720 sqq.). V. Of the Platonic Theory of Ideas. By W. Wurwe1t, D.D. Master of Trinity College, Cambridge. [Read November 10, 1856.] Tuovcn 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 Lectwres 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 least 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 formule, his account is as follows: [Vol. 11. p. 117.] “ς 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. «ς 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 himn— are then by the causative act of creation deposited in things, differencing them one from the other, so that the things partake of them (μετέχουσι), communicate with them (κοινωνοῦσι). “ 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.” 06 Dr 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 Phedo he refers to them, as illustrating the nature of the human mind: in the Republic and the Timeus he again speaks of truths which far transcend anything which the senses can teach, or even adequately exemplify. The senses, he argues in the Theetetus, cannot give us the knowledge which we have; the source of it must therefore be in the mind itself; in the Jdeas 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 Sensei? 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, “1 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 Vou. X. Parr I. 13 98 Dr 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. «ς Tell me,” says he (ὃ 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 ;-—of 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, «ὁ 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, (ἢ 11) “ἴῃ making the same thing be in * Prof. Butler, Lect. ix. Second Series, p. 136, appears to | for the assumption of such Ideas; but I see no trace of think that Plato had sufficient grounds (of a theological kind) |, them, Dr 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 (ᾧ 12): ‘In what way, Socrates, can things participate in Ideas, if they cannot do so either integrally or partibly?” “(ΒΥ 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 Greatuess, 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. (ᾧ 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. 261, 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. 10 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, Partnenides: 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? (ᾧ 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 Jam 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 soif 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 of 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: “Ὁ 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 | be perceptibly weaker, if the common interpretation were the habit of the Greek language. It slightly leans in favour | 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. All 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 Formule, 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 Clazomenx, 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 Dr 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 T’heatetus, Socrates is represented as saying that he when very young had seen Parmenides who was very old}. Athensus, 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. Athenzeus 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 mine to Zeno for his instructions (119 a). tT p. 183 e. t Deipn. x1. c. 15, p. 105. VI. On the Discontinuity of Arbitrary Constants which appear in Divergent Developments. By G. G. Sroxes, M.A., D.C.L., 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 ty “cos = (w% — mw) dw in a form which admits of extremely easy 0 . 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 * Camb, Phil. Trans. Vol. 1X. Part 11. Vou, Ἂς Parr: 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 ΓῚ : 2a 2a)" 2a)? u=2f Sree OO ou seep 0 1 2.3 8.4.5 OF ARBITRARY CONSTANTS, &c. 107 The integral and the series are both convergent for all values of a, and either of them completely defines « for all values real or imaginary of a. We easily find from either the integral or the series d Gi, Ὁ 2au=2 seseiseiseessosicinse ses sesieesee sacs seerven (2) This equation gives, if we observe that u = 0 when a = 0, Wa ayy es ae a a a’ u = 96 nf e* da = 26 fee SG tsstieaate ΡΟ (3) This integral or series like the former gives a determinate and unique value to w for any assigned value of @ 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 uv in a form convenient for calculation when a is large, let us seek to express τὸ 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 2aw and 2, and the leading term of the development is a~'. Assuming a series with arbitrary indices and coefficients, and deter- mining them so as to satisfy the equation, we readily find 1 7 1.3 oe eee a 20 "αὐ 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 = 2 0 --- au ΞΞ in 3 whence we get for the complete integral of (2) 1 1 1.3 1.8.5 Wm Cen 8 oe — oh th oF cect ces eneeereee (4 α΄ 2a°* Ba * gal @) 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 (1) or (3) shews that τὸ is an odd function of a, changing sign with a. But according to (4) τὸ 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@ + of - 1 sin 0). 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 0, which must be increased or diminished (suppose increased) by z. Hence, if we knew what C was for a range π᾿ of 0, suppose from θ = a to 0 =a+7, we should know at once what it was from 0 =a +a to 0=a+ 27, 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, by adding, if need be, a positive or negative multiple of 27, which as appears from (1) or (3) makes no difference in the value of w. 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 farbut 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...(2¢ — 1) gigti+l a the value of τὸ so obtained will satisfy exactly, not (2), but the differential equation 1.8... (21 -- 1) du aa hee en aiqhit® ogee Seep bie (5) Let τρ be the true value of τὸ 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 τὸ ought to satisfy (2), we ought to have U = Up + 2e* ["e"'da, whereas since our approximate expression for τὸ actually satisfies (5) we actually have, putting A; for the last term, τ τε Uy + “πο ["(@-A)) OO ii ὍΡΟΝ (6) ao 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 4; will be very small. Hence in general 4; may be neglected in comparison with (2), and we may use the expression (4), though we stop after ὁ + 1 terms of the series, as a near approximation to w. 5. But to this there is an important restriction, to understand which more readily it will be convenient to suppose the integration from a, to a performed, first by putting da = (cos @ + \/— 1sin θ) dp, and integrating from p, to p, θ᾽ remaining equal to @,, and then da = p(-sin@ + 4/- 1 cos 6) dé, and integrating from @, to θ, p remaining unchanged. This is allowable, since u is a finite, con- tinuous, and determinate function of a, and therefore the mode in which p and @ vary when a passes from its initial value a, to its final value a is a matter of indifference. The modulus of e® will depend on the real part p* cos 20 of the index. Now should cos 20 become a maximum within the limits of integration, we can no longer neglect A; in the integration, For however great may be the value previously assigned to i, the quantity p~*-'e” °° will become, for values of θ 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~*1, yet as the mutual destruction of positive and negative parts may take place quite differently in the two integrals /2e“da and {A,e“da, we can conclude nothing as to. their relative importance. 6. Now cos26@ 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 0 to x or π᾿ to 27, 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 0<@ 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 —(vi¢0V=1 inclusive. Omitting the common factor e » which may be restored in the end, we have for the rest of the series -20V-1 40V=1 pite Mggit e- Mite t eee If we denote by D or 1+ A the operation of passing from μὲ to u;,,, and separate symbols of operation, this becomes (+ ePVY I D+ enV -1 + 2.) Mis or {1 ΞΕ A) 4: ἭΝ ταὶ παρ, : : ξ- 9) ΜΞ: Now Re Co ee ΤΕ ee, ) Ν which reduces the expression to . (2 sin @)~*e Cag - (2 sin@)-te- (νη) γι, or, putting q for (2 sin@)~’, to - oy 8π 1 es gee YT 4g ote-tV-1Ay,4 ge (ET at, Now if p be very large, and μ; belong to the part of the series where the moduli of con- secutive terms are nearly equal, the successive differences Ay; A*n,,... will decrease with great rapidity, Hence if @ have any given value different from zero or a multiple of 7, by taking OF ARBITRARY CONSTANTS, &c. 111 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 @=0 or =z, the above transform- ation fails, since g becomes infinite. In this case if we want to calculate τὲ 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 =u, is made to depend on fu,dw 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 J; 21... from the logarithms of jj;41, mi+2-.., where / is nearly equal to the mean increment of the logarithms at that part of the series, to associate the factor f whose logarithm is ὦ with the symbol D, and take the differences of the numbers, Bis S Mere. fm 42, ὅς, 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 /— 1a for a in (4) after having put for τὸ its first expression in (3), we have 2d" [eda = aah 1 Ce” ἐν τ τ se Ὁ Sas ὁ whenee, putting a=, we have 6 - νι. --᾿ πὸ. Hence we get for the general expression for C in (4), σ- --ιπὸ, when0<0 and τ- It is worthy of remark that in this expression the transcendental quantity πὸ appears as a true radical, admitting of the double sign. Two cases of the integral τῇ “eda occur in actual investigations, namely when θ-Ξ, 0 when the integral leads to f ‘e~"dt, which occurs in the theory of probabilities, and when 0 -Ξ, when it leads to Fresnel’s integrals { * cos (=) ds and J * sin (F) 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 θ + \/=1 sin 6), where @ is a small positive quantity, and after equating the real parts of both sides of the equation make @ vanish, we find, whichever sign be taken, 1 1 ine 1.3.5 ΝΑ 4 + cosvee 207 (Pd ......... (18) 0 pap ae Sa The expression which appears on the second side of this equation may be regarded as a singular value of the sum of the series aig’ t gigi Ὁ cesceeseeenseesees (14) a series which when @ 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 8, 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 χω) be a finite function of w, the value of which changes abruptly from ὦ to ὅ as # increases through the value c, a quantity lying between 0 and π, and f(«) be expanded between the limits 0 and π᾿ in a series of sines of multiples of w, and if @ (m, x) be the sum of m terms of the series, the value of ᾧ (m, x) for an infinitely large value of n and a value of # infinitely near to c is indeterminate, like that of the fraction (w@+y)+a-y (wy tarty’ 0 which takes the form “4 when # and y vanish, but of which the limiting value is wholly indeterminate if w and y are independent. We may enquire, if we please, what is the limit of the fraction when @ first vanishes and then y, or the limit when y first vanishes and then a, for each of these has a perfectly clear and determinate signification. In the former case we have, calling the fraction Ψ (a, y), OF ARBITRARY CONSTANTS, &c. 113 lim. 9 lim. Ψ (ὦ, y) = Lith νοῦ τξ y in the latter a+ Φ a +n = 1, lim, ,-o lim. 9 Ψ (ὦ, y) = lim. ,-5 So in the case of the periodic series if we denote by & a small positive quantity lim. ε. ο lim.,_.. Φ (σι, ὁ — &) = lim. f (ὁ -- ξ) = α, Ἰΐπι. ε. lim.,,_,, p(n, ¢ + ξ) = 1ἴπη..,.007 (6 + ξ) = ὃ; but we know that lim. lim.z_ Φ (m, ὁ © ξ) = lim.,_., Φ (m, ¢) = 4 (a + δ). Similarly in the case of the series (14) if we denote its sum by χ (a) = τσ (ρ» 8), and use the term limit in an extended sense, so as to understand by lim.,_,, F (p) a function of p to which F(p) may be regarded as equal when p is large enough, and if we suppose 6 to be a small positive quantity, we have from (11) lim.g_y lim.,-.. @ (p, 0) = lim.y- fee"* (“eda -/ -1 πὲ} 0 = 805 ' ["e* dp -/f/-1 mie"; 0 Himm.gag lime (py -- 0) = lity ἔϑο τ᾽ [“e*da + ν΄ --ἰ whe} 0 = ae [ edo + ν΄ -ἰ ae, 0 whereas equation (18) may be expressed by Litn.pam limp-g τ (p, +0) = lim,,.x(p) = 207" ["e*dp. 0 There is however this difference between-the two cases, that in the case of the periodic series the series whose general term is Ad (nm, 6) 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 sum 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 Vot, X, Parr I, 15 114 PROFESSOR STOKES, ON THE DISCONTINUITY } then equation (18) 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 established, let us consider the function defined by the equation 7 1.1.8 τι 1 @—-— ὡϑ + +2 2.4 9.4.0 πα ποι πο δ Suppose that we have to deal with such values only of the imaginary variable ἃ 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 « Now we happen to know that the series is the development of (1 + #)*. But this function adinits of one or other of the following developments according to descending powers of # :— 1 ΤΥ ἀρ ess = a £73 — ot ot ἜΗΝ τὴν ἜΣΤΩ +246 ΠΝ tee ition y «nek “pny ED 1 1.1.3 2.4.6 ποτ ἢ Let Φ = p (cos 6 + ν΄ —1 sin @), and let az? denote that square root of # which has 46 for its amplitude. Although the series (16), (17) are divergent when p < 1, they may in general, for a given value of θ, be employed in actual numerical calculation, by subjecting them to the transformation οὔ" Art. 8, provided p do not differ too much from 1. The greater be the accuracy required, @ 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 @ lie between (2i-1)4+a and (2i1+1)*-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 (16) or (17) may by the method of Art. 8 be converted into another, which is at first sufficiently convergent to give w with a sufficient degree of accuracy by employing a finite number only of terms, If m terms be stimmed 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 m +n +1. And jin this ‘case we cannot pass from one to the other of the two series (16), (17) without rendering ον discontinuous. But when @ passes through ah odd multiple of + we may have to pass from one of the two series to the other. Now when @ is increased by ὅπ ‘the series (16) or (17) changes sign, whereas (15) remains unchanged. Therefore in calculating ὃν for two valites of Θ᾽ differing by 27 ‘we must employ the two Seriés (16) ‘and (17), one in each ‘ease. Hence we nist employ one ‘of the series from'0= -- @ 10.0.5 7, the other from Ὁ = 7 to 8 37, and so ‘on; and therefore if we knew which series to take for some one value of ἃ everything would be determined. Now Wwhén ‘p+ 1 the ‘sériés (16) bécomés identical with (16) when 6 ‘thas ‘the particular value 0. Hence (16) ‘and not (17) ‘gives the true value of’ When ~'r < 0'< &. OF ARBITRARY CONSTANTS, &c. 115 13. Let p, @ 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 ὦ = —1, that is, by p= 1, θ-π. To each value of w 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 uw. If P, be any point in the plane, either within the circle or not, and a moveable point P start from P,, and after making any circuit, without passing through S, return to Py again, the function (1 + a)} will regain its primitive value u,, or else become equal to —u,, according as the circuit excludes or includes the point 8, which for the present purpose may be called a singular point. Suppose that we wished to tabulate τώ, using when possible the divergent series (16) in place of the convergent series (15). For a given value of θ, 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 (16). Let OP be the smallest value of p for which the series (16) may be employed ; for which, suppose, it will give τὸ correctly to a certain number of decimal places. The length OP will depend upon @, and the locus of P will be some curve, symmetrical with respect to the diameter through S. As @ 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 (16), but we cannot, keeping within this space, make @ pass through the value 7. The series (16), (17) are convergent, and their sums vary continuously with ἃ, when p>1; and if we employed the same series (16) for the calculation of τὸ for values of # having amplitudes a — β, «+3, corresponding to points P, P’, we should get for the value of τὸ at P’ that into which the value of τὸ 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 τὸ 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 w having moduli less than unity, nor would there be any advantage in using the divergent series (16) 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 9 =0 and θ ='z. 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, — 3a for n » uU τα Ὁ ΤΥ ΎΎΎΎΎΎΣ (18) 15—2 116 PROFESSOR STOKES, ON THE DISCONTINUITY The complete integral of this equation in ascending series, obtained in the usual way, is 9a ρζω" ρ0ὅ 29 u= 441---- ἐνὶ | ἘΦ 5 4.88. δ᾽ 5.85.6. δ. 8:0 Ὁ ᾿ \ Qa 9? a7 gx"? + Bie + — ise 3.4°8.4.6.7138.4.6.7.9.10~ } οὐϊδω αν ἀν aie These series are always convergent, and for any value of w# real or imaginary assign a determinate and unique value to τι. The integral in a form adapted for calculation when z is large, obtained by the method of my former paper, is 1.5 Ye egy AE ἋᾺ 1... 7.11.2... w= Cortef - Ἶ i 1 + "ἢ + “0 9 1.14408 1.2.1447a5 1.2.3. 1445q2 eat s500)(20) 1.5 1.5.7.11. 1.5.7.11418.17 sp ae eee 1.1440¢ 1.4. 1443 1.4.8. 1445. + Da-te™" {: "ἢ The constants C, D must however be discontinuous, since otherwise the value of τὸ deter- mined by this equation would not recur, as it ought, when the amplitude of w is increased by 2m. 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 6s Cat f,(@) + Dat TGS. cascectessccsbebenscegusass COE) and let f(#), when we care only to express its dependance on the amplitude of Φ, be denoted by F(@). We may notice that F, (0 + 2x) = F, (0); Fy, (0 + ξπ) = Fy(O) seceesseeseeeee (22) 15. 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 PMG: ἃ term. In order to represent to the eye the existence and progress of the functions f,(#), f,(v) for different values of 0, draw a circle with any radius, and along a radius vector inclined to the prime radius at the variable angle 0 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, @ alone being supposed to vary, or in other words proportional to cos 39. For greater convenience suppose these distances moderately small compared with the radius. Consider first the function F\(6) 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',(@) is already ε OF ARBITRARY CONSTANTS, ἄς. 117 traced, since F, (0) = F,(0 +2). If now we conceive the curve marked with the proper values of the constants 0, D, it will serve to represent the complete integral of equation (18). In marking the curve we may either assume the amplitude 6 of @ to lie in the interval 0 to 2m, 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 θ 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 Οὐ or D to alter discontinuously as @ 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... (δὲ — 5) (6ὲ -- 1) 1.2... ¢(14408)' and the modulus of this term, expressed by means of the function I’, is Γ (ὁ «-- 2) Γ G+) ΓΑ Γ(Γ( + 1) 4p)” which when ὁ is very large becomes by the transformations employed in Art. 7, very nearly, r/ eS (ἢ - TEE) (4h). 6 Denoting this expression by μ,90 and putting for Γ() (4) its value m cosec = or 27, we have (ni) (29) = (27i τῶν ove ceccwceeenceoucees My π (4pte)! whence for very large values of 7 ΠΕΣ] 4 μι = apt Ce eeees ces vee vos vesces cee seseseees (24) 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 » we have very nearly from (24), (23) is 48, uw = (2ri)-te-t = (2i)-te-*, If the exponential in the expression for μ᾿ be multiplied by the modulus of the exponential in the superior term, the result will be ο΄ 4 2co0s§ 6) pt 3 the sign -- or + being taken according as cos 8. 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 τ 005 80 -- 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 118 PROFESSOR STOKES, ON THE DISCONTINUITY ean 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. 1), we see that it is only at the points a, ὦ, 6, 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 a4b, bBc, 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. , : ς π T ᾿ Since there is no superior term from θ = -- ᾿ to@=+ εν the coefficient of the other term cannot change discontinuously at a (i.e. when @ passes through the value zero); and by what has been already shewn the coefficient must remain unchanged Big 3; throughout the portion bBe 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 remainder of the course*, The second line on the right-hand side of (20) is what the first becomes when the origin of @ is altered by +37, 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. a ως, a In Fig. 3 the uninterrupted interior branch of the curve : ae : π is made to lie in the interval ἧς to π. It would have done equally well to make it lie in the interval τξ to —a; we should thus in fact obtain the same complete integral merely somewhat differently expressed. * A numerical verification of the discontinuity here represented is given 2s an Appendix to this paper. OF ARBITRARY CONSTANTS, &c. 119 The integral (20) may now be conveniently expressed in the following form, in which the discontinuity of the constants is exhibited : 4 4 155 Pe ony 11 τ τ (- lar + =) Ου-ἰο 5 { pp A + 2 = ws} 3 8 1. 144a 1.2. 14479 Qar ΚῚ 1.5 1 5. 7.1 «|-τὸ —#to + 2r)Da-te*® {14+ — + ΞΈΡΕΙ 8 1.14 1.98. 1445 Ξ . : 4qr 4a : ᾿ In this equation the expression (- yt + =) denotes that the function written after it is to be taken whenever an angle in the indefinite series 0-47, O-27, 0, OF+2n0, O4 47... falls within the specified limits, which will be either once or twice according to the value of θ. 17. If we put D =0 in (25), the resulting value of τὸ will be equal to Mr Airy’s integral, multiplied by an arbitrary constant, w being equal to — τ. When @=0 we have the integral belonging to the dark side of the caustic, when θ = πὶ 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 + a, — 7 in succession to the amplitude of #, 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 # by x, 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 4, B with C, D in the two different forms (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. 2 3 Let ν =f aan aA ῶ τ" -¥—e\5(52 4 ex) -- ext dd δ} Ὁ ¢ ο᾽ ,- "οὗ Ξι-- -α ἐὺ; ϑ 3 whence δου οἷ οὗ 120 PROFESSOR STOKES, ON THE DISCONTINUITY In order to make the left-hand member of this equation agree with (18), we must have οὗ =— 27, and therefore c= — 3, or 3a, or 36, a, B being the imaginary cube roots of — 1, of which a will be supposed equal to μά + lsin εἶ cos — - =} . 8 8 Whichever value οὗ ὁ be taken, the right-hand member of equation (26) will be equal to --θ» 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 w=E έ “τ (ὅλα ae) dd + F ‘fs 07 (+ BaP) Orv exsen cnn (27) 0 0 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 EH, 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+Ba= fve*{(1+a)E+(1 +P)F+s[(1-a) E+ - 6) Flan} ar which gives, since foe "ee ae ee aC Ξ a’ = — β, β' « --α, f e*dr=4LQ)s fe λάλ = ἐ Τῷ), A={T@{G+@aDE+(1 feted (28) νον ΔΕ Ρα coe. σὐῤθν 19. We have now to find the relations between H, F and C, D, for which purpose we must compare the expressions (25), (27), supposing « indefinitely large. In order that the exponentials in (25), may be as large as possible, we must have @ ==" in the term multiplied by C, and @=0 in the term multiplied by D. We have therefore for the leading term of u οὐ τν-ρ-ἰρε, when 0 = =; Dp-te, when θ =0. Let us now seek the leading term of wu from the expression (27), taking first the case in which 6=0. It is evident that this must arise from the part of the integral which involves e™ or in this case ¢, which is (E+ Py) [e-* +a, 0 OF ARBITRARY CONSTANTS, ἃς. 121 Now ϑρὰ -- λ΄ is a maximum for δὰ = p> Let A= ρϑ + <3 then ϑρὰ — λῦ = 9ρῇ -- sph? -- Ὁ, and our integral becomes ert ("grt Pat. —pt Put (= 3-49-4&; then the integral becomes s-ip-tet [ “ ec Boshi ας, ΕΞ 3tp? Let now p become infinite; then the last integral becomes { 6. δ᾿ ἀξ or πὸ. For though the index -- £*- 3~§p~3£* 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 — 2p%. Hence then for θ = 0, and for very large values of p, we have ultimately u = 8-ὑπὸ(Ε + F) pte. Next let 0= =. In this case aw = -- ρ, and we get for the leading part of u ak fed, 0 which when p is very large becomes, as before, 373 ort akp=*e**, Comparing the leading terms of τὸ both for θ-- and for θ-- 0, we find, observing that α -- ὁ: 5 6-- ν΄ -13 in E, 2 -- 3-ἐπῖ(Ε + F). ἜΤ ΤΉΝ, Eliminating E, 1᾽ between (28) and (29) we have finally ΤΥ ΤΟ πρ ΤΊ 2} ἀκ ὙΤ C4 ay B= 8x40 (8) {-C + ὦὥντι DR. 20. Asa last example of the principles of this paper, let us take the differential equation a 1 ὩΣ + ee Pee: nada are uae τ εν σον ol) The complete integral of this equation in series according to ascending powers of # 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 Vote X, “Parr. 16 122 PROFESSOR STOKES, ON THE DISCONTINUITY when \/— 1a is written for w, 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 (81) 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 a at a u= (4 + Blog x) (1 Ee re eee +...) ee ae seldecves (Ont a” a ax ἀρ Reet.) where Spat) + 27) grt oe eet, The series contained in this equation are convergent for all real or imaginary values of «, but the value of « determined by the equation is not unique, inasmuch as log Φ 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 2πΒν΄ -ἰ. If p» 9, the modulus and amplitude of a, be supposed to be polar co-ordinates, and the expression (32) be made to vary continuously by giving continuous variations to p and @ without allowing the former to vanish, the value of log # will increase by Qna/—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 af 12.38% 2 ge δὲ iin sk μι ἑῷ τοῖν + vee 2.40 2.4 (4a)* 2.4.6 (4ω)" a + Du-ie* 1+ 1" F! 13. 3? 12, 82 5? weeeesoes Toe 4. 4 (40) * 2.4.6 (4a) These series are ultimately divergent, and the constants C, D are discontinuous. It may be shewn precisely as before that the values of θ for which the constants are discontinuous are oe “ἀπ, —27, 0, 20, 47... for C, coc SH Hy. πὸ By wn for D. * Camb. Phil. Trans, Vol. IX. Part IL. p. [38.] OF ARBITRARY CONSTANTS, &c. 123 Hence the equation (33) may be written, according to the notation employed in Art. 16, as follows: 2 2 1 τ = (0 to 2m) Cu-te-* (1 - Ae Pe we) + (— πο + 7) Dade (1 + 2.40 22. It remains to connect 4, B with O, D. For this purpose we shall require the third form of the integral of (31), namely τ = {EF + F log (a sin’w)} (67° + e775?) dur .......6. 9.6. (35) 0 As to the value of loga 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 «. We have A+Bloga = 2 [1 + Flog (ὦ sin’w)} dw 0 =a (E+ Flog 4) + 2rlog (4). F; whence Pairk. LIOR POE (36) To connect C, D with E, F, we must seek the ultimate value of « when p is infinitely A=rH-2rlog2. ὁ increased. It will be convenient to assume in succession θ =0 and θ-- πσ. We have ulti- mately from (34) u= Dp-te when 9@=0; w=-— A a Cp- ter when 0= πὶ ...... (87) It will be necessary now to specify what value of log z we suppose taken in (35), Let it be log p+ \/-106, @ being supposed reduced within the limits 0 and 2x by adding or subtracting if need be 2iz, where ὁ is an integer. The limiting value of τὸ for θ =0 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 ὼς Cp ΤΟΣ ΟἿΣ Ὁ, of the former paper correspond to By Thy Nee De es. Es of the present. Hence we have for the ultimate value of ὦ for θ = 0 u= (=) ‘¢ {E + (τ᾿ Τ' (3) + log 2) F}. ΓΑ ἐπα ςφςο (98) For θ-- 7, (35) becomes u -[ἶτ ἘπΡν --ια Flog (ρ εἰπ' w)} (ὁ ἐτῦτθιε PS) dey 5 and to find the ultimate value of w we have merely to write E + 7F fmt for Εἰ in the above, which gives ultimately for θ = 7 τ 3 w=(Z) PLE + eF/—1 + {πο Γ) + loge} ΤΣ]. .........ὄ (39) 16—2 194 PROFESSOR STOKES, ON THE DISCONTINUITY Comparing the equations (38), (39) with (37), we get C= (2) eV =7- ται {π΄ Γ΄ (4) + log 2} v=1F}| wagon eee Dez (5) [Εἰ πο Γ' (4) + log 2} ΓΊ. | Eliminating E, Ε΄ between (36) and (40), we get finally C= (Ὡπ)τ [ν΄ - τ 4 + {{π ATG) + log 8) ν΄-ἴ - πὶ eo ele (41) D = (απ) [4 + {π-ὸ Τ΄ (4) + log 8} 81. Conclusion. 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. [Added 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. J subjoin therefore the numerical calculation of the integral to which fig. 2 relates, for two values of w, from the ascending and descending series separately. For this integral D = 0, and I will take C = 1, which gives, (equations 30,) A=nr T(t); B=-38a3T (3); and log 4 = 0°1793878; log (— B) = 03602028. The two values of 2 chosen for calculation have 2 for their common modulus, and 90°, 150°, 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 6. 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 term. Real part. of V1. Real part. of νΞ1. 0 + 1°000000 + 2°000000 ] - 12°000000 + 12°000000 2 — 28°800000 — 20°571429 3 + 28°800000 — 16°457143 4 + 15°709091 + 7595605 sy — 5°3885974 + 2°278681 6 — 1:267288 — 0°479722 7 + 0°217249 — 0°074762 8 + 0°028337 + 0°008971 9 — 0:002906 + 0°000855 10 — 0°000940 — 0:000066 11 + 0°000016 — 0°000004 12 + 0°000001 Sum -- 13'330099 + 11°628385\/— 1 — 2:252373 — 11°446641 / —1 Sum multiplied by 4, -- 20°14750 + 17°57548 \/—1; by B, + 5:16230 + 2628499 γί --Ἰ. 126 PROFESSOR STOKES, ON THE DISCONTINUITY When the amplitude of w becomes 150° in place of 90°, the amplitude of αὖ 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 w 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 # it would be sufficient to multiply by cos 60° +4/—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 = (1 +4/3/ 1), which gives the result — 25:30182 + 8°64681 4/ — 1. Hence we have for the result obtained from the ascending series: for amp. # = 90°, for amp. # = 150°, From first series — 20°14750 + 17°57548 he ge, Ἐ — 20°14750 — 17°57548 ΑΞ 1. From second series + 5'16230 + 96:23499 ν.-- 1 — 25°30182 + 864681 4/— 1 Total -- 14°98520 + 4381047 \/— 1 — 45°44882 — 892867 \/-1 On account of the particular values of amp. # 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. # = 90°, in which case there is no inferior term. Coefficient Order. Modulus. Real part. of V=1. 0 1:0000000 - 1:0000000 1 0°0122762 +.0:0086806 + 0:0086806 2 0°0011604 + 0°0011604 3 0:0002099 -- ΟΟΟΟΙ 484 + 0°0001484 4 0:0000563 -- 0:0000563 5 0°0000200 —0-0000142 — 0:0000142 6 0:0000089 — 0°0000089 7 0°0000047 + 0:0000033 — 0:0000033 8 0:0000029 -- 0:0000029 9 0:0000021 + 0:0000015 + 0:0000015 10 0°0000017 + 0°0000017 11 00000015 —0:0000010 +0-0000010 Remainder — 0°0000007 — 0 0000017 Sum + 10084677 + 0°0099655 \/— 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 OF ARBITRARY CONSTANTS, &c. 127 1°0616, 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®= + 0:0340, At= + 0:0373, &e. 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 form p (cos θ + /—1 sin 6), and thus multiplied by e~** and by wt. We have for series log. mod. = 0°0036832 amp. = + 0° 33’ 58”. 21 for exponential log. mod. = 1°7371779 amp. = + 130° 49’ 0”. 78 for ατὲ log. mod, = 1°9247425 amp. =— 22° 30’ 1:6656036 + 108° 52’ 58”. 99 When the amplitude of Φ is 150°, there are both superior and inferior terms 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. # = 150° and amp. # = — 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 x, or more generally for any two amplitudes equidistant from 120°, the amplitudes of φῇ will be equidistant from 180°, so that for any rational and real function of w? we may pass from the result in the one case to the result in the other by simply changing the sign of \/— 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 at we have simply to replace the amplitude -- 22°30’ by — 37° 80, Hence we have for the superior term log. mod. = 1°6656036 ; amp. = — 168° 52" 58”. 99. When amp. 2 is changed from 150° to — 210°, amp. a? is altered by 3 x 180°, and there- fore the sign of aw? is changed. Hence the log. mod. of the exponential is less than it was by 2 x 1787... 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 + 0'0001 +0°9914 + 0:00764/—1 log. mod, = 19963 ; 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 θ + s/ -- τ sin 6) to the form at /-1 b, we have for the final result, obtained from the descending series: For amp. Φ = 90°. For amp. wv = 150°, From superior term — 14°98520 + 43°810464/—1; -- 45-43360 — 892767 \/— 1 From inferior term — 0°01524 — 000100 Pe gi 1 — 45°44884 — 8°92867 γ΄, — 1 Had the asserted discontinuity in the value of the arbitrary constant not existed, either the inferior term would have been present for amp. # = 90°, or it would have been absent for amp. # = 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 Φ 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 ἃ as 2 the descending series have a decided advantage over the ascending. VII. On the Beats of Imperfect Consonances. By Auaustus Dr Moraan, F.R.A.S. of Trinity College, Professor of Mathematics in University College, London. [Read Nov. 9, 1857.] Tux 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 resultt 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 formule, 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 kp and kg vibrations per second, be & what it may. Consequently, an interval remains con- stant, not with p —q, but with logp—logg. 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 2p vibrations: hence log 2p — log p, 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 | now when it is adopted, the beats were and sometimes are used name of the place where it is printed, not of the place where | in tuning: but when equal temperament is required (and this the publisher sells it. Both these editions are printed for | system has gained ground rapidly) the tuners have nothing to Cambridge publishers (the Merrills). do with beats, except to get perfect octaves by destroying them. + So long as unequal temperament was in use, and even | I speak of the organ, and of this country. Voted PARTS; 17 180. 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 m, the integer in 1000 (log m — logm) is the number of subdivisions contained in the interval, quam proximé. 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 to m vibrations per second, neglect the decimal point in logm —logn, or in logn — log m, whichever is positive, To find the ratio of the numbers of vibrations in an interval of ἃ 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 “ log 2 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 8 -- 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 7:0195, 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. as the value of log2; the one which precedes is a near approximation. of the equation This rule is as accurate Both are consequences 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 for purposes which do not often occur, are of value only when acoustical logarithms; and proposed systems, of which the they save complicated operations. Such tables are not in bases are 2 and ‘2/2. Prony gave both tables in his Instruc- tions Elémentaires sur les moyens de calculer les intervalles musicaux, Paris, 1832, 4to. The second table shows at once, in logm —logm, 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 ae which gives the number of commas in m:n, by logm—logn. The atom which 1 have proposed, which is the 540th part of a comma, gives the commas by division by 60 and 9. I have my . doubts whether any tables will be so convenient as those of common logarithms, used in the way I propose. Special tables, 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 formule 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 553. 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. t Writers are very obscure in their use of the word vidra- 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 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. about a note in the last century. The change can be traced in its progress, 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. nent rise, which the organ-tuners would of course follow, and then the same effect would be This would have a tendency to cause a perma- 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. 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 By a beat, I mean any acoustical cycle derived from name occurs, it will be more convenient to call them T'artini’s beats, because, when they become rapid enough to give a note, that note is the grave harmonic detected by Tartini in or tion; 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, F.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 disproportioned 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 about 1714. tini’s beats. Mr DE MORGAN, ON THE BEATS OF IMPERFECT CONSONANCES. And even when they give a sound, it will still be convenient to call them Tar- These beats are in their perfect theoretical existence when a consonance is quite true, and they owe their usual existence to its approwimate 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 Smith’s 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 Tartiai’s grave har- monic. In the place in which one would have expected him to mention it, namely, when he mentions the flutterings, as he calls them, which I name Tartini’s 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 flutteringst 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 is so precisely what he might have written of Smith, that I quote it. ‘Son livre est écrit d’une maniére si obscure, qu’il nous est impossible d’en porter aucun jugement: et nous apprenons que des Savans illustres en ont pensé de méme. II seroit ἃ souhaiter que lAuteur engagedt quelque homme de lettres versé dans la Musique et dans l’art d’écrire, ἃ développer des idées qu’il n’a pas rendues assez nettement, et dont l’art tireroit peut-étre un grand fruit, si elles étoient 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 the 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, Rameau, 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. T'artini 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 connexion with the truth, no doubt; but the are demands and obtains a great number of combinations which neither system will allow. Tt 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 collec- tion headed Assemblée Publique &c. Ihave never seen this memoir. ‘here is 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 ( Acoustique, 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 découverte, 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, ὥς, 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, Harmonie and Systéme ; Matthew Young’s Enquiry, &e. + Smith does not, so far as I can find, attempt to explain these flutterings ; 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 battemens to both, and quite correctly; for both are battemens, 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. Sc, 1701, Paris, 1719, p. 359) to find the son fixe, as he calls it, which makes 100 He directs us to take organ-pipes, at least two feet long, and to tune Here he speaks of what I call Smith’s beats, of which he clearly knows the negative use, namely, the vibrations in a second. diatonic intervals so perfect that not the smallest battement shall be perceived. 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 Dr Smith (Harmonies, 2nd Ed. p. 96) complains that he cannot count Sauveur’s beats: but, though he used low notes, he took the prominent concords beats which can be easily counted. or discords of the scale, which are not near enough. Dr Young pronounced Smith’s work “ἃ large and obscure volume, which for every pur- pose except the use of an impracticable+ instrument leaves the whole subject precisely where it found it.” been correct: had the volume been larger, it had probably been less difficult; it is a small It leaves the subject where it found the subject If Dr Young had said that the work was largely obscure, he would have volume for the quantity of subject-matter. 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. 1. pp. 83, 84, 93, 134-139; Robison, Mech. Phil., Brewster’s edition, Vol. 1v. 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 “ἃ 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 (1. 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 243, and making the three sound together, gets a three-pipe beat of 48, 49, 50 vibrations. He then inserts 48} and 493, 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! Musique! que de crimes on commet en ton nom ! + 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 (rv. 414—421) goes through the whole phenomena of the octave. [{ 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.” any beats except those which occur in imperfect unisons, in which Tartini’s beat is no other than the vibration of the note itself. fifths, he declines explanation, and (εν. 409) states what “ Dr Smith demonstrates.” In common with a great number of other writers, he ventures on no explanation of When he comes to mention the beats of badly-tuned He calls the method of beats, and to my mind very justly, the greatest discovery (tv. 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 ewelusion of 'Tartini’s har- monic. Young, in replying, writes as follows (1. 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 δ᾿ 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 an organ-pipe by means of the rapidity of the beating with others, Mém. de 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 person 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. τ. 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 Dr Smith’s labours are such 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. + 1 think this is a mistake. I 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 tind 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 the Academy was appointed to inspect their repetition. The ac- count of this experiment is a part of the history of the subject. «ΟΜ, Sauveur en rendit conte luy-méme et avoiia que pour cette fois elle n’avoit pas bien réussi, car d’autres fois, et en présence des plus habiles Musiciens de Paris, elle avoit paru trés juste et trés précise. La difficulté de la recommencer, Vappareil qu’il faut pour cela, d’autres occupations plus pressantes de M. Sauveur, et méme d’autres recherches d’Acoustique, ot il a été obligé de s’engager par la liaison qu’elles avoient avec le Son fixe, ont été cause qu’on en est demeuré 1a, mais on scait qu’en fait d’experiences il ne faut pas se décourager aisément, et qu’elles ont pour ainsi dire, leur caprices que l’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, but 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 the paper 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, trés-desagréables...) 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 ὦ 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 formulz, and then unravelling the demonstration of the same formule 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 > , and let the higher note make m vibrations while the lower note makes m. 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 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 simplicity. * Emerson arrives at the formula which I presently mark as | and having lent it to Emerson as giving a higher probability to ( -- Φ) Mn+x; Mr Woolhouse arrives at (1— δ) 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 £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 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. ΤῸ 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-sounds}, 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 @ 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 w (1 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 - Θ be the time of vibration of the upper note in the imperfect consonance which gives the beats. When @ 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 @ is negative. It is the same thing to the ear whether the beats move forwards or backwards. Let 2 be the ratio of the consonance of the perfect and imperfect upper note; that is, let e=na:na+0. Thus #<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 na na+é0 M ma Nma=1, M(na+6)=1, v= ΒΤ ἥπερ θ- = na; kmna=1, kn=N. a Let 6 be the number of beats in one second. A beat, as shown, lasts through as many : . EOD BG aaa i na -- θ) α of the shorter vibrations as there are units in -- : its time is then (nae ON θ θ : sa that we have 6 1-@ 1-@ (πα. θ᾽ a = (1 - a) kmn = (1 -- 5) mN = ——— nM = mN --Μ. B= Dr Smith does not elicit* any of these formule, the last of which is remarkably simple. Thus if a fifth be tuned imperfectly to 200 and 3014 vibrations per second, we have 200 x 3 — 8013 x2 = — 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 B=(M'’-M)n. Or * Since this paper was written the article ‘Beats’ in the | with two varieties arising out of different modes of expressing Edinburgh Encyclopedia, attributed to Mr John Farey, has | the division of the octave, Emerson’s method, and the formula been pointed out tome. This article contains Smith’s formula,’ | mN-nM, But no explanation is given. Mr DE MORGAN, ON THE BEATS OF IMPERFECT CONSONANCES. 139 thus :—In every consonance of which the lower number is ”, every wrong vibration per second in the upper note is m 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 formule given by Dr Smith are obtained as follows. major and minor tone, being 81 : 80, let @ correspond to the fraction ῳ : p of a comma, Then The comma, or difference of a 4 Oe (=) ' 81 2na 1 -- * es] - ὦ. : j . Now (1 -- α) Py ΤΣ nearly; a being small 8o\ 4 2q Pat bide YS a Oe ; whence # (=) Tree nearly. And B=(1-«)mN= Ἢ Ἢ : 161» Ἐ4 ᾿ 9 ΠΡ = Ξ nM = = n, | x 161p — q which are Smith’s formule (2nd ed. p. 82). When the upper note is too sharp, gy must be made negative, the negative sign of β being neglected. If μ be the fraction of a mean semitone by which the upper note is flat, we have, for the number of beats in a minute, -5 104 1 μ 60 (a -2 3) mN, or ---- μη Ν, or 1θ0άμ (= - —|mN 380 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 — °0000115129a) mN very nearly, . 4 Χχ8Χχ)]18 amlN nearly. 801000 These formule 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 —mM is much more easy. * The passage over the greatest common measure being m cil 2m, or a+ eV oo : My, fairly arrived at, as the time of a beat, the transition to the M pia akata rip formula mN—nM may be very briefly made. We know that, armen ὁ the nares mal of the shorter wave gains the mM. -π m and n being prime to one another, there is, before we arrive at mn, one way and one only in which pm—gn=1; and one way and one only in which gn~pm=1. The ratio N: M of the numbers of vibrations in the erroneous consonance, and also of the lengths of the waves, is not 5 : m, but = of a common measure in every vibration of the higher note, than is mM —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 fraction 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,2N. If m be an odd number, we must change n,m, into m, 2n; but if m be even, m, m, must change into 4m, τῆς since the fundamental ratio must be in its lowest terms. And we must also change the sign of g, neglecting the negative sign of the value of 3, when it occurs. quently, β' being the number of beats of QP! in a second, we have Conse- ἘΠ Gs. | jae ὥ (m odd) β = ως Nm, β τ αν ἐρὴ 2n = 2B; +q py 2q Uypaebiten abe. β 161} - q That is, when the fundamental number (in the ratio m: 2) of the mistuned note is odd, the But when this fundamental number is even, the interval and its octave complement have the (m even) B = M.n = β. ae! ἢ 161p + q interval complemental to the octave beats twice as fast as the lower interval first given. 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- Let N make n vibrations while M makes m: let M make m’ vibrations while Z makes 7: the fractions m:n and ἢ: m’ being in their lowest terms. Let the imperfect consonances NM, ML, NJ, beat severally β, β', B, times per second: (3 being positive when the higher note is flat, and negative when it is ing notes represented by their numbers of vibrations per second. * 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 in sufficiently good descrip- tions of machinery, a point in which an obscure writer is generally most perplexed and perplexing. I take the cause of Dr Smith’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 that 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 ἃ 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 when 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 (>): the third (Ὁ): the sixth (ἢ: the seventh (Fz . The usual major intervals are the tone (Ὁ): the third 4 G) ; the fourth (5)> in which there is a failure; the fifth (5): the sixth (ξ ): the seventh (ἐ *). In the minor and 2 25 major semitone (i i) the rule is inverted ; and also in the minor fifth (3): Keeping, however, to common intervals used in tuning, and calling the fourth a minor to the 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 mi and mn. Then 1B + πβ΄ = 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. Smith’s beats themselves have a long inequality whenever ξ is ποῦ an integer; of which I 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. University Contece, Lonpon, August 12, 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. Different tuners, however excellent their ears, do not agree in their results. Two men, tuning different 149 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 then 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, FH, are m, 2m, 3m, 4m, 5m, 6m, 7m; while those in the keys of CH, GH, D#, Ad, F are 6m, 5m, 4m, 3m, 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 Cyclopedia. 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. Sis 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 the 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 "2846 of a mean semitone. 2. The keys being arranged dominantly, that is, in the order C, G, D, A, E, B, FH, cH, GH, D#, A#, F, C, G, D,...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 ‘2151 of a mean semitone. 7 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 ACH, together with those of AE, EB, BF, FHCH, 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, 75 8, ἢ, 45 7 8, P, ἢ, 75 8, 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, φ-- 891, 7=0, s="0391: 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 "1860 and -1369+°0391, equal temperament giving *1564 to all. If the temperaments of the fifths run in cycles of three, as in p, 9, 7; Ps % 1 Ps U1 Ps HPs 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 § I have made p=0, q='01955 as in equal temperament, r=2q; which satisfies 4 (ρ Ὁ 4 Ἐγ)- 9846, The temperaments of the major thirds in dominantly successive keys are *1564, *1564—g, *1564—2q: 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 5 C 0 0:00000 0°00000 ΟὍΟΘΟΟΟ C CH} 1 1:00000 101955 1196ὅ CH D 2 202444 2700000 201955 DH| 3 298534 3°01955 300000 D¥ E 4 402982 4:00000 401955 F 5 499022501955 δΌΙ05δδ FH] 6 6:01466 6-00000 6΄ΌΟΟΟΟ F¥ G 7 7:01466 7:01955 —7°01955 GH} 8 7:99022 800000 801055 GH A 9 9'02932 9°01955 — 900000 A#| 10 9:98684 10:00000 1001955 ΔΒ Β 11. 11Ὁ2444 11Ὸ105ὅ 11°01955 Vibrations in one Minute. Beats in one Minute. P Q R 5 P Q R 5 144000 144000 144000 14400°0 488 122 000 000 σ CH| 152563 152563 152735 15273°5 517 775 108:4.. 1.57 CH 16163°5 161863 161635 16181°7 547 411 Ο00Ὁ 1095 D D#| 171246 171107) 171489 17124°6 580 580 116ῸὉ =~: 00°0 D¥ E 18142'9 18178.6 181429 18163"4 614 769 000 61 E F 19221°7 192108 19243'4 19243°4 651 825 -—-:130°2-130°2 FE FH} 203647 20381'9 203647 20364°7 690 1207 000 000 G 21575°6 + 21593'9 21600°0 21600°0 730 365 1462 78:2 G GH| 22858°6 228457 22858°6 + 22884"4 774 966 000 154-9 G 24217°S 24258°9 24245°2 24217°8 82:0 891 1641 Ο0Ὁ Α A#| 466579 256362 4566670 256869 868 651 000 = 870 B 271886 272220 272143 27214'3 990 1382 1842 184-2 B The vibrations are calculated from the formula log M=log N+}, log 2 xa, where M and WN are the vibrations in the higher and lower notes, and # the number of mean semitones in the interval. The beats are calculated from the formula mN—nM; for the fifths 3N-2M. The beats are those of each note with the fifth above it: thus Aff F' (the octave above ΕἾ beats 86°8 times in a minute im 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,G, 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, g, 7, s, in R, and p, gq, 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 G Ὁ" (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 FC!, in which CF and FC! beat equally, to CGC!, in which GC! beats twice as fast as CG, 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. De M. January 18, 1858. Vou. X. Parr I. 19 VIII. On the Genuineness of the Sophista of Plato, and on some of its philosophical bearings. By W.H. Tuompson, M.A., Fellow of Trinity College, and Regius Professor of Greek. [Read Nov. 23, 1857.]. 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 1 Cambridge Philosophical Transactions, Vol. 1x. Part 1v. 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 many, 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 φασί τινες 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, 111. 3. 9, we read thus: φανερὸν ὅτι οὐδὲ δόξα μετ᾽ αἰσθήσεως οὐδὲ dt’ αἰσθήσεως, οὐδὲ συμπλοκὴ δόξης Kal αἰσθήσεως φαντασία av εἴη. 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, ᾧ 107, p. 264 8, where we are told that the mental state denoted in a previous sentence by the verb φαίνεται, is “‘a mixture of sensation and judgment,” σύμμιξις αἰσθήσεως καὶ δόξης: 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 φαντασία. Ὅταν μὴ καθ᾽ αὑτὴν ἀλλὰ δ αἰσθήσεως παρῇ τινι τὸ τοιοῦτον αὖ πάθος, ap οἷόν τε ὀρθῶς εἰπεῖν ἕτερόν τι πλὴν φαντασίαν. A φαντασία is, it will be seen, according to Plato a variety of δόξα. 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 συμπλοκή: he wrote σύμ- κμιξις, but it is remarkable that the word συμπλοκὴ does occur two or three times over in this part of the dialogue; hence Aristotle, writing from memory, substitutes it for the σύμμιξις of the original. One of the most learned and trustworthy of his commentators, Simplicius, has the gloss: tov Πλάτωνος ἔν τε τῷ Σοφίστῃ καὶ ἐν τῷ Φιλήβῳ τὴν φαντασίαν ἐν μίξει δόξης τε καὶ αἰσθήσεως τιθεμένου, ἐνίστασθαι πρὸς τὴν θέσιν διὰ τούτων δοκεῖ. Now in the Philebus the definition in question does not occur, though the mental act which Plato calls φαντασία is graphically described, and the cognate ‘participle φανταζόμενον is used in the description (p. 38 9). 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 ἐν τῷ Ἱππίᾳ, ἐν rH | entire system comes under review in that work, of which one Φαίδωνι. It is remarkable that these are the only two dia- | book is appropriated to the theory of ideas alone. The Par- logues quoted by name in the Metaphysics: though Plato’s | menides, which is largely drawn from, is not once named, 19—2 148 PROFESSOR THOMPSON, ON THE GENUINENESS OF 2. The next passage I 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: Ὅσοι μὲν οὖν οἴονται πολιτικὸν καὶ βασιλικὸν καὶ οἰκονομικὸν καὶ δεσποτικὸν εἶναι τὸν αὐτὸν οὐ καλῶς λέγουσιν᾽ πλήθει “γὰρ καὶ ὀλι- ότητι νομίζουσι διαφέρειν ἀλλ᾽ οὐκ εἴδει τούτων ἕκαστον... ὡς οὐδὲν διαφέρουσαν μεγάλην οἰκίαν ἢ σμικρᾶν πόλιν. “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 Ἐ: πότερ᾽ οὖν τὸν πολι- τικὸν καὶ βασιλέα καὶ δεσπότην καὶ ἔτ᾽ οἰκονόμον θήσομεν ws ἕν πάντα ταῦτα προσαγορεύ- οντες, ἦ τοσαύτας τέχνας αὐτὰς εἶναι paper, ὅσαπερ ὀνόματα ἐῤῥήθη. “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 ?” (τί δέ; μεγάλης σχῆμα οἰκήσεως, ἢ σμικρᾶς αὖ πόλεως ὄγκος μῶν τι πρὸς ἀρ- χὴν διοίσετον). ‘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 GEconomic ?” (εἴτε βασιλικὴν εἴτε πολιτικὴν εἴτε οἰκονομικήν τις ὀνομάζει μηδὲν αὐτῷ διαφερώμεθα.) 8. There is a passage in Aristotle’s treatise De Partibus Animalium (1. 6. 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 μεσοτομία or διχοτομία, the method of mesotomy or dichotomy. ‘* Some persons,” says Aristotle, ‘ get at particulars by dividing the genus into two differenti: 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 (διασπᾶν τὸ syévos), for example, to avoid classing birds under two distinct heads, an error is committed in the ‘written divisions’ (γεγραμμέναι διαιρέσεις), in which some birds come under the genus Terrestrial, and some under that of Aquatic Animals (ἐκεῖ γὰρ τοὺς μὲν μετὰ τῶν ἐνύδρων συμβαίνει διῃρῆσθαι τοὺς δ᾽ ἐν ἄλλῳ γένει), 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, διήρητο ξύμπαν τὸ ζῷον τῷ τιθάσῳ καὶ ἀγρίῳ. 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 (ὥσθ᾽ ὁτιοῦν ζῷον ἐν ταύταις (ταῖς διαιρέσεσιν) ὑπάρχειν), 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, 1 Soph. 220 a: τὸ μὲν πεζοῦ γένους τὸ δ᾽ ἕτερον νευστικοῦ | tonic ‘Divisions’ similar perhaps to that of the ‘ Definitions’ ζῴου. Politic. 264 c: τῆς μὲν ἀγελαίων τροφῆς ἔστι μὲν | attributed by some to Speusippus, and compiled partly from ἔνυδρον, ἔστι δὲ ξηροβατικόν. The words ‘written divisions’ | the Dialogues and partly from Plato’s oral teaching. are supposed to refer to a work now lost, a collection of Pla- THE SOPHISTA OF PLATO, &c. 149 κύνες ἐν τῇ ᾿Ινδικῇ, ὕες, αἶγες, πρόβατα. 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 Politicus 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. Ina work similar in its scope to the Sophista, the curious treatise περὶ Σοφιστικῶν ἐλέγχων, occurs a definition of ““ Sophistic,” which to my ear is an echo of the Platonic Dialogue, I allude to the often repeated definition, ἔστιν 4 σοφιστικὴ φαινομένη σοφία ἀλλ᾽ οὐκ οὖσα, καὶ ὁ σοφιστὴς χρηματιστὴς ἀπὸ φαινομένης σοφίας ἀλλ᾽ οὐκ οὔσης (S. E. 1. 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 διαιρετικὸς λόγος of the Sophista, a definition identical with the νέων καὶ πλουσίων ἔμμισθος Onpevtys— 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 ? i 5. In the same treatise, c. 5, § 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 7s, is not, if you tell them that any entity is mot so and so—say that A is not aman. 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 μη ὃν the contrary of the dv, when we dare to affirm that the μὴ ov is: the truth being, that we altogether decline to say anything about the contrary of the ov, whether any such contrary is or is not conceivable by the reason.” ἡμεῖς μὲν “γὰρ περὶ ἐναντίου τινὸς αὐτῷ (sc. τῷ ὄντι) χαίρειν πολλὰ λέγομεν, εἴτ᾽ ἔστιν εἴτε μὴ λόγον ἔχον ἢ καὶ παντάπασιν ἄλογον. p- 258 Ε. To this same passage I suppose Aristotle to allude in the Metuphysica (vi. 4. 13, Bekk. Oxon.) ἀλλ᾽ ὥσπερ ἐκ τοῦ μὴ ὄντος λογικῶς φασί τινες εἶναι TO μὴ ὃν οὐχ ἁπλῶς ἀλλὰ μὴ ὄν, κι τ. Δ. (Where λογικῶς = ‘ sensu dialectico,’ as distinguished from φυσικῶς.) 6. I shall have more to say on these passages hereafter : for the present they are mentioned for the sake of the coincidence. The φασί τινες, as already observed, is Aristotle’s frequent formula of acknowledgment. If any one doubt that the τινὲς are in this instance a τίς, or if he doubt who the τίς may be, let him hear Aristotle in another part of the same work; διὸ Πλάτων τρόπον τινὰ οὐ κακῶς τὴν σοφιστικὴν περὶ τὸ μὴ ὃν ἔταξεν᾽, Met. v. 9, ᾧ 8, and then turn to the Sophista, pp. 235 a, 237 a, 258 B, 264 p, 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,” (ἀποδιδράσκων εἰς τὴν τοῦ μὴ ὄντος σκοτεινότητα), taking 1 ἁπλῶς τόδε ἢ πῇ λέγεσθαι καὶ μὴ κυρίως, ὅταν τὸ ἐν | τι μή ἐστιν; οἷον εἰ μὴ ἄνθρωπος. μέρει λεγόμενον ὡς ἁπλῶς εἰρημένον ληφθῇ, οἷον εἰ τὸ μὴ ὄν 2 ¢ Plato was right to a certain extent, when he represented ἐστι δοξαστὸν, ὅτι τὸ μὴ ὃν ἔστιν" ob γὰρ ταὐτὸν εἶναί τέ τι | the Non-ens as the province of the Sophist.” καὶ εἶναι ἁπλῶς. ἢ πάλιν ὅτι τὸ ὃν οὐκ ἔστιν ὃν εἰ τῶν Burm 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 their 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 We need have little scruple in rejecting a dialogue so poor in matter and dry in treatment as the Second weight, except when they are supported by considerations purely philological. Alcibiades, when we find the evidence of its spuriousness strengthened by the occurrence of But it would be rash criticism to condemn the Second Hippias, in which no such irregularities occur, merely grammatical forms which no writer of the best times would have used’. 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. more thoroughly Platonic. So far as the mere style is concerned, there is no dialogue in the whole series 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 dryeone; 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 1 T cannot but think that had the Master of Trinity exa- mined the Politicus with the same care which he has bestowed 2 The Sophista is also recognized, as we have seen, by the vigilant and profoundly learned Simplicius, also by Porphyry 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- 266. 99) which exposed the philosopher to a well-known practical jest: the somewhat wild but highly imaginative mythus, redolent of the Timaus, (p. 269 foll.): 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!”’ (ap. Simp. ad Phys. p. 335, Brandis), Clemens Alexandrinus and Eusebius quote it as Plato’s. If it is not named by Cicero, neither are the Philebus and Theetetus. The omission of any mention of this latter dialogue by the Author of the Academic Questions is really remarkable. 3 e.g, ἀποκριθῆναι for ἀποκρίνασθαι, σκέπτεσθαι for σκο- πεῖσθαι. The latter barbarism, I presume, would be defended from Laches, Ὁ. 1858. τί aor’ ἔστι περὶ οὗ βουλευόμεθα και σκεππτόμεθα, but to me it seems clear that σκεπτόμεθα is an interpretamentum of βουλευόμεθα, which is used in a sense not strictly its own, as in the same passage, paulo supra; el ἔστι τις τεχνικὸς περὶ οὗ βουλενόμεθα. 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 Thestetus 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 Timeus, 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 Theetetus. 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 Theetetus Socrates bespeaks an interview for the following day, of which he is reminded by Theodorus in the opening sentence of the Sophista. The Politicws or Statesman is, in like manner, a professed continuation of the Sophist. that between the Theetetus and the Sophista. 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 The connexion, however, between these two is on the surface much closer than In the Theetetus we are not informed what is 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 Φιλόσοῴφος, 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 Theetetus and that of one of those ““ Schleiermachersche Grillen”’ which contribute to the amusement even of his admirers. Stallbaum seems to think ' Asa specimen of this, take the argument with the yn- γενεῖς, 246.D, seg., and the mock solemnity with which the * Ens’ of the εἰδῶν φίλοι is described, 249 a. 2 Pp. 217 a. 3 Schleiermacher, for instance, conceives that the omission is intentional, and that we must look for the missing portrait in the Symposiwm and Phedo; of which the first teaches us how a philosopher should live, the latter how he should die. This is that the title of the Parmenides may originally have been Φιλόσοφος, 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 Theetetus. 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 Phedrus 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 συναγωγὴ, Collection: by Aristotle ἐπαγωγὴ, Induction: the second is called by both Plato and Aristotle διαίρεσις, or the διαιρετικὴ μέθοδος, 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 Phadrus a Platonic organon in miniature. Now it so happens, that the Thectetus and the Sophista pretend, each of them, to be an exemplification of one of these two dialectical methods: the Theatetus of a συναγωγὴ, the Sophista of a διαίρεσις. It is this fiction which gives life and unity of purpose to the Thee- 1 It was first re-instated, so far as I know, by Mr Mill. 3 See Appendix I. Phedr. 265p, foll. 3 Those who are unskilled in the application of these pro- cesses are termed épiotixol in the Philebus, 16 Ἑ. οἱ δὲ viv τῶν ἀνθρώπων σοφοὶ ἕν μὲν, ὅπως dv τύχωσι, Kal θᾶττον Kal βραδύτερον ποιοῦσι ποῦ δέοντος μετὰ δὲ τὸ ἕν ἄπειρα εὐθύς" τὰ δὲ μέσα αὐτοὺς ἐκφεύγει" οἷς διακεχώρισται τό τε διαλεκ- τικῶς πάλιν καὶ τὸ ἐριστικῶς ἡμᾶς ποιεῖσθαι πρὸς ἀλλήλους τοὺς λόγους. 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 Phedrus now under review. In the received text we read καὶ πολλὰ θᾶττον, κατιλ. The sense manifestly requires the omission of πολλά. 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. 4 Compare Theet. 145D—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, 216 Ε, 253 Ὁ, the true philosopher must adopt in his search for scientific truth. The one is a hunt after the true conception of ém- στήμη or science, the other an investigation of the genus and differentize 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 Add to this, that the Sophista helps materially towards a solution of the question, What is Science? which is the professed aim of of philosophy the vivacity and interest of a drama. the dialogue which precedes it. It attains this object in two ways. First, by enlarging the conception of that which is mot Science, treating the subject on its logical or dialectical, as the Theetetus 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 Thea@tetus it is shewn that the Protagorean dictum, that Truth exists only relatively to its percipient (πάντων μέτρον ἄνθρωπος), and the kindred, though not identical Cyrenaic dogma, that sense is knowledge, and the sensations the sole criteria of truth (κριτήρια ta πάθη), 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. the dialogue first in order. It may conduce to greater clearness if I take this critical portion of 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. 1 The theory that “ Science is right Opinion combined with Sensation” is given by Zeller to Antisthenes on grounds which seem highly probable. 51 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 πρόφασις---ἃ natural and probable occasion for the intro- duction of important controversial discussions. It constitutes Vou. X. Part I. 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. 3 Comp. Phadr. 264 Cc: dei πάντα λόγον ὥσπερ ζῷον cuve- ordvat, κι τ.λ. 20 1δ4 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 Thectetus, and ἃ fortiori to the Sophista, is about 898. 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*. Thestetus assents to this mechanically, but the Stranger, doubting the sincerity of his assent, explains his meaning more fully. The word φανταστικὸς 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, “να fly in the face of my Master, the great Parmenides, who both in oral prose and written metre adjured his disciples to beware of committing themselves to this contradiction®. To extricate ourselves then from the ἀπορία 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- 1 Soph. 9421): τὸ δὲ wap’ μῶν ᾿Ελεατικὸν ἔθνος ἀπὸ Ξενοφάνους...ἀρξάμενον. 3 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 ἢ 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 Timeus 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 Theetetus? 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 ? 8 Ἐπεὶ καὶ viv μάλ᾽ εὖ καὶ κομψῶς els ἄπορον εἶδος διερευ- νήσασθαι καταπέφευγεν. 236 D. 4 Τετόλμηκεν ὁ λόγος οὗτος ὑποθέσθαι τὸ py ὃν εἶναι" ψεῦδος γὰρ οὐκ ἂν ἄλλως ἐγίγνετο ὄν. 287 Α. 5 Ἀπεμαρτύρατο πεζῇ τε ὧδε ἑκάστοτε λέγων καὶ μετὰ μέτρων" οὐ γὰρ μήποτε τοῦτο δαῇς, εἶναι μιὶ ἐόντα, ἀλλὰ σὺ τῆσδ᾽ ἀφ᾽ ὁδοῦ διζήσιος εἶργε νόημα. Ib. THE SOPHISTA’ OF PLATO, &c. 155 diction is in fact no contradiction, but that there is a sense in which the μὴ ὃν is, and in which the ὃν is not'.” 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- says > ing of this Eleatic denial of the conceivableness of non-entia. ‘ You can never learn,’ 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 thinker. 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,” τὸ yap αὐτὸ νοεῖν ἐστίν τε καὶ εἶναι : and, χρὴ τὸ λέγειν τε νοεῖν τ᾿ ἐὸν ἔμμεναι", “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 was to Parmenides Metaphysic, and Metaphysic Logic. That which is conceivable alone is, and that is which is conceivable. ‘The abstraction ““ ΤῸ 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: ἕν τὰ πάντα, ‘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®. Andon 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.” (ἄνθρωπός ἐστι μὴ θήριον, as Parmenides would have expressed it.) But a μὴ θήριον is, according to his logic, a μη dv; therefore all so- called ὄντα are at the same time μὴ ὄντα : non-existent, and therefore inconceivable, and so altogether out of the domain of Science. 2 Tov τοῦ πατρὸς Παρμενίδου λόγον ἀναγκαῖον ἡμῖν | v. 94, Mullach. ἀμυνομένοις ἔσται βασανίζειν, καὶ βιάζεσθαι τό τε μὴ dv 4 Frag. v. 48, ed. Mullach. ὡς ἔστι κατά τι; καὶ τὸ ὃν αὖ πάλιν ὡς ἔστι πῃ. p. 341}. Bnxwete tine ἔστι γὰρ εἶναι, μηδὲν δ᾽ οὐκ εἶναι. Comp. Arist. Soph. El, c. 5, 8.1, quoted above. sense seeeeee OU0ED γὰρ ἢ ἔστιν ἢ ἔσται 3 οὐ γὰρ μήποτε τοῦτο δαῇς, εἶναι μὴ ἐόντα. Ἄλλο παρὲκ τοῦ ἐόντος. Ibid. 3 ταὐτὸν δ᾽ ἔστι νοεῖν τε καὶ οὕνεκέν ἐστι νόημα. Frag. 5 Ibid. ν. 110. 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 ὃν is after all a pitiful μὴ ὄν! 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. 2468.) 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 ὃν and the μὴ 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 1 Soph. 245c, 964 Bekk.: μὴ ὄντος δέ ye τὸ παράπαν τοῦ ὅλου, ταὐτά τε ταὐτὰ ὑπάρχει TH ὄντι, Kal πρὸς TH μὴ εἶναι μηδ᾽ dv γενέσθαι ποτὲ ὄν. 3 πάντοθεν εὐκύκλου σφαίρης ἐναλίγκιον ὄγκῳ. Parm. v. 103. 3 ἕν᾽ ἐκ πάντων ἴδωμεν ὅτι τὸ ὃν τοῦ μὴ ὄντος οὐδὲν εὐπορώτερον" εἰπεῖν ὅ τί ποτ᾽ ἔστιν. p. 345 ῈἙ. 4 This reticence, of which it is not 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, such 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 those living personages with whom he stood in no relation of common pur- suits or common friendships, e.g. Lysias, Gorgias, &c. The Pythagoreans, though remote in place, were his friends and correspondents, and in speaking of them he observes the same tule as in the case of his living Athenian contemporaries, in- dicating without expressly naming them. Thus, in the Poli- ticus, p. 285, they are merely denoted as cou οὶ, “ 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, &e. 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 (εἴδη or ὄντα or οὐσία). They are complimented in the dialogue as ἡμερώτεροι, “ more civilized” or “more humane,” than their rude materialistic antagonists: but they are at the same time taken sharply to task by the Eleatic Stranger: and for what? For the absence, from their scheme of Idealism, of that 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 :” οὐ συγχωροῦσιν ἡμῖν τὸ viv δὴ ῥηθὲν πρὸς τοὺς “γηγενεῖς οὐσίας πέρι, 248B; the thing they refuse to admit being neither more nor less than that κοινωνία or μέθεξις τῶν εἰδῶν", which Aristotle cannot or will not under- Like Plato, they distinguish the two worlds of sense and pure ideas, the “γένεσις from the ovata (γένεσιν τὴν δὲ οὐσίαν χωρίς πον stand in his critique of the Platonic Doctrine of Ideas. διελόμενοι λέγετε, 248 A), but, unlike him, they deny that the one acts or is acted upon by the other: they even deny that Being (εἴδη or οὐσία) 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 arguments by which the “Friends of Forms” (εἰδῶν φίλοι, 248 a) are pushed to this admission may not ring sound to a modern ear; but my the capacity of being known’, 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, κοινωνία, 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 1 Soph. 246 a, § 65 Bekk. * Aristotle objects to the term μέθεξις οὐ the ground that it is metaphorical. Now as a logical term, the Platonic μέθεξις is but the counterpart of ὕπαρξις, 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 (ὑπάρχει) to B” and “B partakes of Α ᾽ (μετέχει) are both 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 ὑπάρχει of Aristotle. 3 Ty οὐσίαν δὶ κατὰ τὸν λόγον τοῦτον γιγνωσκομένην ὑπὸ τῆς γνώσεως, καθ᾽ ὅσον γιγνώσκεται κατὰ τοσοῦτον κινεῖσθαι διὰ τὸ πάσχειν, ὃ δή φαμεν οὐκ ἄν γενέσθαι περὶ τὸ ἠρεμοῦν". p. 248 E. 4 Compare 249 D, §75: τῷ δι φιλοσόφῳ καὶ ταῦτα μάλι- στα τιμῶντι πᾶσα ὡς ἔοικεν ἀνάγκη διὰ ταῦτα μήτε τῶν ἕν ἢ καὶ τὰ πολλὰ εἴδη λεγόντων τὸ πᾶν ἑστηκὸς ἀποδέχεσθαι, τῶν 7 αὖ πανταχῇ τὸ ἕν κινούντων μηδὲ τὸ παράπαν ἀκούειν, ἀλλὰ κατὰ τὴν τῶν παίδων εὐχὴν, ὅσα (Ws?) ἀκίνητα καὶ κεκινημένα, τὸ ὄν τε καὶ τὸ πᾶν, ξυναμφότερα λέγειν. This passage, as I 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 Theetetus. This is not the only echo of that dialogue heard in the Sophista, 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, μάλα εὐλαβῶς ἄνωθεν ἐξ ἀοράτου ποθὲν ἀμυνομένων, 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,” ταῖς χερσὶν ἀτεχνῶς πέτρας καὶ δρῦς περιλαμ- Bavovres, 246 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 Theatetus, p. 155 5, 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 Theztetus 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,” (πᾶν τὸ ἀόρατον οὐκ ἀποδεχόμενοι ws ἐν οὐσίας μέρει). These persons are garnished with the epithets “hard,” “stubborn,” “thoroughly illiterate,” σκληροὶ---ἀντίτυποι---μάλ᾽ εὖ ἄμουσοι. 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 ἀντίτυπος of the Thewtetus would be felt to convey an allusion “intelligible to the intelligent.” The μάλ᾽ ev ἄμουσοι finds its echo in the synonymous epithet ἀπαίδευτοι, which Aristotle in the Metaphysica bestows on Antisthenes and his followers+. Every one, however, must see, without further argument, that the description in the T'heetetus 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 1 This epithet I conceive to be justified by Cicero’s notice, ‘* Hi quoque (sc. Megarici) multa a Platone,” Acad. Qu, τι. 42, and also by the brief statement of the Megaric dogmas which Cicero gives us in the context of this passage. 5 In the Philebus—a dialogue which treats of therelation of οὐσία to γένεσι: 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 ptinciple he denominates γένεσις els οὐσίαν, the possibility of which process is precisely what the εἰδῶν pior—the pure idealists of this dialogue—deny. Philed. p. 24, foll. 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, 88 109, 110. ® Compare Soph. 247 c: διατείνοιντ᾽ ἂν wav ὃ μὴ δυνατοὶ ταῖς χερσὶ ξυμπιέζειν εἰσὶν ws dpa τοῦτο οὐδὲν τὸ παράπαν ἐστίν. 4 vit. 8. 97: οἱ ᾿Αντισθένειοι καὶ οἱ οὕτως ἀπαίδευτος. 5.1 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!. Many other stinging pleasantries were interchanged by the leaders of the * Your body has eyes, your soul has none,” was the curt reply of Plato. two schools: and Antisthenes, less guarded than his antagonist, wrote a dialogue “in three parts,” entitled Σάθων, which was avowedly directed against Plato in revenge for a biting reply (Diog. Laert. 111. ὃ 35; vr. ᾧ 16). 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, ὅτε οὐκ ἔστιν ἀντιλέγειν. If we add, that he also wrote four books on Opinion and Science (περὶ δόξης καὶ ἐπιστήμης)» 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 “ἐ 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 1 Tzetzes, Chil. v11. 605; Schol. in Arist. Categ.ed. Brandis, | and brought with him as many of his pupils as he could induce Ρ. 666, 45 and 684, 26; Zeller, G. P. 11. p. 116, note 1. 2 Metaph. v. 29: ᾿Αντισθένης wero εὐήθως μηθὲν ἀξιῶν λέγεσθαι πλὴν τῷ οἰκείῳ λόγῳ ev ἐφ᾽ ἑνός" EE ὧν συνέβαινε μὴ εἶναι ἀντιλέγειν, σχεδὸν δ᾽ οὐδὲ ψεύδεσθαι. Plat. Soph. lL: οὐκ ἐῶντες ἀγαθὸν λέγειν ἄνθρωπον, ἀλλὰ τὸ μὲν ἀγαθὸν ἀγαθὸν τὸν δὲ ἄνθρωπον ἄνθρωπον. The latter passage explains the οἰκείῳ λόγῳ of Aristotle, and the allusion is further deter- mined by the ἀμούσου τινὸς καὶ ἀφιλοσόφου applied to the upholder of the similar sophisms noted at p.259D. In the latter passage occur the following words: οὔ τέ τις ἔλεγχος οὗτος ἀληθινὸς, ἄρτι τε τῶν ὄντων τινὸς ἐφαπτομένου δῆλος νεογενης ὦν. * This is no genuine or legitimate confutation : but theinfant progeny of a brain new to philosophical discussion.” This hangs together with the γερόντων τοῖς ὀψιμαθέσι---““1Π6 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, to follow his example. A similar sarcasm is hurled at Diony- sodorus and Euthydemus, in the Euthyd. p. 272 ο, which not improbably was designed to glance off from them upon some contemporary Eristic. .Antisthenes, we know, was present at the battle of Tanagra, in s.c. 426. He may therefore have been Plato’s senior by some 20 years. 8 καὶ γὰρ ὦ ’yabé, τό γε πᾶν ἀπὸ παντὸς ἐπιχειρεῖν ἀποχωρίζειν, ἄλλως τε οὐκ ἐμμελὲς καὶ δι καὶ παντάπασιν ἀμούσου τινὸς καὶ ἀφιλοσόφου. Θ, τί δή; BH. τελειοτάτη πάντων λόγων ἐστὶν ἀφάνισις τὸ διαλύειν ἕκαστον ἀπὸ πάντων" διὰ γὰρ τὶν ἀλλήλων τῶν εἰδῶν συμπλοκὴν ὁ λόγος γέγονεν ἡμῖν. Soph. 268 Ὁ. 4 διάλογος ἄνευ φωνῆς γιγνόμενος ἐπωνομάσθη διάνοια. Soph. 208 Ὁ. Van Heusde first pointed out the infamous ety- mology lurking in this passage (διάνοια τεδε ἄλογος ἄνευ) The sentiment, without the etymology, occurs in Theat. 189 E: (τὸ δὲ διανοεῖσθαι καλῶ) λόγον Sv αὐτὴ πρὸς αὑτὴν κιἱ ψυχὴ διεξέρχεται περὶ ὧν ἂν σκοπῇ. 160 PROFESSOR THOMPSON, ON THE GENUINENESS OF epitomized by Aristotle, in which 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 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. usually called ‘the Sophists.” It is remarkable, however, that when Plato handles the Cynical Ethics, he treats their author with far more In comparing it with the Pleasure Theory of Aristippus, he speaks of the Cynical system with qualified approbation. leniency than in this dialogue. Avoxepis”, ‘austere or morose,” is the hardest epithet he flings at Antisthenes in the Philebus: he even attributes to him a certain nobleness of character (φύσιν οὐκ ἀγεννῆ), 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 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 logomachy of earlier or inferior thinkers. 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 That the διαρετικοὶ λόγοι, the “amphiblestrie organa‘,” in which he endeavours to catch and land Plato takes occasion to dispose of the doctrines of certain formidable antagonists. first the Sophist and then the Statesman, were regarded by Plato himself in this light, we «ς Ts it,” asks the Eleatic Stranger, ‘‘for the Statesman’s sake alone, that this long quest has been instituted, or is it not learn from his own testimony in the Politicus, 285 D, § 26 Bekk. rather for our own sake, that we may strengthen our powers of dialectical enquiry upon 5. J. Ε, 5. How much less then would a man of sense have submitted to a tedious enquiry into the definition of the subjects in general ? It was doubtless for this general purpose. 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 1 Hence the explanation of Philebus, 448: καὶ μάλα δει- vols λεγομένους τὰ περὶ φύσιν. 2 Phil. 44. : μαντευομένοις οὐ τέχνῃ ἀλλά τινι δυσχε- ρείᾳ φύσεως οὐκ ἀγεννοῦς, λίαν μεμισηκότων τὴν τῆς ἡ- δονῆς δύναμιν, καὶ νενομικότων οὐδὲν ὑγιές....... σκεψάμενος ἔτι καὶ τἄλλα αὐτῶν δυσχεράσματα. 1. Ὁ: κατὰ τὸ τῆς δυσχερείας αὐτῶν ἴχνος. The accomplished and unfortu- nate Sydenham first pointed out the reference in these epithets to the Cynics and their master. The od τέχνῃ of Plato tallies with the ἀπαίδευτοι of Aristotle, and with his own ἄμουσοι, ἅς. 3. P. 202 Ὁ. Simple as the analysis of the Proposition into ὄνομα καὶ ῥῆμα (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. 1. 108, Wyttenb.), Apuleius, Doctr. Plat, 111. p. 203. Comp. Plat. Crat. 431 5. 4 Soph, 235 5. 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 Athenwus*. In this passage we are introduced into the interior of the Academic halls, and the curtain rises upon a group of youths who are 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. 15 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,” ‘improving their dialectical powers” by a lesson in botanical classification. Is a pumpkin a herb? Is it a grass? 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 διαιρετικοὶ λόγοι of the Sophista and Politicus represent what really occurred within the walls of. the 1 ὡς βραχύτερα ἂν yevouéva τοὺς σύνοντας ἀπειργάζετο A. καὶ τί ποτ’ ἄρ᾽ ὡρίσαντο καὶ τίνος γένους διαλεκτικωτέρους καὶ τῆς τῶν ὄντων λόγῳ δηλώσεως εἶναι τὸ φυτόν; δήλωσον, εἰ κάτοισθά τι. εὑρετικωτέρους. Polit. 286 Ἑ. B. πρώτιστα μὲν οὖν πάντες ἀναυδεῖς 211. p. ὅθ. As this fragment has not yet received the τότ᾽ ἐπέστησαν, καὶ κύψαντες attention it deserves, it is printed in full. χρόνον οὐκ ὀλίγον διεφρόντιζον. A, Τί Πλάτων Kar’ ἐξαίφνης ἔτι κυπτόντων καὶ Σπεύσιππος καὶ Μενέδημος, καὶ ζητούντων τῶν μειρακίων πρὸς τίσι νυνὶ διατρίβουσιν ; λάχανόν τις ἔφη στρογγύλον εἶναι, ποία φροντίς, ποῖος δὲ λόγος ποίαν δ᾽ ἄλλος, δένδρον δ᾽ ἕτερος. διερευνᾶται παρὰ τοῖσιν ; ταῦτα δ᾽ ἀκούων ἰατρός τις τάδε μοι πινυτῶς, εἴ τι κατειδώς Σικελᾶς ἀπὸ γᾶς κατέπαρδ᾽ αὐτῶν ἥκεις, λέξον, πρὸς γᾶς ὃς ws ws ληρούντων. B. ἀλλ᾽ oléa λέγειν περὶ τῶνδε σαφῶς" A, ἢ που δεινῶς ὠργίσθησαν Παναθηναίοις γὰρ ἰδὼν ἀγέλην χλευάαζεσθαί τ᾽ ἐβόησαν" μειρακίων Ἂν " τὸ γὰρ ἐν λέσχαις ταῖσδε τοιαυτί ἐν γυμνασίοις ᾿Ακαδημίας ποιεῖν ἀπρεπές. ἤκουσα λόγων ἀφάτων ἀτόπων" B. οὐδ᾽ ἐμέλησεν τοῖς μειρακίοις" περὶ γὰρ φύσεως ἀφοριζόμενοι ὁ Πλάτων δὲ παρὼν καὶ μάλα πράως, διεχώριζον ζῴων τε βίον οὐδὲν ὀρινθείς, ἐπέταξ᾽ αὐτοῖς δένδρων τε φύσιν λαχάνων τε γένη. πάλιν [ἐξ ἀρχῆς τὴν κολοκύντην) KG?’ ἐν τούτοις τὴν κολοκύντην ἀφορίζεσθαι τίνος ἐστὶ γένους" ἐξήταζον τίνος ἐστὶ γένους. οἱ δὲ διήρουν. Com. Grec. Fragm, v. 111. p. 370, ed. Meineke. Vou... Paar |, 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: be 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 discussed is that which we have been considering, for Aristotle describes it as Division by contradictory Differentie’. 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. This 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. 11. ¢. x11. § 6, and Schol. in loc. So Abelard (Ouvrages Inédits. Op. 569, ed. Cousin: coll. pp. 451, 461), distinguishes between those divisions which imply di- chotomy and those which do not: e.g. παντὸς ἑκάστοτε θεμένους ζητεῖν...ἐὰν οὖν [μετα]λάβωμεν, :μετὰ μίαν δύο, εἴ πως εἰσί, σκοπεῖν, εἰ δὲ μή, τρεῖς ἢ στιν ἄλλον ἀριθμόν, καὶ τῶν ἕν ἐκείνων ἕκαστον πάλιν ὡσαύτως μέχριπερ ἂν τὸ κατ᾽ ἀρχὰς ἕν μὴ ὅτι ἕν animal. animal. καὶ ἄπειρά ἐστι μόνον ἴδῃ τις, ἀλλὰ καὶ ὅποσα. 1 understand ᾿ this passage as conveying Plato’s distinction between his own method and that of the Eleatics and their Eristic successors, who acknowledged only a ἕν and an ἄπειρον. 2 For the length of the process will evidently depend on the distance, so to speak, 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 Sophista, as will be seen from the following scheme: 5 1 cH man. horse. ox, &c. man. not man. Porphyry attributes the latter or dichotomous method to Plato. It could not be “ Eleatic,” for each of the contraries would be in that scheme a ‘‘non-ens.” It is remarkable that a similar Divisio Divisionum occurs in the Politicus, 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 Phedrus, describes as a μελοτομία. Κατὰ μέλη τοίνυν αὐτὰς οἷον ἱερεῖον διαιρώμεθα, ἐπειδι δίχα ἀδυνατοῦμεν, δεῖ γὰρ εἰς τὸν ἐγγύτατα ὅτι μάλιστα τέμνειν ἀριθμὸν ἀεί. The division he proceeds to make, is a distribution of “accessory arts” συναιτίοι τέχναι; into seven i ats rer 1 ψυχῆς co-ordinate groups. A similar relaxation is permitted in the πε τὰ οὐ irda creat) : εἰς εν ὡς Subt = Philebus, p. 16D: Δεῖ οὖν ἡμᾶς.....«ἀεὶ μίαν ἰδέαν περὶ | γυμναστική. ἰατρική. νομοθετική. δικαστική. Θεραπεία. δὶ We 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. tinguish the Sophist from the Philosopher, the trader in knowledge from its disinterested In the Sophista it is Plato’s professed intention to dis- 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 Phedrus 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 Phedrus, 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. 16, 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 the physician, and all tendance of the soul in those of 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, 3025, the dichotomy is com- poised in 8 single step: ἐν ταύταις dj τὸ παράνομον καὶ ἔννομον ἑκάστην διχοτομεῖ τούτων. I trust 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 1 comprehend his views they do not seem necessarily inconsistent with my own. The typical 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. Be 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.” Qi1—2 164 Pheedrus, 265 e, ὃ 110, MAI. To δ᾽ ἕτερον dy εἶδος τί λέγεις ὦ Σώκρατες; ΣΩ. Τὸ παλιν κατ᾽ εἴδη δύνασθαι τέμνειν, κατ᾽ Ν La , A , 3 ~ ‘ , ἄρθρα, ἣ πέφυκε, καὶ μὴ ἐπιχειρεῖν καταγνύναι μέρος a7 -~ , , , ν᾽. > e μηὸεν, κακου μαγείρου τρόπῳ χρώμενον" GLA ὥσπερ La ‘ , ‘ }. ” ~ , e ~ ἄρτι τὼ λόγω TO μὲν ἄφρον τῆς διανοίας ἕν τι κοινῇ εἶδος ἐλαβέτην, ὥσπερ δὲ σώματος ἐξ ἑνὸς διπλᾶ καὶ ὁμώνυμα πέφυκε, σκαιά, τὰ δὲ δεξιὰ κληθέντα, οὕτω 4 A ~ , e a tc ας ‘ a e καὶ τὸ τῆς παρανοίας ws ἕν ἡμῖν πεφυκὸς εἶδος ἡγη-- ΄ Ν ΄ e s pate! > ‘ σαμένω TW λόγω, ὁ μὲν TO ἐπ᾿ ἀριστερὰ τεμ- νόμενος μέρος, πάλιν τοῦτο τέμνων οὐκ ἐπανῆκε, ᾿ > eT 4} 5 ‘ ᾽ , , ᾽ πρὶν ἐν αὐτοῖς ἐφευρὼν ὀνομαζόμενον σκαιόν τιν ἔρωτα ἐλοιδόρησε μάλ᾽ ἐν δίκῃ. ὁ δ᾽ εἰς τὰ ἐν δεξιᾷ = , > 4 ¢ a e , ‘ > , ~ τῆς μανίας ἀγαγὼν ἡμᾶς, ὁμώνυμον μὲν ἐκείνῳ θεῖον δ᾽ 4 ΕΣ > 4 s ΄ > αὖ tw ἔρωτα ἐφευρὼν καὶ προτεινάμενος ἐπ- , « ’ ” ok aa! a ἥνεσεν ὡς μέγιστον αἴτιον ἡμῖν ἀγαθῶν. ΦΑΙ. ΣΏ. Τούτων δι ἔγωγε αὐτός τε ἐραστής, ὦ s ᾿Αληθέστατα λέγεις. τὸ ~ ὃ , A “ wv? er > Φαῖδρε, τῶν διαιρέσεων καὶ συναγωγῶν, ἵν᾿ οἷός τ᾽ ὦ ΄ ~ λέγειν τε Kat φρονεῖν: ἐάν τέ Tw ἄλλον ἡγήσωμαι , a ~ - δύνατον εἰς ἕν καὶ ἐπὶ πολλὰ πεφυκόθ᾽ ὁρᾶν, τοῦ- { \ τον διώκω ““κατόπισθε pet’ ἴχνιον wore θεοῖο. Καὶ , ‘ \ ’ 6k a ν \ ᾿ a“ μέντοι Kat τοὺς δυναμένους αὐτὸ δρᾶν εἰ μὲν ὀρθῶς Ἅ ‘ , ‘ > -~ > s , ~ ἢ μὴ προσαγορεύω θεὸς οἶδε, καλῶ δ᾽ οὖν μέχρι τοῦδε ͵ διαχεκτικούς. PROFESSOR THOMPSON, ON THE GENUINENESS OF Sophista, 264 π. EE, Πάλιν τοίνυν ἐπιχειρῶμεν, oxi Covres διχῇ͵ ‘ ‘ ΄ Ul ν ᾽ A ‘ al τὸ προτεθὲν γένος, πορεύεσθαι κατὰ τοὐπὶ δεξιὰ ἀεὶ μέρος τοῦ τμηθέντος ἐχόμενοι τῆς τοῦ σοφιστοῦ κοινωνίας, ἕως ἂν αὐτοῦ τὰ κοινὰ παντὰ περιελόντες, ‘ Pree ΄ ͵ ᾽ ΄ , τὴν οἰκείαν λιπόντες φύσιν ἐπιδείξωμεν μάλιστα μὲν ἡμῖν αὐτοῖς, ἔπειτα δὲ καὶ τοῖς ἐγγυτάτω γένει τῆς τοιαύτης μεθόδον πεφυκόσιν. Jb. 253 νυ, ὃ 82. Τὸ κατὰ γένη διαιρεῖσθαι, καὶ Ω 3 A “- “ ε , ‘@ ef ” μήτε ταὐτὸν εἶδος ἕτερον ἡγήσασθαι μήθ᾽ ἕτερον ὃν ταὐτὸν μῶν οὐ τῆς διαλεκτικῆς φήσομεν ἐπιστή- μης εἶναι; Θ. [Ναί,] φήσομεν... ΖΞ. ἀλλὰ μὴν τό γε διαλεκτικὸν οὐκ ἄλλῳ δώσεις, εἷς ἐγῷμαι, πλὴν τῷ καθαρῶς τε καὶ δικαίως φιλοσόφῳ. Ib. 229 5, 881. Τὴν ἄγνοιαν ἰδόντες εἴ πῇ κατὰ μέσον αὐτῆς τομὴν ἔχει τινά, διπλῆ γὰρ αὑτηὶ ’ bed e A ‘ ν , 9 γιγνομένη δῆλον ὅτι καὶ τὴν διδασκαλικὴν δύο ἀναγ- ΄ , ” a v.09 ey ΄ a co ee , ka Cer μόρια ἔχειν, ἕν ἐφ᾽ ἑνὶ γένει τῶν αὑτῆς ἑκατέρῳ. Politicus, 263 B. - »"- - -» ca μέρος αὐτὸ ἀναγκαῖον ειναι TOU πραγματος οτου περ a> ‘ a a ΕἾ Εἶδος μὲν ὅταν ἢ τον, καὶ ἂν εἶδος λέγηται: μέρος δὲ εἶδος οὐδεμία ἀνάγκη. (This explains the κατ᾽ ἄρθρα 4 πέφυκε of the Pheedrus.) ’ 10. 265 a. Καὶ μὴν ἐφ᾽ 6 ye μέρος ὥρμηκεν ὁ λόγος ἐπ᾽ ἐκεῖνο δυο τινὲ καθορᾶν ὁδω τεταμένα φαίνεται, τὴν μὲν θάττω, πρὸς μέγα μέρος σμικρὸν διαιρούμενον, τὴν δ᾽ ὅπερ ἐν τῷ πρόσθεν ἐλέγομεν, ὅτι δεῖ μεσοτομεῖν ὅτι μάλιστα, τοῦτ᾽ ἔχουσαν μᾶλ- λον, μακροτέραν γε μήν. Ib. 202 p, occurs ἃ specimen of the “unskilful carving” (κακοῦ μαγείρου τρόπον) of the Phadrus. Ei τις τἀνθρώπινον ἐπιχειρήσας δίχα διελέσθαι γένος διαιροίη καθάπερ οἱ πολλοὶ...τὸ μὲν Ἑλληνικὸν (τὸ δὲ) βάρβαρον. ..ἢ τὸν ἄριθμόν τις αὖ νομίζοι κατ᾽ εἴδη δύο διαιρεῖν μυριάδα ἀποτεμνόμενος ἀπὸ πάντων, εἷς ἕν εἶδος ἀποχωρίζων, κιτ.λ, In allusion to Xen. Mem. tv. § 11, a passage noticed by the Master of Trinity, p. 595 of his paper, I may observe that the etymology of Dialectic, ἀπὸ τοῦ dtadéryerv, is undoubtedly vicious, and is nowhere countenanced by Plato. On the contrary, Dialectic is described in the Philebus, 58 &, as ἡ τοῦ διαλέγεσθαι δύναμις. He could not have adopted Xenophon’s etymology, for as we have seen, the Platonic Dialectic includes cvvaywyy as well as διαίρεσις. The etymology was tempting, and Xenophon, who writes very mféch at random upon philosophical subjects, was unable to resist the temptation. who in his History of Philosophy, derives σοφιστὴς from σοφίζειν instead of σοφίζεσθαι, an error in which he has been followed by English scholars who ought to have known better. A similar error is that of Hegel, THE SOPHISTA OF PLATO, &c. 165 APPENDIX II. On the Earth-born (γηγενεῖς) 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, τοῦτο μόνον 1. Democritus, on the contrary, says, νόμῳ ἔστιν, ὃ παρέχει προσβολὴν καὶ ἐπαφήν τινα. πάντα τὰ αἰσθητά, ἐπ ἕῃ ἄτομα καὶ Kevov.—Frag. ed. Mullach. p. 204. 2. ταὐτὸν σῶμα καὶ οὐσίαν wpiCovro: they defined 2. Democritus denies that the sense of touch “substance” to mean corporeal substance only. conveys any true knowledge. Ἡμεῖ τῷ μὲν ἐόντι οὐδὲν ἀτρεκὲς Evvieuev, μεταπῖπτον δὲ κατά τε σώματος διαθιγὴν καὶ τῶν ἐπεισιόντων καὶ τῶν ἀντιστηριζόντων. 3. They despised τοὺς φάσκοντας μὴ σῶμα ἔχον 3. Democritus held “87: οὐθὲν μᾶλλον τὸ ὃν τοῦ εἶναι. μὴ ὄντος ἔστιν, ὅτι οὐδὲ τὸ κενὸν τοῦ σώματος.---- Arist. Met. τ. 4. In other words, that vacuum (his μὴ ὄν) was in every respect as real as corporeal sub- stance. The Cyrenaics are not the “γηγενεῖς, for they admit nothing to be real except the affection (πάθος), 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. vit. 191: Φασὶν οἱ Kupnvaixot κριτήρια εἶναι τὰ πάθης καὶ μόνα καταλαμβάνεσθαι καὶ ἀδιαψευστὰ τυγχάνειν" τῶν δὲ πεποιηκότων τὰ πάθη μηδὲν εἶναι καταληπτὸν inde ἀδιαψευστόν. The case of the Ephesian ῥέοντες is not worth considering, for they acknowledged no οὐσία, as the Earth-born know nothing of γένεσις, which they properly class with the ἀόρατον. The view I have adopted, that the passages in the Theetetus 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 T'heetetus 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 nondum satis animadversa.” Some of the allusions to this philosopher which Winckelmann detects in the Theetetus appear to me doubtful, but I observe with pleasure that he acknowledges the double bearing of the epithet ἀντίτυπος, 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 Dec. 7, 1857.] Tue 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 memory, and I know but one instance (that of the proposition ex equali in ordine perturbatd, 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 ew @quali, and of the doctrine of similar triangles, I beg leave to 6. Β. 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 ὁ, 4; and if these rectangles be so applied together that the sides a and ὦ shall be in a straight line and that the side B shall meet the side 4; the two rectangles will be the complements of the rectangles on the diameter of a rectangle. F ΄ D B a E b G A K L H IT 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 GF'D, EDF, is less than two right angles, produce the lines FG, DE, till they meet in H; and draw HI parallel to FD or GE. If the rectangle under ὃ and 4 is not terminated in the line HJ, let it be terminated by the line KL. Since KL is parallel to 6 or GE and therefore parallel to 177, it will be entirely above or below HI. Now by Euclid, the complements FZ, ET, are equal; but, by hypothesis, FE, EL, are equal; therefore EL is equal to ZI, which is impossible if the line KZ is above or below HJ; therefore KL coincides with HJ, and the rectangle ὦ, A, coincides with the complement FI, and the two given rectangles therefore are the complements, &c, @.E.D. ‘ProrosrTion (B). If the rectangle contained under the lines a, B, is equal to the rect- angle contained under the lines 4, ὃ; and if the rectangle contained under the lines ὃ, C, is equal to the rectangle contained under the lines B,c; then will the rectangle contained under the lines a, C, be equal to the rectangle contained under the lines J, δ. [This is equivalent to the ordinary ex equali theorem, If Gils Ot Ἀπ ἢ), and δι Bo G5 Then will a: ¢:: 4: C] 168 G. B. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS " LM A P a Bae 8 Σ 2] BR BIG T IT ὃ aa ο c E v Cc R Q x 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 ὃ of one be in the same straight line with HE or ὃ of the other. In the right-angled triangles [DH, HFK, the sides including the right angles are equal, therefore the angle FHK is equal to the angle DIH, and is the complement of the angle DHT; therefore JH 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 EI. 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 4; by Proposition (A), LH and HO are the complements of the parallelograms about the diameter of the rectangle LO; therefore IH and consequently JHK (which are in the same straight line) are in the diameter; therefore IHKP 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 apply the rectangle HS whose side F'\S' or HT’ is equal to C; and produce ST and QR to meet in V; and join JV. Then, proceeding from the hypothesis that the rect- angle contained under ὁ and B is equal to the rectangle contained under 6 and C, it will be shewn in like manner that XHIV is a straight line. Therefore PK HIV 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, 0, and HX is the rectangle contained under 6, 4; therefore the rectangle contained under the lines a, C, is equal to the rectangle contained under the lines A, 56. @£E.D. Corottary. 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 6, B, homonymous sides; the rectangle contained under the lines a, B, is equal to the rectangle contained under the lines A, ὃ. FOUNDED ON ORDINARY GEOMETRY, &c. 169 [The equivalent theorem in νι φϑ λοι is ct 3B] eel 4 ye] Apply one ils upon the other as in the right-hand diagram, so that the side 6 meets the hypothenuse 4 at right angles, and the vertex of the angle opposite b meets the vertex of the angle included by 4 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 GF-D. Therefore the rectangle under a and B 15 equal to the rectangle under A and ὅ. @.x.D. Prorosition (D). If a, 6, and A, C, are homonymous sides of equiangular triangles, the rectangle contained under a, C, will be equal to the rectangle contained under ὁ, A. σ ce ΑΝ “4 From the angles included by the sides A, C, and a, ¢, let fall the perpendiculars B, 6, 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 4, ὃ. Again, Rectangle under 8, C, is equal to rectangle under B, c. Therefore by Proposition (B), Rectangle under a, C, is equal to rectangle under A,c. a@.£.D. Proposition (E). If ὃ, ο, and B, C, are homonymous sides including the right angles of two equiangular right-angled triangles, the rectangle contained under ὁ, C, will be equal to the rectangle contained under ὁ, B. This may be considered a case of the last proposition, or it may be treated independently thus. Vou. X. Parr I. 22 170 6. Β. 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 /-DG is equal to the triangle EDH. But the rectangle under ὃ, 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 ¢, B. @.£.D. Prorosition (F). If the rectangle contained under the lines a, B, is equal to the rect- angle contained under the lines 4, 6; the parallelogram contained under the lines a, B, will be equal to the equiangular parallelogram contained under the lines A, ὃ. [This is equivalent to the proposition, διαὶ b τ: By Thena: 6:: A.cosa : 8.005 α.} F D E B B a α H Τ εἰ 5 A ἘΠ L 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 DG is equal to the rectangle HZ, or that the rectangle under a, HG, is equal to the rectangle under ὁ, 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 JL, B. But by hypothesis, the rectangle under B, a, is equal to the rectangle under A, 6; therefore by Proposition (B), the rectangle under EG, a, is equal to the rectangle under IL, 6. Or, the parallelogram under the lines B, a, is equal to the equiangular paral- lelogram under the lines 4, ὃ. @.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. 1. 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. Tueorem. 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. Let A, B, C, be the centers of the given circles. Let N be the center of the circle whose radius WO is equal to the radius BX, 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 WF, 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, MC. ‘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—2 172 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 FJ is parallel to NB. It will be shewn in exactly the same way that the parallelogram under AF, BJ, with the angle FAI, is equal to the parallelogram under AJ, NF, with the angle FAJ, But the parallelogram under AF’, BI, with the angle 241, is the excess of the parallelogram under AF, AI, with the angle F'AJ, above the parallelogram under AF, AB, with the same angle; or is the excess of double the triangle 4.11 above double the triangle AFB, or is double the triangle BFJ. Similarly the parallelogram under 47, NF, with the angle 7.41, is double the triangle NFJ. Therefore the triangles BFI, ΝΕ, are equal; therefore FT is parallel to VB. And as MF and FT are both parallel to NB, MF and FT are in the same straight line. ᾳ. E. ἢ. ADDENDUM. I am permitted by Professor De Morgan to transcribe the simple process for demon- strating the theorem of ew @quali in ordine perturbata, to which allusion is made above. If 2.2 6.262 2G, and 6:¢%# A: B, Then will a: ¢ Ὁ 4: Οἱ To exhibit the process more clearly to the eye, use the connecting mark ——~ for one ratio and <> for the other; then the theorem stands thus, If a-~b te 2e, oe pee 1] then a: ὃ " 43 Ὁ. To prove it, take a fourth quantity d, such thata : ὃ :: ¢ : d. Then b= >Re -—wd. But A= BRC, Therefore, ex equali, ὃ : ἃ :: A: C. But, because a : ὃ :: 6 : d, therefore alternando, a: ¢ :: ὃ : d. Substituting there- fore the ratio a : ὁ for ὃ : din the analogy just found, ΓΕ 33. At GG, ᾳ. E. Ὁ. G. Β. AIRY. Rorat OpsErvatory, GREENWICH. September 2, 1857. X. On the Syllogism, No. ΠΠ, and on Logic in general. By Avaustus Dz Moraan, F.R.AS., of Trinity College, Professor of Mathematics in University College, London. [Read Feb. 8, 1858.] I pur 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. vitr. Part 3; Vol. 1x. Part 1). the object of logic; on its present state; on the opinion of the world with respect to it; on It will best be disposed under two heads, [I shall first put together remarks 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, exceptt in perspicuity, accuracy of expression, and the like. I did not know that very high authority was then teaching its alwmni to assert that logic had always » There is an intelligible translation of Kant’s logic, and, as I judge by comparison with Tissot, a good one, by John Richardson. London, 1819, 8vo. + Of Lambert’s additions Kant says that like all legitimate subtilties, they sharpen the intellect, but are of no material use. Logic thinks about thought: what for? that we may think the better, that we may sharpen the intellect, Conse- quently, every part of logic which makes us think more acutely conduces to the very use of logic itself. No part of logic is of any material use, in Kant’s sense of the word. The scaffolding by which the house was built is of no use to the inhabitants, except indeed when repairs or additions are wanted. But the main question of the utility of logic refers to education, during which the scaffolding is up. 174 Mr DE MORGAN, ON THE SYLLOGISM, No. III, been one sided, deprived of much scientific truth, encumbered with much scientific falshood, perverted and erroneous in form, and given, in some of its doctrines, to impeach the truth of the laws of thought on which it is founded, In one extreme of opinion, logic, language, and common sense are never at variance: in another, Aristotle exhibits the truth partially, not always correctly, in complexity, and even in confusion. Between these opinions I am not obliged to choose. I am satisfied, with the satisfaction of one long used to the distinction between demonstrated and probable conclusion, that the old logic is, so far as it goes, accurate in method and true in result; that is, as to the quod semper, quod ubique, quod ab omnibus : but without affirming that all that is called necessary is necessary, or that all that is called natural is natural. I feel equally sure that it is only a beginning; that it contains but a small part of the whole which it arrogates to itself in its old aspirations and its modern defi- nition; and that the low estimation in which a large part of the educated world now holds it is to be traced to consequences of this incompleteness. II. Logic inquires into the form of thought, as separable from and independent of the matter thought on. To every proposal for a new introduction there is but one answer ;—-You outstep the bounds of logic, you introduce material considerations. On this point the first question is, What is the distinction of form and matter ?—the second, Who are best able to judge of it ? The form or law of thought—asserted differences between these words being of no im- portance here—is detected when we watch the machine in operation without attending to the matter operated on. The form may again be separable into form of form and matter of form : and even the matter into form of matter and matter of matter; and so on. The modus ope- randi first detected may be one case of a limited or unlimited number, from all of which can be extracted one common and higher principle, by separation from details which are still differences of form. Take a nut-cracker, two levers on a common hinge. Put a bit of wood between the levers to represent filberd, walnut, beechnut, almond, or any other kind of nut. We have here what a logician would call the form of nut-cracking: and, imitating his practice of in- sisting that he has obtained pure form so soon as he has effected one separation, we may say that we have got the pure form of nut-cracking. But when we come to consider the screw, the hammer, the teeth, &c. we begin to apprehend that the pure form of nut-cracking is strong pressure applied to opposite sides of the nut, no matter how; and this even though we may detect in all the instruments the principle, as we call it, of the hinged levers. The logician is not much accustomed to the working presence of his own great distinction : the mathematician deals with it unceasingly, though with little apprehension of its existence, in most cases. Though logic has been in waking life for at least fifteen hundred. years, its real definition has not been in recognised existence during the fifteenth part of that time: this definition has indeed been obeyed in many points, it has been caught for a minute and let go again, it has been seen through a glass darkly,—at any time from Aristotle inclusive: it is only in very modern days that it has been seized, stript of its coverings, and firmly fixed in its place, And the first imperfect introduction, and the perfect recognition, have been the work of mathematicians. AND ON LOGIC IN GENERAL. 175 Of the two philosophers who might have made the distinction of form and matter exercise a strong influence over their systems, Aristotle did it, and Plato did not. Plato’s writings do not convince any mathematician that their author was strongly addicted to geometry; they shew at most that he may have been well versed in it: I have no objection to say that geo- metry helped him in his colouring. We know that he encouraged mathematics, that his followers form a school, and that the reputation of the school has given the character of a geometer to the founder. But if—which nobody believes—the μηδείς ἀγεωμέτρητος εἰσίτω of Tzetzes had been written over his gate, it would no more have indicated the geometry within than a warning not to forget to bring a packet of sandwiches would now give promise of a good dinner. But Aristotle was a mathematician, versed in that science and addicted to it: geometry aided him in the tracing of his outline. even after rejection of those which are doubtful, some of which, supposing him to be only a putative father, show that a very positive mathematical character was assigned to him by his This appears throughout his writings, successors. that many, including some who should have known better, have assigned the form of thought But the definition was To him we owe such perpetual indication of the distinction of form and matter to him as his definition of logic, giving him the word into the bargain. never distinctly conceived in that character until the last century, when it was propounded by a philosopher whose earliest studies had been in mathematics, which he had taught in conjunc- tion with logic for fifteen years before he gave himself up to the study of the pure reason. If, between Kant* and Aristotle, there were one leader of philosophical opinion who more nearly And the history of man in species analogises with what we have seen of man in individuals: we trace our mathe- than another caught the conception, it was the mathematician Leibnitz. matics to the Greeks and the Hindoos, the two independent cultivators of systems of logic in which form is investigated for its own sake, though the separation is indistinctly conceived -by both. Of the Romans, we only know that they originated nothing, either in mathematics or in logic: and it is just worth notice here that Boethius, the only Roman who gave us a summary of Aristotle, was the only Roman who gave us a summary of Euclid. The separation of mathematics and logic which has gradually arrived in modern times has been accompanied, as separations between near relations generally are, with a good deal of adverse feeling. Great names in each have written} and spoken contemptuously of the other ; while those who have attended to both are aware that they have a joint as well as a separate value. This alienation of the two sciences has furnished two magazines to those who would put down all education except that which immediately conduces to production of wealth: in * It is only of late years that, in this country at least, Kant’s | distinction.” definition has been clearly apprehended, and its truth sincerely felt. If the inquirer will look out for English works preceding 1848, or thereabouts, which state Kant’s definition as an exist- ing thing, not to speak of adopting it, he will have some diffi- culty in finding one. In some old notes of my own, made after comparison of Aristotle, some of the medievals, and Kant, I find the following sentence: “I should say [of formal and material] that the great leader saw the distinction, that the .Schoolmen made the distinction, and that Kant bwilt wpon the + There is no occasion to refer to any of the ordinary exhi-. bitions, whether dissertations in favour of ignorance, or orations in contempt of knowledge. But there is one which deserves preservation for its humour, and which may be lost with an ephemeral pamphlet, if not elsewhere recorded. An Oxford M.A. writing on education, about ten years ago, advocated some pursuit of mathematics: for, said he, man is an arith- metical, geometrical, and mechanical animal, as well as a ratio- nal soul, 170 Mr DE MORGAN, ON THE SYLLOGISM, No. III, fact, if what either party has advanced against the other be true, the common opponent has a good case against both, provided only mathematics enough for a higher kind of land-surveyor be exempted from the common doom, and made a part of professional education. There never was in history the time at which mathematics, in any branch, wanted a pal- pable separation of form and matter: and mathematicians have always seen the separation, though they have not always rightly apprehended the relation of the components. They have spoken much of abstraction, a word truly applied to their function: but they have not duly distinguished between abstraction of colleague qualities from each other, and abstraction of the instrument from the material. They have also dwelt much on generalisation, a word so truly descriptive of what is always taking place within the precinct, that they have oftentimes made it give name to the fence. The first element of mathematical process is the separation of space from matter filling it, and quantity from the material guantuwm: whence spring geometry and arithmetic, the studies of the laws of space and number. Distinctions which are of form in arithmetic become material in algebra. The lower forms of algebra become material in the algebra of the functional The functional form becomes material in the differential calculus, most visibly when this last is merged in the calculus of operations. But, though the distinction of form and matter be very certainly present to those who can see it, it is equally certain that many fol- lowers of the mathematics have their ideas of the distinction as dark as those of any of the old logicians. The difference is that the mathematician cannot help dealing with the thing in question, though under a name of too little intension: he cannot but be sensible of abstraction; but he may be unused to remember that he abstracts form from matter. ‘The logician on the other hand may, as often was the case, have his system cast in so material a mould, that he is hardly sensible even of abstraction: and when the fault is not palpably committed in the treatise, the individual reader may, of his own inaptitude to abstract except under symbolic compulsion, convert formal logic into material. symbol. Accordingly, the separation of form is often learned language to the logical student, with a bad dictionary to read it by: to the mathema- tician it is as often M. Jourdain’s prose, and nothing more. To the logician it is a collect for certain holidays; it is the paternoster of the mathematician, who may run it over without thinking of the meaning, if he ever knew it. And these tendencies, large in amount in the learner, have their sway even in the books he learns from, and in the discussions of the highly informed: the great distinction of form and matter is more in the theory* of the logi- cian than in his practice, more in the practice of the mathematician than in his theory. “ lam fully aware of the boldness of my comparison of the logician and mathematician, and of the audacious appearance which it is likely to present to a class of inquirers who have hitherto been allowed to distribute functions to the branches of human knowledge pretty nearly in their own way. My aver- ments are of that kind which nothing but success will justify: and about which controversy is useless. It is not competent to those who are only logicians, and to those who are only mathe- maticians, to settle a question in which the alleged unfitness of either to decide is a part of the matter to be decided: still less is it competent to the few who unite both characters to demand of the others that they shall see this. Time must settle it; and I believe this will be the way. As joint attention to logic and mathematics increases, a logic will grow up among the mathematicians, distinguished from the logic of the logi- cians by having the mathematical element properly subordinated to the rest. This mathematical logic—so called quasi ἴωσι a non nimis lucendo—will commend ‘itself to the educated world by showing an actual representation of their form of thought— a representation the truth of which they recognise—instead of a mutilated and onesided fragment, founded upon canons of which they neither feel the force nor see the utility. AND ON LOGIC IN GENERAL. 177 111, nothing but the form. must be taken into account, and consequently be overtly expressible, in logic: for logic must Logic bears on its modern banner, The form of thought, the whole form, and It has been excellently well said that whatever is operative in thought be, as to be it professes, an unexclusive reflex of thought, and not merely an arbitrary selec- tion,—a series of elegant extracts,—out of the forms of thinking. Whether the form that it exhibits be stronger or weaker, be more or less frequently applied, that, as a material and contingent consideration, is beyond its purview. Nevertheless, so soon as a form of thought is exhibited which does not come within the arbitrary selection, the series of elegant extracts, it is forthwith pronounced material :— : St. Aristotle! what wild notions ! Serve a ne eweat regno on him! The proper reply to every accusation of introducing the material where all should be formal, is as follows. its form: therefore this thought has its form. Logic is to consider the whole form of thought : your logic either contains the form of this thought, or it does not. If it contain* the form of this thought, shew it: if not, introduce it. I shall now state three instances of the objection. In my last paper, as in my work on Formal Logic, I separated form from matter in the The copula performs certain functions ; it is competent to You say this thought or process is material: now every material thinking has copula of the common syllogism. those functions; it is competent because it has certain properties, which are sufficient to vali- date its use, and, all cases considered, not more than sufficient. The word ‘is,’ which identifies, does not do its work because it identifies, exceptt in so far as identification is a transitive and convertible notion: ‘A is that which is Β΄ means ‘A is B’; and ‘A is B’ means “ B is A’. Hence every transitive and convertible relation is as fit to validate the syllogism as the copula ‘is, and by the same proof in each case. Some forms are valid when the relation is only transitive and not convertible; as in ‘give.’ Thus if X—-Y represent X and Y connected by a transitive copula, Camestres in the second figure is valid, as in Every Z—Y, No X—Y, therefore No X—2Z. * When I see a chapter in a book of logic headed On ma- terial and formal consequence, distinguishing “4 =B, B=C, therefore d=C” as material from “4 is B, B is C, therefore A is C” as formal, I am at first inclined to think that the distinction of formal and material is that of contained and not contained—in Aristotle. But when the title-page shews me an author whose mind is as free from the sway of that distinction as my own, I am compelled to have recourse to the difference between the ideas of form belonging to the mathematician and to the logician. Is there any consequence without form? Is not consequence an action of the machinery? Is not logic the science of the action of this machinery ? Consequence is always an act of the mind: on every consequence logic ought to ask, What kind of act? what is the act, as distinguished from the acted on, and from any inessential concomitants of the action ? For these are of the form, as distinguished from the matter. What is the difference of the two syllogisms above? In the first case the mind acts through its sense of the transi- tiveness of ‘equals :’ in the second, through its sense of the transitiveness of ‘is.’ Transitiveness is the common form: the Vou. X. Parr I. difference between equality and identity is the difference of matter. But the logician who hugs identity for its transitive- ness, cannot hug transitiveness: let him learn abstraction. + I again call the reader’s attention to the pure form of nut- cracking, with which I began. The syllogism is the nut to be cracked. I believe I have got to the pure form, which equally applies to two levers, a screw forced into a receptacle, Nas- myth’s steam-hammer, the collision of a couple of planets, as the case may be: the common form of all being pressure enough applied to opposite sides of the nut, The logician in- sists upon it that the pure form is a couple of metallic levers, with friction-studs, if that be the proper name, to prevent the nut from slipping aside, and such a hinge that, according to the way we turn it, the levers give convenient entrance to a common nut or a walnut. All his additions to the pure form I admit to be usual and convenient: but I affirm and main- tain that whatever can crack a nut, and does crack a nut, is a nut-cracker; and being a nut-cracker, must be considered as a nut-cracker, and included among nut-crackers, in every trea- tise on the whole form of nut-cracking. 23 178 Mr DE MORGAN, ON THE SYLLOGISM, No. III, For if any one X—Z, this with Z—Y, gives X—Y, which is excluded by the second premise. To this the objection is that the process is material, for that it is of the matter of the proposition whether give will or will not do: that towch, for instance, will not do, Does not this,—from a living writer who in combination of logical learning and logical acumen is second to none—corroborate my assertion that the logician has the distinction of form and matter’ more in his theory than in his practice? I might as well say that ‘Every X is Y’ is a mate- rial proposition: it is of the matter of X and Y whether it be true or no. In the following chain of propositions, there is exclusion of matter, form being preserved, at every step :— Hypothesis. (Positively true) Every man is animal Every man is Y Y has existence ——_—__—_——— Every X is Y X_ has existence Every X — Y is a transitive relation a of X—Y a a fraction < or = 1. (Probability 8) a of X—Y B a fraction < or =1. The last is nearly the purely formal judgment, with not a single material point about it, except the transitiveness of the copula. But ‘is’ is more intense than the symbol ——, which means only transitive copula: for ‘is’ has transitiveness, and more. Strike out the word transitive, and the last line shews the pure form of the judgment. The same objection has been raised to the law of inference when the middle term is definitely quantified. If the fractions a and β of the Ys be severally As and Bs, and if a+ be greater than unity, it follows that some As are Bs. To this it is objected that whether a+ be or be not greater than unity, is material. No doubt it is; and so is the case of the logician’s canon of syllogism, that the middle term must be universal in one or both premises. The logician demands a=1, or 8=1, or both: he can then infer; but only because he knows that when more in number have been named than there are separate things to name, some must have been named twice. But he does not know this better of 1+/3 than of 2+ (more than 3): or if he did, the difference of form and matter is not merely difference of arithmetical facility. The writer against whom I am contending declares that, as a logician, he cannot know that 2 and 2 make 4. I-do not ask him for so much: I do not ask him to know that there are cases in whicha+$>1. What I say is this, that in every case in which it shall happen (if ever it do happen, which is by hypothesis more than we know) that a+ > 1, in each of these cases he is bound, as a logician, to infer that some As are Bs. And this instance is another corroboration of my assertion that the distinction of form and matter is more in the theory of the logician than in his practice. As a third instance, I note that the limited universe, and its division into two contraries, are pronounced material, because it is mot by logic we learn that when property is the universe, real and personal are contraries, Neither by logic do we learn that every man is animal; but by logic we analyse our use of this proposition in conversion, in inference &c. Similarly, by logic we learn how we use contraries in inference &c. But what things are contraries, logic no more needs to inquire than law needed to inquire who wore the crown AND ON LOGIC IN GENERAL. 179 before she settled whether writs should run in the name of the King de facto or of the Pre- tender. ‘ A little consideration will shew us that every inference which is anything more than pure symbolic representation of inference is due to the presence of something material: even a derived or compound symbol, representing inference, shews the presence of something material. Here are two purely formal propositions, in which P, Q, R, S, represent individual objects of thought, and —A-— indicates a relation A :— P -A- 9 R -B- 8 P stands in A-relation to Q and R in B-relation to S. What are we to infer? Now rub out R, and for it write Q. This is material: it is now seen to be of the matter of our system that the second subject is the first predicate. And now we have P --Α-- Q, and Q -B- S. Can we infer anything? With the form of combination of relations in our thoughts, we may symbolise it, and say P ~-AB- S. Now make the relations material: let -- Α-- and —B-— each be ‘is.’ then we have a material inference; P is Ὁ, Q is S, therefore PisS. In common logic, the objects of inference, being terms expressed in general symbols, are void of matter ; the relations between them, and the modes of inference, are material: I speak of logic as it is. Many relations have a common form: the logician cannot yet see that when many cases, no matter what, proceed upon a common principle, his concern is with that principle. It is his business to apprehend the principle and to shew, as to the modus operandi of the mind, how containing cases severally contain it, and apply it. I am charged with maintaining that thought is a branch of algebra, instead of algebra a branch of thought.. The answer is easy enough. Logic considers, not thought, but the form of thought, the law of action of its machinery. Psychology herself does not know what thought is: and the odds are that if she did she would not feel bound to tell logic. Thought, the genus, has parts of its machinery, usually under cover, which work by daylight in algebra, the species, to every one who has meditated on the principles of algebra. He who makes me confound all other thought with algebra, because I call attention to what is more visible in algebra than in other thought, though it exists in all thought, must make his own logic responsible for the inference, not mine. He may hire a soldier to cook his victuals, because both soldier and cook cut flesh with steel: but neither Mr Boole, the greater culprit, nor I, the lesser* one, have done anything to deserve an invitation to the feast. I might with much more justice charge the logician with affirming all thought a branch of geometry, instead of geometry a branch of thought. By processes nearly resembling those which led Des Cartes to affirm that space is all the essence of matter, he reduces all thought of comparison to the assertion or denial of containing and contained. These are originally terms of space-relation: and his only syllogism, his universal includent of all argument, can be fully symbolised by areas: a practice which many logicians dislike, and with reason, for it tells tales, I have pointed out, in my second paper, the syllogism in which the copulee may be any relations whatever. The copula of cause and effect, of motive and action, of all in which post hoc is of the form and propter hoc (perhaps) of the matter, will one day be carefully ἘΠ Not meant for extenuation: I wish I were the greater one. 23—2 180 Mr DE MORGAN, ON THE SYLLOGISM, No. III, considered in a more complete system of logic. The cases in which A, simultaneous with B, is either cause or effect according to the attribute considered, will be duly symbolised. For instance, it is disputed whether men dive for pearls because pearls fetch a high price, or whether pearls fetch a high price because men dive for them: it is one or the other, according to the attribute of the actions held in view. Considered as volitions, the diver is willing to dive because the lady is willing to pay dear for her necklace. As necessities, the lady must pay dear because the diver must dive. The word because is the heading of a chapter in the form of thought, of which many a complexity is yet unanalysed, simply because it is possible to reduce relation to class, by throwing ‘X has A-relation to Y’ into the form ‘ X is in the class of objects having A-relation to Y.’ Hence, to the world at large, logic is neither the form of their thought, nor the matter, nor the junction of both. The judgment of the logician is only one of the judgments of mankind. When a common person says ‘Achilles killed Hector,’ his objects of thought are the two heroes: his mode of thinking them is in the relation of slayer and slain in time past. The logician demands that he shall think himself to be identifying by the verb ‘is’—either Achilles with the former slayer of Hector, or Hector with the former slain of Achilles, or slaughter with the former action of Achilles on Hector, or time past with the date of that action. All these forms are unquestionably coexistent and coextensive with the relation affirmed: out of any one all the others may be evolved; they are different dichotomies and reintegrations of a coexistence of four things. But neither reintegration represents the manner in which the relation is held in thought. Each dichotomy makes it possible that a contradiction may step in, which the reintegration denies: one of them shews a front to the assertion that Patroclus killed Hector, another to the assertion that Achilles was Hector’s defender, &c. And so it always happens: a person who wants to signify that ‘ Achilles was the person who killed Hector’ will take care, on the principle of not saying one thing when another is meant, to avoid the phrase ‘Achilles killed Hector, or else to supply ‘was the person who’ by emphasis on Achilles: unless it be a person who has been long in the hands of Giant* Maul. In all pro- positions, existence is predicated of the terms in the fact of predication. When I say X is Y, I do not mean ‘if X exist and Y exist, then X is Y:’ I mean that X and Y do exist, and that they are the same. Accordingly, when I am told that ‘Achilles ¢s the former slayer of Hector,’ it is as if it could not be disputed that Hector-was slain, so that the only question remaining is, Who killed him? For the books of logic give no way of denying ‘X is Y’ except ‘ X is not Y.’ But should I be told ‘Achilles killed Hector’ I should not receive it in this way, nor should I believe it was so intended. I should receive it as an equally balanced combination of elements, in which the dichotomy is left to myself, to be made according to my own mode of assent or denial, including a right given to me to preserve the existing balance. I see great difference in the propositum between ‘ This house was built by Jack’ and ‘ This is the [or even a] house that Jack built.’ Granting it true that either of the logician’s forms will give as much * According to incomparable John Bunyan, this worthy | begin to fight, and the opponent objects to the respondent, lived at the end of a dark valley and “did use to spoil young | “These be but generals, man, come to particulars.” Maul pilgrims with sophistry.” What was hinted at appears in this, | was the most difficult giant to kill of Bunyan’s whole troop. that Mr Greatheart and the giant settle the guestio before they AND ON LOGIC IN GENERAL. 181 inference as the simple relation, it does not follow that the logician’s form is the form of thought we actually employ in inference. It is one thing to say, I can shew you by such and such reductions how to demonstrate the only inference these premises will give; and quite another thing to add, Therefore this is the way you infer. IV. Logic is both science and art: and the art, the logica utens, ought to be a prepara- tion for sure and rapid material application. The proposition of the world at large is highly complex: it is loaded with what I shall call charges. disjunctive ; it introduces allusions, for reinforcement, for explanation, for justification of its It has complex terms, conjunctive and appearance, for colouring and effect. It gives reasons, takes syllogisms into the description of terms, and implies assertions in giving reasons, leaving the assertions to be supplied from their reasons. It is a tapestry, of which the logical form is only the original web. It undergoes conversions in which idiom demands synonymes: but the logica docens keeps clear of the whole theory of complex terms by throwing the proposition into disjunctive or dilemmatic forms which the actual form of thought does not recognise, Is the student of logic, gene- rally speaking, prepared rapidly to analyse the two following propositions, and to say whether or no they must be identical, if the identity of synonymes be granted ? The suspicion of a nation is easily ex- cited, as well against its more civilised as against its more warlike neighbours ; and such suspicion is with difficulty removed. When we see a nation either backward to suspect its neighbour, or apt to be satisfied by explanations, we may rely upon it that the neighbour is neither the more civilised nor the more warlike of the two. This, under the symbols I have used and shall use, is the conversion of the form A, B))CD into c, d))ab, The world would have treated logic with more respect, if it had led up to such conversions as the above. But it lands us and leaves us, as to conversion, in ‘ Some tyrant is cruel’ turned into ‘Some cruel is tyrant,’ or the like: a needful commencement, but a lame and impotent conclusion. I will now take a syllogism, one syllogism, well charged* certainly, but only with charges * Ofall the writers on logic whom I have examined, John Milton is the one who delights in extracting the syllogism from its loading: his instances are almost entirely from the Latin poets, which he probably needed no sight to recall. Milton’s logic was published two years before his death. ‘ Joannis Miltoni, Angli, Artis Logice plenior institutio, ad Petri Rami methodum concinnata’ (London, Impensis Spencer Hickman, Societatis Regalis Typographi, 1672, 12mo, portrait). The logic of Ramus was adopted by the University of Cambridge, pro- bably in the sixteenth century. George Downame, or Downam, who died Bishop of Derry in 1634, was prelector of logic at Cambridge in 1590, His ‘Commentariiin P. Rami... Dialec- ticam..,.’ (Frankfort, 1616, 8vo,) is an excellent work. The Cambridge book then most in use was the Dialectica of John Seton, first published (Ames) in 1563, and repeated down to 1611 at least: it is noticed by Dr Peacock as the book to some editions of which (from 1570 onwards, if not before, I find) Buckley’s arithmetical verses are appended. It is not a Ramist book: the presumption is that Downam was the Cambridge apostle of his doctrine. Ramism fixed a mark upon Cam- bridge which it has never lost to this day; that is, if the acts in divinity, &c. be still kept in the old form. The distri- bution of the syllogism into three conditionals, ‘Si A sit B, cadit questio; sed A est B, ergo cadit questio, &c.’ is pure Ramism, both as to form and phrase. Never having paid any attention to Ramist logic, I never could understand this form. No one could inform me: even a question sent to the Notes and Queries produced no reply except an ingenious conjecture that the casus guestionis explains Shakspeare’s meaning of the obscure words ‘loss of question’? in Measure for Measure, act ii. scene 4: a phrase on which commentators were so far to seek that Johnson proposed ‘‘toss of question.’’ And so it stood until 1 happened to propose the difficulty to Prof. Spald- ing of St Andrews, who replied that an explanation might be presumed if we knew, or could assume, that this form was intro- duced by Ramists. Though cognisant of Cambridge Ramism, I had never had the sense to put the two things together. I greatly regret the abolition of the act for the B.A. degree. It was the most useful of the exercises, and the most trying. 182 Mr DE MORGAN, ON THE SYLLOGISM, No. III, which are incessantly used. I insert it for the consideration of those who, for want of advice to the contrary, imagine that the logical gymnastic can afford no higher exercise than the per- ception of ‘No cruel is kind, some cruel is tyrant, therefore some tyrant is not kind’, duly chronicled as Ferison* of the third figure, cousin by the conversion side to Ferio of the first. The following single, though not simple, syllogism is an extract from a letter to a person who had supposed, from some circumstances of character and fact, that a common friend of his own and of the writer must have been the person who had figured in the narrative of a very silly proceeding :— “We both see clearly enough that he [the hero of the narrative] must have been rich, and if not absolutely mad, was weakness itself subjected either to bad advice, or to most unfavourable circumstances. How then can‘you persist in identifying him with the friend of whom we are now speaking; who was indeed very rich, and easily swayed, and so far, we will say, not distinguishable from our hero; but who was conspicuous for clearness of head and sobriety of fancy; who never sought serious counsel except from his father’s old friends, and you know what men they were ; and who passed his youth in severe study varied only by useful exertion, and his manhood in domestic life and country occupations.” Says the man of the world to the logician, I am very clear that two men who are proved to be different cannot be the same: but all I learnt at college about identity and difference, and excluded middle into the bargain, has done nothing towards putting me into a condition rapidly to assert or deny that the advocate has put the principle of difference between the rich fool and his rich friend. Here are two complex descriptions one of which contradicts the other. The description of the rich fool excludes him from either of three classes: the descrip- tion of the rich friend places him in one of those classes: the two cannot then be the same. In the symbols I use—and symbols will one day be the scaffolding of logical education, though useless then, as now, to all who have not mastered them—the argument is expressed as follows. Η is the rich fool; h any other person; H’ the rich friend; R rich, r not rich; W weak, w not weak; A badly advised, a not so; C unfavourably circumstanced, c not so. H)) R[M, W(A, C)]; contrapositively, r, m (w, ac) )) h; or r, mw, mac)) h; but H’)) mac; whence H’)) h; or H’): (H. The syllogism itself is the web of an argument, on which the tapestry of thought is woven; the primed canvas on which the picture is painted. The logician presents it to the world as the tapestry or the picture: he does this in effect by the position he makes it occupy; for he sends the primed canvas to the exhibition. And the world does not see that, though the syllogism be a mere canvas, it stands to the thinker in a very different position from that in which the canvas stands to the painter. Call the historian or the moralist a practised artist at a thousand a year, and I am well content that his structure of the canvas shall be valued at ten shillings a week: it would not hurt my argument if it were valued at a halfpenny. For the painter can and does delegate the preparation of the canvas; the historian cannot put out his logic. He must do it himself as he goes on; and he must do it well, or his whole work is spoiled. * I think as I always did of the admirable ingenuity of these words, for their purpose: they are the most meaning words ever made, AND ON LOGIC IN GENERAL. 183 I will take an example from one of the unusual forms of syllogism. Say “ The time is past in which the transmission of news can be measured by the speed of animals or even of steam; for the telegraph is not approached by either.” Is this a syllogism? Many would say it is not; but wrongly. Throw out the charges, the modal reference to past falsehood and present truth, the advantage of the telegraph, its superior speed, the reference to progress conveyed in even—and we rub off the whole design of the picture. But the ground which carried the design is a syllogism. In old form it is Darapti, awkwardly. All telegraph speed is (not steam speed) All telegraph speed is (not animal speed) Therefore Some (not animal speed) is (not steam speed). In the system which admits contraries it is a syllogism with two negative premises, and a form of conclusion unknown to Aristotle: it is, in the symbols I use, the deduction of )( from )* ()" ( No animal speed is telegraph speed No steam speed is telegraph speed Therefore Some speed is neither animal nor steam speed. When this is presented, a person would naturally ask, What then? The answer to this question is seen when the charges are restored, and the sentence takes its proper place in the whole argument. : V. A great objection has been raised to the employment of mathematical symbols: and it seems to be taken for granted that any symbols used by me must be mathematical. The truth is that I have not made much use of symbols actually employed in algebra; and the use which I have made is in one instance seriously objectionable, and must be discontinued. But it has been left to me to discover this mistake, into which I was led, as I shall shew, by the ordinary school of logicians. If A and B be the premises of a syllogism, and C the conclusion, the representation A+B=C is faulty in two points, The premises are compounded, not aggre- gated; and AB should have been written: the relation of joint premises to conclusion is that (speaking in extension) of contained and containing, and AB