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VOLUME X. PART L— 2 CAMBRIDGE: PRINTHBY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS ; ἶ AND SOLD BY DEIGHTON, BE] AND CO. AND MACMILLAN AND (Ὁ. CAMBRIDGE; BELL AND DALDY, LONDON. M. DCCC. LVIIL. — δ, ‘ALT ya ΠῚ VW ὉΠ ΙΝ ἡ Δ... _ fate ΔΝ GRR A 1, Wear "ἰδ νι ἡ ΝΟ ἢ: 1. Π|. ΝΠ. VIII. ΙΧ. CONTENTS OF νου. Son aR Tn Ls On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space, mith Applications. By R. B. Haywarp, M.A., Fellow of St John’s College, Reader in Natural Philosophy in the University of Durham. .....++++++- On the question, What is the Solution of a Differential Equation? A Sup- plement to the third section of a paper, On some points of the Integral Calculus, printed in Vol. IX. Part Il. By Augustus De Morean, of Trinity College, Vice-President of the Royal Astronomical Society, and Professor of Mathematics in University College, London. .....+++++-+++- On Faraday's Lines of Force. By J. Cunrk ΜΆΧΕΣ, B.A., Fellow of Trinity College, Cambridge... ..+ +++ +++seecee cre σεν cesses sense rset sss The Structure of the Athenian Trireme; considered with reference to certain difficulties of interpretation. By J. ΝΥ. Donapson, D.D., late Fellow of Trinity College, Cambridge. «1. +++seeece veer esc eecee sete ees esses: Of the Platonic Theory of Ideas. By W. WHEWELL, D.D., Master of Trinity College, Cambridge. ....++.++++0+-+2eeerc ents enes stents On the Discontinuity of Arbitrary Constants which appear in Divergent Developments. By Ὁ. G. Sroxes, M.A., D.C.L., See. R.S., Fellow of Pembroke College, and Lucasian Professor of Mathematics in the Univer- sity of Cambridge. .....--.-0c.enceeeese one ceccmatascsccntiece sc nas On the Beats of Imperfect -Consonances. By Aucustus De Morean, F.R.A.S., of Trinity College, Professor of Mathematics in University Gollemest Bandome cs < -(-/e'-J:=12)s » sia\eie ore emacs reies rece aac alas On the Genuineness of the Sophista of Plato, and on some of its philosophical bearings. By ΝΥ. H. Tuompson, M.A., Fellow of Trinity College, and Regius Professor of Greek. ..++-++eesssceseee erect cet eet testes ee On the Substitution of Methods founded on Ordinary Geometry for Methods based on the General Doctrine of Proportions, in the Treatment of some Geometrical Problems. By G. B. Arry, Esq., Astronomer Royal. ....-- On the Syllogism, No. III., and on Logic in general. By Aueustus DE Morean, F.R.A.S., of Trinity College, Professor of Mathematics in University College, THT ec coc cdos cd0504 τ F000 00 O00 BS000R On the Statue of Solon mentioned by EEschines and Demosthenes. By J. W. Donatpson, D.D., Vice-President of the Society ...+++++++++res seers Instances of remarkable Abnormities in the Voluntary Muscles. By G. E. Pacer, M.D., F.R.C.P., late Fellow of Gonville and Caius College......- On Organic Polarity. By H. F. Baxrer, Esq., M.R.C.S.L...-- +--+ +++: A Proof of the Existence of a Root in every Algebraic Equation: with an examination and extension of Cauchy's Theorem on Imaginary Roots, and Remarks on the Proofs of the existence of Roots given by Argand and by Mourey. By Avcustus Dr Morean, F.R.A.S., of Trinity College, Professor of Mathematics in University College, London. ....++++++++- PAGE ina) - 84 94 129 146 166 ADVERTISEMENT. THe Society as a body is not to be considered responsible for any Jacts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. Tue Socrery takes this opportunity of expressing its grateful acknowledgements to the Synpics of the University Press, for their liberality in taking upon themselves the expense of printing this Part of the Transactions. I. On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space, with Appli- cations. By R. B. Haywarp, M.A. Fellow of St John’s College, Reader in Natural Philosophy in the University of Durham. [Read Feb. 25, 1856.] “*...gardons-nous de croive qu'une science soit faite quand on l’a réduite ἃ des formules analytiques. Rien ne nous dispense d’étudier les choses en elles-mémes, et de nous bien rendre compte des idées qui font l’objet de nos spéculations.” Pornsor. “..cest ume remarque que nous pouvons faire dans toutes nos recherches mathématiques; ces quantités auxiliaires, ces calculs longs et difficiles οὐ l’on se trouve entrainé, y sont presque toujours la preuve que notre esprit n’a point, dés le commencement, considéré les choses en elles-mémes et d'une yue assez directe, puisqu’il nous faut tant d'artifices et de détours pour y arriver; tandis que tout s’abrége et se simplifie sitét qu’on se place au yrai point de vue.” vid. Tue general principles, which I have endeavoured to keep in view in the investigations of this paper, are those contained in the above quotations from Poinsot. My object is not so much to obtain new results, as to regard old ones from a point of view which renders all our equations directly significant, and to develop a corresponding method, by which these equations result directly from one central principle instead of being (as is commonly the case) deduced by long processes of transformation and elimination from certain fundamental equations, in which that principle has been embodied. The frequent occurrence of exactly corresponding equations, (though this correspondence is sometimes disguised under a different mode of expression) in many investigations of Kinematics and Dynamics suggests the inquiry whether they do not result from some common principle, from which they may be deduced once for all. An investigation based on this idea forms the first part of this paper, in which it will be shewn how the variations of any magnitude, which is capable of representation by a line of definite length in a definite direction and is subject to the parallelogrammic law of combination, may be simply and directly estimated relatively to any axes whatever. The second part is devoted to the general problem of the dynamics of a material system, treated in that form which the previous Calculus suggests, together with a development of the solution in the case of a body of invariable form. Since whatever novelty of view is contained in this paper consists rather in the relation of the details to the general method than in the details themselves, much that is familiar to every student of Dynamics must be repeated in its proper place, but it is hoped that such repetition will in general be compensated by a new or fuller significance being obtained. As regards the problem of rotation, M. Poinsot’s solution in the ‘‘ Théorie de la Rotation” is so Won SCF IN a 1 1 ῷ Mr R. Β. HAYWARD, ON A DIRECT METHOD OF ESTIMATING complete and so entirely satisfies the conditions expressed in our quotations above from that work, as to leave nothing to be desired. But it does not appear to me that his method, which depends essentially on the summation of the centrifugal forces, is so widely applicable beyond the limits of this particular problem as that by which the same results are obtained in this paper: but be this as it may, any new point of view, if a true one (‘vrai point de vue’) has its special advantages, and on this ground may claim some attention. SECTION I. The Method, with some kinematical Applications. 1. As we shall here be concerned only with the directions of lines in space, and not with their absolute positions, it will be convenient to suppose them all to pass through a common origin O, and to define the inclination of two lines as OP, OQ by the are PQ of the great circle, in which the plane POQ meets a sphere whose centre is O and radius constant. We shall also suppose any linear velocity, acceleration or force, represented by a length along OP, to tend from O towards P, and any angular velocity or the like, represented in like manner, to tend in such a direction about OP that, if OP were directed to the north pole, the direction of rotation would coincide with that of the diurnal motion of the heavens. 2. Let τὸ denote any magnitude, which can be completely represented by a certain length along the line OU, and which can be combined with a similar magnitude v along OV by means of a parallelogram, like the parallelogram of forces or velocities. Then of course w may be resolved in different directions by the same principles, and thus if we adopt rectangular resolution, the resolved part of τὸ along OP will be τί οοβ UP, which may be denoted by τέ. We proceed to inquire how zw, varies by a change in the position of OP. 2 3. Suppose OP to be a line moving in any manner about O, and that it shifts from OP to a consecutive position OP’ in the time dt; and conceive that this motion arises from an angular velocity Q about an instantaneous axis OJ. Resolve Q into its components Q cos 1U about OU and Qsin JU about a line in the plane JOU, perpendicular to OU: and farther resolve this latter component in the plane perpendicular to OU into the components Q sin JU .cos JUP in the plane POU and Qsin JU. sin TUP perpendicular to the same plane. Then the component in OU and that perpendicular to it in the plane POU produce displacements of P perpendicular to the are UP, and consequently do not ultimately alter the length of the are UP, so that w, remains ultimately unchanged so far as the motion of OP is due to these components: but the component perpendicular to the plane POU increases UP by the are Qsin JU. sin 17}. dt, and therefore the increment of τὲ, from this component (being equal to -- w.sin UP.d.UP) is — uQsin UP.sin JU.sin JUP . dt. But the other increments being zero, this is the ¢otal increment of w,, wherefore we have te . : i ee - uQsin JU.sin UP. sin JUP...(A). VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 3 Up It is well to observe that vanishes, when the three axes OL, OU, OP lie in the same plane, and in particular when two of them ccincide, as is evident from the above equations, or from considerations similar to those by which it was obtained. 4. Inthe above investigation we have supposed τὲ to be constant both in direction and intensity ; let us now suppose w to vary in both respects with the time (¢). The change in 2 in the time dt may be conceived to arise from its composition with the quantity fdt in the line OF, and f may properly be called the acceleration of τὸ at the time ¢. Now fat may be resolved in the plane UOF into 7". dt cos FU along OU and f. dt sin FU perpendicular to OU, and the components of w + dw will therefore be w+ f.dtcos FU along OU and f.dt sin FU perpendicular to OU; whence, if d@ denote the angle through which OU shifts in the time dé towards OF, it will readily be seen that ultimately du dp : Ξ; =fcosFU, “u ΝΣ =fsin FU,..(B). If then the acceleration f be known both as to direction and intensity at every instant, the motion of OU and the variation in the intensity of τί may be determined by these last equations. In fact, the point 77 on the sphere of reference continually follows the point F d with the velocity = so that the problem of determining U’s path is the same as the old problem of the path described by a dog always running towards his master who is himself in motion, the only difference being that the path is here on a sphere instead of a plane. 5. Next for the variation of wu,, when w varies with the time. It is plain that w, varies from two causes; first, by reason of the acceleration f, and secondly, by reason of the motion of OP due to the angular velocity Q about OJ, and that the total variation will be the sum of these two partial variations. Now the latter has been calculated above, and the former is obviously the resolved part of fd¢ along OP or f.dt cos FP, therefore we obtain the equation* Z = = f cos FP -- uQsin IU. sin UP. sin IUP...(C). This equation of course contains the previous equations (B): thus, if OP and OU coincide always, UP is always zero and the second side of (C) reduces to its first term: and again if εἰ ΟΡ be always in the plane FOU and perpendicular to OU, w, is always zero, Q = 2, IU and UP are quadrants, JUP a right angle, and FP the complement of F'U, and therefore, as above, Wala aoe 0 =f sin Bese: 6. We may farther illustrate the application of equation (C) by supposing OP to coincide with certain other lines specially connected with OU and OF. *Tt should be remarked here that the angle UP must be | Q about OJ causes the motion of P, resolved in the are (7 P, to considered positive or negative, according as the positive rotation | be from or towards U. 1—2 ᾿ 4 Mr R. Β. HAYWARD, ON A DIRECT METHOD OF ESTIMATING Let U, U’, U" and F, F’, F” be three consecutive positions of U and F respectively, and K, K’ those poles of FF’, Ε΄ ΕΠ respectively, (considered as ares of great circles) about which positive rotation brings F to F’, and F’ to F”. We know that U’ lies on the are UF between U and F, and {77 on the are U’F" between U’ and F’. Also it is plain that F” is the pole of ΚΑ΄. and therefore that Καὶ Κ΄ measures the angle of contingence between the consecutive elements FF’, F'F”: in fact, the loci of K and F are so connected that the elementary ares of the one are equal to the angles of contingence of the other, and vice versa. Suppose the locus of F to be defined by ele- ments, corresponding to what Dr Whewell has named in plane curves intrinsic elements, that is, by elements a, ε such that the elementary are FF” = da, and the angle of contingence between FF’ and 11 =de: and suppose the locus of U defined in like manner, so that UU’ = dg, and the angle of contingence FU’ {7 =dy. Also let UF =n, and angle UF F’ = v. Now let OP coincide always with OF. Then will w,=wcosp, and J being taken to : d coincide with k, Q= FP and therefore equation (3) becomes d da ., πον... ὭΣ (wcosp) Ξ ἢ" - τ ς᾽ sing.sin Καὶ οἷν KUP. But sinkU.sinkUF = sinkFU = sin (= —y ) = cos r, du and — = fcosp; Prego therefore we obtain after reduction du da Α -- — —cosy + ~s = Vsaqq00 5 dt dt ἀν} Ὁ ᾿ (1) Again let OP coincide always with OK, then up =ucos UK =usinu.cos UF K = using. sin ν, and I may be taken to coincide with #” or ultimately with F, so that (3) becomes, d (2 being =<), Las Salsa Se, sa HORT tene dt μ. μν)Ξ Poa Be -sin OA. sin = —u—sin p.cos = i μ᾿ .COS ν, or after reduction aet a τι 7 + —sin« = 0......(2). “tany τ The equations (1) and (2) together with tle two equations (B) serve to determine (after eae ἐτ du d da d eliminating μ and νὴ ΕΣ and ΠΝ when f, = = are given, that is, when the intensity and du ( a) tan % i VELOCITIES, ἃς. WITH RESPECT TO AXES MOVEABLE IN SPACE. 5 variation in direction of the acceleration of w are given for every instant. And we have also from the triangle FU’F" ultimately ._ an : to determine πε and therefore we have equations to determine the intensity and variation in direction of τὲ 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 u,, uy, τε, denote the resolved parts of τὲ along the moveable rectangular axes Ox, Oy Oz, 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,dt: also Ox changes its position by reason of the rotations Q,, Q., the first of which shifts it in the plane of za through the angle Q,d¢ from O., and the latter in the plane of ay through the angle Q.dt towards O,; and from the first of these causes w, receives the increment π Uw, COS (Ξ τῶ, dt) + u, cos (Q,dt) — τι,» or — τ ἀξ ultimately, while from the second it receives the increment u, cos (— — Ω dt} + u, cos (Q.dt) -- τι,» y fo} z Ζ Ὁ or uv,Q.dt ultimately. Hence the total increment of w,, being the sum of these partial increments, we obtain the equation y dt = fy + U,Q2 — Uz Q, du ae =f, + Uz Q, — Uy Qz, du, ὭΣ =f, + u,Q, — Uz Qy | Similarly for w,, τὸ, we should obtain | du eee ee (CB): | | | 8. To illustrate the applicability of these last obtained equations, we will select a few particular kinematical problems. a. Relative velocities of a point in motion with respect to revolving axes. From the nature of the quantity w, it will be seen that it may be taken to denote the radius vector OP of a point P, and u,, u,, u, may then be replaced by the co-ordinates, ὡς y, x: also f, denoting the acceleration of τι, will in this case denote the absolute velocity of P, and f,, f,, fz the absolute velocities resolved in the directions of the axes, which we will denote by Ὁ.» v,, 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 dx dt =0U,+ ψῶ. = 20, dx dy dz — of the point with respect to the co-or- d which determine the relative velocities — an which ete dt > de. dé dinate axes. If the point be fixed relatively to the axes, and a, Yo) %) be its co-ordinates, the above equation becomes Ὁ, =Qy. 30 — Ὡς - Yo 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 x, so that y, % both vanish, 0= vy - «2Q,, O=VU,+ xQ,,. In these, if 2, 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 συ, 0, ᾧ denote radius vector, altitude and azimuth, then v=7, a, = - Pos, Ὁ whence dr dé Caan ie bere v,=rcosO—, the common expressions for the components, relatively to polar co-ordinates, of the velocity of a point. b. Accelerations, radial, transversal in the vertical plane, and perpendicular to that plane. In our general formule wu will now denote a velocity, and f an acceleration strictly so called. And in this case d d ω, τ τ, Oy = Tee =reoso σε, d Q, = —-— sin @, Q, = — =, cos 8, Cee dt wherefore, by equations (1) Ε ‘ a ν d 2 d 2 radial acceleration =f, = = = (« : *| + 7 cos’ 0 τ ): transversal acceleration in the vertical plane =f, ἀ ( do φ dr dé = f(r.) - (-- sin θ. cos 0S = ΕἾΠΕΝ =) bes [- - rsin@ cos", dt dt : ἡ azimuthal acceleration = f, = Ξ ( r cos @ “e) - (- S = 0050 + 7 sin 3 ἜΣ a @ VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 7 1 d ἐφ = 22 2 ef ΝΜ (: cos θ t ) ἡ cos θ᾽ dt 6. 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 7 ae’ Uy = 0, u, = 0, dr de ae a ω, dt’ where de, dr denote respectively the angle between consecutive tangents, and that between consecutive osculating planes. Hence : : γ᾽ tangential acceleration =f, = ἘΣ : ae ἃ κίας. ds de ds\2 de 1 /ds\? acceleration in principal normal =f, = —.— = (5) .π-Ὄ- κε - (5) " dt dé \dt/ ‘dsp \dt acceleration in normal to osculating plane =,f, = 0. SECTION II. Dynamical Applications. 9. 1] 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 w in the previous section, the Calculus there developed is applicable to them. 8 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING 11. Consider a material system at any instant of its motion. Each particle is moving with a definite momentum in a definite direction, which may be resolved into components in given directions in the same manner as a velocity or a force. Let this momentum be resolved in the direction of a given axis OP, and its moment about that axis taken, the resolved part may be called the linear momentum, and the moment the angular momentum, of the particle relatively to the axis OP. Let the same be done for every particle of the system, and the sums of their linear and angular momenta taken, these sums may then be called respectively the linear and angular momenta of the system relatively to the axis OP. 12. Let the linear momenta relatively to the three axes Ox, Oy, Ox be denoted by u,, Uys u,, 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, n, 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 ὅτι," + το ἢ + u?t3, when Lim: Mi Uz? Uy 2 U3 and if this be denoted by τι, it is plain that the linear momentum along any line inclined to the direction of τὲ at an angle @ will be uw cos @. 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 Ὁ. 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 τι. but changes, as O moves in any other direction; and finally, that its intensity will be a minimum and its direction coincident with that of wu, 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 dé in the direction of the resultant force R of the forces (P), and an angular momentum (df relatively to the axis of the resultant couple G of the same forces. Since however the internal forces consist of pairs of equal and opposite forces in the same straight line, by the nature of action and reaction, the momenta produced by them will vanish in the VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE, 9 summation over the whole system; we may therefore regard R and G as the resultant force and resultant couple of the external forces. Then the linear momentum u along the line Ὁ 17 must be compounded with the linear momentum Rdé in the line OR in order to obtain its value at the time ¢ + dt: and in like manner the angular momentum h relatively to the axis OH must be compounded with the angular momentum Gd¢t 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 d d oe come ie Pee it RUD dt dt where d@ is the are through which J moves towards # in the time dt: and dh d — = Goos GH, pals ΠΕ ΣΙ dt dt 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 du, du du —= X, —= MW = = ᾽ dt dt dt Ζ dh, dh dh. Sea κα —_= pad ἘΞ dt 3 dt ; dt a 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, τ = Αι τ ῶς -- uQ,, dh, — = ἢ + h,Q, = Vo Oop dt 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 Vou. X, Parr. I. 2 10 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING a minimum: also that for any other position of the origin the direction of the invariable axis and the intensity of the momentum about it will depend upon the position of the line, parallel to the central axis, in which the origin lies, just as in the corresponding propositions relative to couples. 17. Any one of the different sets of equations in ᾧ (15) may be used to determine completely « and h, when the forces are given or vice versa. It is to be observed that the equations involving h, refer either to a fixed origin, or to an origin, whose motion is always in the instantaneous direction of τ the linear momentum, for, as we saw, a change of the origin in this direction does not produce a change in ἢ, 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,a, denote the linear velocities of the origin in the directions of the axes, the equation for h, becomes dh, ai = L+h,Q, —hQ, + τινας -— tay . The equations involving τὲ, 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 eaternal 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. ΤῸ 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 Ἧ 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 ἢ 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 ἢ and the arrangement of its mass ; for the direction of motion of the centre of gravity being that of the linear momentum, ἢ referred to that point as origin will be independent of uw. 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, ἃς. 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. of some point invariably connected with it combined with a motion of rotation about a certain The motion of an invariable system is always reducible to the motion of translation axis through that point. Let v,, v,, v, denote the resolved velocities along Ox, Oy, Ox of the point O, to which the translation is referred, and let w,, ὦ,» w. denote the resolved angular velocities about the same lines; then the velocity of any particle m, whose co-ordinates are @, y, %, is V, + w,% — wy in the direction of Ow, with similar expressions for the directions Oy, Ox. system, we find U, = X(m) .v, + ων. Hence summing the linear and angular momenta of the several particles of the =(mz) -- w.=(my), * I 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 detini- 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 fictitiows 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 actwai 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 external 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,, = =m(y.vz + wy — wt που, + WX — W,2) = (my) .v, — =(mz).v, + ΣΟ. y? +2). ὦ, — S(may)w, — L(mz2) . ὦ, with similar expressions for w,, τί; and h,, Az. From these equations it appears that, when the linear and angular velocities of the system are referred to an arbitrary point O, 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 S(mz), =(my), (mz) all vanish, and the equations become those, of which the types are Uy = =(M) . Urs h, = S(my? + “)w, — =(mey) .e, — =(mz2)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 w, y, x respectively) hi = Ao, hg = Bos, hs = Ca, ; hence the axis of angular momentum OH, whose equation is cae eT) ἢ hula τ τς is parallel to the normal to the central ellipsoid * [t 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, ἃς. WITH RESPECT TO AXES MOVEABLE IN SPACE. 13 Ax* + By? + Cz* = 1, at the point, where the axis of angular velocity OJ, whose equation is meets it. Also reciprocally OJ is parallel to the normal to the ellipsoid, whose equation is a y ‘ - ᾿ -- -- -- = A BES 2. 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 w be the angular velocity about OJ, and 7 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 OZ is zero; hence Iw =h.cos HI, an equation connecting ὦ and w, the quantities J and HZ 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 w 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 ὦ will change by reason of the variation of cos HI Σ : : Sar ue 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 ἡ 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 ΠΟΙ, 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. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING If the moveable rectangular axes in § (15) be supposed fixed in the body and coincident with the principal] axes, we must substitute W}; Ws ὡς for Q,, Q,, Q., and hy hy, h,, or Aw, Bw,, Cw; for ἢ.» h,, h., and then we obtain three equations, of which the type is, either dh, eh lgl Sy) & — — — }jh.h,, dt rs - τ ahs 1 or 12! = 1 + (B- 0). wary 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 Kd¢ about an axis OK related to OG, just as Of 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 ὦ about OL; and apply equation (C) of section I., putting ἡ for uw; therefore dh = —hw.sinJH.sin HP .sin LHP, c which determines the acceleration of momentum for any line OP. This acceleration will be zero, if OP bein the plane ΠΟΙ͂, and a maximum, if OP be perpendicular to ΠΟ], 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 HOJ, and equal to + hw sin HJ, when OF is taken on that side of ΠΟΙ͂ 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 (X) 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 (hk) 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 (@), 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 ‘ cowple 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 HOY, 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 ὦ, «, A, Iw =h cos HI, Kr = GeosGk, Τὰ =f cos FL, where f = hw sin HJ, 1, K, 1, denoting the moments of inertia about OZ, 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 HOTZ 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 OZ is conveniently referred to its motion in the plane HOJ and the motion of that plane about OH. tT ] ae κί oO \ er a FF Z Let the conjugate radii OZ, OL, OM be denoted by r, γ΄, γ΄, then the moments of inertia about them are =. ae za by the property of the central ellipsoid : also let the angles HOJ, ἘΞ 57 γ FOL be denoted by @, 6’: then our last equations become (1) w=hrcos0, (2) «=Grcos@, (3) A= (hwsin 8). γ᾽" cos θ΄. Resolve w, x, \ along the axes OH, OM, OF; the component velocities are then ὦ cos@ along OH, wsin@ along OM, and zero along OF, while the component accelerations are «cos@ along OH, «sin @ +) sin 0 along OM, and ) cos 6’ along OF ; whence, by applying either the equation (C) or the equations (£), Sw cos @) = κ cos 0 = Gr? 605) O,....2...0.2+eeerere serene ον (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 H OJ is diametral. 10 Mr R. Β. HAYWARD, ON A DIRECT METHOD OF ESTIMATING = (w sin 0) = «sin @ + Asin θ' = Gr* cos θ sin 8 + (hw sin 8) . x” cos 6 sin 6’,,..... (5) at ω sin@.Q =X cos θ' = (hwsin 8) . 7? cos? 6',.......200eeee-e0-+-(6) where © is the angular velocity of OM (i. e. of the plane HOJ) about OH. Also we have ΞΞ ΠΕ aco mans socanccg cosancascasadl (7) Let p, p’ denote the perpendiculars from O on the tangent planes to the central ellipsoid at I, LZ respectively, then p = r cos 0, p Ξ τ΄ cos θ΄. 1 : : Equation (4) becomes by (1) = (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(tan@) d/wsinO\ δ 51η θ΄ : : _= = Se > ἢ at οἷν Seeereeeee 8 dt = 5 cos 7) w cos 0 Bien G.itan ¢ (8) and from (6) ὈΞΎ τ᾿ ΠῚ (0) 31. Now r, τ΄, r” being conjugate radii of the central ellipsoid, there exist three relations between them and the conjugate axes; these are, (putting p sec 0, ρ΄ sec θ' for 7, σ΄ respectively and denoting the angle JOL by x) 1 1 1 BtG 9,,,.2 "2, 12 3..,,3 2 2A! on? 1 1 1 pr”? + p*r”? + pp” sec 0 sec’ θ΄. sin X= Rat GA ἘΣ τ’ suppose, p’ sec? 0 + p® sec? +7"? = = E, suppose, 9 fo 40 1 SST Tc and by reason of the rectangularity of the planes JOM, LOM, we have cos x = sin @ sin θ΄. = G, suppose, Eliminating r” and y, we obtain G * sec? @ + p” sec? 0’ + —_. = E, » » ae G (= + =) + pip(sec* @ + sec’ θ' -- 1) =F. Pp Pp From these eliminating sec? θ΄, we obtain Ἔν. Ὁ 2 2d 4 (1-2 +4 - S)cort}, » Pp P Pp which, (remembering what E, Ε΄, G denote, and putting a, 8, y for the three quantities 1 : 1 1 1 2 respectively) ΡῈ Ap*’ Ἢ Bp’ Cp* P y is equivalent to p” = ρα + αβγ cot* 6); VELOCITIES, ἄς. WITH RESPECT TO AXES MOVEABLE IN SPACE. 17 also, since ρ΄, θ΄ are involved in precisely the same manner as p, 9, it follows that p* = p'(1 + a Bir cot’ 6) ; where a’, β΄, “γ΄ are.what a, 8, Ὑ become, when ρ΄ is put for p. From these equations we obtain t? cot? θ΄ — ani q 2 8 9 3 a By 1+ ary cot 6 1 1 1 1 t? but ἘΞ. rn ey ἘἸΘΎ ΘΟΕ ες Ap”* Ap? 1+ aPycot’é 1 + aBry cot’ @ whence, with the corresponding expressions for ΒΞ γ΄, (1 + aBry cot? 0) t? 6’ = — cot’. ν co cot” 0 (1 + Bry cot’ θ)( + γα cot® θ}(1 + αβ cot’ 6) hence ρ΄, θ΄ are known in terms of p, 0. 32. Substituting now for μ΄, θ΄ in terms of p, 0, we obtain from equation (8) = + hp’$ -- (1 + By cot’ @)(1 + ya cot” O)(1 + αβ cot* 6) }3,.........(10) 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, vy, are constants. 1 aoe : If @ denote the azimuth at any instant, = =Q, and dividing the last equation by the preceding, we obtain a relation involving ᾧ and Θ᾽ 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 = Ap? sec @, from which ὦ is known when @ is known by means of equation (10), and thus the velocity about OZ is known completely as well as its position at any time. 33. If there be no forces acting, i. e. if G=0, h is constant, as is also wcos @, the re- solved angular velocity of the body about OH. Also the vis viva of the body ν ω. and is therefore constant; and hence — is constant, or ὦ < 7; both well known results. It may r Vou. X. Parr I. 3 18 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING vis viva : ; also be well to note that p* = - ~, even if @ do not vanish, and therefore (angular momentum)? : 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, OF 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. ε 35. When the polar axis of the central spheroid always lies in the plane of the meridian, let 6 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 : : - d about its axis, the motions of 04, OB, OC will be due to the velocities Q cos θ, Q sin 6, - about them respectively: also the actual velocities of the body about the same axes are : : dé F de dé respectively w, Q sin 0, ae’ and the consequent angular momenta 4w, BQsin 0, B We where ὦ, ae? are reckoned positive when the motion about their axes is in the same direction as the earth’s about its axis. VELOCITIES, ἄς. 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 (1) in the first section applied to the case before us give d dd Ἶ dé — (40) = B —.Qsin§ -- BOsin9.— = τι ω) dt Q sin@ — BQ sin 0 di 0, ORue dt. dog dé —(BQ = ai au Qsin 0) = G + Aw di B at Q cos 8, d de : 2 Ε (8 = = BQ sin@.Qcos@ — Aw. Q sin @; dt dt from the first equation, w is constant, and from the last V0 A ae (Ge -- Qeos 8) asin ds 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, whence the motion of the axis OA is precisely similar to that of the circular pendulum, whose ; a A Bg ᾿ξ τὰς ; F Σ length is 7, where 7s Bow and therefore J= A? the direction of the earth’s axis taking the WAL place of the direction of the force of gravity. 2 Also since τῇ, = 0, when sin @ = 0, there are two positions of equilibrium of the axis O4, namely, when @ = 0, and @ = 7: the former is stable and the latter unstable, when ὦ 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 Q cose vertical and Q sin horizontal in the north direction: hence the angular velocities by which the axes move, are relatively to OA, OB, OC respectively ; : d — Qsine cos Φ; — Qsine sin @, - + Q cose, and the corresponding angular momenta are d Aw, -- BQsin csin @, a(t + Q cose), 40 Mr R. Β. HAYWARD, ON A DIRECT METHOD, ἂς. whence as before, ἀ(4.) 0 dt ; if d Sic BQ sine sing) = G + ἴω (‘f+ cos οἹ + B(Z + Qcose) .Qsin ὁ cos ᾧ, d Ig : ἔ : ater + Qcose) = BQsine sing. Qsine cos @ + Aw. Qsine sin p: and therefore w is constant, and a A : ΣΑΣ : aa = Boe sine sin @ + Q? 5ἰη" δ. sin Φ cos φ, or approximately a A OE = 5 aQsine. 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 ———-—, AwQ sin ¢ + Duruam, Feb. 19, 1856. Rk. B. H. Π. 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. TX. Part Il. By Aveusrus 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.] Trustiné that it will be sufficient excuse for a very elementary paper, that writers of the highest character are not agreed with each other on a very elementary point, I beg to offer some remarks upon the usual solution of such an equation as dy? — a°dx® = 0, to which Euler assigns the integral form (ψ -- αὐ -- ὁ) (y+ax+e)=0, where b and ¢ are independent constants. Most other writers insist on the condition b = ec. 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 (ῳ -- αὐ +b) (yt+au +5) for (ν -- αὐ -- δὴ) (y+ax+c), makes two remarks. The first,— chacun de ses facteurs doit tre 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... ἢ and grounds the right to do this on the possibility of thus obtaining αἰΐ 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 2, the equation (y — x) (y — αὖ = 0 implies power of choice between the relations y = 2’, y=e. The equation (y — αὖ) (ὦ -- 1) = 0 implies the relation y = οὐ with a dispensation from all relation in the case of ἃ = 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. i 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 egwation can have two related arbitrary constants in its primitive. Let γα, y, y’) = 0 involve any number of relations between 2, y, y’: and let p(x, y, a,b) = 0 be a relation between a and ὃ, or any number of relations. Consequently, selecting one relation by which to satisfy @ = 0, values of a and 6 can be found to satisfy both (a, y, a,b) =0, and also f(a + h, y + k, a, δ) = 0, for any values of 2, y,h,k. Hence, for any values of a and y, y’ may have any value whatever: and this is incompatible with f(z, y,y’)=0. But this is no argument against any form of @(v,y, a, ὃ.) = 0, in which the constants are not in relation; as ψίω, y, 2) - x(2, 4,6) = 0. For we cannot pretend to satisfy ψία, Y, α). χίυ, y, Ὁ) = 0, W(v +h,y +k, a). x(v +h,y +k, δ) =0, for any values of w,y,h,k, except by W(x,y,a) = 0, and y(vw+h,y+k,b) =0, or else by W(w +h, y + k, a) = 0, x(a, y, ὃ) =0. And from neither set can we deduce y’. If W(a, y, a) = 0 be a primitive of f(#,y, ν΄) = 0, there appears nothing ἃ priori to prevent our saying that ψίω, y, a). (a, y, δ) = 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'=/(2,y) be satisfied by different relations @(v,y, @) =0, ψίυ, y,b) = 0, then, taking a and ὦ so as to satisfy both at a given point (w,y), we find, generally, two values of ψ' at (7,y). But y'=/(#,y) may give these two values; irreducibly connected, as in y’ = 1 + ,/y, or reducibly, as in ψ' =1+4/y*. The great point of algebraical interest, namely, that when the two values of ψ' are irreducibly connected Φ = 0 and ψ =0 are the alternatives of an equation which can be rationalised or otherwise inverted into y= 0, where y 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 = αὖ, which is certainly one relation for determination of y from x, 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 /etfer carries in its signification are of the valwe, so called. Accordingly, permanence of form does not necessarily give continuity of value. The immediate passage of Es uf sinav.v~'do from +4 to — 4a, as # passes through 0, might be discovered by the 9 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 ἡ τ which ends at a =0 joins the branch of y=a@ + e-* which begins at a =0 with acontact of the order ¢ , 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 + o to + co be counted under this term. I will not, on this point, give more than an expression of my conviction that the word continuity must, by that dictation which has turned wnity into a number, and its factor into a multiplier, be extended to contain the usual passage through infinity. Let m-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 v3 an interminable continuity, and a change of form answering to, and indeed derived from, the change of form seen in (+ 2)? and (— a). We have, in truth, all the quantitative properties of ome relation, and all the formal properties of two. The attainment of a reducible case is the loss of the quantitative properties also: thus (x* + a)! is non-ordinally continuous, and not so much as primordinally, when a = 0. We are now in a condition to answer the question, What is the solution of a differential equation ?—at least so far as having a clear view of the imperfect manner in which the question is put. We are obliged to ask in return, what requirements as to continuity are conveyed in the word solution ? 1. The word solution may require the most absolute notion of permanence of form, not granting even the passage from ( -- ᾧ) to(+ 2)’. 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* = θη. Μ΄ we announce three solutions. To γ΄ = 0, any parallel to the axis of w. To y’ =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 γ =2x the negative value of ,/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 ἡ = (a — 8)", which we please, but only one, for the solution of y* = 24/y «α΄. 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 w, and of the other factor for others. The isolation of the factors is the postulation of a certain permanence of form. 94 Mr DE MORGAN, ON THE QUESTION, 3. In asolution we may allow change of form, witha given kind of continuity at the junction. If we mean to stipulate nothing whatever about continuity, we may at any value of x leave one curve, and proceed upon another. If we require non-ordinal continuity, we can only do this where two curves join each other. If we require ordinal continuity or continuity of the same order as the equation, we may propound as a solution of y’=2\/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(x, y). Av, with a very small value of Ax, which may be as small as we please in the reasoning, a solution of ψ' = y(«, 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 = (Az)’. It may now be affirmed that (ῳ -- αὦ - Ὁ) (y+ aa+c)=0, ὃ and e being perfectly independent constants, is a solution of γ΄ -- α΄ Ξ Ο; nothing in the general theory of the primordinal differential relation in any way withstanding. It remains to examine the assertion that the generality of this solution is not restricted by the supposition ὦ = e. To a certain extent this assertion is true: no more curves are obtained or included before the limitation than after it. Beyond this point the assertion is not true. The condition b = 6 belongs to one mode of grouping a solution of ψ' = a with a solution of y’= —a: but there is an infinite number of modes in ὦ = ge. If ordinal continuity be held sufficient, and if Pla, Ys b) = 0, ψίω, y, ὁ) = 0 be independent relations satisfying f(z, y, y’) = 0, and if P =0 be the most complete singular solution, then Ῥ. φία, Ys Dy). φίω, Yy be) oe oe WU, Ys C1) « ψίω, Ys 65)... = 0 is the most general solution, where 6,, b:,...€,, Cs,--. are in any number, and of any values. This however is but equivalent to P. p(, y, 6) . (2, y, 6) = 0 with the usual addition ‘for any values whatever of ὃ and ec’, 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, 6 and ὁ being in relation in p(x, y, ὃ, 9) = 0, we may designate all the curves contained in φίω, y, fe, 6) = 0 as the group (fe,c). Generally speaking, the curves of the group (76,6) are different from those of (Fe, ὁ). The unlimited number of cases of (716, 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 φᾷ. y, Ὁ). 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 (P -- 8) (Q-c) + R=0, P,Q, R, being each a function of w and y: and let P’ represent P, + P,.y', &c, When ὁ = fe, the primordinal equation of the group (fe, ο) is Ro +4/(R? + 4P’'QVR) R-4/(R?+4P' QR) Q+ oP = if ἢ + 2" | : Let Πὲ = nV, 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 = ΚΡ, the algebraic result of eliminating ὁ between (P — δ) (Q -- ο) =0, and (P — fe) (Aya (Q -- ο) P’=0. And the biordinal equation is determined by differentiating R+f(R2+4P'VR) ΦΡ' ἱ Do this fully, clear the result of fractions, and write »V for R: it will then appear that γ΄" is seen only in terms multiplied by positive powers of μι; and so that ~ = 0 gives PQ =0in 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 # and y, cannot have any differential equation clear of those constants under the xth order; and an equation of single and irreducible relation between x,y, y',...y must have a primitive containing m constants in relation to wand y. But a primitive equation in which 2 constants are contained in alternative relations, m, in one relation, m ina second, ἅς. does not require a differential equation of the mth order; but has an equation of alternative relations, one of the m,th order, one of the n,th order, &c. From a primitive having ” constants, in relation with a and y, no constants can be eliminated in favour of y’, ψ΄, &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 φίω, ψ, α) =0 give a = O(a,y,), and therefore @,+,.y' = 0 for a differential equation, in which a has disappeared and ψ' is introduced, it is easy to give this differential Vor. X. Parr I. 4 26 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, P(x, y) + Ay $(a, y)t* + ... = 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 A, + A, P(a,y) + ... = 0 is but a transformation cf some case of (2, y) = F(A, As...) or of Pia, y, f(Aq, 4j5---)} =0. All we have done, then, amounts to no more than use of the obvious theorem that a single arbitrary constant is equivalent to an arbitrary function of as many arbitrary constants as we please. Moreover, we may prove that P+ y’ can only be a factor in the differential of one class of forms. If {F(x,y)}’ give M(P+y’), nothing but ἐψ ζω, y)}’ can give N(P + y/): and F(a, y)=const. and WF («,y ) = const. are the same equations. But it is otherwise with P + y’, P being a function of x,y,y’. This occurs, as previously shewn, in the differentiations of two distinct classes of forms. Thus 0+y” is a factor in (ων -- y)}' and in }Fy'}’. The equation 7ίαν' -- y) = 4, + A, Fy + 4. {Fy}? +... is one which contains in every sense, formal and quantitative, as many arbitrary constants as we please; and an alteration in the value of one of them, is an alteration in the character of the relation subsisting between wy’ — y and y’. Nevertheless, it is impossible to get rid of any one constant in favour of y” in any way except one which results in y” = 0, an equation from which all the constants have disappeared. Considerations similar to those which have been applied to primordinal equations might also be applied to equations of any order. A. DE MORGAN. University Cottrecre, Lonnon, March 29, 1856. Ill. On Faraday’s Lines of Force. By J. Currk 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 ¢emperature 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 formule, there would be nothing to distinguish between the one set of phenomena and the other. Mr. MAXWELL, ΟΝ 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, 1882. 90 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 ean 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. he 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 Jluid motion. If the motion of the fluid be what is called steady motion, that is, if the direction and velocity of the motion at any fixed point be independent of the time, these curves will repre- sent the paths of individual particles of the fluid, but if the motion be variable this will not generally be the case. ‘The cases of motion which will come under our notice will be those of steady motion. (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, ON FARADAY’S LINES OF FORCE. the lines drawn at intervals depending on the assumed unit, and afterwards to assume the unit as small as we please by taking a small submultiple of the standard unit. (6) To define the motion of the whole fluid by means of a system of unit tubes. Take any fixed surface which cuts all the lines of fluid motion, and draw upon it any system of curves not intersecting one another. On the same surface draw a second system of curves intersecting the first system, and so arranged that the quantity of fluid which crosses the surface within each of the quadrilaterals formed by the intersection of the two systems of curves shall be unity in unit of time. From every point in a curve of the first system let a line of fluid motion be drawn. These lines will form a surface through which no fluid passes. Similar impermeable surfaces may be drawn for all the curves of the first system. The curves of the second system will give rise to a second system of impermeable surfaces, which, by their intersection with the first system, will form quadrilateral tubes, which will be tubes of fluid motion. Since each quadrilateral of the cutting surface transmits unity of fluid in unity of time, every tube in the system will transmit unity of fluid through any of its™ sections in unit of time. The motion of the fluid at every part of the space it occupies is determined by this system of unit tubes; for the direction of motion is that of the tube through the point in question, and the velocity is the reciprocal of the area of the section of the unit tube at that point. (7) We have now obtained a geometrical construction which completely defines the motion of the fluid by dividing the space it occupies into a system of unit tubes. We have next to shew how by means of these tubes we may ascertain various points relating to the motion of the fluid, A unit tube may either return into itself, or may begin and end at 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 sowrce within that space, capable of creating and emit- ting unity of fluid in unity of time, and to be afterwards swallowed up by a sink capable of receiving and destroying the same amount continually, There is nothing self-contradictory in the conception of these sources where the fluid is created, and sinks where it is annihilated. The properties of the fluid are at our disposal, we have made it incompressible, and now we suppose it produced from nothing at certain points and reduced to nothing at others, The places of production will be called sources, and their numerical value will be the number of units of fluid which they produce in unit of time. The places of reduction will, for want of a better name, be called sinks, and will be estimated by the number of units of fluid absorbed in unit of time. Both places will sometimes be called sources, a source being understood to be a sink when its sign is negative. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 33 (8) It is evident that the amount of fluid which passes any fixed surface is measured by the number of unit tubes which cut it, and the direction in which the fluid passes is determined by that of its motion in the tubes. If the surface be a closed one, then any tube whose ter- minations lie on the same side of the surface must cross the surface as many times in the one direction as in the other, and therefore must carry as much fluid out of the surface as it carries in. A tube which begins within the surface and ends without it will carry out unity of fluid; and one which enters the surface and terminates within it will carry in the same quantity. In order therefore to estimate the amount of fluid which flows out of the closed surface, we must subtract the number of tubes which end within the surface from the number of tubes which begin there. If the result is negative the fluid will on the whole flow inwards, If we call the beginning of a unit tube a unit source, and its termination a unit sink, then the quantity of fluid produced within the surface is estimated by the number of unit sources minus the number of unit sinks, and this must flow out of the surface on account of the incompressibility of the fluid. In speaking of these unit tubes, sources and sinks, we must remember what was stated in (5) as to the magnitude of the unit, and how by diminishing their size and increasing their number we may distribute them according to any law however complicated. (9) If we know the direction and velocity of the fluid at any point in two different cases, and if we conceive a third case in which the direction and velocity of the fluid at any point is the resultant of the velocities in the two former cases at corresponding points, then the amount of fluid which passes a given fixed surface in the third case will be the algebraic sum of the quantities which pass the same surface in the two former cases. For the rate at which the fluid crosses any surface is the resolved part of the velocity normal to the surface, and the resolved part of the resultant is equal to the sum of the resolved parts of the com- ponents. Hence the number of unit tubes which cross the surface outwards in the third case must be the algebraical sum of the numbers which cross it in the two former cases, and the number of sources within any closed surface will be the sum of the numbers in the two former cases. Since the closed surface may be taken as small as we please, it is evident that the distribution of sources and sinks in the third case arises from the simple superposition of the distributions in the two former cases. 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. νον ΡΒ 1 5 94 Mr MAXWELL, ΟΝ 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 in a direction perpendicular to two of its faces. Then the resistance will be ku, 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 ὦ units, the dif- ference of pressure at its extremities will be kuh. (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 swrface (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 8. 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 / be the distance between two consecutive surfaces of equal pressure mea- sured along a line of motion, then since the difference of pressures = 1, kvh = 1, which determines the relation of v to h, so that one can be found when the other is known. Let s be the sectional area of a unit tube measured on a surface of equal pressure, then since by the definition of a unit tube : vs = 1, we find by the last equation δ. = 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. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 35 ; (14) If the surfaces of equal pressure are known, the direction and magnitude of the velocity of the fluid at any point may be found, after which the complete system of unit tubes may be constructed, and the beginnings and endings of these tubes ascertained and marked out as the sources whence the fluid is derived, and the sinks where it disappears. In order to prove the converse of this, that if the distribution of sources be given, the pressure at every point may be found, we must lay down certain preliminary propositions. (15) If we know the pressures at every point in the fluid in two different cases, and if we take a third case in which the pressure at any point is the sum of the pressures at corresponding points in the two former cases, then the velocity at any point in the third case is the resultant of the velocities in the other two, and the distribution of sources is that due to the simple superposition of the sources in the two former cases. For the velocity in any direction is proportional to the rate of decrease of the pressure in that direction; so that if two 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 sien 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 should be 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 dor?" : k : The rate of decrease of pressure is therefore ku or ἀπ and since the pressure = 0 7 ΣΝ παν ess : : k when 7 is infinite, the actual pressure at any point will be Pe 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 Aor If we have a source formed by the coalition of S unit sources, then the resulting : ki : - , : pressure will be p = ramet so that the pressure at a given distance varies as the resistance πΥ 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, ON FARADAY’S LINES OF FORCE. 37 (21) If the system of pressures and the distribution of sources which produce them be known in a medium whose resistance is measured by Δ, 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 Κ΄. 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 ἢ. (23) Let the space within a given closed surface be filled with a medium different from that exterior to it, and let the pressures at any point of this compound system due to a given distribution of sources within and without the surface be given; it is required to determine a distribution of sources which would produce the same system of pressures in a medium whose coefficient of resistance is unity. Construct the tubes of fluid motion, and wherever a unit tube enters either medium place a unit source, and wherever it leaves it place a unit sink, Then if we make the surface impermeable all will go on as before. Let the resistance of the exterior medium be measured by k, and that of the interior by k’. Then if we multiply the rate of production of all the sources in the exterior medium (including those in the surface), by ἄν and make the coefficient of resistance unity, the pressures will remain as before, and the same will be true of the interior medium if we multiply all the sources in it by k’, including those in the surface, and make its resistance unity. Since the pressures on both sides of the surface are now equal, we may suppose it permeable if we please. We have now the original system of pressures produced in a uniform medium by a combination of three systems of sources. The first of these is the given external system multiplied by &, the second is the given internal system multiplied by Δ΄, 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. 98 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. (24) Let there be no resistance in the medium within the closed surface, that is, let κ' = 0, then the pressure within the closed surface is uniform and equal to p, and the pressure at the surface itself is also p. If by assuming any distribution of pairs of sources and sinks within the surface in addition to the given external and internal sources, and by supposing the medium the same within and without the surface, we can render the pressure at the surface uniform, the pressures so found for the external medium, together with the uniform pressure p in the internal medium, will be the true and only distribution of pressures which is possible. For if two such distributions could be found by taking different imaginary distributions of pairs of sources and sinks within the medium, then by taking the difference of the two for a third distribution, we should have the pressure of the bounding surface constant in the new system and as many sources as sinks within it, and therefore whatever fluid flows in at any point of the surface, an equal quantity must flow out at some other point. In the external medium all the sources destroy one another, and we have an infinite medium without sources surrounding the internal medium. ‘The pressure at infinity is zero, that at the surface is constant. If the pressure at the surface is positive, the motion of the fluid must be outwards from every point of the surface; if it be negative, it must flow inwards towards the surface. But it has been shewn that neither of these cases is possible, because if any fluid enters the surface an equal quantity must escape, and therefore the pressure at the surface is zero in the third system. The pressure at all points in the boundary of the internal medium in the third case is therefore zero, and there are no sources, and therefore the pressure is everywhere zero, by (16). The pressure in the bounding surface of the internal medium is also zero, and there is no resistance, therefore it is 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 j : :- 1΄ --κα conceive a source producing fluid at a rate — , and wherever a tube leaves it we must , place a sink annihilating fluid at the rate » then calculating pressures on the supposition that the resistance in both media is & 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, b, ὁ be the components of the velocity at any point, and a, β, y the components of the resistance at the same point, these quantities will be connected by the following system of linear equations, which may be called ‘‘ equations of conduction,” and will be referred to by that name. a= Piat+ Ω.β + Ry, b= P,B+Q7+ R,a, ΟΞ Pyy + Qia + R, B. In these equations there are nine independent coefficients of conductivity. In order to simplify the equations, let us put ΩΝ ΞΞ 25; Q,-#,=2lT, Ποῦ δ ὙΠ OCS Soccoc cen ose: where 41" = (Q,- Rf)’ + (Q,- R,)? + (Q,- R,)’, and 7, m, m are direction cosines of a certain fixed line in space. The equations then become a=Pa+tS8,B+ δὲ +B - my)T, b= P,B+ 8. + S,a + (ly — na)T, e= ΡΟ + S,a + S\B -- (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= PB + (ly - na)T, c = Py +(m'a - UB)T, where 1, m’, π΄ are the direction cosines of the fixed line with reference to the new axes. If we make dp d dp Bae Yay’ andz=— the equation of continuity da db_ de Ἂς, dx dy dz ᾽ becomes ἢ dp , d P ,d Ρ -Ξ —— ξΞΞ ἢ τ dx Ps dy” 5. dz? a and if we make aaV/Pt, y= V Pa Ὁ Ξ V PSG ip dp dp then ae = de SF a ate dé = 0, the ordinary equation of conduction. It appears therefore that the distribution of pressures is not altered by the existence of the coefficient 7. 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,, Ῥ 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, ON FARADAY’S LINES OF FORCE. 41 source to a place where the pressure is zero, the number of cells would have been p, and if the tube had come from the sink to zero, the number would have been γ΄, and the true number is the difference of these. Therefore if we find the pressure at a source § from which .§ tubes proceed to be p, Sp is the number of cells due to the source §; but if S’ of the tubes terminate in a sink at a pressure p’, then we must cut off $“p’ cells from the number previously obtained. Now if we denote the source of S' tubes by S, the sink of S tubes may be written — ,5΄, sinks always being reckoned negative, and the general expression for the number of cells in the system will be Σ (Sp). (80) The same conclusion may be arrived at by observing that unity of work is done on each cell. Now in each source S$, S' units of fluid are expelled against a pressure p, so that the work done by the fluid in overcoming resistance is Sp. At each sink in which S’ tubes terminate, 8’ 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 -- XS"p’, or more. concisely, considering sinks as negative sources, W= =(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 # is a coefficient depending on the nature of the medium and on the positions of the source and the given point. In ἃ uniform medium whose resistance is measured by k, kS k p= ie dh S Aes ’ 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). (82) Ina uniform medium the pressure due to a source S$ kt S Peet At another source ,8 at a distance 7 we shall have ; k SS’ ; Sa ae. copy aiSoe if p’ be the pressure at §' due to S’. If therefore there be two systems of sources Σ(.) and =(S’), and if the pressures due to the first be p and to the second ρ΄, then 3(S'p) = X(Sp’). For every term S‘p has a term Sp’ equal to it. Vor, X° Parr I: 6 42 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. (33) Suppose that in a uniform medium the motion of the fluid is everywhere parallel to one plane, then the surfaces of equal pressure will be perpendicular to this plane.° If we take two parallel planes at a distance equal to k from each other, we can divide the space between these planes into unit tubes by means of cylindric surfaces perpendicular to the planes, and these together with the surfaces of equal pressure will divide the space into cells of which the length is equal to the breadth. For if h be the distance between consecutive surfaces of equal pressure and ¢ the section of the unit tube, we have by (13) 8 = kh. But s is the product of the breadth and depth; but the depth is &, therefore the breadth is h and equal to the length. If two systems of plane curves cut each other at right angles so as to divide the plane into little areas of which the length and breadth are equal, then by taking another plane at distance k from the first and erecting cylindric surfaces on the plane curves as bases, a system of cells will be formed which will satisfy the conditions whether we suppose the fluid to run along the first set of cutting lines or the second *. Application of the Idea of Lines of Force. I have now to shew how the idea of lines of fluid motion as described above may be modified so as to be applicable to the sciences of statical electricity, permanent magnetism, magnetism of induction, and uniform galvanic currents, reserving the laws of electro-magnetism for special consideration. I shall assume that the phenomena of statical electricity have been already explained by the mutual action of two opposite kinds of matter. If we consider one of these as positive electricity and the other as negative, then any two particles of electricity repel one another with a force which is measured by the product of the masses of the particles divided by the square of their distance. Now we found in (18) that the velocity of our imaginary fluid due to a source § at a distance y 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 resolyed 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, ΟΝ FARADAY’S LINES OF FORCE. 43 By this assumption we find that if V be the potential, dV = Xdu + Ydy + Zdz = — dp, or since at an infinite distance V = 0 and p =0, V = —p. In the electrical problem we have In the fluid p == (= = 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 r ' = (pdm') =—-, =(pS’)=—_, =(p'S) = = (pdm), 4π 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 X=- τς Ξ ku, k dm = — S, 4π whole potential of a system = -- 2Vdm = -- W, where W is the work done by the fluid in over- Tv 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 5 Series XI. At Mr MAXWELL, ON FARADAY’S LINES OF FORCE. of one substance having a greater inductive capacity, or conducting the lines of inductive action more freely than another, If we suppose that in our analogy of a fluid in a resisting medium the resistance is different in different media, then by making the resistance less we obtain the analogue to a dielectric which more easily conducts Faraday’s lines. It is evident from (23) that in this case there will always be an apparent distribution of electricity on the surface of the dielectric, there being negative electricity where the lines enter and positive electricity where they emerge. In the case of the fluid there are no real sources on the surface, but we use them merely for purposes of calculation. In the dielectric there may be no real charge of electricity, but only an apparent electric action due to the surface. If the dielectric had been of less conductivity than the surrounding medium, we should have had precisely opposite 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 Theory of Magnetism, Chapters III. ἃ V. Phil. Trans, 1851. + Experimental Researches (3292). : Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE. 45 and diamagnetic bodies worse, than the surrounding medium. By referring to (23) and (26), and supposing sources to represent north magnetic matter, and sinks south magnetic matter, then if a paramagnetic body be in the neighbourhood of a north pole, the lines of force on entering it will produce south magnetic matter, and on leaving it they will produce an equal amount of north magnetic matter. Since the quantities of magnetic matter on the whole are equal, but the southern matter is nearest to the north pole, the result will be attraction. If on the other hand the body be diamagnetic, or a worse conductor of lines of force than the surrounding medium, there will be an imaginary distribution of northern magnetic matter where the lines pass into the worse conductor, and of southern where they pass out, so that on the whole there will be repulsion. We may obtain a more general law from the consideration that the potential of the whole system is proportional to the amount of work done by the fluid in overcoming resistance. The introduction of a second medium increases or diminishes the work done according as the resist- ance is greater or less than that of the first medium. The amount of this increase or diminu- tion will vary as the square of the velocity of the fluid. Now, by the theory of potentials, the moving force in any direction is measured by the rate of decrease of the potential of the system in passing along that direction, therefore when κ΄, the resistance within the second medium, is greater than ὦ, 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 kh’ is less than k&, so that the force is now from points of less to points of greater resultant magnetic force. Since these results depend only on the relative values of k and k’, it is evident that by changing the surrounding medium, the behaviour of a body may be changed from paramagnetic to diamagnetic at pleasure. It is evident that we should obtain the same mathematical results if we had supposed that the magnetic force had a power of exciting a polarity in bodies which is in the same direction as the lines in paramagnetic bodies, and in the reverse direction in diamagnetic bodies +, 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 * 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. P t Exp. Res. (2836), &c. 40 Mr MAXWELL, ΟΝ 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. xut. 496, and of M. Quincke, xtv1r. 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 fo 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 7. 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. IIL. p. 513. Mr MAXWELL, ΟΝ 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 A 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 ρ΄ 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΄. » --ρ΄, 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 IK, where K is the resistance of the whole circuit. The same idea of quantity and intensity may be applied to the case of magnetismt. The quantity of magnetization in any section of a magnetic body is measured by the number of lines of magnetic foree 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 § 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 =Ff= τς : 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 k , F=I [du= 1K =p-p. * Exp. Res. Vol. 111. p.519. + Exp. Res. (2870), (3293). 48 Mr MAXWELL, ΟΝ 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 = Didydz, 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 = (fdz), where da is the element of length in the circuit, and the summation is extended round the entire circuit. In the same circuit we have always / = 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 Ampere, and his results have stood the test of subsequent experiments. From the single assumption, that the action of an element of one current upon an element of another current is an attractive or repulsive force acting in the direction of the line joining the two elements, he has determined by the simplest experiments the mathematical form of the law of attraction, and has put this law into several most elegant and useful forms. We must recollect however that no experiments have been made on these elements of currents except under the form of closed currents either in rigid conductors or in fluids, and that the laws of closed currents only can be deduced from such experiments. Hence if 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 Xdv+ Ydy+Zdz a complete differential. ͵ Mr MAXWELL, ΟΝ 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). Vor. X. Parr I. ῆ — δ0 Mr MAXWELL, ΟΝ 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 circuit will be in equilibrium, and there will be no current. Hence the electro-motive forces depend on the number of lines which are cut by the conductor during the motion, If the motion be such that a greater number of lines pass through the circuit formed by the conductor after than before the motion, then the electro-motive force will be measured by the increase of the number of lines, and will generate a current the reverse of that which would have produced the additional lines. When the number of lines of inductive magnetic action through the circuit is increased, the induced current will tend to diminish the number of the lines, and when the number is diminished the induced current will tend to increase them. That this is the true expression for the law of induced currents is shewn from the fact that, in whatever way the number of lines of magnetic induction passing through the circuit be increased, the electro-motive effect is the same, whether the increase take place by the motion of the conductor itself, or of other conductors, or of magnets, or by the change of intensity of other currents, or by the magnetization or demagnetization of neighbouring magnetic bodies, or lastly by the change of intensity of the current itself. In all these cases the electro-motive force depends on the change in the number of lines of inductive magnetic action which pass through the circuit +. * Exp. Res. (3077), ἄς. In this case we have electro-magnetic forces acting on the + 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. 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, ON FARADAY’S LINES OF FORCE. 51 It is natural to suppose that a force of this kind, which depends on a change in the number of lines, is due to a change of state which is measured by the number of these lines. A closed conductor in a magnetic field may be supposed to be in a certain state arising from the magnetic action. As long as this state remains unchanged no effect takes place, but, when the state changes, electro-motive forces arise, depending as to their intensity and direction on this change of state. I cannot do better here than quote a passage from the first series of Faraday’s Experimental Researches, Art. (60). “‘While the wire is subject to either 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 II, On Faraday’s “ Electro-tonic State.” When a conductor moves in the neighbourhood of a current of electricity, or of a magnet, or when a current or magnet near the conductor is moved, or altered in intensity, then a force * (3172) (3269). tion of Electric, Magnetic and Galvanic Forces. Camb, and T 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, ON 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 (vyz) estimated in the directions of the axes a, y, ¥ respectively will be denoted by a, ὦ, ὁ. The quantity of electricity which flows in unit of time through the elementary area dS = dS' (la, + mb, + πο.) where /mn 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, β. γε be the effective electro-motive forces οὐσία του βι- Ya - ΤΡ | (A) ΞΖ, - --- 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 ἄς, a, = ἤρα,, β. = kbs, Y2 = kee, (8) but if the resistance be different in different directions, the law will be more complicated. These quantities a, (3, Ὑ5 may be considered as representing the intensity of the electric action in the directions of vyz. 54 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. The intensity measured along an element do of a curve e=la+mB+ny, where Jm are the direction-cosines of the tangent. The integral fed> 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 = [(Xdx + 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 Jedc = {(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 JedS, where e=la+mb + ne, Imn being the direction-cosines of the normal, ἐν fedS = [αὐγὰς + [φάω + [fededy, the integrations being effected over the given surface. When the surface is a closed one, then we may find by integration by parts da db de dS = — + —+— dz. fe 2 wii ἐς dy ai =) oy If we make da db dec Soe tf - - An oeaeeenneeene (© a tap eee (C) fedS = 47 [[fededydz, 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, ΟΝ 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, b, ὁ, k, a, 3, y; p, and p, are the same in form as those which we have just given. a, b, ὁ are the symbols of magnetic induction with respect to quantity ; &, denotes the resistance to magnetic induction, and may be different in different directions; a, B, ry, are the effective magnetizing forces, con- nected with a, ὁ, 6, 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, ὁ 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, 11. 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 laws shew that currents are to be combined like velocities or forces. The third law is the expression of a property of all attractions which may be conceived of as depending on the inverse square of the distance from a fixed system of points; and the fourth shews that the electro-magnetic forces may always be reduced to the attractions and repulsions of imaginary matter properly distributed. In fact, the action of a very small electric circuit on a point in its neighbourhood is identical with that of a small magnetic element on a point outside it. If we divide any given portion of a surface into elementary areas, and cause equal currents to flow in the same direction round all these little areas, the effect on a point not in the surface will be the 56 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. same as that of a shell coinciding with the surface, and uniformly magnetized normal to its surface. But by the first law all the currents forming the little circuits will destroy one another, and leave a single current running round the bounding line. So that the magnetic effect of a uniformly magnetized shell is equivalent to that of an electric current round the edge of the shell. If the direction of the current coincide with that of the apparent motion of the sun, then the direction of magnetization of the imaginary shell will be the same as that of the real magnetization of the earth*. The total intensity of magnetizing force in a closed curve passing through and embracing the closed current is constant, and may therefore be made a measure of the quantity of the current. As this intensity is independent of the form of the closed curve and depends only on the quantity of the current which passes through it, we may consider the elementary case of the current which flows through the elementary area dydz. Let the axis of x 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 dydz, then the total intensity measured round the four sides of the element is dB, dz + (5, + a dy, dz 2 dy, dy ( γι πὶρ dy 2) dz, dp, dz τ (8: ~ dz S| dy, dry, dy Ὁ τ 1 d Total intensity = -- = “δ dy dz. od y 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 curye embracing it, we shall have a= Ἐτατι ΠΣ dy’ ee dry, _ day » da dz da, dB, OO ee These equations enable us to deduce the distribution of the currents of electricity whenever we know the values of a, 2, y, the magnetic intensities. Ifa, B, Ὑ be exact differentials of a function of xyz with respect to w, y and x respectively, then the values of a, ὃς ὁ. 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, αν, dey dx ta dz which is the equation of continuity for closed currents. Our investigations are therefore for —=0, 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 εἰ ate Ela eV dx? cea 5 (where V and p are functions of wyx never Wee and vanishing for all points at an infinite + 4πρ-Ξ ΞΞ 0, distance,) can be satisfied by one, and only one, value of V. See Art. (17) above. Ἢ Tueorem II, The value of V which will satisfy the above conditions is found by integrating the expression » pdvdydz iN = a’ | ty yt t+2- z/) j where the limits of wyz 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 Taylor’s Scientific Memoirs. Tueorem III. Let U and V be two functions of wyz, then Vv Ev a dV OE dUdV dUdV [fe Ξε - dy? ara =] dadydz = -- ae re Sr ὩΣ ae ap + == dadydz «αἰδοῦς rant where the integrations are supposed to a 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 Vortec, Parr 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, 18), and therefore the number of cells in a given space is directly as the square of the velocity. Tueorem LY. 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 xyz, then the equation in p d 1 d d i dp d 1 d δ παρ cies 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 af3¥ 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 daz db, de, dz dy dz and the theorem shews that when the electro-motive forces and the rate of production of electricity at every part of space are given, the value of the electric tension is determinate. Since the mathematical laws of magnetism are identical with those of electricity, as far as we now consider them, we may regard a/3y as magnetizing forces, p as magnetic tension, and p as real magnetic density, k being the coefficient of resistance to magnetic induction. The proof of this theorem rests on the determination of the minimum value of dp ὌΝ erl dp CA al dp dV Ἴ = OS Ren ies Pee ho oe ee pe ee ἘΣ a= fff [ες FE Ξ τ dy zi aly de | ease where V is got from the equation EV ΟΥ̓ ἀπ da τὰ dy? ὡ dz* + 4p = 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 : : dV ; : dV “ quantity” resolved in ἃ; would be simply ae and the intensity & aa But the actual quan- ἢ hy : ; 1 d d tity and intensity are κ(« - «ΕἸ and a -- = , and the parts due to the distribution of media 1 oi tf alone are therefore d pepe lh Saeed a dell 1 - dV dV ἘΣ ΩΝ dx da ἡ Mr MAXWELL, ΟΝ 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. TurorEem V. If a, b, ¢ be three functions of w, y, z satisfying the equation da αὖ de ἢ da Ἔ dy ‘ dea it is always possible to find three functions a, 8, Ὑ which shall satisfy the equations d3 dy --- — -π-Ξ a, dz αν dy da de de” : da dp dy dx Let A = edy, where the integration is to be performed upon ὁ considered as a function of y, treating w and x as constants. Let B= jadz, C =fbdv, A’ = fbdz, Β' = fedex, C’ = fady, integrated in the same way. Then «ῳ | Q | 5 will satisfy the given equations; for dp dy da de db κα -- - -“-| -- dz -- | — da = | — d: —d dz dy Hise fe” lage daa ie a la db de fz ὙΠ τὺ. “πη ὌΠ; ween dz at τ “- εἰ avs [τ αν = and 00 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. In the same way it may be shewn that the values of a, (3, Ὑ satisfy the other given Bia Ue The function \, may be considered at present as perfectly indeterminate. The method here given is taken from Prof. W. Thomson’s memoir on Magnetism (Phil. Trans. 1851, p. 283). As we cannot perform the required integrations when a, ὃ, 6 are discontinuous functions of x, 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 @A dA @aA da * dy + a ae re LDR bbe ap dst wae τὰ ee 4ΦῸ dat ἀν ἀκ" +ce=0, by the methods of Theorems I. and II., so that 4, B, C are never infinite, and vanish when a, y, or καὶ is infinite. Also let _dB dC dy ema fs dy ar? da’ dC dA d β er dug ᾽ \ dx ἀξ dy 44. 48 ἂψ ἂν ἀν dz’ then d3 dy ἃ (dd dB aC (iA eA GA = = ( oF oF Ὶ = || SS τ- aE :) dz ἂν dx \dw~ ἂν ᾿ ἀπ) λα dy’ dz of d (— dB dC p ~ dw \dx ε dy ie πὶ ᾿ If we find similar equations in y and x, and differentiate the first by x, the second by y, and the third by x, remembering the equation between a, ὃ, 6, we shall have = a zn dA ΑΒ dy _ da? * dy? * dz eS τὴ) - ' 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, β, Ὑ have been rightly determined. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 61 The function Ψ is to be determined from the condition da dB dy ;@ @& @ de * a (ie at ae) da * dy ee if the left-hand side of this equation be always zero, Ψ must be zero also. Tueorem VI. Let a, ὃ, c be any three functions of 2, y, z, it is possible to find three functions a, B, y and a fourth V, so that da τς ΟΥε αν dy Ὁ dB dy dV and a= Pre dy Gr aig? _ dy da dV dw dx dy” da dB dV c= — + — dy ἀν dz Let da db de 4 dx dy dz hg and let V be found from the equation @2V dV av da* Τ αν: ci at ee then ; dV a i) dx’ ; dV 6 =6 ==, dy Ἡ dV ec =c- ae 3 satisfy the condition dd db de Af ‘dx dy dz a and therefore we can find three functions 4, B, C, and from these a, β, y, so as to satisfy the given equations. Tueorem VII. The integral throughout infinity Q = fff (aa, + διβι + cyy:)dadydz, 62 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. where αἱ δ᾽ ¢,, a 3, γι are any functions whatsoever, is capable of transformation into Q=+ Sf τ ἄπρρι - (aoa, + Bobs + Yot2) | dudydx, in which the quantities are found from the equations da, db, ε de, Ae ὃ a a ae da, dB, a eae fates a eA = 0; a, By yo V are determined from a, ὃ, οι by the last theorem, so that dp, dy, fi, dV dz ἀψ dx’ : dB, _ dy, &e., 3 dy and p is found from the equation ap ap P Ξ-- — +— + 4 =0 da” dy” εἰ dz eae For, if we put a, in the form dB, dy dV dz ἂν dx’ and treat 6, and ¢, similarly, then we have by integration by parts through infinity, remem- bering that all the functions vanish at the limits, da, = dy, dB, dy dy, da, da, df, = “7 - τς a) + a ( = n)* B, (= - =) + Y0 ι-- Ta) fede or Q=+ fff[{(4r Vp’) = (ads + Bobo + γι.) }duvdydz, and by Theorem ITI. J[[Vp'dudydz = |{[ppdedydz, so that finally Q = {{Π{4πΡρ -- (αγας + Byb2 = yor2)} dadydz. If a,b,c, represent the components of magnetic quantity, and a, B, yi those of magnetic intensity, then p will represent the real magnetic density, and p the magnetic potential or tension. a, b, ον will be the components of quantity of electric currents, and a, By γο Will be three functions deduced from αἱ δι οι» 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 αἱ 6,¢, are the components resolved in the directions of w, y, ¢ of the Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 63 quantity of magnetic induction acting through a given point, and a,(3,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 4. RS ἘΝ ~~ ale dy When there are no electric currents, then ada + B,dy + yidz =dp, a perfect differential of a function of w, 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 RS 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 = f{/p:p,dedydz throughout the system. Let us now call Q the total potential of the system on itself. The increase or decrease of Q will measure the work lost or gained by any displacement of any part of the system, and will therefore enable us to determine the forces acting on that part of the system. By Theorem III. Q may be put under the form Q=+ τ [[f@a + 6,2, + ery )dedydz, in which a, 3, γι are the differential coefficients of p, with respect to a, y, ¥ 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. ἢ 1 Q= Sf {ne = a (aes + Body + για) ἡαιάναν. 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 a, 3, y, the means of avoiding the consideration of the quantity of magnetic induction which passes through the circuit. Instead of this artificial method we have the natural one of considering the current with reference to quantities existing in the same space with the current itself. To these I give the name of Electro-tonic functions, or components of the Electro-tonic intensity. θ4 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, ὃ; 65 and their intensity by a, Bo yx Then the amount of work due to this cause in the time dé is dt {ff (a,a. + b.3, + esry2)dadydz in the form of resistance overcome, and dt d = - ff; (asa) + 6.35 + Csry))dvdydz 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 - dt d dt | ih { (da, + b.B, + eyy.)dedyds+ τ — ib [[( aa, + b.B, + exy)dedyde, where the integrals are taken through any arbitrary space. We must therefore have 1d α;ας + ἢ... + Cry. = — —(Aray + 0.39 + Coryo) 47 dt for every point of space; and it must be remembered that the variation of Q is supposed due to variations of a, β. γυ: and not of a,b,c,. We must therefore treat a,b,c, as constants, and the equation becomes 1 da, i Π55 1 dy, α,[«.. Ὁ ) +0.(8.+ 7 Oe. (γι + fo) -ο. 4ἀπ dt 4c dt In order that this equation may be independent of the values of a, ἢ. c,, 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, 5 ἘΠ πα : It appears from experiment that the expression =r refers to the change of electro-tonie 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 t, and if «, y, s be the co-ordinates of a moving article, then the electro-motive force measured in the direction of Φ is a=— l fda, da de, dy da, dz da, ( ΞΞ et + ) 4a \da dt dy dt dz αὐ dt * Translated in Taylor’s New Scientific Memoirs, Part II. Mr MAXWELL, ON 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 formulz, 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 rownd 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 11. 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). γοι. Χ. DAR Tle 9 06 Mr MAXWELL, ON FARADAY’'S LINES OF FORCE. Law III. The entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes through that surface. Law IV. The quantity and intensity of electric currents are connected by a system of equations of conduction. By these four laws the magnetic and electric quantity and intensity may be deduced from the values of the electro-tonic functions. I have not discussed the values of the units, as that will be better done with reference to actual experiments. We come next to the attraction of conductors of currents, and to the induction of currents within conductors. Law V. The total electro-magnetic potential of a closed current is measured by the product of the quantity of the current multiplied by the entire electro-tonic intensity estimated in the same direction round the circuit. Any displacement of the conductors which would cause an increase in the potential will be assisted by a force measured by the rate of increase of the potential, so that the mechanical work done during the displacement will be measured by the increase of potential. Although in certain cases a displacement in direction or alteration of intensity of 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 Experimental Researches. 1 do not think that it contains even the shadow of a true physical theory; in fact, its chief merit as a temporary instrument of research is that it does not, even in appearance, account for anything. There exists however a professedly physical theory of electro-dynamics, which is so elegant, so mathematical, and so entirely different from anything in this paper, that I must state its axioms, at the risk of repeating what ought to be well known. It is contained in M. W. Weber’s Electro-dynamic Measurements, and may be found in the Transactions of the Leibnitz Society, and of the Royal Society of Sciences of Saxony *. The assumptions are, (1) That two particles of electricity when in motion do not repel each other with the same force as when at rest, but that the force is altered by a quantity depending on the relative motion of the two particles, so that the expression for the repulsion at distance r is * When this was written, I was not aware that part of M. | 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. xiv. The value of his researches, both experimen- Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE, 67 ee (: a b =) γΣ aaa: sae dt?) ° (2) That when electricity is moving in a conductor, the velocity of the positive fluid relatively to the matter of the conductor is equal and opposite to that of the negative fluid. (3) The total action of one conducting element on another is the resultant of the mutual actions of the masses of electricity of both kinds which are in each, (4) The electro-motive force at any point is the difference of the forces acting on the positive and negative fluids. From these axioms are deducible 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? 1 would answer, that it is a good thing to have two ways of looking at a subject, and to admit that there ave two ways of looking at it. Besides, I do not think that we have any right at present to understand the action of electricity, and I hold that the chief merit of a temporary theory is, that it shall guide experiment, without impeding the progress of the true theory when it appears. There are also objections to making any ultimate forces in nature depend on the velocity of the bodies between which they act. If the forces in nature are to be reduced to forces acting between particles, the principle of the Conservation of Force re- quires that these forces should be in the line joining the particles and functions of the distance only. The experiments of M. Weber on the reverse polarity of diamagnetics, which have been recently repeated by Professor Tyndall, establish a fact which is equally a consequence of M. Weber’s theory of electricity and of the theory of lines of force. With respect to the history of the present theory, I may state that the recognition of certain mathematical functions as expressing the “ electro-tonic state” of Faraday, and the use of them in determining electro-dynamic potentials and electro-motive forces, is, as far as I am aware, original; but the distinct conception of the possibility of the mathematical expressions arose in my mind 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 1. 1851, Art.78, ἅς. As an instance of the help which may be derived from other physical investigations, I may state that after I had investigated the Vheorems 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. Q—2 68 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. ExaMPLes. I. Theory of Electrical Images. The method of Electrical Images, due to Prof. W. Thomson*, by which the theory of spherical conductors has been reduced to great geometrical simplicity, becomes even more simple when we see its connexion with the methods of this paper. We have seen that the pressure at any point in a uniform medium, due to a spherical shell (radius = a) giving out 2 fluid at the rate of 47Pa? units in unit of time, is ΚΙ — outside the shell, and #Pa inside it, γ 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 47P’a”, then the expression for the pressure will be, outside the shells, ΟῚ I a p= 4nP — --ἀπρ' where 7 and 7’ 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 Now the surface, for which the distances to two fixed points have a given ratio, is a sphere of which the centre O is in the line joining the centres of the shells CC’ produced, so that CO=CC eek =: “ὦ τ᾿ and its radius τοῦ Pa?. Pa? If at the centre of this sphere we place another source of the fluid, then the pressure due to this source must be added to that due to the other two; and since this additional pressure depends only on the distance from the centre, it will be constant at the surface of the sphere, where the pressure due to the two other sources is zero. We have now the means of arranging a system of sources within a given sphere, so that when combined with a given system of sources outside the sphere, they shall produce a given constant pressure at the surface of the sphere. * See a series of papers “On the Mathematical Theory of Electricity,” in the Cambridge and Dublin Math. Jour., begin- ning March, 1848. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 69 Let a be the radius of the sphere, and p the given pressure, and let the given sources be at distances b, ὃ; &c. from the centre, and let their rates of production be 47P,, 4π|Ρ, &c. 2 2 δ ° a @ 2 : ᾿ Then if at distances hk &c. (measured in the same direction as 6,6, &c. from the 1 2 centre) we place negative sources whose rates are : a the pressure at the surface r=a will be reduced to zero. Now placing a source 47 at the centre, the pressure at the surface will be uniform and equal to p. The whole amount of fluid emitted by the surface r = a may be found by adding the rates of production of the sources within it. The result is To apply this result to the case of a conducting sphere, let us suppose the external sources 4a P,, 42 P, to be small electrified bodies, containing e, 65 of positive electricity. Let us also sup- pose that the whole charge of the conducting sphere is = F 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 6 from the centre we must find a point on the same radius at a distance 2 a - - α : : . b? and at that point we must place a quantity =—e mn of imaginary electricity. 1 1 At the centre we must put a quantity Z’ such that a a E'= E+e,— +e, —+ &e.; δι bs then if R be the distance from the centre, 7,7, &c. the distances from the electrified points, and r’,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 lire ὦ line @ R=a and —=-, LCS rr 5.8 TT ‘To so that at the surface J Ge =—+—4+—4+kKe, eer δ᾽ Ὁ ΟΣ 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 = =I, and 6, is infinitely distant along axis of x, 1 and E=0. The value p outside the sphere becomes then a νας) r and inside p=0. II. 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 @ is d (m a — (=) =—-—Im roe The effect of the sphere in disturbing the lines of force may be supposed as a first hypothesis to be similar to that of a small magnet at the origin, whose strength is to be determined. (We shall find this to be accurately true.) Let the value of the potential undisturbed by the presence of the sphere be p=. Let the sphere produce an additional potential, which for external points is 3 p=A 2k +h =| SIS 2 dp k-k αὖ dr --. Cae 1 dp k-k αϑ dp eal ee. τῷ ἘΠῚ Ξ Ὁ r 49 ( ok + {' 5 ΟΝ ὦ, , ap? 1 ZL 1 Ἢ" k-K αϑ k—# [ὦ v= -- — ------ -- || = F111 + = (1- ; ——,} — ΘΝ: ξ dr Ὁ r dO i γ᾽ sin’@ dp Ε 2k +k! γ a Er erry pu os 9) 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 πο τῆν δ vl μμώεδίςι, τ : --- k—K Fs k-K ὦ ey a ne ] eae ae 2k + k 4 eee This expression, which must be positive, since b is greater than a, gives the moment of a’) 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 (& — k’), and is therefore quite insensible except in iron *. V. Two Spheres between the poles of a Magnet. Let us next take the case of the same balls placed not in a uniform field but between a north and a south pole, + M, distant 96 from each other in the direction of 2. * See Prof. Thomson in Phil. Mag. March, 1851. Vor. X. Parr 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 1 1 P=M Gao 7 Fae) From this we find as the value of 1". , AM? r r P= = (1-55 +95 costo); dI ΠΡ os lag - 18 oe r sin 20, and the moment to turn a pair of spheres (radius a, distance 2b) in the direction in which @ is increased is k-Kk M*db? ok aK 8 sin 20. — 36 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 ὦ — Δ΄, 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 ts 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,, k., 3, 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 7, m, n. Let us now take the case of a homogeneous sphere whose coefficient is 4, placed in a uniform magnetic field whose intensity is 11 in the direction of x. The resultant potential outside the sphere would be k,— ke 2) 2) eit hee ἘΞΗ [Ξε Σ Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 75 and for internal points 3h, p= lL] ——— a. Pr 2k, +k So that in the interior of the sphere the magnetization is entirely in the direction of w. It is therefore quite independent of the coefficients of resistance in the directions of # and ἡ, 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 7, but we must use a different coefficient for each. We find for external points kak one oie Τ᾿ ns |—?, ἘΞ: στε, ΤΣ , Ska nee Ξε par {tes my +n + ( γ3 and for internal points -1( 38k, ge 3k, ἜΤΙ 3k ) tee NGRCn kK! eae) The external effect is the same as that which would have been produced if the small magnet whose moments are k,-k ko— hr ἴ-- k : Wa’, ——__ mIa'’, ἘΞ Ε᾿ n 2k.+ ke 3 2k, +k’ 2h, + he’ fa had been placed at the origin with their directions coinciding with the axes of x,y,z. The effect of the original force J in turning the sphere about the axis of x may be found by taking the moments of the components of that force on these equivalent magnets. The moment of the force in the direction of y acting on the third magnet is , 3 ots ——, mnI"*a 2ka+ ke : and that of the force in x on the second magnet is The whole couple about the axis of w is therefore 3k’ (ks — ke) - I? 3 Gh, + K)(2k, +h)" 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 w is vertical, and let J be horizontal, then if @ be the angle which the axis of y makes with the direction of J, m= cos θ, n = —sin@, and the expression for the moment becomes Ki (ka— ks) Ta? sin 20 2 (2k, +k)Qhjt kK) - tending to increase 6. 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 76 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. the coefficient of this quantity can be but little altered by changing the value of k’ tok, the The expression then becomes ; pee ΑΝ F 1 ἘΞ: TI?a’ sin 20, value in space. 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. were away, whether the substance of the shell be diamagnetic or paramagnetic. In the case of no resistance the entire effect of the shell on any point, internal or external, may be represented by supposing a superficial stratum of magnetic matter spread over the outer surface, the density being given by the equation p= 31 cosé. 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 ginary mag Ρ 3 τ 12 p = — I=, cos, 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 #, the resistance in the external medium being hk’. The thickness of the Let the magnetic moment of the permanent magnetism Then magnetized shell may be neglected. be Za’, and that of the imaginary superficial distribution due to the medium k= Aa’. the potentials are 3 external p’= (I + A) = cos@, internal p, = (1 + A) 7 cos 0. The distribution of real magnetism is the same before and after the introduction of the medium ὦ, so that ΣῈ 20 1 i Q τις Ξτί + 4) +7 (+4), k-K or πον τ," The external effect of the magnetized shell is increased or diminished according as & is greater or less than Δ΄. It is therefore increased by filling up the shell with diamagnetic matter, and diminished by filling it with paramagnetic matter, such as iron. * Taking the more general case of magnetic induction re- ferred to in Art. (28), we find, in the expression for the moment of the magnetic forces, a constant term depending on 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 nature, we must admit that 7'=0 in all substances, with respect to magnetic induction. This argument does not hold in the case of electric conduction, or in the case of a body through which heat or electricity is passing, for such states are maintained by the continual expenditure of work. See Prof. Thomson, Phil, Mag. March, 1881, p. 186. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. {{{| 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 t, and let its external effect be that of a magnet whose moment is Za*. 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, poly p= 3 395, Pp, = Ar cos θ, - ἀρ’ dp, and since there is no permanent magnetism, —- = ——, when r =a, dr dr 4- -- 41. 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 — 3a sin θάθ, so that the quantity of the current referred to unit of area of section is as - 351 a sin θ. 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 : . - : . 326 so that the distance between successive coils measured along the axis of # is —, then n there will be m coils altogether, and the value of J, for the resulting electro-magnet will be n 1-τπ--Ἴ. 6a ~ The potentials, external and internal, will be n a? nr p = I, — —cos0 p, =— 21, — -- cos@. P =6 7 = Pi “δ τῶ 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 ἢ 830 αἢ n 581: ἵ — cos 0, (DSSS SS ern ---- οὐ πε . 0 Diese ors Pi ἀρ αστε hah ee 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 73 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, Ba = wW2, Yo= — &Y where ὦ is some function of 7. Where there are no electric currents, we must have ay, by, cs each = 0, and this implies d (s =) ὃ ἜΣ, ἂς | oC the solution of which is G = ὍΣ + ἘΠ) 5 Within the shell w cannot become infinite; therefore w = (Ἷ is the solution, and outside α must vanish at an infinite distance, so that C, o= — 7: is the solution outside. The magnetic quantity within the shell is found by last article to be —2 pe ee Oe θα 2k +k dr dy therefore within the sphere Tel Ἐπ ΤΠ Outside the sphere we must determine w so as to coincide at the surface with the internal value. The external value is therefore fn 1 a 2a Sk+k ν where the shell containing the currents is made up of x 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 π΄; then the total electro-tonic intensity EJ, round the second coil is found by integrating o=- 9 2 EI, = 7 “wa sin Ods, Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE. along the whole length of the wire. The equation of the wire is φ cos = —_, 7) π where 7) is a large number ; and therefore ds = asin 0d, = — an’ sin*0d0, 4π Qa 1 *, ἘΠ)Ξ — wan’ = -- —ann' I——.. Ja ΕΜΙ 3 Sk +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 πῇ or η΄" for nn’. 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 J and I’. 2 Let XW stand for pees Ss , then the electro-motive force of the first wire on the second is 8 (8k+k) pail | — Nnn an That of the second on itself is dl’ — Nn? —. πῶς The equation of the current in the second wire is therefore ,al dl’ — Ninn! — — ΝΗ ΞΕ SRE -ς...0.6ππὸι τς (1) dt dt The equation of the current in the first wire is ἃ] dl’ = Nit 5, ~ Nan’, LC DLE Ht ERC CROEE Eliminating the differential coefficients, we get Ὡς ee TF ἘΞ ΕΠ) 7) 1) n in. eT αὐ d N #2. SS} | ΞΕ: ΞΞ --- ᾿ξ ita atal ata 88 G » 9) ie Δ; ἢ (3) from which to find-T and 1. For this purpose we require to know the value of F in terms of ¢. Let us first take the case in which F is constant and J 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. 80 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. ; n= mn” Putting 4 τίου N FG =f =) we find ‘ : Ἢ F The primary current increases very rapidly from O to π᾿ and the secondary commences at , F - ” and speedily vanishes, owing to the value of + being generally very small. n The whole work done by either current in heating the wire or in any other kind of action is found from the expression Hi ” 1? Rat. 0 The total quantity of current is tae 0 For the secondary current we find = Fn? + Fi τ I?Rdt= = he i, de Rn? 4’ fi The work done and the quantity of the current are therefore the same as if a current Fn’ of quantity I’ = ——— had passed through the wire for a time 7, where IR'n 8 'R ne 5,2 = ΟΝ ΞΞΞ + = . δὲ ie z) 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 #' suddenly cease while the current in the primary wire is J, and in the secondary =0. Then we shall have for the subsequent time -Ξ (hah ΞΞ 0 1- ive ῳ ᾽ ΞΞ R’ eae , : Rn . Tc The equivalent currents are 1 7. and 1 J, Rn? 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 τ, 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 NW depends on the form of the machine, and may be determined by experiment for a machine of any shape. Mr MAXWELL, ON FARADAY’S LINES OF FORCE. 81 2 XII. Spherical shell revolving in magnetic field. Let us next take the case of a revolving shell of conducting matter under the influence of a uniform field of magnetic force. The phenomena are explained by Faraday in his Experimental Researches, Series II., and references are there given to previous experiments, Let the axis of x 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 z, in the plane of za. Let # be the radius of the spherical shell, and 7 the thickness. Let the quantities a, Bo, Yo be the electro-tonic functions at any point of space; a, δ᾽» 6)» a4, βι» γι symbols of magnetic quantity and intensity; a, by, Cy, ay, By, Y2 of electric quantity and intensity. Let p, be the electric tension at any point, ag = “P+ bay Bo = Te + By penises £@) aan + key πος ae ren The expressions for ap, 3, yo due to the magnetism of*the field are iE ay = Ay + | y cos θ, By = By + = = (xsin @ — wos 6), YS oie Y= Cy Gis y sin 0, A,, B,, C, being constants; and the velocities of the particles of the revolving sphere are dx dy dx —=0. ae = — wy, FF = Wt, dt We have therefore for the electro-motive forces ih ἃ i 7 Gee - 95. — = cos Qlac An dt 4π 2 1 dB, WoL Be a a ae 3 “95 Fwy, 1 dy, 1 Ut . y= -- et gam Gas sin Owe. Vout. X. Part I. 11 89 Mr MAXWELL, ON FARADAY’S LINES OF FORCE. Returning to equations (1), we get i (= os) da, 4, ie dy dz dy da From which with equation (2) we find gle Sy ὅν, vain ΤῊ 4π 4 ὯΝ b, = 0, tt a oc Cc, = — — — sin Owa, k 4π 4 Po = ee Tw ἢ (x* + y®) cos @ -- xz sin 0}. 167r 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 of a small magnet whose moment is wl sin @, with its direction along the axis of y, τ. 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 en themselves. Let the final result of this action be a system of currents about an axis in the plane of ay inclinéd to the axis of # at an angle ᾧ and producing an external effect equal to that of a magnet whose moment is J’ 1, The magnetic inductive components within the shell are [,sin @ - 2I’ cos ᾧ in a, - 21 sin @ 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, (Z, sin 6 — 97΄ cos Φ) in w produces 2 ie (ὦ, sin 8 -- 2T’ cos Φ) in y, τ. (- 27’ βἷπ Φ) in y produces 2 a (2I’ sin φ) ina; T, cos 8 in x produces no currents. “ The expression for p, 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, I sin @ — 21’ cos ᾧ = ἢ sin Ὁ ss w2I' sin ᾧ “ἀπ. δ τὸ T A - 21’ sin @ = 7 w(Z, sin @ — 2T' cos φ), whence Al TR: Q4qrke cot ᾧ = ogee.” l-)—_—_ V1 ai 24ark "| 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 6, then Tcos @ is the horizontal component in the direction of magnetic north. «ἦι sin 6. The result of the rotation is to produce currents in the shell about an axis inclined at a small angle = tan7} ao to the south of magnetic west, and the external effect of these 247 currents is the same as that of a magnet whose moment is 1 Tw 2 J 24ark | + 1" The moment of the couple due to terrestrial magnetism tending to stop the rotation is Q4ark To a 94π|} + Τὼ and the loss of work due to this in unit of time is 24k Tw 8. β4πι| + To? RTI cos 0. BF cos® Θ, R'I? cos? 0. This loss of work is made up by an evolution of heat in the substance of the shell, as is proved by a recent experiment of M. Foucault, (see Comptes Rendus, xt. p. 450). 11—2 IV. The Structure of the Athenian Trireme ; considered with reference to certain difficulties of interpretation. By J. W. Donawpson, D.D. late Fellow of Trinity College, Cambridge. [Read November 6, 1856.] Tue 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 8.c. The elementary form, of which it Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 85 was an extension, and which kept its place by the side of the trireme, was the penteconter or single-banked galley with fifty rowers. The short flat-bottomed barges of the earliest sea- men were not adapted either for rapid navigation or for warfare. And as soon as the Greek mariners put out to sea either to trade with or to plunder distant cities, they seem to have adopted the long sharp-prowed vessel with its twenty-five rowers on each side. Herodotus says expressly that the Phocwans, who navigated the Archipelago, the Adriatic, and the western Mediterranean as far as 'Tartessus, used for this purpose οὐ στρογγύλησι νηυσί, ἀλλὰ πεντηκοντόροισι (I. 163), and the mythical Argo, which represents the first of those voyages, half piratical, half commercial, which the Thessalians made into the Black Sea, was undoubt- edly regarded as a penteconter. The tradition generally reckons fifty Argonauts, and it was not without a distinct reference to this, that Pindar describes the dragon killed by Jason as “bigger in length and breadth than a penteconter, which blows of steel have perfected” (Pyth. 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 (J/. 11. 227). Whether the ships of the Beeotians, to which Homer gives a complement of 120 men (J/. 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 Erythraans 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 to 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 ἐπέβαται or soldiers on board. Now in the nautical inscriptions published by Béckh, we have a particular class of oars called by this name, αἱ mepivew 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 Opavira, or bench- rowers, in the highest tier, 54 ζυγῖται or cross-bit-rowers, on the second tier, and the same number of θαλαμῖται, 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 86 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 epibatw, of four pentekostyes, which was the Lacedemonian arrangement at the first battle of Mantineia (Thuc. v. 68). or it was two lochi of 100 men each, if we prefer Xenophon’s subdivision (Rep. Lac. 11. 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.) “5 ΕΞ a. Oar of thalamite seated on deck. ὦ. Oar of zygite seated on stool on deck, c. Oar of thranite seated on stool on gangway. Besides the objection stated in the text, that this arrangement will not explain the Greek names of the three tiers of rowers, it is impossible to 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. 243 sqq., but we can infer from it that the ribs were always bound together with cross-beams before they were covered with planks. These cross-beams or cross-bits are called 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 «Aides 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. a _ Sere a Σὲ ἽΝ, = 18 | ll δ δ a. Thranus, or long stool, placed on the alternate zygon, or supported by the se/is, and extending 7 feet amidships. ὁ. Zygon, or cross-plank, running athwart the vessel at intervals. 6. Thalamitic seat, 4ft. 6in. d. Thalamos, or hold leading to anilos. e. Platform for Epibate running along the traphex, and 6 feet wide, with bulwarks of 3 feet. Ff. Selis, or gangway, fore-and-aft, 4 feet wide. A—B, (Breadth of beam) =18 feet. C—D, (Depth) =12 feet. 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 κληῖδες OF 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 cé\uara, 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 ἃ 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 ¢ranstra or cross-planks of the vessel. Boéckh 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 rpadnyt or bulwark, And in a subsequent part of the same inscription (p. 291) we have the phrase τῶν ζυγῶν κεπώ- πήνται 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. Il. 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 Timeeus (ap. Athen. p. 37) of the young men at Agrigentum who fancied that their house was a trireme at sea, one of - = , \ \ , τ ὃν, , , them says ὑπὸ TOU δέους καταβαλὼν ἐμαυτὸν ὑπὸ τους θαλάμους: ως ἐενι μαλιστα κατωτάτω Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 89 exeiunv, “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. 11. 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 @pavos are other forms of θρόνος, but this seems very unlikely. It would be more reasonable to connect θρόνος with the root otop-, and to understand an original form στρόνος, but to recognize in θρᾶνος or θρῆνυς the root of θραύω; 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. Τί 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 ζυγίτης 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 ep ἑπταπόδην), and ‘he left the deck of the equal ship” (λίπε 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 (J/. xv. 674731). 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 Vor, XS) Pann 1. 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 Pirgeus, 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. vy. 33), and to expose the rowers to missiles from boats rowing along-side (Thucyd. vir, 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. J have no doubt that the thranitic oars were Jonger than this, and the epithet δολιχήρετμος which Pindar applies to A’gina (Ol. v111. 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. JJ. xxi. 155: δολιχεγκής: 111. 846, ἧτο. : δολιχόσκιον éryxos). 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 (v1. 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 partium corporis humani, I. 24, Vol. 111. p.85, Kuhn): καθάπερ οἶμαι Kav ταῖς τριήρεσι τὰ πείρατα τῶν κωπῶν εἰς ἴσον ἐξικνεῖται καί τοι γ᾽ οὖν οὐκ ἰσῶν ἁπασῶν οὐσῶν, καὶ yap οὖν κἀκεῖ Tas μέσας μεγίστας. Aristotle makes a similar comparison (de partibus animalium, 1v. 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 fulerum—and having asserted the principle, he says: ἐν μέσῃ δὲ TH νηὶ πλεῖστον τῆς κώπης ἐντός ἐστιν Kal γὰρ ἢ ναῦς ͵ ᾽ , , ΓΞ “ x a 3, Ἐπ > , , δέ θ , “ ΄ ε , TavTyH ευρυτατὴ εστιν. WOTE πλείον ET augotepa εν! EXET at MEpos τῆς κωτπῆς EKATEPOU Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 91 τοίχου ἐντὸς εἶναι τῆς vews,—and at the end he adds: διὰ τοῦτο of μεσόνεοι μάλιστα κινοῦσιν" μέγιστον yap ἐν μέσῃ νηὶ 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, Ρ. 4) that each xygite sat between the ¢hranite and thalamite immediately next to him, and the words of Pollux (above, p. 7) that the ζυγά were τὰ μέσα τῆς vews (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. pecoveot, 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 ¢halamites 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, 553). The interval between two oar-ports on the same tier was two cubits (Vitruv. τ. 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 epibata. It is not impossible that the ¢thranus rested on the selis, so that there were syga or cross planks only where the zygites sat. This seems to be suggested by the explanation in Julius Pollux (1. 87): πὸ δὲ περὶ TO κατάστρωμα Opavos, ov οἱ 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 σελίδας as Ta 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 cedis 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. Aneed. 62, 27): σελὶς βιβλίου" λέγεται δὲ καὶ cers 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 κερκίδες 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 σελίδες, because they were flanked on each side by seated spectators, just as the σελίδες in the trireme 19—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 (Equwites 546), which has been found unintelligible : αἴρεσθ᾽ αὐτῷ πολὺ τὸ ῥόθιον, παραπέμψατ᾽ ep 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 Aschylus, 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): σὺ ταῦτα φωνεῖς νερτέρᾳ προσήμενος κωπη. κρατούντων τῶν ἐπὶ ζυγῷ δορός: ςς ΤΉρβθ 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 Clytemnestra the ζυγῖται, and the murdered Agamemnon the 6pavirys! 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 AXschylus 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 Guyer, 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 ἐπὲ 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 νερτέρᾳ 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. Pheniss. 74: ἐπεὶ δ᾽ ἐπὶ ζυγοῖς καθέζετ᾽ ἀρχῆς» and Eustathius tells us that the Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. 93 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 ΖἜβομγ- 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 vy. 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 Aischylus had in his recollection the lines in the Zdiad, where Agamemnon says that old age shall find her: ἱστὸν ἐποιχομένην 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, ceena sit in transtro ?” 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. τ. 87: ἐκεῖ που καὶ σκήνη ὀνομάζεται τὸ πηγνύμενον στρατηγῷ ὴ 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 Tparyos ἐν πρύμνῃ πόλεως οἴακα νωμῶν, βλέφαρα μὴ κοιμῶν ὕπνῳ). (4) To the practice of moving fore and aft along these cross-planks with frequent intervals, at least where the rowers sat, even if the selis was planked, I also refer the proverbial expression of warning, that ‘ we must take care not to step into the bilge-water, or put our foot into the hold” (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 Aischylus 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. Wuewett, D.D. Master of Trinity College, Cambridge. [Read November 10, 1856.] Tuovcu 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 ἃ phi- losophy which can be expressed in our own language;—that his system, if we hold it to be well founded, shall compel us to deny the opposite systems, modern as well as ancient;—and that, so far as we hold Plato’s doctrines to be satisfactorily established, we should be able to produce the arguments for them, and to refute the arguments against them. These seem reasonable requirements of the adherents of any philosophy, and therefore, of Plato’s. I regard it as a fortunate circumstance, that we have recently. had presented to us an exposition of Plato’s philosophy which does conform to those reasonable conditions; and we may discuss this exposition with the less reserve, since its accomplished author, though belonging to this generation, is no longer alive. I refer to the Lectures on the History of Ancient Philosophy, by the late Professor Butler of Dublin. In these Lectures, we find an account of the Platonic Philosophy which shews that the writer had considered it as, what it is, an attempt to solve large problems, which in all ages force themselves upon the notice of thoughtful men. In Lectures VIII. and X., of the Second Series, especially, we have a statement of the Platonic Theory of Ideas, which may be made a convenient starting point for such remarks as I wish at present to make, I will transcribe this account; omitting, as I do so, the expressions which Professor Butler uses, in order to present the theory, not as a dogmatical assertion, but as a view, at 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. 1. p. 117.] **Man’s soul is made to contain not merely a consistent scheme of its own notions, but a direct apprehension of real and eternal laws beyond it. These real and eternal laws are things intelligible, and not things sensible. «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 him— are then by the causative act of creation deposited in things, differencing them one from the other, so that the things partake of them (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. ςς 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.” 8 4 Ρ 90 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 minds 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 Zdeas which it possesses. The impressions of sense are constantly varying, and incapable of giving any certainty: but the Ideas on which real truth depends are constant and invariable, and the certainty which arises from these is firm and indestructible. Ideas are the permanent, perfect objects, with which the mind deals when it contemplates necessary and eternal truths. They belong to a region superior to the material world, the world of sense. They are the objects which make up the furniture of the Intelligible World: with which the Reason deals, as the Senses deal each with its appropriate Sensation. But, it will naturally be asked, what is the Relation of Ideas to the Objects of Sense? Some connexion, or relation, it is plain, there must be. The objects of sense can suggest, and can illustrate real truths. ‘Though these truths of geometry cannot be proved, cannot even be exactly exemplified, by drawing diagrams, yet diagrams are of use in helping ordinary minds to see the proof; and to all minds, may represent and illustrate it. And though our conclusions with regard to objects of sense may be insecure and imperfect, they have some shew of truth, and therefore some resemblance to truth. What does this arise from? How is it explained, if there is no truth except concerning Ideas? To this the Platonist replied, that the phenomena which present themselves to the senses partake, in a certain manner, of Ideas, and thus include so much of the nature of Ideas, that they include also an element of Truth. The geometrical diagram of Triangles and Squares which is drawn in the sand of the floor of the Gymnasium, partakes of the nature of the true Ideal Triangles and Squares, so that it presents an imitation and suggestion of the truths which are true of them. The real triangles and squares are in the mind: they are, as we have said, objects, not in the Visible, but in the Intelligible World. But the Visible Triangles and Squares make us call to mind the Intelligible; and thus the objects of sense suggest, and, in a way, exemplify the eternal truths. Dr WHEWELL, ON THE PLATONIC THEORY OF IDEAS. 97 This I conceive to be the simplest and directest ground of two primary parts of the Theory of Ideas;—The Eternal Ideas constituting an Intelligible World; and the Partici- pation in these Ideas ascribed to the objects of the world of sense. And it is plain that so far, the Theory meets what, I conceive, was its primary purpose; it answers the questions, How can we have certain knowledge, though we cannot get it from Sense? and, How can we have knowledge, at least apparent, though imperfect, about the world of sense ? But is this the ground on which Plato himself rests the truth of his Theory of Ideas ? As I have said, I have no doubt that these were the questions which suggested the Theory; and it is perpetually applied in such a manner as to shew that it was held by Plato in this sense. But his applications of the Theory refer very often to another part of it;—to the Ideas, not of Triangles and Squares, ‘of space and its affections; but to the Ideas of Relations— as the Relations of Like and Unlike, Greater and Less; or to things quite different from the things of which geometry treats, for instance, to Tables and Chairs, and other matters, with regard to which no demonstration is possible, and no general truth (still less necessary and eternal truth) capable of being asserted. I conceive that the Theory of Ideas, thus asserted and thus supported, stands upon very much weaker ground than it does, when it is asserted concerning the objects of thought, about which necessary and demonstrable truths are attainable. And in order to devise argu- ments against this part of the Theory, and to trace the contradictions to which it leads, we have no occasion to task our own ingenuity. We find it done to our hands, not only in Aristotle, the open opponent of the Theory of Ideas, but in works which stand among the Platonic Dialogues themselves. And I wish especially to point out some of the arguments against the Ideal Theory, which are given in one of the most noted of the Platonic Dialogues, the Parmenides. The Parmenides contains a narrative of a Dialogue held between Parmenides and Zeno, the Eleatic Philosophers, on the one side, and Socrates, along with several other persons, on the other. It may be regarded as divided into two main portions; the first, in which the Theory of Ideas is attacked by Parmenides, and defended by Socrates; the second, in which Parmenides discusses, at length, the Eleatic doctrine that All things are One. It is the former part, the discussion of the Theory of Ideas, to which I especially wish to direct attention at present: and in the first place, to that extension of the Theory of Ideas, to things of which no general truth is possible; such as I have mentioned, tables and chairs. Plato often speaks of a Table, by way of example, as a thing of which there must be an Idea, not taken from any special Table or assemblage of Tables; but an Ideal Table, such that all Tables are Tables by participating in the nature of this Idea. Now the question is, whether there is any force, or indeed any sense, in this assumption; and this question is discussed in the Parmenides. Socrates is there represented as very confident in the existence of Ideas of the highest and largest kind, the Just, the Fair, the Good, and the like, Parmenides asks him how far he follows his theory. Is there, he asks, an Idea of Man, which is distinct from us men? an Idea of Fire? of Water? ‘In truth,” replies Socrates, ‘* I have often hesitated, Parmenides, about these, whether we are to allow such Ideas.” When Plato had proceeded to teach that there is an Idea of a Table, of course he could not reject Vion. τ: ΡῬΆ τ Te 19 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, ‘can there be any other way of participation than these two?” Socrates replies by a simili- tude: “ὙΠῸ Idea, though One, may be wholly in each object, as the Day, one and the same, is wholly in each place.” The physical illustration, Parmenides damages by making it more physical still. “You are ingenious, Socrates,” he says, (§ 11) “in making the same thing be in * Prof. Butler, Lect. ix. Second Series, p. 136, appears to | 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: “Tf 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, “Tf 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. “Tf 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,” tell.” Parmenides, who completely takes the conduct of the Dialogue, then turns to another says Socrates, “it does not seem easy to part of the subject and propounds other arguments. ‘* What do you say to this?” he asks. “There is an Ideal Greatness, and there are many things, separate from it, and Great by virtue of it. But now if you look at Greatness and the Great things together, since they are all Great, they must be Great in virtue of some higher Idea of Greatness which includes both. And thus you have a Second Idea of Greatness; and in like manner you will have a third, and so on indefinitely.” This also, as an argument against the separate existence of Ideas, Socrates is represented as unable to answer. He replies interrogatively : “ Why, Parmenides, is not each of these Ideas a Thought, which, by its nature, cannot exist in anything except in the Mind? In that case your consequences would not follow.” This is an answer which changes the course of the reasoning: but still, not much to the advantage of the Ideal Theory. Parmenides is still ready with very perplexing argu- ments. (§ 13.) “The Idea, then,” he says, “are Thoughts. They must be Thoughts of something. They are Thoughts of something, then, which exists in all the special things; some one thing which the Thought perceives in all the special things; and this one Thought thus involved in all, is the Jdea. 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. It may seem bold to maintain an opinion different from that of so many eminent scholars; but I think that the solution which I offer, will derive confirmation from a consi- deration of the whole Dialogue; and therefore I shall venture to propound it in a distinct and positive form. It is this: I conceive that the Parmenides is not a Platonic Dialogue at all; but Antiplatonic, or more properly, Eleatic: written, not by Plato, in order to explain and prove his Theory of Dr WHEWELL, ON THE PLATONIC THEORY OF IDEAS, 101 Ideas, but by some one, probably an admirer of Parmenides and Zeno, in order to shew how strong were his master’s arguments against the Platonists, and how weak their objections to the Eleatice 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, Parmenides: 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 (ᾧ 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 7 am at the same time one and many, where is the wonder? For I partake of maniness: my right side is different from my left side, my upper from my under parts. But I also partake of Oneness, 109 Dr WHEWELL, ON THE PLATONIC THEORY OF IDEAS. for I am here One of us seven. So that both are true. And so if any one say that stocks and stones, and the like, are both one and many,—not saying that Oneness is Maniness, nor Mani- ness Oneness, he says nothing wonderful: he says what all will allow. (ἢ 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 (ὗ 19) has acknowledged that he is δὲ 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 Pirzeus, of whom we have so charming a picture in the opening of the Republic. He is from Clazomenz, 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’heetetws, Socrates is represented as saying that he when very young had seen Parmenides who was very old}. Athenzus, however ἢ, 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). ἡ 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., See. R.S., Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cam- bridge. [Read May 11, 1857.] In a paper “On the Numerical Calculation of a class of Definite Integrals and Infinite Series,” printed in the ninth volume of the Transactions of this Society, I succeeded in developing the integral [ cos” (w — mw) dw in a form which admits of extremely easy ὃ Q 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. IX. Part II. Vou. X. 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 alveady 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, @ 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 oad 2a (Ξα)" (2a)° u=2 e-* sin 2aadxz = - = eee 6) vo 1 Was 3.4.5 OF ARBITRARY CONSTANTS, ἄο. 107 The integral and the series are both convergent for all values of a, and either of them completely defines τὸ for all values real or imaginary of a. We easily find from either the integral or the series This equation gives, if we observe that τ = 0 when a = 0, i ge [eda = 20" ta + ἐς + Zi + Ξ: -- Ἢ ΘΟ ὅν (3) ᾿ τ δὴν ie eR oi 2 5:0 This integral or series like the former gives a determinate and unique value to wu 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 w in a form convenient for calculation when a is large, let us seek to express w 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 ὦ Assuming a series with arbitrary indices and coefficients, and deter- mining them so as to satisfy the equation, we readily find 1 ] 1.3 “u=—- +— +—— a 20 2*a° Hi vee This series can be only a particular integral of (2), since it wants an arbitrary constant. To complete the integral we must add the complete integral of du 4 ὃ — + 24u = da : whence we get for the complete integral of (2) Aa de RIPE NT u=Ce* +-+—_+—— + Ξ Gh ΟΣ ἢ a’ ἜΠΕΑ ΠΕ σα. NCD) 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) w 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 + ν΄ — 1sin 8). 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 @, which must be increased or diminished (suppose increased) by π. 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+ 7 to @=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 @ included in the range a to α τ 3π, 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 τι. 4, When p is large the series (4) is at first rapidly convergent, but be p ever so great it ends by diverging with increasing rapidity. Nevertheless it may be employed in calculation provided we do not push the series too far, but stop before the terms get large again. To shew in a general way the legitimacy of this, we may observe that if we stop with the term 1: 9. 5.- (ον 1 ὃν OF 3 gig the value of τὸ so obtained will satisfy exactly, not (2), but the differential equation du τ ει 1.5 Ὁ Ἰ ΟΣ ἘΠῚ eS. em ἘῸΝ Ὁ [5) Let πο be the true value οὗ τὸ for a large value a, of a, and suppose that we pass from ἂρ 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 τ = Uy ὁ ze" [εἰ Δα, αρ whereas since our approximate exprossion for w actually satisfies (5) we actually have, putting A, for the last term, u =U, +e-* ["(e-A)e*da ποτ... (θὴ αο 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 i+ 1 terms of the series, as a near approximation to τέ. 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 8) dp, and integrating from p, to p, θ᾽ remaining equal to θ.. and then da = p(-sin@ + ν΄ — 1cos 6) dé, and integrating from 8, to 6, p remaining unchanged. This is allowable, since w 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 90 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” 558 will become, for values of @ 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 ρ΄ 1“, yet as the mutual destruction of positive and negative parts may take place quite differently in the two integrals /2eda and [A,e"da, we can conclude nothing as to their relative importance. 6. Now cos2@ will continually increase or decrease from one limit to the other, or else will become a maximum, according as the two limits 00 and @ lie in the same interval 0 to 3 or 7 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<@ Did fenle'a)se's\e)eicivis stesso (15) Suppose that we have to deal with such values only of the imaginary variable xv 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 ἴο τ. Now we happen to know that the series is the development of (1 +)’. But this function admits of one or other of the following developments according to descending powers of & :— ΤΠ ΜΕ DEE De Neer ΕΣ SR ΕΒ ἐπ 2 2.4 2.4.6 1 5] Tig Tl Ὁ Se fie ASI -§ be oem Seah να 5.4 6 ἢ ceetetetetsseee (17) Let wv = p (cos 0 + ν΄ -1 sin 6), and let αὐ denote that square root of 2 which has 4@ 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 of 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 (θὲ -- 1) π᾿ Ἔα and (θὲ 1) π -- α, 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 summed directly, and in the formula of Art. 8 the n'" difference be the last which yields significant figures, the number of terms actually employed in some way or other in the summation will be m+ +1. And in this case we cannot pass from one to the other of the two series (16), (17) without rendering τὸ discontinuous. But when @ passes through an odd multiple of w we may have to pass from one of the two series to the other. Now when @ is increased by 27m the series (16) or (17) changes sign, whereas (15) remains unchanged. Therefore in calculating w for two values of @ differing by 27 we must employ the two series (16) and (17), one in each case. Hence we must employ one of the series from @ = — a to @=7, the other from @=7 to @= 3m, and so on; and therefore if we knew which series to take for some one value of w everything would be determined. Now when p=1 the series (15) becomes identical with (16) when @ has the particular value 0, Hence (16) and not (17) gives the true value of τὸ when — π᾿ < θ « π. OF ARBITRARY CONSTANTS, &c. 115 13. Let p, 6 be the polar co-ordinates of a point in a plane, O the origin, C a circle described round O with radius unity, ,ϑ' the point determined by # = -- 1, that is, by p= 1, @=7. ‘To each value of w corresponds a point in the plane; and the restriction laid down as to the moduli of w 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 JS, return to Py again, the function (1 + a)! will regain its primitive value τι. or else become equal to -- τέ.» 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 «, using when possible the divergent series (16) in place of the convergent series (15). For a given value of 0, 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 w correctly to a certain number of decimal places. The length OP will depend upon 6, 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 S. For points lying between the curve and the circle we may employ the series (16), but we cannot, keeping within this space, make 9 pass through the value πὶ 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 ax — B, «+3, corresponding to points P, P’, we should get for the value of τὸ at δ΄ 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 # 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 0 = π. 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 τὸ for U, — 3x for n ay IS eee mee meee ee eee POO eee ereeeenee (18) 15—2 116 PROFESSOR STOKES, ON THE DISCONTINUITY The complete integral of this equation in ascending series, obtained in the usual way, is χὰ χὰ 9*2° 2.3 ΠΤ Ε 6? 5. 51 ΓΒ ΤΕ ΤΟ Me ] ᾿ Bios ΟΝ ο᾽ gar? Sa 5.46.7 | 3.4.6. ΟΝ These series are always convergent, and for any value of @ real or imaginary assign a determinate and unique value to w. The integral in a form adapted for calculation when ἃ: is large, obtained by the method of my former paper, is . 1 1.5:.7.11. 1 7 oul mee u = Ca-te-* Us τ ay Στ 9 + .-. 1.144 1.2.144a 1.2.3. 144322 Ξε 58 τοῖς (90) ; 15 1.5.7 .41π|ὰ αἰ Πα 8. “ἢ + Da-te* 1+ F + + —3 + ... 1.14407 1.2. 1443 1.9.8. 1445.5 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 « is increased by 27. 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 abana tf. (we) Da saafa Omari ese oe escieea once τ δὴ and let 314), when we care only to express its dependance on the amplitude of w, be denoted by F(@). We may notice that (0 Ξε = Fy (0) 5) OO Ὁ Be) = FAO) cs... te oeoees (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 Fig. 1. term, In order to represent to the eye the existence and AL progress of the functions f,(v), ζ. (0) for different values of ue ine = θ, draw a circle with any radius, and along a radius vector / “| : BOX. inclined to the prime radius at the variable angle @ take two [i Ζ \ * distances, measured respectively outwards and inwards from V | ἢ \ a4 } 3B the circumference of the circle, proportional to the real part ΙΝ of the index of the exponential in the superior and inferior | AON / f terms, 6 alone being supposed to vary, or in other words N FON 3 72 proportional to cos 30. For greater convenience suppose NK ee τ these distances moderately small compared with the radius. Consider first the function #\(8) 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 OF ARBITRARY CONSTANTS, &c. 117 traced, since F, (0) = F, (θ +27). If now we conceive the curve marked with the proper values of the constants (Οὐ, D, it will serve to represent the complete integral of equation (18). In marking the curve we may either assume the amplitude @ of w to lie in the interval 0 to 27, 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 @ 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) (δὲ -- 1). 1.2... ἡ (14408)! and the modulus of this term, expressed by means of the function [, is r@+2)0 @+ 7) Γ()Γ() Γ( + 1) 4p" which when 7 is very large becomes by the transformations employed in Art. 7, very nearly, Aye (2) + PAE) (pe. ζ . . . . . Tv Denoting this expression by y»;, and putting for Γ() P(8) its value COREE: ΟΣ ΟΡ 2ῶπ, we have εἴ a; = (2 Gato ΡΟΣ ΣΡ ον τ {93} whence for very large values of 7 Αι t Fa; 1 ict a Share (a) 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 μ᾽ we have very nearly from (24), (23) — 48, u = (@7i)-e-* = (2ri)~2e7". If the exponential in the expression for » be multiplied by the modulus of the exponential in the superior term, the result will be —(4=2c0s3 6) pt e (42 co0s$6)p 3 the sign -- or + being taken according as cos 36 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 + cos 30 = 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 can the coefficient of the inferior term alter discontinuously, and the coefficient of the other term cannot change so long as that term remains the superior term. Referring for conve- nience to the figure (Fig. 1), we see that it is only at the points a, 5, 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 adb, 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. . . Β T Tv . Since there is no superior term from θ = -- Ξ to @=+ ric the coefficient of the other term cannot change discontinuously at a (i.e. when @ passes through the value zero); and by what Fig. 2. has been already shewn the coefficient must remain unchanged throughout the portion bBe of the curve, and therefore be equal to zero; and again the coefficient must remain unchanged throughout the portion eCaAb, 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 +27, and the arbitrary constant changed, Hence if we take the term corresponding to the curve represented in Fig. 8, and having a constant coefficient throughout the portion there repre- sented, we shall get another particular integral with one arbitrary constant, and the sum of these two particular integrals will be the complete integral. In Fig. 3 the uninterrupted interior branch of the curve ἢ ‘ ΤᾺΣ \ ἱ Xs b ἌΣ π . ~ ΕΖ is made to lie in the i = A y ers δδ fo” e interval 5 to x. It would have done ee ieee (Ag - “Ὁ δ equally well to make it lie in the interval -- to --ππ; we should thus in fact obtain the same complete integral merely somewhat differently expressed. * A numerical verification of the discontinuity here represented is given as an Appendix to this paper. OF ARBITRARY CONSTANTS, &c. 119 The integral (20) may now be conveniently expressed in the following form, in which the discontinuity of the constants is exhibited ; 4π 4 π᾿ 1.5 Woe ΤΙ τὸ = (- — to + *) Comber yg ee Ἷ τ Ξε: 8 8 1.144¢# 1.8. 1445 Qa Η ΠΟ ἢ Ἐπ a fg La! +(- ἘΠ to + 2m) De-te* [ + — + gail saat ᾿ 1. 144g! 1. 2. 144228 : 5 : 4π 4π 3 : - In this equation the expression (- ΤΣ to + Ἢ denotes that the function written after it is to be taken whenever an angle in the indefinite series 0-47, 0-27, 0, O42, 0: ἀπ... 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 w will be equal to Mr Airy’s integral, multiplied by an arbitrary constant, ὦ being equal to — (=) = When @=0 we have the “a integral belonging to the dark side of the caustic, when θ = 7 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 + 7, — @ in succession to the amplitude of x, and take the sum of the results. If we have got the analytical expression belonging to the bright side, we must alter the ampli- tude of w by 7, and reject the superior function in the resulting expression. It is shewn in Art. 9 of my paper “On the Numerical Calculation, &ec.” 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. ‘Let υ Ξ| πὰ, then τ: τι = | IGS + ex) — cx} dr cles sri cit whence See a eee we tC) 190 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, 8 being the imaginary cube roots of —1, of which a will be supposed equal to ω isin — cos — = —_ 3 nF 3 Whichever value of ¢ be taken, the right-hand member of equation (26) will be equal to —9> and therefore will disappear on taking the difference of any two functions ev corresponding to two different values of ce. This difference multiplied by an arbitrary constant will be an integral of (18), and accordingly we shall have for the complete integral us Ef e~* (6 ae) dd + Ff 6 Gee Ns onan 60) 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 ΚΕ, 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+ Be= fe {(1 +a)E+(14+B)F+3[(1- a?) E+ (i - 6) ΕἸ aX} ad which gives, since a= -—G, β᾽ Ξ -- a, a Te = 1 r@Q), i “e-* rd = 4TQ), Ag f{a+a)E+(1 aes ἔζων; ον (88) B= ΤΩ {(τ-:β) Ἐπ τ a) ΕἾ. 19. We have now to find the relations between EZ, F and C, D, for which purpose we must compare the expressions (25), (27), supposing a indefinitely large. 2 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 2 = Ξ 27 Ce 6 * -ρτἐ ον, when θ = = 3 ρ΄ τε, when θ -- Ο. Let us now seek the leading term of u from the expression (27), taking first the case in which 8=0. It is evident that this-must arise from the part of the integral which involves ΟἾΔ or in this case e*, which is (E + Fy f-e-* +a. OF ARBITRARY CONSTANTS, &c. 121 Now ϑρὰ -- λ΄ is a maximum for ) = pis Let A= ρ' + ((8 then ϑρλ —- = 2p! - 32? - G and our integral becomes opt [° φ- apt e fe dg. Put (= 8-ρ ᾿ξ: then the integral becomes sprite! [ ὁ δτονειδας, +3 J =3tp? Let now p become infinite; then the last integral becomes | e-“dé or 73. For though the index — £°- 3-2) £° 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 — 23. Hence then for @ = 0, and for very large values of p, we have ultimately w= 3-343 (E + δ) pte. to π Next let θ- In this case az = — p, and we get for the leading part of u «ΕΒ ΤῊΝ 0 which when p is very large becomes, as before, 3-3atakp te, : : oar : Comparing the leading terms of w both for @=— and for @=0, we find, observing that a= ev? i τ ρων ( io Ely a en fammmmemsurse oe oeaees aa Eliminating E, F between (28) and (29) we have finally ὄπ ΕΠ τ te e- De a #1 (4)} we a τοις (30) B= 3-40 (8) §-C + YD}. 20. As a last example of the principles of this paper, let us take the differential equation σ idu ἢ (31 ee efi —— — = ON cociveciore\ccaivesiveciesa [9 dz? Ὁ dx Ἢ 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 ΝΟΥΣ OS, IPAR 1 16 122 PROFESSOR STOKES, ON THE DISCONTINUITY when \/— 12 is written for 2, 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 ($1) 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 2 p= a 6 & u = (A+ Blog a) τ Bo πο πς τς-ς Ι pe ye Fal Paces (9 \ ) -B (S84 SoS es || + ΠΟ OF ae 6° where Spa bo) = omen ee The series contained in this equation are convergent for all real or imaginary values of a, but the value of τὸ determined by the equation is not unique, inasmuch as log a has an infinite number of values. ΤῸ pass from one of these to another comes to the same thing as changing the constant 4 by some multiple of Oa Bry aula 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 4 will increase by Qnr/—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 ΓΤ le 12/535 12:8? δὲ “= Caer ar al - Soe es 2.4v 2.4(44)° 2.4.6 (42)* (33) ‘ Pere) - 1: ἘΞ 9: 12 155}. ἘΞ 41 + —— τ΄ - τ - eee 2.40 2.4(42)? 2.4.6 (42)° These series are ultimately divergent, and the constants C, D are discontinuous. It may be shewn precisely as before that the values of @ for which the constants are discontinuous are Soe 2 ©; Sam eee an ΤΟΣ Ὁ: ... — STs) -- TT 5πὸ-ν for D. * Camb. Phil. Trans. Vol. 1X. Part 11. p. [38.] OF ARBITRARY CONSTANTS, &e. 123 Hence the equation (33) may be written, according to the notation employed in Art. 16, as follows: 1 2 u = (0 to 27) Ca-te-* (1 -——_ +...) + (- to + 7) Date (1 + 2.40 2.40 1 Bec) Goagoe (34) 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 -[ + F log (a sin’w)} (δέ αν" ΡΠ 7%") ΠΩΣ «τι -----. γος (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 4, B with KH, F, it will be sufficient to compare (32) and (35), expanding the exponentials, and rejecting all powers of w, We have =a (E+ Flog x) + 27rlog (4). F; whence aa ee ey Se eR 073) B=7cF. To connect C, D with E, F, we must seek the ultimate value of « when p is infinitely increased. It will be convenient to assume in succession θ =0 and @=7. We have ulti- mately from (34) u = 7)ρ 360 when 0=0; w=-— A ge Co-3eP ΠΟ — "7 cesses (G7) It will be necessary now to specify what value of log # we suppose taken in (35). Let it be log p+ v/s @ being supposed reduced within the limits 0 and 27 by adding or subtracting if need be 2iz, where ὁ is an integer. The limiting value of « for @=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 CoD, Cy ΤΟ ΣΕ of the former paper correspond to Ἢ: Ἢ Cie ame of the present. Hence we have for the ultimate value of u for θ = 0 τ -ε (Z)e ees (τ ΓΕ og ΣΕ Rack SE ἘΝ (88) For θ- 7, (35) becomes u -Γ {E+ wFW/-14+F log (p sin? w)} (e-P 5° + eS) dw ; and to find the ultimate value of τὸ we have merely to write Εἰ + π 1 We 1 for Ε in the above, which gives ultimately for 9 = π u=(=) [b+ πε νίξι. ὑπ (Δ + log2} Fy]. ......... (39) 2p 194 PROFESSOR STOKES, ON THE DISCONTINUITY Comparing the equations (38), (39) with (37), we get Cale \2 Jeevatnars trie ebm vate Ce ττ . . . ᾿ = eS o ~~ } Ὁ - (ΞῚΤΕ «πο Γ΄ 4) + log 9} FI. Eliminating E, F between (36) and (40), we get finally C = (@r)-[/ = 1A + {(πτΓ() + logs) = 1-2} BY, ) D = (2n)-4[A + {a 31" (4) + log 8} 8]. 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 APPEND EX: [Added since the reading of the Paper. ] Own 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=7iT(); B= -— 38m? (2); and log A = 0°1793878; log (— B) = 0°3602028. The two values of x 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 V—1. Real part. of V—1. 0 + 1°000000 + 2°000000 ] — 12°000000 + 12°000000 2 — 28°800000 — 20°571429 3 + 28°800000 — 16°457143 4 + 15°709091 + 77595605 5 — 5'385974 + 2°'278681 6 — 1:267288 — 0°479722 i + 0°217249 — 0:074762 8 + 0°028357 + 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 of -1 Sum multiplied by A, — 20°14750 + 17°57548 \/—1; by B, + 5°16230 + 26-23499/-1. 126 PROFESSOR STOKES, ON THE DISCONTINUITY When the amplitude of # becomes 150° in place of 90°, the amplitude of αὐ 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 2 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 ὦ it would be sufficient to multiply by cos 60° Bay Aa sin 60°. But since the amplitude of # 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 1 ΞΟ. product of the sum of the second series by B, and multiply by = (1 + /3/—1), which gives the result — 25°30132 + 8°64681 4/ — 1. Hence we have for the result obtained from the ascending series: for amp. ἃ = 90°, for amp. ἃ = 150°, From first series — 20°14750 + 17°57548 \/— 1 — 20°14750 — 17°57548 \/—1 From second series + 5°16230 + 26:23499 / al — 25°30182 + 864681 »/ al; Total -- 1498520 + 43°81047 \/— 1 — 45°44882 — 8-92867 \/—1 On account of the particular values of amp. w 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 00011604 + 0°0011604 3 00002099 — 00001484 + 0:0001484 4 0:0000563 -- 0:0000563 5 00000200 —0-:0000142 -- 0:0000142 6 00000089 — 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 -- 00000010 + 0:0000010 Remainder — 0'0000007 — 00000017 Sum + 1:0084677 + 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, A*= + 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 4 = 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 + ν΄ isin 8), 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. = 17371779 amp. = + 130° 49’ 0”, 78 for αἱ log. mod, = 1°9247425 amp. = — 22° 30’ 16656036 + 108° 52’ 58”. 99 When the amplitude of w 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. v = 150° and amp. ὦ = — 210°, and to take the sum of the results. The first will give the superior, the second the inferior term. For the amplitudes 90°, 150° of «, or more generally for any two amplitudes equidistant from 120°, the amplitudes of 2? will be equidistant from 180°, so that for any rational and real function of #? we may pass from the result in the one case to the result in the other by simply changing the sign of ν΄ = 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 2 we have simply to replace the amplitude — 22°30’ by — 37° 8.0. Hence we have for the superior term log. mod. = 16656036 ; amp. = — 168° 52’ 58”. 99. When amp. # is changed from 150° to — 210°, amp. a? is altered by 3 x 180°, and there- fore the sign of x! is changed. Hence the log. mod. of the exponential is less than it was by 2 x 1737... 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 — 00087 + 0°:0087 — 0°0012 +0°0001 + ΟὍΟΟΙ +0:9914 + ΟὍΟΤΟ νι. -- log. mod. = 1:9963 ; amp. = + 96΄.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΄.δ Hence reducing each imaginary result from the form p (cos @ + ν΄ — 1 sin 0) to the form a +/—16, we have for the final result, obtained from the descending series: For amp. # = 009, For amp. 2 = 1509, From superior term — 14°98520 + 43°810464/—15; -- 45-43360 — 8-92767\/— 1 From inferior term — 0°01524 — 0:00100+/ — 1 — 45°44884 — 8:92867 \/ — 1 Had the asserted discontinuity in the value of the arbitrary constant not existed, either the inferior term would have been present for amp. 2 = 90°, or it would have been absent for amp. ἃ; = 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 oceur in consequence of the special values of the amplitude of a 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 w as 2 the descending series have a decided advantage over the ascending. VII. On the Beats of Imperfect Consonances. By Aucustus De Morcan, F.R.AS. of Trinity College, Professor of Mathematics in University College, London. {Read Nov. 9, 1857.] Tue 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 result{ 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 kq vibrations per second, be & what it may. Consequently, an interval remains con- stant, not with p—q, but with logp—logqg. 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 ἴῃ 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 toe 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. \Wieit, 28 Iai Τ᾿ 17 130 Mr DE MORGAN, ON THE BEATS OF IMPERFECT CONSONANCES. Many writers, from Sauveur downwards, have seen the convenience of using the figures of “3010300, the common®* logarithm of 95. Thus Sauveur, for one method, divides the octave into $01 parts, so that if the higher of two notes make m vibrations while the lower makes n, the integer in 1000 (log m — log) is the number of subdivisions contained in the interval, quam proxime. The tuner of the pianoforte is required to estimate half a subdivision: for the fifth of equal temperament is 175°60 subdivisions, and the perfect fifth is 176:09 subdivisions. Even in practice, then, a smaller subdivision is required: and theory will hardly be content without the representation of the 50th part of the smallest interval in common practical use. I should propose to divide the octave into 30103 equal parts, 2508°6 to a mean semitone. Each part may be called an atom; and we have the following easy rules, which suppose the use of a table of five-figure logarithms. To find the number of atoms in the interval from m to m vibrations per second, neglect the decimal point in log m —logn, or in logn — log m, whichever is positive. To find the ratio of the numbers of vibrations in an interval of & 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 3 — log 2) atoms, or 17609, which multiplied by 4 is 70436. The 300th part of this is 235, and the 10,000th part is 7, of which one-ninth may be called 1. And 70436 — 241 is 70195, whence 7010, say 7°020, is the number of mean semitones in a perfect fifth. Τὸ find the atoms in a number of mean semitones, multiply by 10,000; add to the result its 300th part and its 10,000th part, and divide by 4. Thus 12 mean semitones gives 120,000 increased by 400 + 12, or 120412, which divided by 4 gives 30103. This rule is as accurate as the value of log 2; the one which precedes is a near approximation. Both are consequences of the equation 1 1 1 x “30103 = (: + + ) : 12 40 300 10,000 Dr Smith found that the D of his organ, the first space below the lines of the treble, gave 254, 262, 268, double vibrationst in the common temperatures of November, September, and August. * Euler, and after him Lambert, suggested the use of acoustical logarithms; and proposed systems, of which the bases are 2 and '/2. Prony gave both tables in his Instruc- tions Elémentaires sur les moyens de calculer les intervalles musicaux, Paris, 1832, 4to. The second table shows at once, in log m—logn, 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 8l Ξ A 5 A 30° which gives the number of commas in m:n, by log m—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, for purposes which do not often occur, are of yalue only when they save complicated operations. Such tables are not in the way when wanted; and when they are found, their struc- ture and rationale have to be remembered. It is a sufficient proof of the state of knowledge of the theory of beats that a work which goes so deeply into the 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 63 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 80, altogether ‘93, of a mean semitone. Mr Woolhouse’s experiment gives 254 double vibrations to the C immediately below; and other experiments give nearly the same, for our day. The common tradition is that concert-pitch has risen about a note in the last century. The change can be traced in its progress. Robison, at the end of the last century, found the ordinary tuning-forks gave 240 vibrations for C, that is, 270 vibrations for D, a little higher than Dr Smith’s organ at its warmest. Possibly some of this effect may have arisen as follows. The organs being tuned in the cold to the usual pitch of the day, the orchestras, on tuning with them after the air had been warmed by a crowd, would find it necessary to raise their pitch. This would have a tendency to cause a perma- nent rise, which the organ-tuners would of course follow, and then the same effect would be repeated. The convenience of representing the Cs by powers of 2 has led many writers to choose 256 as the number of double vibrations in the first C below the lines of the treble: I trust this power of 2 will be enough to prevent the pitch from making any further ascent. The subject to which I now come has been perplexed from the beginning by a confusion of different things under one word. By a beat, I mean any acoustical cycle derived from composition of ordinary vibrations; whether the returns can be distinguished by the ear as separate occurrences, or whether they are rapid enough to cause a sound. The first kind* of beats were used by Sauveur: but as there is a confused discussion about them in which his name occurs, it will be more convenient to call them TJ'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 ynison 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) :— “ΕἼΤ 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—2 182 Mr DE MORGAN, ON THE BEATS OF IMPERFECT CONSONANCES. about 1714. And even when they give a sound, it will still be convenient to call them Tar- tini’s beats. These beats are in their perfect theoretical existence when a consonance is quite true, and they owe their usual existence to its approximate truth. Tartini* used to tell his pupils that their thirds could not be in tune when they played or sang together, unless they heard the low note: assuming, doubtless, that their perceptions were as acute as his own. The second kind of beats I shall call 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 Tartini’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 flutterings? 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 | which appears independent of Tartini’s, is that of a physical 1754. D’Alembert’s account of this work is so precisely what | philosopher, and is developed in a masterly manner. He gave he might have written of Smith, that I quote it. ‘Son livre | the theory, and detected the beats which occur when the grave est écrit d'une maniere si obscure, qu'il nous est impossible | harmonic becomes inaudible by lowness. His memoiz was pub- d’en porter aucun jugement: et nous apprenons que des Savans | lished by the Royal Society of Montpellier in 1751, in a collec- illustres en ont pensé de méme. ΠῚ seroit ἃ souhaiter que | tion headed Assemblée Publique &c. Ihave never seen this l’Auteur engageat quelque homme de lettres versé dans la | memoir. ‘There is a long extract from it in a curious and ex- Musique et dans l’art d’écrire, ἃ développer des idées qu’il | cellent work, which I never see quoted, the Essai sur lamusique n’a pas rendues assez nettement, et dont l'art tireroit peut-étre | ancienne et moderne, Paris, 1780, 4 vols. 4to, attributed by un grand fruit, si elles étoient mises dans le jour convenable.” | Brunet to Jean Benjamin de la Borde. M. Romieu, of Montpellier, published a memoir in 1751, in | Chladni (Acoustigue, p. 253) says that the first mention of which he described Tartini’s grave harmonic: and hence some | the grave harmonic which he knew of is by G. A. Sorge (An- have made him the first discoverer. But Tartini had been | weisung zur Stimmung der Orgelwerke, Hamburg, 1744), who teaching the violin, on which instrument he was the head of a | asks why fifths always give a third sound, the lower octave of celebrated school, a great many years: that he should not have | the lower note, and concludes that nature will put 1 before 2, 3, published the grave harmonic to every pupil whom he taught | that the order may be perfect. If Tartini’s evidence in his own to tune by fifths, is incredible. He himself affirms in his | favour be disallowed, then Sorge becomes the first observer. work that he always did so from 1728, when he established | But to me the uncontradicted assertion of a teacher whose his school: and further, that he made the discovery on his | pupils were scattered through Europe, and included men so violin, at Ancona, in 1714; this was the year after he dreamed | well known on the violin as Nardini, Pugnani, Lahoussaye, the Devil’s Sonata. As it is stated that he told how the devil | &c. &c., that he had pointed out the third sound to all his played to him in his sleep, many years after, to Lalande, who | school from 1728 to 1750, is real evidence. Chladni’s mention could make astronomical gossip of any thing, I should not be | of Tartini is as uncandid as possible :—‘ Tartini, auquel on at all surprised if a certain four-volume work contained evidence | ἃ voulu attribuer cette découverte, en fait mention dans son of the date of the grave harmonic. Tratiato...’ Mentions it! No one knew better than Chladni Rameau, not Romieu, is the natural counterpart of Tartini. | himself (as he proceeds to show, the moment the paragraph In 1750 he published his celebrated treatise on harmony, the | about priority is finished) that Tartini’s whole book 15 ἃ system completion of a system which he had sketched in previous | foundedupon it. D’Alembert, La Borde, Rousseau, &c. do not works: and he and Tartini are thus related. Tartini makes | dispute Tartini’s claim; and the common voice of Europe his grave note the natural and necessary bass to the consonance | gives no other name to the discovery. which produces it: Rameau makes the harmonics of any given On this subject in general see the Article Fondamental in note the natural and necessary treble of the given note as a bass. | the Encyclopedia, by D’Alembert ; Rousseau’s Musical Dic- These contemporary counter-systems are now exploded: they | tionary, Harmonie and Systéme ; Matthew Young’s Enquiry, have an uncertain connexion with the truth, no doubt; but the | &c. are demands and obtains a great number of combinations which + Smith does not, so far as I can find, attempt to explain neither system will allow. these flutterings; though I think it may be collected that he It is due, however, to Rameau to observe that his discovery, | 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 But it is not true that Sauveur confounds the phenomena by imagining them to be the same, by put- quite correctly; for both are battemens, though arising from different sorts of cycles. 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 vibrations in a second. He directs us to take organ-pipes, at least two feet long, and to tune diatonic intervals so perfect that not the smallest battement shall be perceived. Here he speaks of what I call Smith’s beats, of which he clearly knows the negative use, namely, the acquisition of perfect concords by avoiding them. Having thus procured a perfect major and minor third to one note, he sounds them together, the interval being 25 : 24 in ratio of vibra- tions, and thus procures a battement (but this is Tartini’s beat) at each 25th vibration of the 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 upper note. beats which can be easily counted. or discords of the scale, which are not near enough. Dr Young pronounced Smith’s work ‘a large and obscure volume, which for every pur- pose except the use of an 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 If Dr Young had said that the work was largely obscure, he would have volume for the quantity of subject-matter. It leaves the subject where it found the subject only in the minds of those who do not master it; in which number we must place Young (Peacock, Life of Young, pp. 128, 129; Works, Vol. 1. pp. 83, 84, 93, 134139; Robison, Mech. Phil., Brewster’s edition, Vol. 1v. pp. 408, 411, 412). make it clear that he confounded Tartini’s beat with Smith’s, though Smith had distinctly One sentence from Young will 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 483 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! t+ 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. 194 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 (1v. 414—421) goes through the whole phenomena of the octave. It is clear that Young confounded the two kinds of beat: and even Robison, while animadverting on Young’s opinion of Smith, gives strong reason to think that he does not make the distinction. He informs us (rv. 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 In common with a great number of other writers, he ventures on no explanation of than the vibration of the note itself. 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 (1v. 411) made in the fifths, he declines explanation, and (1v. 409) states what “ Dr Smith demonstrates.” 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 evelusion of Tartini’s har- monic. Young, in replying, writes as follows (1. 186): ** Why then are we obliged to call it Dr Smith’s discovery, or indeed any discovery at all? Sauveur had already given directions for tuning an organ-pipe by means of the rapidity of the beating with others, Mém. de V Acad. 1701, 475, ed. Amst. Dr Smith ingeniously enough extended the method; but it appears to me that the extension was perfectly obvious, and wholly undeserving the name either of a discovery or of a theory.” This amply proves that neither Robison nor Young had read Smith’s theory; and I have very strong doubts that any person who has written on the subject He tells us (4Acous- ever did read it. Chladni makes precisely the same mistake as Young. * Except, perhaps, Young’s reiteration of his own mistake, several years after, in the Course of Lectures (London, 2 vols. 4to, 1807, Vol. 1. 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. 1 find nothing in Sauveur’s memoir of 1701 about beats, except what I have described. Lagrange, in his celebrated Turin memoir on sound, refers (p. 75) to Sauveur’s memoir of 1700 (not 1701) in so vague a manner that he might be supposed to have Smith’s beats in view. On looking at the volume for 1700, I find, not a memoir by Sauveur, but the description of one, forming part of the abstracts called Histoire. Here we 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. “ M. 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, ou 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, wni- sons ought to give a grave sound when perfectly tuned; to say nothing of his appearing to believe in some system of tuning a whole instrument in which there are no beats. All the modern writers with whom I am acquainted content themselves, at the utmost, with describing the phenomenon, and giving some account of the beats of imperfect unisons, except as I proceed to mention. Some time ago, after detecting the explanation from the formule, 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. writers in whom 1 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. The only 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 2 be two numbers prime to one another, m > m, and let the higher note make m vibrations while the lower note makes m. In the diagram I shall suppose m = 5, = 3, or the interval a major sixth. I shall also suppose each whole wave to be one of condensation, for simplicity. And first, let two zeros of condensation, one in each wave, be synchronous. The following diagram represents the whole of one wave of Tartini’s beat, whether it be the * Emerson arrives at the formula which I presently mark as (1-2) Mn~2x; Mr Woolhouse arrives at (1 --- ὡἋ) 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 ccunty 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 and having lent it to Emerson as giving a higher probability to Emerson having seen it than anything I can create from com- parison of the two methods. It is very likely, then, that he had not seen Smith’s Harmonics. The amusing biography of Emerson, which is prefixed to his collected works, and which appears to have been written by some one who had ample in- formation, states that he was a very desultory student till after thirty years of age. Having been treated with contempt by his wife’s uncle, he determined to gain a name, that he might prove himself the better man of the two. This he has done: if the name of his relative were now worth inserting, it would only be in connexion with the statement, true or false, that, though possessed of two livings and a stall, he made a large income by the practice of surgery. Emerson died in 1782, in his 8lst 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 To others it consists in alternate perception of the two sounds of the conso- nance; and this also is intelligible, as the stronger parts of the two waves alternate. For myself, though I can perceive both the effects above mentioned when I look out for them, the phenomenon which forces itself on my ear is an alternation of vowel-sounds{, as in u-a u-a condensation. 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 wnisons, 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 ὦ 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 xv (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 +@ be the time of vibration of the upper note in the imperfect consonance which gives the beats. When Θ᾽ 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 # be the ratio of the consonance of the perfect and imperfect upper note; that is, let a=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 m M na Pa νυ Nma=1, M (ma + 8) =1, Sie ap 1 x = na, kmna=1, kn=N. a Let ( be the number of beats in one second. A beat, as shown, lasts through as many “ἃ SG) eee } πα -- θ) α of the shorter vibrations as there are units in a: its time is then a ; sa that we have θ 1 -- 1 -- Φ B= = = (1 -- 2) kmn = (1 -- 2) mN = —— nM =™mN -- mM. (na + θ) α a v 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 8 — 8011 x 2 = — 8, 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 octaye, 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 :—TIn every consonance of which the lower number is , every wrong vibration per second in the upper note is n beats per second*, With this theorem as a key, a rationale can be obtained without difficulty; but it does not connect the two beats, and would, I think, be subject to the doubt I have cast on Emerson’s method. The formule given by Dr Smith are obtained as follows. The comma, or difference of a major and minor tone, being 81 : 80, let 6 correspond to the fraction q: p of a comma, Then 4 80 c= ae > 81 2na . Now ([ -- a)"=1———_—\—— nearly; a being small: 2+(n-1)a 8o\ 5 2q whence ὦ = (= )?- 1 ———— nearly. 161p + q 81 2q A d =(1-@7@ Nes nd B=(1-.2)m sip a" = ΒΡῈ Pe Tae nM x 161p — q which are Smith’s formule (2nd ed. p. 82). When the upper note is too sharp, g must be made negative, the negative sign of 3 being neglected. If » be the fraction of a mean semitone by which the upper note is flat, we have. for the number of beats in a minute, a 104 1 60 (a -2 =) mN, or ——umlN, or 1042 ι- - aN 30 80 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) mV very nearly, 4x8x13 amiNV nearly. 301000 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—mnM is much more easy. * The passage over the greatest common measure being m vi =m, Οὐ πὶ sad : mM, fairly arrived at, as the time of a beat, the transition to the mM: ee : formula mN—nM may be very briefly made. We know that, consequently the commencement of the shorter wave gains the m and n being prime to one another, there is, before we arrive | fraction ἬΝ i of a common measure in every vibration of 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 Δ: M of the numbers of vibrations in the erroneous consonance, and also of the lengths of the waves, is not 2 : m, but the higher note, than is mN —nM common measures in one second, or in M higher vibrations; and each gain of a com- mon measure is a beat. This demonstration, a little more developed, will be, 1 should think, the best that can be given. 18—2 140 Mr DE MORGAN, ON THE BEATS OF IMPERFECT CONSONANCES. Let the notes of the imperfect consonance be P, Q, and let P' be the octave above P. If the interval PQ be tuned too flat, then QP! is two sharp, and vice versa. All remaining as above for PQ, in passing from PQ to QP! we must change NV, M into M,2N. If m be an odd number, we must change n, m, into m, 2; but if m be even, m, m, must change into 3m, τς 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 8, when it occurs. Conse- quently, β' being the number of beats of QP! in a second, we have (m odd) B = (Gin @9 (m even) 3 = ἘΣΣΙ That is, when the fundamental number (in the interval complemental to the octave beats twice as fast as the lower interval first given. fo) a δῈ Τα Nun, β' = _*1_ Nm, [53] 2q mae = 28; 29 = NT ys = rrr ee = B. ratio m:m) of the mistuned note is odd, the But when this fundamental number is even, the interval and its octave complement have the same rate of beating. one. This is one of Smith’s* experimental verifications, and is a very easy 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 WV, M, L, be three ascend- ing notes represented by their numbers of vibrations per second, Let N make n vibrations while M makes m: let M make m’ vibrations while Z makes 7: the fractions m:m and 7: m’ being in their lowest terms. Let the imperfect consonances NM, ML, NZ, beat severally (3, β΄, B, times per second: (3 being positive when the higher note is flat, and negative when it is * There must needs be some way of explaining the excessive difficulty of this one work of Dr Smith’s. His Optics, if not a model of perspicuity, is by no means notable for obscurity ; on the contrary, I find it abounding 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 whick 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 6 tone (3): the third (5): the sixth (ἢ: the seventh 16 ie 9 . (5): The usual major intervals are the tone (G): the third 4 (3) . the fourth (:) in which there is a failure; the fifth 2 5 Ξ (5) ; the sixth (3) ; the seventh (ὦ. In the minor and - : 25 16 cox major semitone [ » 16) the rule is inverted ; and also in the - 45 es 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 ΟΟΝΘΟΝΑΝΟΕΞΒ. 141 sharp; and the same of the rest. Let g be the greatest common measure of ml and m’n. Then 1B + ηβ΄ = gB. From this we may obtain such theorems as the following. The beats of a minor third exceed those of the following major third by twice the beats of the whole fifth which they make up. Twice the beats of a minor third exceed three times the beats of the major third which it follows by five times the beats of the fifth they make up. = < a, : 5 Smith’s beats themselves have a long inequality whenever Ὁ is not 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 Coniecr, LoNnpDon, August 12, 1857. POSTSCRIPT. A Frew 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 duleimers, 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, Ὁ, A, E, B, FH, are m, 2m, 3m, 4m, 5m, 6m, 7m: while those in the keys of CH, GX, DE, 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 ereat as they can then be. g 8 y 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 ‘2346 of a mean semitone. 2. The keys being arranged dominantly, that is, in the order C, G, D, A, E, B, FH, CH, Gd, DH, Ad, 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 sweceeding keys will always amount to a comma, or ‘2151 of a mean semitone. 3. The temperament of the minor third in any key, together with the temperaments of the fifths in the three preceding keys, will always amount to a comma. Thus in all systems, the temperament of 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, 7, 8, 295 7. 7. 8: P, ἢ, > 89 in their fifths, the temperament of every major third will be p+q+7+s less than a comma, or ‘0782 of a mean semitone less than a comma. In the system R, I have taken p=0, g="0391, r=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 1369 and +1369+'0391, equal temperament giving "1564 to all. If the temperaments of the fifths run in cycles of three, as in p, ἢ. 7; DP, 5. 7, Ps GT Ps V7 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, g="01955 as in equal temperament, r=2q; which satisfies 4(p+q+r)=:2346. The temperaments of the major thirds in dominantly successive keys are “1604, *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 τ ὦ 0:00000 0°00000 000000 | σ CH) i 100000 10106 1105 C# D 2 9:02444 200000 = 201955 D DH} 3 298534 3°01955 ϑ8ΌΟΟΟΟ D¥ E 4 402932 4:00000 401955 F 5 499022 5°01955 501955 F FH!) 6 601466 600000 —_ 600000 ΤῈ εἰ 1 ἢ τοΙ4066 701955 7°01955 G GH| 8s 7:99022 800000 801955 GE A 9 9:02932 9:01955 ΟὍΟΟΟΟ A A#| 10 9°98534 10:00000 1001955 ΑΒ Β 11 1109444 11Ὸ1055 11 Ὁ10δδ Vibrations in one Minute. Beats in one Minute. P Q R 5 P Q R S 14400°0 14400°0 144000 144000 | 488 122 000 000 | CH| 152563 152563 152735 15273°5 PITTS 1058. SU le ae D 16163°5 16186°3 16163°5 16181°7 547 411 00:0 109°5 D#| 171246 171101 171489 171246 58:0 58:0 116°0 000 D¥ 18142°9 18173°6 18142°9 18163°4 614 769 00:0 615 F 19221°7 19210°8 19243'4 10948:4 65:1 32°5 1809 1302 FE FH| 203647 20381°9 203647 203647 600 1207 00:0 000 F G 21575°6 21593°9 21600°0 21600°0 73°0 865 146°2 73°2 G GH] 228586 22845°7 22858°6 22884"4 174 966 000 1549 GH A 24217°S 242589 24245°2 24217°8 82:0 821 1641 ooo | A A$} 256579 256862 25657:9 25686-9 868 6551 000 870 | A B 27183°6 272220 272143 272143 92:0 1382 1842 1842 | B The vibrations are calculated from the formula log M=log N+ 1, log 2 xa, where M and WN are the vibrations in the higher and lower notes, and z the number of mean semitones in the interval. The beats are calculated from the formula mN—nWM/; for the fifths 3N-2M. The beats are those of each note with the fifth above it: thus A#f ΕἸ (the octave above ΕἾ beats 86°8 times in a minute in equal temperament (P), The vibrations are taken as in the pitch frequently used for organs, when not wanted to combine with the orchestra, that is, a diatonic semitone (15 : 16) below the ordinary concert- pitch of our day, in which C (on the first line below the treble) gives 256 double vibrations per second. In tuning to the concert-pitch, each number in the lower table, be it of vibra- tions or of beats, must be increased by its 15th part. For an octave above, the number of beats must be doubled: for an octave below, it must be halved. Thus, CG beating 48'8 times in a minute, C,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 #, in which the difference between dominantly successive keys is greater than in the others, would be the best of all. But by making p, q, 7, s, in R, and p, q, 7, in S, more nearly equal than in the instance given, any less amount of adherence to the distinctive feature might be secured. It is useless to speculate on systems with any view of materially diminishing the number of beats in the thirds and sixths. In equal temperament, the consonance G A# beats more than 1150 times in a minute, while GD!’ (D! the octave above D) beats only 73 times. Nor can the beats be reduced, in the different consonances of a chord, either to equality, or to near commensurability, throughout any considerable portion of the scale. It is the irregularity of the beating which is its chief disadvantage: regularity would give merely the effect of a faint drum-accompaniment; but such change as that from CF ΟἹ, 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 VILL. On the Genuineness of the Sophista of Plato, and on some of its philosophical bearings. By W.H. Tuomeson, 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 net 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 αν. 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. 8. 9, we read thus: φανερὸν ὅτι οὐδὲ δόξα μετ᾽ αἰσθήσεως οὐδὲ Ot αἰσθήσεως, οὐδὲ συμπλοκὴ δόξης καὶ αἰσθήσεως φαντασία ἂν εἴη. - ν ΚΜ , ae in the darkness of the Non Ens,” (ἀποδιδράσκων εἰς τὴν τοῦ μὴ ὄντος σκοτεινότητα), taking 1 ὡπλῶς τόδε ἢ πῇ λέγεσθαι καὶ μη 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" οὐ 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 ONT" 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 secruple 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 used3. because it contains paradoxes apparently inconsistent with other parts of Plato’s writings. Tried by this test, the Lysis and the Laches, and perhaps the Charmides, would fare but ill. Yet in them, those who have eyes to see have not failed to recognize the touches of the Master’s hand, and the perfection of the form has outweighed the doubtfulness of the matter. Now I am not aware that any philological objections have been urged against the Sophista. So far as the mere style is concerned, there is no dialogue in the whole series more thoroughly Platonic. In their structure the periods are those of Plato, and they are unlike those of any other writer. Throughout, as it seems to me, the author is writing his very best. His subject is a dry one; and he strives to make it palatable by a more than ordinary neatness of phrase, and by a sustained tone of pleasantry. His style is terse or fluent, as terseness or fluency is required: but the fluency never degenerates into laxity, nor the terseness into harshness. The most arid dialectical wastes are refreshed by his humour: Few besides Plato would have thought of describing the endless wrangling of two sects who had no and bloom in more places than one with imagery of rare brilliancy and felicity. Δ T cannot but think that had the Master of Trinity exa- mined the Politicus with the same care which he has bestowed on the Sophista, he would have formed a different opinion of the genuineness of the two dialogues. The Politicus contains passages full not only of Platonic doctrine, but of Platonic idiosyncrasy. I may mention, as a few out of many, the 2 The Sophista is also recognized, as we have seen, by the vigilant and profoundly learned Simplicius, also by Porphyry (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 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 Timeus, (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!”’ Questions is really remarkable. 3 e.g. ἀποκριθῆναι for ἀποκρίνασθαι, σκέπτεσθαι for cKo- πεῖσθαι. The latter barbarism, I presume, would be defended from Laches, Ὁ. 1858. 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 Tis τεχνικὸς περὶ οὗ βουλευόμεθα. τί ποτ᾽ ἔστι περὶ οὗ βουλευόμεθα και 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 Theextetus 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 same interlocutors meet, with an addition in the person of an Eleatic Stranger, and they meet The Sophista, it is well known, is professedly a continuation of the Theetetus. by appointment: for at the conclusion of the Thecetetus Socrates bespeaks an interview for the following day, of which he is reminded by Theodorus in the opening sentence of the Sophista. The Politicus or Statesman is, in like manner, a professed continuation of the Sophist. that between the T'hecetetus and the Sophista. The connexion, however, between these two is on the surface much closer than In the Theetetus we are not informed what is to be the subject of the next day’s talk, but in the Sophista* three subjects are proposed for consideration—the Sophist, the Philosopher, and the Statesman; and the choice is left to the The third day is devoted to the Statesman, who is made the subject of an investigation similar to that pursued new-comer, who selects the Sophist as the theme of that day’s conversation. 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 We are left to infer that the Philosopher Instead of this projected Tetralogy, we the Schoolmen after them, the method of Division. was to be handled on the fourth day in like fashion. have only a Trilogy. No dialogue exists under the title of Φιλόσοφος, 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 ' Asa specimen of this, take the argument with the y- , one of those ‘‘Schleiermachersche Grillen” which contribute to yevets, 240 Ὁ. seg., and the mock solemnity with which the ‘Ens’ of the εἰδῶν φίλοι is described, 249 a, 5. Β᾿ 917.Ὰ: 3. Schleiermacher, for instance, conceives that the omission is intentional, and that we must look for the missing portrait in the Symposium and Phedo; of which the first teaches us how a philosopher should live, the latter how he should die. This is the amusement even of his admirers. Stallbaum seems to think 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 Either, it is said, he may start from particulars, and from these rise to generals: or he may assume a are marked out for the dialectician to pursue in searching for definitions’. 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*. in miniature. Now it so happens, that the Theetetus and the Sophista pretend, each of them, to be an exemplification of one of these two dialectical methods: the Theatetus of a συναγωγὴ, the We have, therefore, in this passage of the Phadrus a Platonic organon 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. 2 See Appendix I. Phedr. 265, foll. 3 Those who are unskilled in the application of these pro- cesses are termed épiotixot in the Philebus, 16 Ἑ. οἱ δὲ νῦν τῶν ἀνθρώπων σοφοὶ ἕν μὲν, ὅπως av τύχωσι. Kai θᾶττον Kai βραδύτερον ποιοῦσι Tov δέοντος μετὰ δὲ τὸ ἕν ἄπειρα εὐθύς" τὰ δὲ μέσα αὐτοὺς ἐκφεύγει" οἷς διακεχώρισται τό Te 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, κιτιλ. 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. 145 D—148, with Sophista, init, and 253, §§ 82, 83, Bekk. THE SOPHISTA OF PLATO, &c. 153 tetus, a dialogue which is in reality a critical history of Greek psychology as it existed down to the fourth century, just as the Sophista is virtually a critique of the logic or dialectic of the same and previous eras. ‘The one dialogue exposes the unsoundness or incompleteness of the mental theories of Protagoras, of the Cyrenaics, whose founder Aristippus was Plato’s con- temporary and rival, and perhaps of certain other schools whose history is less known to us!. The Sophista, in like manner, passes under review the logical schemes of the Eleatics, of their Both dia- logues, as I have said, profess to be at the same time exemplifications of the processes which admirers, the semi-Platonic Megarians, and finally of Antisthenes and the Cynics. 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 differenti of the conception implied in the term Sophist ; and this fiction* serves in both cases to bind together the critical It lends to each the unity of an organic whole*; and infuses into a critical treatise on an abstfuse branch Add to this, that the Sophista helps materially towards a solution of the question, What is Science ? which is the professed aim of and polemical investigations which make up the main body of either dialogue. 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 Thewtetus it is shewn that the Protagorean dictum, that Truth exists only relatively to its pereipient (πάντων μέτρον ἄνθρωπος); and the kindred, though not identical Cyrenaic dogma, that sense is knowledge, and the sensations the sole criteria of truth (κριτήρια τὰ 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. At the same time I 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 Vor. X. Parr I. the framework or “plot” of the drama. 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. Phedr. 264 ο: δεῖ πάντα λόγον ὥσπερ ζῶον συνε- στάναι, κι T.A. 20 154 PROFESSOR THOMPSON, ON THE GENUINENESS OF The oldest, and in the history of Speculation the most important, of these three schools was the Eleatic, founded, as the Stranger from Elea tells us in this dialogue, by Xenophanes’, though its doctrines underwent some modification, and received extensive development in the hands of Parmenides and Zeno, his successors. When Plato wrote this dialogue, there is every reason to suppose that the Eleatic school had ceased to exist. The latest known successor of Parmenides, Melissus, flourished, as the phrase is, about the year 8.0, 440, and Zeno is placed a few years earlier. ‘The earliest date which it is possible to assign to the Thewtetus, and ἃ fortiori to the Sophista, is about 393.2 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. is inadmissible in the case of his disciple. Socrates may have had such opponents, though we read of none; but the hypothesis 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 Theztetus an admissidn 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*. Theztetus 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, “νὰ 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. 242D: τὸ δὲ παρ᾽ ἡμῶν ᾿Ελεατικὸν ἔθνος ἀπὸ Ξενοφάνους... ἀρξάμενον. 3. Apuleius, de Dogm. Plat. 569, says that Plato took up the study of Parmenides and Zeno (inyenta 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 thetime. Does this imply that he visited Elea instead ? If so, and if he composed the Sophista and its sister-dialogues on his return, we obtain a clue to the fiction of an Eleatic Stranger. He was Plato, on his return from a sojourn at Elea, laden, it may be, with Eleatic lore. The circumstance that the conduct of the dialogue devolves upon this Stranger is pointed to as one proof that the Sophista was not written by Plato, whose custom is to make Socrates his Protagonist. The secondary part which Socrates plays in the 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 ? 3 "Ἐπεὶ καὶ νῦν μάλ᾽ εὖ καὶ κομψῶς els ἄπορον εἶδος διερευ- νήσασθαι καταπέφευγεν. 390 D. 4 Τετόλμηκεν ὁ λόγος οὗτος ὑποθέσθαι τὸ μιὶ ὃν εἶναι" ψεῦδος γὰρ οὐκ ἂν ἄλλως ἐγίγνετο ὄν. 237 A. 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 dv 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- ing of this Eleatic denial of the conceivableness of non-entia. ‘ You can never learn,” says Parmenides, “Ὁ 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 οὐ 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 Kal εἶναι: and, χρὴ τὸ λέγειν τε νοεῖν T ἐὸν Eupevac’, “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 “To Be” is the same as Absolute Existence. The “Ens logicum” and the ‘Ens reale” are the same thing. The only certain proposition is the identical one ‘‘ Being is,” for “not-Being is Nothing®.” Hence the Formula which served as the Eleatic watchword: 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 ¢his 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 ὄν; 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. ν. 94, Mullach. 1 Tov τοῦ πατρὸς Tlapmevidov λόγον ἀναγκαῖον ἡμῖ ἀμυνομένοις ἔσται βασανίζειν, καὶ βιάζεσθαι τό TE μὴ ὃν * Frag. v. 43, ed. Mullach, ὡς ἔστι κατώ τι; Kai TO ὃν ad πάλιν ὡς ἔστι wy. p. 241d τ τ Τ᾿ ἔστι γὰρ εἶναι, μηδὲν δ᾽ οὐκ εἶναι. 2 οὐ γὰρ Baez τοῦτο δαῇς, εἶναι μὴ ἐόντα. *AXNo 4 παρὲκ τοῦ ἐόντος. Ibid. Comp. Arist. Soph. El. c. 5, § 1, quoted above. “| - seen 0U0Ev γὰρ ἢ ἔστιν ἢ ἔσται ] Ϊ “5 Ibid. v. 110. 3 ταὐτὸν δ᾽ ἔστι νοεῖν τε καὶ οὕνεκέν ἐστι νόημα. Frag. 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 uy ov'! He is enabled to effect this reductio ad absurdum by the incautious proceeding of Parmenides, who instead of entrenching himself in the safe ground of an identical proposition, and thence defying the world to eject him, must needs invest his Ens with a variety of attributes calculated to exalt it in dignity and importance. It is ‘*unbegotten,” it is ““ solitary,” it is ‘“immoveable,” it is “a whole,” it is even ‘‘ like unto a massive orbed sphere®.” (Soph. 246 £.) and by a series of well-concerted dialectical operations, succeeds, as we have seen, in carrying the citadel. In one of these unguarded outworks the Stranger effects a lodgment, 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 Sopk. 245 c, 964 Bekk.: μη ὄντος δέ ye τὸ παράπαν τοῦ ὅλου, ταὐτά Te ταὐτὰ ὑπάρχει TH ὄντι, Kal πρὸς τῷ μὴ εἶναι μηδ᾽ ὧν γενέσθαι ποτὲ ov. 2 πάντοθεν εὐκύκλου σφαίρης ἐναλίγκιον ὄγκῳ. Parm. ν. 103. 3 ἵν᾽ ἐκ πάντων ἴδωμεν ὅτι τὸ ὃν τοῦ μὴ ὄντος οὐδὲν εὐπορώτερον εἰπεῖν ὅ τί ποτ᾽ ἔστιν. p. 245 8Ε. 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, kc. The Py:thagoreans, though remote in place, were his friends and correspondents, and in speaking of them he observes the same rule as in the case of his living Athenian contemporaries, in- dicating without expressly naming them. Thus, in the Polt- ticus, p. 285, they are merely denoted as couyoi, “ ingenious persons.” his, by the way, is a passage of great importance, as indicating the limits within which Plato ‘*pythagorized,”” and the particulars in which he dissented from his Italic friends. THE SOPHISTA OF PLATO, &c. 157 contemporary or nearly contemporary with the philosopher himself, many have been embalmed in the writings of Eusebius and Sextus Empiricus, the Aristotelian Commentators, Cicero, and others: not to mention the vast store of undigested learning amassed by Diogenes Laertius. Now of the two sects who here come under revision, and who enact the part of Gods and of Giants in the famed Gigantomachy’, which is familiar to most readers of Plato, the occupants of the celestial regions are rightly, as I think, judged to mean the contemporary sect of the Megarics. They are idealists in a sense, but their idealism is not that of Plato. They so far relax the rigid Eleatic formula “unum omnia” as to admit a plurality of forms (εἴδη or ὄντα or οὐσία). They are complimented in the dialogue as ἡμερώτεροι, “more civilized” or 3 “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 νῦν δὴ ῥηθὲν πρὸς Tous “γηγενεῖς οὐσίας πέρι, 2488; 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- stand in his critique of the Platonic Doctrine of Ideas. Like Plato, they distinguish the two worlds of sense and pure ideas, the “γένεσις from the οὐσία (γένεσιν τὴν δὲ οὐσίαν χωρίς που διελόμενοι λέγετε, 948 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 capacity of being known®, The arguments by which the ‘Friends of Forms” (εἰδῶν φίλοι, 248.) are pushed to this admission may not ring sound to a modern ear; but my business is not with the soundness of Plato’s opinions, but with their history: and it would be easy to produce overwhelming evidence both from his own writings and those of Aristotle to the truth of the statement, that however the phrase is to be interpreted, there is, according to Plato, a fellowship, 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, 8 65 Bekk. | ® Aristotle objects to the term μέθεξις on 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 ““Β partakes of A”’ (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 mot 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 Τὴν οὐσίαν δι κατὰ τὸν λόγον τοῦτον γιγνωσκομένην ὑπὸ τῆς γνώσεως, καθ’ ὕσον γιγνώσκεται κατὰ τοσοῦτον κινεῖσθαι διὰ τὸ πάσχειν, ὃ δή φαμεν οὐκ dv γενέσθαι περὶ τὸ ἠρεμοῦν. p. 2488. 4 Compare 249 Ὁ, 8. 75: τῷ Oj 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 T αὖ πανταχῇ τὸ ἕν κινούντων μηδὲ τὸ παράπαν ἀκούειν, ἀλλὰ κατὰ tiv τῶν παίδων εὐχὴν, ὅσα (ws?) ἀκίνητα καὶ κεκινημένα, τὸ ὄν πε 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 Thee/etus. 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 Kal δρῦς περιλαμ- βάνοντες, 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 T'hewtetus, p. 1558. 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 ὡς ἐν οὐσίας μέρει). These persons are garnished with the epithets “hard,” “stubborn,” * thoroughly illiterate,” σκληροὶ---αἀντίτυποι--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 Theatetus 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, | idealists of this dialogue—deny. Philed. p. 24, foll. The dis- “Hi quoque (sc. Megarici) multa a Platone,’’ Acad. Qu. 11. 42, and also by the brief statement of the Megaric dogmas which Cicero gives us in the context of this passage. 2 In the Philebus—a dialogue which treats of the relation 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 principle he denominates γένεσις els οὐσίαν, the possibility of which process is precisely what the εἰδῶν pidov—the pure 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. 8 Compare Soph. 347 c: διατείνοιντ᾽ ἂν πᾶν ὃ μὴ δυνατοὶ ταῖς χερσὶ ξυμπιέζειν εἰσὶν ὡς ἄρα τοῦτο οὐδὲν τὸ παράπαν ἐστίν. 4 vit. 3. 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, &e. 159 any manual of the history of philosophy. For our present purpose it were to be wished that some portion of the voluminous writings of Antisthenes had been preserved, in addition to the meagre declamations, if they are really his, which are commonly printed with the Oratores Attici. The notices, however, which Aristotle and his commentators have preserved to us, countenance the assumption just made, that the Earth-born are the Cynics, Hatred of Plato and the Idealists seems to have been the ruling passion of Antisthenes, and this passion drove him into the anti-Platonic extremes of Materialism in Physics, and an exaggerated Nominalism in Dialectic. <‘* He could not see Humanity, but he could see a Man,’ is one of his recorded sarcasms upon the doctrine of ideas’. ‘ Your body has eyes, your soul has none,” was the curt reply of Plato. Many other stinging pleasantries were interchanged by the leaders of the two schools: and Antisthenes, less guarded than his antagonist, wrote a dialogue “in three parts,” entitled Za@wv, which was avowedly directed against Plato in revenge for a biting reply (Diog. Laert. 111. ὃ 353; v1. § 16). it is not a little curious that it was written to disprove the very position which Plato devotes a The subject of this dialogue has been recorded, and 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, ὅτι 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 Another paradox of this school, closely connected with the last, is recorded by Aristotle*, and sarcasti- the remainder of this dialogue is, in the main, a critique of the Cynical Logic. cally noticed at page 251 8 of the Sophista, in terms which leave little doubt as to the object of Plato’s satire. If Antisthenes really pushed this paradox to its legitimate results—and from the character of the man it is not unlikely he did—he must be understood as maintaining that identical propositions are the only propositions which do not involve a contradiction: a theory which, as Plato shews, renders language itself impossible*, as well as that inward *¢ discourse of reason*,” of which language is the antitype. The resemblance of the Cynical Logic to the Eleatic is usually accounted for by the cir- cumstance that Antisthenes had been a hearer of Gorgias, who wrote a treatise, preserved or and brought with him as many of his pupils as he could induce 1 Tzetzes, Chil. vit. 605; Schol. in Arist. Categ. ed. Brandis, to follow his example. A similar sarcasm is hurled at Diony- p. 664, 45 and 686, 26; Zeller, G. P. 11. p. 116, note 1. 2 Metaph. v. 29: Ἀντισθένης wero εὐήθως μηθὲν ἀξιῶν λέγεσθαι πλὴν τῷ οἰκείῳ λόγω ἕν ἐφ᾽ ἑνός" EE ὧν συνέβαινε μὴ εἶναι ἀντιλέγειν, σχεδὸν δ᾽ οὐδὲ ψεύδεσθαι. Plat. Soph. 1.1.: 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 Kal ἀφιλοσόφου applied to the upholder of the similar sophisms noted at p.259p. In the latter passage occur the following words: od τέ τις ἔλεγχος οὗτος ἀληθινὸς, ἄρτι τε τῶν ὄντων τινὸς ἐφαπτομένου δῆλος νεογενης ὧν. * This is no genuine or legitimate confutation : but theinfant progeny of a brainnew to philosophical discussion.” This hangs together with the γερόντων τοῖς dyiynabéo.—* the old gentlemen who have gone to school late in life,” p. 251 B, and both passages are illustrated by a notice in Diog. Laert, v1. 1, init. that Antisthenes, having been originally a hearer of Gorgias, became at a later period a disciple of Socrates, sodorus and Euthydemus, in the Eudhyd. p. 272. c, 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 p.c. 426. He may therefore have been Plato’s senior by some 20 years. 3 καὶ yap ὦ ’yabé, TO ye πᾶν ἀπὸ παντὸς ἐπιχειρεῖν ἀποχωρίζειν, ἄλλως τε οὐκ ἐμμελὲς Kal δι) καὶ παντάπασιν ἀμούσου τινὸς καὶ ἀφιλοσόφου. Θ. τί δή: B. τελειοτάτη πάντων λόγων ἐστὶν ἀφάνισις τὸ διαλύειν ἕκαστον ἀπὸ πάντων" διὰ γὰρ τὴν ἀλλήλων τῶν εἰδῶν συμπλοκὴν ὁ λόγος γέγονεν ἡμῖν. Soph. 258 Ὁ. * διάλογος ἄνευ φωνῆς γιγνόμενος ἐπωνομάσθη διάνοια. Soph. 263". Van Heusde first pointed out the infamous ety- mology lurking in this passage (διάνοια --δι ἄλογος ἄνευ) The sentiment, without the etymology, occurs in Theet. 189E: (τὸ δὲ διανοεῖσθαι καλῶ) λόγον ὃν αὐτὴ πρὸς αὑτὴν 1] Ψυχὴ διεξέρχεται περὶ ὧν ἂν σκοπῇ. 100 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 usually called ‘*the Sophists.” His ethical opinions on the other hand were borrowed from Socrates; but in passing through his mind they took the tinge of the soil, and seem to the common sense of mankind as startling as any of his dialectical paradoxes. It is remarkable, however, that when Plato handles the Cynical Ethics, he treats their author with far more leniency than in this dialogue. In comparing it with the Pleasure Theory of Aristippus, he speaks of the Cynical system with qualified approbation. Δυσχερὴς", “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 isa work of wider range and profounder bearings than the Sophista, but the dialogues have this in common, that in both the broad daylight of reason is shed on regions which had been darkened by the one-sided speculations or the wilful logomachy of earlier or inferior thinkers. The way in which Antisthenes is dragged from his hiding-place among the intricacies of the Non-existent into the light of common sense, at the close of the present dialogue, appears to me an admirable specimen of controversial ability ; and the broad and simple principles on which Plato founds the twin sciences of Logic and Grammar*® stand in favourable contrast to the sophistical subtlety of his predecessors and contemporaries. At this point of the discussion I would gladly stop: but I feel bound to say a few words ” on what I have ventured to call the ‘‘ logical exercise,” which is the pretext under which Plato takes occasion to dispose of the doctrines of certain formidable antagonists. That the διαρετικοὶ λόγοι, the “ amphiblestric organa*,” in which he endeavours to catch and land first the Sophist and then the Statesman, were regarded by Plato himself in this light, we learn from his own testimony in the Politicus, 285 Ῥ, § 26 Bekk. ‘Is it,” asks the Eleatic Stranger, ‘for the Statesman’s sake alone, that this long quest has been instituted, or is it not rather for our own sake, that we may strengthen our powers of dialectical enquiry upon subjects in general? S. J. It was doubtless for this general purpose. E.S. How much less then would a man of sense have submitted to a tedious enquiry into the definition of the art of weaving, if he had no higher object than that!” He then proceeds to apologize for the prolixity of this method of classification: but adds, “ The method which enables us to distinguish according to species, is in itself worthy of all honour ; ; nay, the very prolixity of an investigation of this kind becomes respectable, if it render the hearer more inventive. In that 1 Hence the explanation of Philebus, 448: καὶ μάλα δει- | with the ὠπαίδευτοι of Aristotle, and with his own ἄμουσοι, νοὺς λεγομένους τὼ περὶ φύσιν. &e. 2? Phil. 440: μαντευομένοις ob τέχνῃ ἀλλά τινι δυσχε- 5. Ῥ 302 τ. Simple as the analysis of the Proposition into peta φύσεως οὐκ ἀγεννοῦς, λίαν μεμισηκότων τὴν τῆς ἡ- ὄνομα καὶ ῥῆμα (Subject and predicate in logic, noun and verb δονῆς δύναμιν, καὶ νενομικότων οὐδὲν ὑγιές...... σκεψάμενος ἔτει | in grammar) may seem to a modern reader, it appears to have καὶ τᾶλλα aitav-éucyxepdonata. Ib. Ὁ: κατὰ τὸ τῆς | heena novelty to Plato’s contemporaries. Plutarch expressly δυσχερείας αὐτῶν ἴχνος. The accomplished and unfortu- | attributes the discovery to Plato (Plat. Qu. v. 1. 108, nate Sydenham first pointed out the reference in these epithets | Wyttenb.), Apuleius, Doctr. Plat. 111. p. 203. Comp. Plat. to the Cynics and their master, The οὐ τέχνῃ of Plato tallies | Crat. 431 B. + Soph. 235 B. THE SOPHISTA OF PLATO, &c. 161 case we ought not to be impatient, be the enquiry short or long.” If we say it is too long, “we are bound to shew that a shorter discussion would have been more effectual in improving the dialectical powers of the student, and helping him to the discovery and explanation of the essential properties of things'.” “Praise or blame, founded on any other consideration, we may dismiss with contempt.” This passage, the importance of which for the appreciation of these two dialogues it is superfluous to point out, derives unexpected illustration from an amusing fragment of a contemporary comic poet, preserved by Athenzeus*, 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 ‘improving their dialectical powers” by a lesson in botanical classification. genus that natural production is to be referred. Is a pumpkin a herb? Is it a grass? Is it a tree? The young gentlemen are divided in opinion—each genus having its sup- porters. Their enquiries, however, are rudely interrupted by a “ physician from Sicily,” who happened to be present, and who displays his contempt for their proceedings in a manner more expressive than delicate. ‘*They must have been furious at this,” says the second speaker. “Oh!” says the other, “the lads thought nothing of it: and Plato, who was looking on, quite unruffled, mildly bade them resume their task of defining the pumpkin and its genus. So they set to work dividing.” In this transaction it is possible that the Sicilian physician may have been in the right, and the philosopher and his pupils in the wrong. And probably the result of their researches, could it be recovered, would add little or nothing to our knowledge of pumpkins. But one thing the passage proves; and that one thing is enough for my purpose. The διαιρετικοὶ λόγοι of the Sophista and Politicus represent what really occurred within the walls of the : 5 i ἃ ; f 1 ὡς βραχύτερα dv γενομένα τοὺς σύνοντας ἀπειργάζετο A, διαλεκτικωτέρους Kai τῆς τῶν ὄντων λόγῳ δηλώσεως καὶ τί ποτ᾽ ἄρ᾽ wpicavto καὶ Tivos γένους εἶναι τὸ φυτόν; δήλωσον, εἰ κατοισθά τι. εὑρετικωτέρους. Polit, 380 π. B. πρώτιστα μὲν οὖν πάντες ἀναυδεῖς * 11. Ρ. 59. As this fragment has not yet received the attention it deserves, it is printed in full. A. Ti Πλάτων καὶ Σπεύσιππος καὶ Μενέδημος, πότ᾽ ἐπέστησαν, καὶ κύψαντες χρόνον οὐκ ὀλίγον διεφρόντιζον. Kar ἐξαίφνης ἔτι κυπτόντων καὶ ζητούντων τῶν μειρακίων πρὸς τίσι νυνὶ διατρίβουσιν; λάχανόν τις ἔφη στρογγύλον εἶναι, 5 F Ric) Be οτεν, 2 au 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 ἥκεις, λέξον, πρὸς yas = * ᾿ Σικελᾶς ἀπὸ γᾶς κατέπαρδ᾽ αὐτῶν ὡς ληρούντων. B. ἀλλ᾽ οἵδα λέγειν περὶ τῶνδε σαφῶς" A. ἢ που δεινῶς ὠργίσθησαν Παναθηναίοις γὰρ ἰδὼν ἀγέλην χλευαζεσθαί τ᾽ ἐβόησαν" μειρακίων = " τὸ γὰρ ἐν λέσχαις ταῖσδε τοιαυτί ἐν γυμνασίοις ᾿Ακαδημίας ποιεῖν ἀπρεπές. ΜΕ XS Sh eas a) ae τ ΣΤΥ ΤΣ ἤκουσα λόγων ἀφάτων ἀτόπων; B. οὐδ᾽ ἐμέλησεν Tots μειρακίοις περὶ γὰρ φύσεως ἀφοριζόμενοι διεχώριζον ζῴων τε βίον δένδρων τε φύσιν λαχάνων τε γένη. Ka?’ ἐν τούτοις τὴν κολοκύντην ἐξήταζον τίνος ἐστὶ γένους. Vou. X. Parr I, 6 Πλάτων δὲ παρὼν καὶ μάλα πράως, οὐδὲν ὀρινθείς, ἐπέταξ᾽ αὐτοῖς πάλιν [ἐξ ἀρχῆς τὴν κολοκύντην) ἀφορίζεσθαι τίνος ἐστὶ γένους" οἱ δὲ διήρουν. Com, Grec. ἔτασπι. vy. 111. p. 370, ed. Meineke. 21 162 PROFESSOR THOMPSON, ON THE GENUINENESS OF Academy: and we can have no doubt that Plato regarded such long-drawn chains of dis- tinctions in the light of a useful exercise for his pupils. They became ‘‘ more inventive” and more dialectical”——may we not say, clearer-headed—by the process. I may add that the Invention of the Divisive Method is traditionally attributed to Plato by the Greek historians of philosophy. Aristotle devotes several chapters of his Posterior Analytics to the discussion of this method: he points out its uses and abuses, and defends it against the cavils of Plato’s successor Speusippus, who abandoned the method because, as he alleged, it supposed universal knowledge on the part of the person employing it. The method 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 Anistotle’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 Angient 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. c. x11. § 6, and Schol. in loc. So Abelard (Ouvrages Inédits. Op. 569, ed. Cousin: coll. pp. 451, | μετὰ μίαν δύο, εἴ πως εἰσί, σκοπεῖν, εἰ δὲ μή, τρεῖς 461), distinguishes between those divisions which imply di- | ἢ τιν ἄλλον chotomy and those which do not: e.g. παντὸς ἑκάστοτε θεμένους ζητεῖν...ἐὰν οὖν [μετα]λάβωμεν, ἀριθμόν, καὶ τῶν ἕν ἐκείνων ἕκαστον πάλιν ὡσαύτως μέχριπερ av τὸ κατ᾽ apxas ἕν μὴ OTL ἕν animal. animal. Kat ἄπειρά ἐστι μόνον ἴδῃ τις, ἀλλὰ καὶ ὅπυσα. 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 /ength 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: 1 SSS = man. horse. ox, ἄς. 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 ἃ μελοτομία. Κατὰ μέλη τοίνυν αὑτὰς οἷον ἱερεῖον διαιρώμεθα, ἐπειδιὴ | δίχα ἀδυνατοῦμεν, δεῖ yap els τὸν ἐγγύτατα ὅτι μάλιστα τέμνειν ἀριθμὸν ἀεί. The division he proceeds to make, is ἃ Θεραπεία. distribution of “accessory arts” συναιτίοι τέχναι, into seven νος —— 2 5 ae ΕΟ : Η ἢ τοῦ σώματος. ἢ τῆς Ψυχῆς. co-ordinate groups. A similar relaxation is permitted in the a ee τ---: τ οὦἜὔἅῦὉἽ - Philebus, p. l6p: Act οὖν ὑἡμᾶς..... «ἀεὶ μίαν ἰδέαν περὶ | γυμναστική. ἰατρική. νομοθετική. δικαστικὴ. THE SOPHISTA OF PLATO, &c. 163 as Plato acknowledges, be more or less successful in proportion to the insight and knowledge of the person employing it. The specimens with which he favours us in these dialogues may be arbitrary, injudicious, or even grotesque: but as logical exercises they are regular—and logic looks to regularity of form rather than to truth of matter, which must be ascertained by other faculties than the discursive. And even in judging of these particular divisions, we must bear in mind the object in view. In the Sophista it is Plato’s professed intention to dis- tinguish the Sophist from the Philosopher, the trader in knowledge from its disinterested seeker: surely no unimportant distinction, nor one without its counterpart in reality, either in Plato’s day or in our own, The ludicrous minuteness with which the successive genera and sub-genera of the “acquisitive class” are made out in detail, would not sound so strange to ears accustomed to the exercises of the Schools; while it subserves a purpose which the philosophic satirist takes no pains to conceal, that, namely, of lowering in the estimation of his readers classes or sects for which he harboured a not wholly unjust or unfounded dislike and contempt. It serves, at the same time, to heighten by contrast the dignity and importance of the philosophic vocation, and in either point of view must be regarded as a legitimate artifice of controversy in a dialogue unmistakeably polemical. APPENDIX I. In the foregoing discussions it is assumed that the method of Division sketched in the 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, 302, the dichotomy is com- prised in a single step: ἐν ταύταις δι τὸ παράνομον Kal ἐννόμον 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 I 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.” Q1—2 164 PROFESSOR THOMPSON, ON THE GENUINENESS OF Phedrus, 265 e, ὃ 110. ane av o> , , “- > ’ = MAI. To δ᾽ ἕτερον δη εἶδος τί λέγεις ὦ Σώκρατες; ν᾿ τ > ὡς. , , =Q. To παλιν κατ᾽ εἴδη δύνασθαι τέμνειν, KaT ~ = , 4 Wye ~ , ΄ ἄρθρα, ἡ πεῴυκε, καὶ μὴ επιχειρεῖν καταγνῦύναι μέρος yt ~ , ͵ ͵ , μηῦεν, κακον μαγείρου τρόπῳ γχρωμενον" ἀλλ ὥσπερ » Nur δ; a ets , « a ἄρτι τῷ λόγω TO μεν ἀῴρον τῆς Ciavoias ἕν τι κοινῇ aN ν᾽ ΄ ᾿ >. ~ \ εἰὸος ἐλαβέτην, ὥσπερ δὲ σώματος ἐξ ἑνὸς διπλά καὶ Hie ͵ Εν τος, ᾿ , ” ὁμώνυμα πέφυκε, σκαιά, τὰ δὲ δεξιὰ κληθέντα, οὕτω ν᾿, Set es ᾿ τ ἐς τ Ree γον τ τς καὶ τὸ τῆς παρανοίας ws εν ἡμῖν πεῴυκος ELdos ἡγη- ͵ ehh Gy ais ἢ ἣ σαμενω TW λόγω, O μὲν τὸ ET αρισπερα τεμ- , ΄ = ΄ - VOMEVOS μέρος, πάλιν τοῦτο τέμνων οὐκ ἐπανῆκε, ‘ ’ ᾽ ~ . ‘ , ‘ ’ , πρὶν ἐν αὐτοῖς ἐφευρὼν ὀνομαζόμενον σκαιόν τιν “ ᾿ , ἌΡ Θὰ eas eS = ερωτα ἐλοιδόρησε μαλ εν ὀίκῃ. oo εἰς Ta Ev ὃ εξιᾷᾳ - , > Nes Gann Neeley - τῆς μανίας ἀγαγὼν ἡμᾶς, ὁμώνυμον μὲν ἐκείνῳ θεῖον NJ 4 ᾽ν» ᾽ ‘ ‘ ’ > ὃ av TW Epwra εφευρων και προπτειναμενος επ- , δ Ἢ isan - ἥνεσεν ὡς μέγιστον αἴτιον ἡμῖν ἀγαθῶν. ͵ , ΦΑΙ. ᾿Αληθέστατα λέγεις. η af ᾿ δὴ "» > ΄ Φ ΣΩ. Tovtwy δι ἔγωγε αὐτός τε ἐραστής, ὦ = Εις ? \ = - Φαῖδρε, τῶν διαιρέσεων καὶ συναγωγῶν, ἵν᾿ οἷος τ᾿ ὦ ἢ ι = On 1 ͵ λέγειν τε καὶ φρονεῖν: eav TE τιν᾽ ἄλλον ἡγήσωμαι we a ἣ ἣ , = = δύνατον εἰς ἕν καὶ ἐπὶ πολλὰ πεφυκόθ' ὁρᾶν, του- ΄ τον διώκω ς ' 5c , - " “κατόπισθε μετ᾽ ἴχνιον date θεοῖο. Καὶ , we ὦ τὰ , es EC abe μέντοι καὶ τοὺς δυναμένους αὐτὸ Cpay εἰ μὲν ὀρθῶς ” ‘ , θ ΝᾺ a> = a= 99 ὩΣ , “- ἡ μὴ προσαγορεύω ὕεος ove, καλω ὁ οὖν μεχρι τουὸε > ; ὀιαλεκτικοῦυς. Sophista, 264 ε. SE. Πάλιν τοίνυν ἐπιχειρῶμεν, σχί Covres διχῇ ‘ ‘ , , ν ᾿ δὶ 4 ‘ Bey τὸ προτεθὲν γένος, πορεύεσθαι κατὰ τοὐπὶ δεξιὰ ἀεὶ μέρος τοῦ τμηθέντος ἐχόμενοι τῆς τοῦ σοφιστοῦ κοινωνίας, ἕως ἂν αὐτοῦ τὰ κοινὰ παντὰ περιελόντες, » ᾿ , , , > - , ‘ τήν οἰκείαν λιπόντες ucw ἐπιδείξωμεν μαλιστα μὲν ἡμῖν αὐτοῖς, ἔπειτα δὲ καὶ τοῖς ἐγγυτάτω γένει τῆς τοιαύτης μεθόδου πεφυκόσιν. Ib. 253 ν, ὃ 82. Τὸ κατα γένη διαιρεῖσθαι, καὶ : fee ae ΠΡΟΣ Pale τὶ μήτε ταὐτὸν εἶδος ἕτερον ἡγήσασθαι μηθ᾽ ἕτερον ὃν ταὐτὸν μῶν οὐ τῆς διαλεκτικῆς φήσομεν ἐπιστή- μης εἶναι; Θ. [Nai,| φήσομεν... Ξ. ἀλλὰ μὴν τό γε διαλεκτικὸν οὐκ ἄλλῳ δώσεις, εἷς ἐγῷμαι, πλὴν τῷ καθαρῶς τε καὶ δικαίως φιλοσόφῳ. Ib. 229 B, $31. Ἱ ἣν ἄγνοιαν ἰδόντες εἴ πῇ κατὰ 2 ae ὦ ͵ = \ ey μέσον αὐτῆς τομὴν ἔχει τινά. διπλῆ yap αὑτὴ γιγνομένη δῆλον ὅτι καὶ τὴν διδασκαλικὴν δύο ἀναγ- ἢ Re ety Δ πὴ One ΑΘ ce ούτε. Ὁ κάζει μόρια ἔχειν, ἕν ἐφ᾽ ἑνὶ γένει τῶν αὐτῆς ἑκατέρῳ. ΠΩ SPN 4 a S Politicus, 263 ΒΚ. Εἶδος μὲν ὅταν ἢ Tov, καὶ μέρος αὐτὸ ἀναγκαῖον εἶναι τοῦ πράγματος ὅτου περ av εἶδος λέγηται: μέρος δὲ εἶδος οὐδεμία ἀνάγκη. 5 . , cla - ΄ (This explains the κατ᾽ ἄρθρα ἡ πέφυκε of the Phedrus.) » . ΄ τ Ib. 265 a. Καὶ μὴν ἐφ᾽ & γε μέρος ὥρμηκεν e , 5519 ν᾿ ~ iN ‘ - «» , o λογος ET εκεῖνο OVO TLE καθορᾶν oow τετάμενα 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 YE μήν. Ib. 262 p, occurs a specimen of the “ unskilful carving” (κακοῦ μαγείρου τρόπον) of the Phedrus. Ei τις τἀνθρώπινον ἐπιχειρήσας δίχα διελέσθαι γένος διαιροίη καθάπερ οἱ πολλοὶ...τὸ μὲν Ἑλληνικὸν (τὸ δὲ) βάρβαρον...ἢ Tov ἀριθμόν τις αὖ νομίζοι κατ᾽ εἴδη δύο διαιρεῖν μυριάδα ἀποπτεμνόμενος ἀπὸ πάντων, ὡς ἕν εἶδος ἀποχωρίζων, κιτ.λ, In allusion to Xen. Mem. 1v. § 11, a passage noticed by the Master of Trinity, p. 595 of his paper, I may observe that the etymology of Dialectic, ἀπὸ τοῦ διαλέγ εἰν» is undoubtedly vicious, and is nowhere countenanced by Plato. On the contrary, Dialectic is described in the Philebus, 58 Ἐς as 9 τοῦ διαλέγεσθαι δύναμις. He could not have adopted Xenophon’s etymology, for as we have seen, the Platonic Dialectic includes συναγωγὴ as well as διαίρεσις. The etymology was tempting, and Xenophon, who writes very much at random upon philosophical subjects, was unable to resist the temptation. A similar error is that of Hegel, who in his History of Philosophy, derives σοφιστὴς from σοφίζειν instead of σοφίζεσθαι, an error in which he has been followed by English scholars who ought to have known better. THE SOPHISTA OF PLATO, &c. 165 APPENDIX. 11. 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. J. The sect in question held that, τοῦτο μόνον 1. Democritus, on the contrary, says, νόμῳ ἔστιν, ὃ παρέχει προσβολὴν καὶ ἐπαφήν τινα. πάντα τὰ αἰσθηπα, ἐπέῃ ἄτομα καὶ κενόν.---- Frag. ed. Mullach. p. 204. 2. ταὐτὸν σῶμα Kat οὐσίαν wpiCovto they defined 2. Democritus denies that the sense of touch “substance” to mean corporeal substance only. conveys any true knowledge. Ἡμεῖς τῷ μὲν ἐόντι οὐδὲν ἀτρεκὲς ξυνίεμεν, μεταπῖπτον δὲ κατά TE σώματος διαθιγὴν καὶ τῶν ἐπεισιόντων καὶ τῶν ἀντιστηριζόντων. 3. They despised τοὺς φάσκοντας μὴ σῶμα ἔχον 3. Democritus held “ὅτι 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 pa) Ov) 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 olyects of sense are to them as unreal as they were to Berkeley. Sext. Empir. adv. Matth. vit. 191: Φασὶν ot Κυρηναϊκοὶ κριτήρια εἶναι τὰ πάθη; καὶ μόνα καταλαμβάνεσθαι καὶ ἀδιαψευστὰ τυγχάνειν" τῶν δὲ πεποιηκότων τὰ πάθη μηδὲν εἶναι καταληπτὸν μηδὲ 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 7'’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, Kuthydemo, 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. Atry, 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 ex equali, and of the doctrine of similar triangles. I beg leave to 6. B. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS, ἅς. 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 b, 4; and if these rectangles be so applied together that the sides a and 6 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. a F —— τ =; D B « Ε bv α : A K L Ζῇ -- - I 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 6 and A is not terminated in the line HZ, 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, LJ, are equal; but, by hypothesis, FE, EL, are equal; therefore EL is equal to HJ, which is impossible if the line AL is above or below HI; therefore KL coincides with HJ, and the rectangle ὦ, A, coincides with the complement £J, and the two given rectangles therefore are the complements, ἅς. @.E.D. Proposition (B). If the rectangle contained under the lines a, B, is equal to the rect- angle contained under the lines A, 6; and if the rectangle contained under the lines ὦ, 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, e. [This is equivalent to the ordinary ev equali theorem, If Geb se Ze Th and δ᾽ δ ne eG Mhten will ais cua onl 108 G. Β. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS Ww Ι M A | Ss = ΞΞ 6 Γ΄ ΠΝ B Hy Bl¢ b ein | M ¢ R Q 7 x Construct the similar and equal rectangles DE, FG, with sides ὃ 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 ΣῊ or ὃ of one be in the same straight line with HZ or ὁ of the other. In the right-angled triangles JDH, HFK, the sides including the right angles are equal, therefore the angle FHK is equal to the angle DIH, and is the complemen. 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 EL, Produce LM, ON, to intersect in P, and join KP. Then, because the rectangle LH, which is the rectangle contained under a and B, is equal to HO, which is the rectangle contained under 6 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 ΚΟ (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 FS 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 K HIV 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, C, and 7X is the rectangle contained under c, 4; therefore the rectangle contained under the lines a, C, is equal to the rectangle contained under the lines 4, 6. @.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. Prorosition (0). If two right-angled triangles are equiangular, and if a, A, are their hypothenuses, and ὦ, 6, homonymous sides; the rectangle contained under the lines a, B, is equal to the rectangle contained under the lines J, ὃ. FOUNDED ON ORDINARY GEOMETRY, &c. 169 {The equivalent theorem in proportions is δι be Gan δὴ me al A wesc E B Apply one triangle upon the other as in the right-hand diagram, so that the side ὁ meets the hypothenuse 4 at right angles, and the vertex of the angle opposite 6 meets the vertex of the angle included by 4 and B. Since the angle GFA 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 4 is the double of the triangle GFD. But because GF is parallel to DE, the triangle GFE is equal to the triangle GFD. Therefore the rectangle under a and B is equal to the rectangle under 4 and ὁ. α.π.Ὁ. Proposition (D). If a, 65 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. ye From the angles included by the sides A, C, and a, 65 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, ὁ. Again, Rectangle under ὦ, C, is equal to rectangle under B, c. Therefore by Proposition (B), Rectangle under a, C, is equal to rectangle under A, c. α.Ε.Ὁ. Proposition (E). If 6, 6, and B, C, are homonymous sides including the right angles of two equiangular right-angled triangles, the rectangle contained under 6, C, will be equal to the rectangle contained under 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 G. 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 FDG is equal to the triangle EDH. But the rectangle under 6, C, is double of the triangle LDH; and the rectangle under c, B, is double of the triangle #DG. Therefore the rectangle under ὦ, C; is equal to the rectangle under c, B. @.E.D. Proposition (F). If the rectangle contained under the lines a, B, is equal to the rect- angle contained under the lines 4, ὃ; the parallelogram contained under the lines a, B, will be equal to the equiangular parallelogram contained under the lines 4, ὁ. [This is equivalent to the proposition, iva δ᾽ =: Ave eee Then a: 6:: A.cosa : B.cosa.] ὃ 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, EG, is equal to the rectangle under ὁ, ZL. 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 JZ, B. But by hypothesis, the rectangle under B, a, is equal to the rectangle under A, b; therefore by Proposition (B), the rectangle under EG, a, is equal to the rectangle under IL, 6. Or, the parallelogram under the lines B, a, is equal to the equiangular paral- lelogram under the lines A, δ. @.£.D. FOUNDED ON ORDINARY GEOMETRY, &c. 171 These Propositions, I believe, will suffice for treatment of the first thirteen Propositions of Euclid’s Sixth Book (Prop. I. excepted), and for all the Theorems and Problems appa- rently involving proportions of straight lines (not of areas, &c.) which usually present them- selves. As an instance of their application, I will take the theorem to which I alluded at the beginning of this paper. 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. eS A Wen Let 4, B, C, be the centers of the given circles. Let N be the center of the circle whose radius NO is equal to the radius BK, and which is touched at O by the tangent DE. Join NB, MF, FI, MN, NI, FB. First we shall prove that MF is parallel to NB. The triangles NOF, CEF, have each one right angle, and they have another angle common ; hence they are equiangular; and by Proposition (C), the rectangle under CF, NO, is equal to the rectangle under NF, EC; or, the rectangle under CF, BK, is equal to the rectangle under 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 WF, 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 17 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 AI, NF, with the angle FAZ. But the parallelogram under 41, BJ, with the angle F'AJ, 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 AFJ above double the triangle AFB, or is double the triangle BFJ. Similarly the parallelogram under 47, NF, with the angle 1.41, is double the triangle NFJ. Therefore the triangles BFJ, NFIJ, are equal; therefore FJ is parallel to VB. And as MF and FJ are both parallel to NB, MF and FJ are in the same straight line. 9. E. ἢ. ADDENDUM. I am permitted by Professor De Morgan to transcribe the simple process for demon- strating the theorem of ex @quali in ordine perturbata, to which allusion is made above. Livan: (OM: aac, Mh (eg a ce als ΝΣ hen) willl (as) σ᾽ Ὁ Ἢ : Ὁ To exhibit the process more clearly to the eye, use the connecting mark γι for one ratio and <> for the other; then the theorem stands thus, ZX eee πη α- Ὁ: Φ - C: To prove it, take a fourth quantity d, such that a : 6 :: ὁ : d. Then ὃ ΖΞΞΞε ὁ σ-- α. But AZ ἜΣ: Ὁ: Therefore, ex equali, ὃ : d :: A: C. But, because a : ὃ τ ὁ : d, therefore alternando, a : ¢ :: ὃ : d. Substituting there- fore the ratio a : ὁ for b : din the analogy just found, G83 38 δ: (΄. Q. E. D. GaP B. ATR NG Royan Ogservatory, GREENWICH. September 2, 1857. X. On the Syllogism, No. ΠΠ|, and on Logie in general. By Avaustus Dr Morean, F.RAS., 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. vitt. Part $; 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 1. General Considerations. if during several previous years, I had not the encouragement which would have arisen from Eleven years ago, when I began to put together details on which I had been thinking 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 I did not know that very high authority was then teaching its alwmni to assert that logic had always nature can improve, except in perspicuity, accuracy of expression, and the like. + 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-eracking: 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 pwre 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. 170 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 οἵ 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. This appears throughout his writings, 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 successors. ΤῸ him we owe such perpetual indication of the distinction of form and matter that many, including some who should have known better, have assigned the form of thought to him as his definition of logic, giving him the word into the bargain. But the definition was 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 than another caught the conception, it was the mathematician Leibnitz. And the history of man in species analogises with what we have seen of man in individuals: we trace our mathe- 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 g, either in mathematics or in logic: and it is just worth notice here that Boethius, the only Roman who gave us a summary both. Of the Romans, we only know that they originated nothin 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 | + There is no occasion to refer to any of the ordinary exhi- felt. If the inquirer will look out for English works preceding | bitions, whether dissertations in favour of ignorance, or orations 1848, or thereabouts, which state Kant’s definition as an exist- | in contempt of knowledge. But there is one which deserves ing thing, not to speak of adopting it, he will have some diffi- | preservation for its humour, and which may be lost with an culty in finding one. In some old notes of my own, made after | ephemeral pamphlet, if not elsewhere recorded. An Oxford comparison of Aristotle. some of the medizyvals, and Kant, I | M.A. writing on education, about ten years ago, advocated | | find the following sentence: “I should say fof formal and | some pursuit of mathematics: for, said he, man is an arith- material] that the great leader saw the distinction, that the | metical, geometrical, and mechanical animal, as well as a ratio- schoolmen made the distinction, and that Kant bwilt wpon the | nal soul. Eero 176 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 quantum: 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 symbol. 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. 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 Jogic—so called quasi lucus a non nimis /ucendo—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 OT: nothing but the form. 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 must be taken into account, and consequently be overtly expressible, in logic: for logic must be, as to be it professes, an unexclusive reflex of thought, and not merely an arbitrary selec- Whether the form that it exhibits be stronger or weaker, be more or less frequently applied, that, as a material and tion,—a series of elegant extracts,—out of the forms of thinking. 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 exeat regno on him ! The proper reply to every accusation of introducing the material where all should be formal, is as follows. You say this thought or process is material: now every material thinking has 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 copula of the common syllogism. The copula performs certain functions ; it is competent to 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, except+ in so far as identification is a transitive and convertible notion: ‘A is that which is B”’ 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 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—Z. transitive and not convertible; as in ‘give.’ * 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 “ A 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 Aristoule. But when the title-page shews me an author whose mind is as free from the sway of that distinction as my own, 1 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. Part 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- sisis 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—2Z, 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 touch, 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 β 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 ‘7s’ 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+ Κα 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 3=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 2): 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+(3>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- Q R -B- 5 ‘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 —A— 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 —A— and —B— each be ‘is then we have a material inference; P is Q, 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. Logie 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 * Not meant for extenuation: I wish I were the greater one. 293—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, ἄς. 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 1 am told that ‘Achilles 7s 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 kilied him? For the books of logic give no way of denying ‘X is Y° except ‘ X is not Ὑ. 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. It has complex terms, conjunctive and disjunctive ; it introduces allusions, for reinforcement, for explanation, for justification of its 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 undergoes It is a tapestry, of which the logical form is only the original web. 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 ὃ 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 * Of all 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 ‘Commentarii in P. Rami... Dialec- ticam....’ (Frankfort, 1616, 8vyo,) 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 questionis explains Shakspeare’s meaning of the obscure words “lossof 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 ¢rying. 182 Mr DE MORGAN, ON THE SYLLOGISM, No. IIL, 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. H is the rich fool; ἢ 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, ¢ not so. H)) R[M, W(A, C)]; contrapositively, r, m (w, ac) )) h; or τ, 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 )* 0)" ( 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 shail 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)» tay nD Pe (ὦ CC κως NNF! 7) 0)f °° GOOG) 2 CO γε: OO a) One 1) JO" NNF τὴ τοὺς 3 ἰ(8 ἢ “ἢ ͵ ἘΝΝ ΒΝ Ὁ ΟΝ τὰ KO} Te ck NFN| (.) 7 2) CC: DCO) = 1). NFN CCG λον ἀν 9595...» δ VVV VPP ὝΕΣ ΤΣ eS viva: as that it should enter once universally. And it would have | twice; and vice versé. Consequently the rule must be, if a been seen that if there be any rule at all on the subject, it must | rule there be, that the middle term has one universal and one be this one. For every syllogism may have the middle term | particular entry. changed into its contrary : it cannot be the rule, then, that the * This was communicated to me by my friend Sir William middle term should enter universally twice; for then there | Rowan Hamilton (Feb. 14, 1856), as a consequence of my would be a class of syllogisms in which it enters particularly | mode of expressing the connexion of premises and conclusion. AND ON LOGIC IN GENERAL. 219 F, affirmative; N, negative; V, universal; P, particular. ‘The compartment NFN contains all the syllogisms in which the first (minor) premise negative and the second premise affirmative give a negative conclusion, The columns (PVP), (VPP) contain all in which the one pre- mise is particular and the other universal. The middle column contains all the universal syllogisms. It is fanked by four compartments of six each: and each one compartment con- tains all the syllogisms of one particular conclusion, Thus all in the upper left flank give the conclusion (). The canon* of validity is as follows :—Every pair of universals gives a con- clusion: and every universal and particular in which the middle terms are of different quantities. The canon of inference is:—Erase the symbols of the middle term, and what is left shews the conclusion. Thus () )*( gives (*(, by which I signify that the copartient of an external is exient : or reading metaphysically, [ ] |:[ gives [-[, the irrepugnant of a repugnant is inde- pendent. Supplying the terms, we have X () Y )-( Z gives X (*( Z; or, on the abacus, ‘Some Xs are Ys, no Y is Z; therefore some Xs are not Zs.’ XXXVIII. Thirty-two combinations give valid syllogisms; and as many are invalid, Sixteen of these invalid combinations, of which eight repeated twice, in conjunction with eight of the valid forms, thirty-two in all, have a meaning of their own, as follows. The form of our syllogism is, A, B, C being relations :— Every A of B is a C. Now there are{ also thirty-two truths of this form, derivable as follows :— Every A is a C of every (converse of B). Thus every complement of every species is a complement: therefore every complement is a complement of every genus. Again, every genus of every partient is a partient: therefore every genus is a partient of every partient. ‘The symbolic rule is as follows :—Choose any one of the thirty-two combinations in which the middle spicule are of different quantities. Reject a universal followed by a particular. In any other case, strike out the middle spicula, and if the result be a universal, either let it stand, or change the second spicula: but if the result be a particular, there is no choice but to change the second spicula. Thus (Π) () is inoperative: there is no relation A of which we can say—Every A is a Let P+ Q=R express that P and Ὁ, coexisting, give R: let | neighbours, give all the eight universal syllogisms in consecu- —P represent the contrary of P (or contradictory); let 0 be | tive pairs, if we read both backwards and forwards, And the symbol of impossibility of coexistence. If then P,Q, R be | under the same rule, eight particular syllogisms are seen in three propositions which cannot coexist, so that P+Q+R=0, | each of the two following cycles: we have three modes of inference P+Q=—R, Q+R=-P, De) :G40 R+P=-Q. Now Barbara may be expressed thus ] ὦ X))Y+Y )) Z4+X(-(Z=0 The following conceit gives a kind of zodiac of syllogism. whence X )) Y+Y )) Z=X))Z Barbara 1. Put round a circle the twelve symbols here consecutively writ- Y ))Z4+X((Z=XK((Y Baroko 11. ten, distinguishing the universals by the thicker parentheses ; X)Y+X(-(Z=Y¥((Z Bokardo 111. ΕΝ 9 This process is carried through all the syllogisms of the first Any two consecutive universals give a universal syllogism : figure. any universal with a contiguous particular gives a particular * There are various ways in which the symbols may be | Syllogism. And these whether we read forwards or back- put together so as to give all the syllogistic forms by consecu- wards. tive pairs. Thus the following set + If pwa prevails in each organ of the body;” that as the stomach and kidneys secreted an acid for example, whilst the liver secreted an alkaline compound, the two former might indicate a positive electric state or condition, and the latter a negative state or condition, Prourf cau- tiously advanced a somewhat similar opinion, and says, “" Admitting that the decomposition of the salt of the blood, & c is owing to the immediate agency of a modification of electricity, we have in the principal digestive organs a kind of galvanic apparatus, of which the mucous mem- brane of the stomach and intestinal canal, generally, may be considered as the acid or positive pole, * Philosophical Magazine, Vol, xxx111. p. 488. + On Stomach and Urinary Diseases, 3rd. edit. p. xxv. Mr BAXTER, ON ORGANIC POLARITY. 249 while the hepatic system may, on the same view, be considered as the alkaline or negative pole, He also quotes an experiment of Marreucctr as, in some degree, confirming his opinion. Doxne*, upon applying one of the electrodes of a galvanometer to the stomach and the other to the liver, obtained an effect upon the needle, and the result of this experiment was subsequently confirmed by Marrevccrt. The suggestion thus thrown out, that the stomach and liver formed poles similar to those of a galvanic pile, having apparently received some confirmation from experimental evidence, it now became of some importance to trace out the circuit, the path of the current; and, if possible, the origin of the power, so as to complete the whole evidence necessary for the proof of the truth of the suggestion. Reasoning upon these facts, and assuming that the stomach and liver did actually form the two poles similar to those of a galvanic circle, it was reasonable to suppose that the electric current would pass from the stomach to the liver by the blood in the portal vein. To ascer- tain the truth of this supposition I now inserted the two platinum extremities of the elec- trodes of a galvanometer into the portal vein, and as far apart as possible, in order to obtain the supposed diverted current ; but no effect was observed. The electrodes were then inserted one into the portal vein, the other into the hepatic vein, still no effect, Povitier} and Mutrer ᾧ, it may be observed, had previously ascertained that no effect occurred when they inserted one electrode into an artery, and the other into a vein, of a living animal. ; No evidence could be obtained from these experiments indicative of the path of the cur- rent; the galvanic circle was therefore not complete ; and some of the essential conditions were evidently wanting, Repeating the experiments of Marrerucci upon other animals than rabbits, the effects observed by Marreuccr were not always obtained; as these results will again come under consideration, they need not now detain us. Pondering over these failures it soon became evident that more correct notions in regard to the origin of the power in the voltaic circle were requisite; the term current also, with its ordinary assocjations (of something flowing in one direction), was a source of great embarrassment, and it was thus found that a deeper insight into a knowledge of Farapay’s|| opinions in respect both to the origin of the power in the voltaic circle, and to that of current force in particular, as AN AXIS OF POWER HAVING CONTRARY FORCES EXACTLY EQUAL IN AMOUNT IN To enter upon these points, however, would viz. CONTRARY DIRECTIONS, was absolutely essential. far exceed the limits of this paper, and it is to the admirable memoirs of this distinguished individual that I must therefore refer for the requisite information]. “ BecauEReEL, Traité de’ Electricité, Tom. τ. p. 327. t Ibid. Tom. rv. p. 300. Δ Journal de Physiologie, Tom. v. p. 5. ὃ MutxeEr’s Physiology, translated by Baty, Vol. 1. p. 148. 2nd edit. || Experimental Researches in Electricity. 1 The title of the papers in the Transactions of the Royal Society was so worded as to imply the notion, that these investi- gations were undertaken for the purpose of applying some of the discoveries of Farapay to Physiology. ΤῸ avoid this mean- Vou. X. Parr I, ing a note was appended to point out in what manner the word to apply was intended to be understood, viz. as shewing the ne- cessity of a thorough acquaintance with FARADAY’s views in regard to voltaic action and his definition of current force. No reason has as yet occurred to lead me to alter this opinion, but on the contrary ; and whatever value may be assigned to Pro- fessor GROVE’S views, as advanced in his Essay, On the Cor- relation of Physical Forces, I am still of opinion, without wishing to detract from the merits of the latter philosopher that the views of Farapay are by far the most philosophic. 32 450 Mr BAXTER, ON ORGANIC POLARITY. Dismissing the notion that the stomach and liver are related to each other in the same manner as the poles of a galvanic circle are mutually dependent, and with a more correct knowledge of the origin of the power in the galvanic circle derived from Farapay’s memoirs, the thought arose that it might be during the formation of the secretions where the changes were actually going on, that the evidence sought for could possibly be obtained. How far these surmises were correct will now be seen. Secr. I. On the Manifestation of Current Force during the formation of the Secretions in the mucous membrane of the alimentary canal, viz. the stomach and intestines. As the mode of employing the galvanometer and of conducting the experiments, together with the precautions necessary to be observed, have already appeared in the Philosophical Transactions of the Royal Society* for the years 1848 and 1852, it will not be necessary to enter into a minute detail of these particulars. The results also of the experiments in the present paper need only be related, as it is my intention to enter more deeply into the theo- retical part of the question than could have been prudently attempted on the former occasion ; for the time has now arrived, when, considering the great development that has taken place in regard to electrical science in general, we may reasonably hope to be enabled by means οὗ. scientific discussion, combined*with experimental observation, to reduce the mass of unconnected facts with which the science of Animal Electricity abounds within some more general laws. Although experiments performed upon the living animal may be considered as affording more satisfactory results, nevertheless, as the results can be obtained, when sensibility is de- stroyed, the following mode may be adopted in preference to the use of chloroform. Let a few drops of strong prussic acid be dropped on the nose, insensibility is thus quickly produced; or let the animal be pithed, and upon laying open the chest or abdomen the heart will be found to beat and the circulation to continue. Under these circumstances, if the platinum electrodes of a galvanometer are placed one in contact with the mucous surface of the small or large intestine, the other in contact with the blood in a vein from the same part, a deflection of the needle will be obtained indicating a current through the instrument, the electrode in contact with the blood being positive to the other in contact with the mucous surface. If the same experiment be repeated with the mucous membrane of the stomach, the effects may vary. If the stomach be empty, then the electrode in contact with the blood of the vein coming from the same pait will also be positive, but if there be any food in the stomach and should it contain much acid, then the electrode in contact with its mucous surface will most probably indicate a positive state. Now these are the fundamental facts and the results, which are readily obtained with proper precautions, and may be thus stated: when the electrodes of a galvanometer are brought into contact one with the mucous surface of the intestine in a living or recently-killed animal, and the other with the venous blood from the same part, an effect oceurs upon the needle indicating the secreted product and the venous blood to be in opposite electric states. * Philosophical Transactions, 1848 p. 243, 1852 p. 279. Mr BAXTER, ON ORGANIC POLARITY. 251 The amount of deflection of the needle would vary according to the delicacy of the instrument employed; with an ordinary galvanometer, consisting of but few coils, the deflec- tion was from 3° to 8° or 10°. When the electrode, instead of being in contact with the venous blood, is in contact with the arterial blood, or the surface of the mesentery, the effects upon the needle are the same, as far as the direction of the current is concerned, but the amount of deflection may not be so great. Let us now endeavour to explain these results according to known actions, such as the chemical reaction of two fluids upon each other, or to the heterogeneity of fluids, as it is sometimes called. If, for example, a glass cell be taken having a porous diaphragm in its middle, such as a piece of membrane, so as to divide it into two cells, and into one compartment we pour an acid solution, and into the other an alkaline solution, and then dip the platinum electrodes of a galvanometer into each of these cells, an effect upon the needle is produced indicating the electrode dipping in the acid solution to be positive to the other. These facts, which have been well worked out by Brecauerrt*, may be enunciated in the following proposition: during the reaction of two fluids upon each other, that which per- forms the part of an acid takes positive electricity, and that of am alkali, negative electricity. In experiments upon animals, as just related, it was found that the electrode in contact with the venous blood was positive to the other, excepting when there was much acid in the stomach, and then the electrode in contact with the mucous surface of the stomach was positive to the other in contact with the blood. Now in order to explain these results, under the supposition that they arise from the chemical reactions of the fluids upon each other, it must be supposed that when the electrode in contact with the venous blood is positive to the other, that then the blood acts as an acid, and not only so, but combines with the substances or fluids in the intestines. When it is found, however, that the electrode in contact with the stomach is positive, then it may be sup- posed, and rightly so, that the results are due to the chemical reactions which occur in that organ between the acids and other fluids that are there found. But should we be justi- fied in supposing that when the electrode in contact with the blood is positive to the other in the stomach, the stomach being empty or containing but little acid, that then the blood is acting as an acid? Here, as in the intestines, it would be necessary to assume that imme- diately after the separation of the secreted product (the acid) from the blood had taken place, that they then immediately recombined, and not only so, but that the blood, in direct opposi- tion to the well-known fact of its alkaline characters, must be acid in order to account for the effects produced. It would, therefore, appear that no grounds exist for believing that the results obtained in the living animal can be considered as entirely dependent upon the mere reaction of the heterogeneous fluids upon each other, upon their combination for example; and with out stopping to adduce more arguments against this supposition, let us now proceed to com- * Loe. cit. Vol. 11. p. 77. 32—2 252 Mr BAXTER, ON ORGANIC POLARITY. pare the results with another class of phenomena, viz. with those actions which take place in a voltaic circle where decomposition is effected. It will be better to confine our attention to the actions which take place in the ewciting cell of a voltaic circle where the power originates, and withdraw our minds for the present entirely from the changes which take place in the decomposing cell of the battery where polar decompositions are effected: the principal object being to ascertain whether, during the decomposition of a compound, or during the separation of an acid from an alkali, the same effects are produced upon the galvanometer as occurs during the combination of an acid with an alkali, Let us take an elementary circle, zinc, platinum, and a dilute solution of muriate of soda, and consider the two metals as forming the terminations of the electrodes of the galva- nometer, one of zinc and the other of platinum, instead of having two platinum electrodes as heretofore. When the electrodes are dipped into the solution, the actions which take place are the following: the muriate of soda is decomposed by the attraction of the zine for the chlorine or muriatic acid, whilst the soda is evolved on the surface of the platinum; now under these circumstances the platinum electrode, in contact with the soda, is positive to the other, and, according to common phraseology, the direction of the current is in the same direction as the cation (the alkaline compound, the soda) is supposed to travel. Here then is a case of decomposition, a separation of an acid from an alkali, effected by chemical agency, and the electrode in contact with the alkali is positive to the other in contact with the acid; the effect being contrary to that observed during the combination of an acid with an alkali, as has been just shewn. Let us now compare the results which occur in the animal with those which take place in the voltaic circle. When the electrode is brought into contact with the venous blood, it is positive to the other in contact with the secreting surface of the intestine; if it be now supposed that the blood is alkaline, and there is every ground for so doing, the electrode in contact with the blood is exactly similar to that in contact with the alkali in the voltaic circle; but instead of the secreted product com- bining with the other electrode, as the acid does in the voltaic circle, it passes away. In the animal the current may be supposed to be dependent upon the decomposition—if I may so term it—of the arterial blood, being as it were separated into its two elements, the secreted product and venous blood, just as the muriate of soda is decomposed and separated into its two elements, muriatic acid and soda. At present, it may be remarked, that no opinion as to the mode in which the secretions are effected is being given; I am only endeavouring to ascertain now what does occur, and to what class of phenomena these actions, those of secretion, bear the greatest resemblance. This subject will again come under our consideration. Before proceeding to shew that in other organs there exists the same manifestation of cur- rent force during secretion, I cannot omit noticing the opinion that WoLLaston entertained in regard to the question now under consideration, and shall therefore quote his own words : “At the time,” says Wotraston*, ‘*when Mr Davy first communicated to me _ his * Loc. cit. Mr BAXTER, ON ORGANIC POLARITY. 253 important experiments on the separation and transfer of chemical agents by means of the voltaic apparatus, which was in the autumn of 1806, I was forcibly struck with the probability that animal secretions were effected by the agency of a similar electric power; since the existence of this power in some animals was fully proved by the phenomena of the Torpedo and of the Gymnotus Electricus; and since the universal prevalence of similar powers of lower intensity in other animals was rendered highly probable by the extreme suddenness with which the nervous influence is communicated from one point of the living system to another. ** And though the separation of chemical agents, as well as their transfer to a distance, and their transition through solids and through liquids which might be expected to oppose their progress, had not then been effected but by powerful batteries; yet it appeared highly pro- bable that the weakest electric energies might be capable of producing the same effects, though more slowly in proportion to the weakness of the power employed. “1 accordingly at that time made an experiment for the elucidating this hypothesis, and communicated it to Mr Davy and to others of my friends. But though it was conclusive with regard to the sufficiency of very feeble powers, it did not appear deserving of publication until I could adduce some evidence of the actual employment of such means in the animal economy. τ ¥ ἘΞ iy “The experiment was conducted as follows: I took a piece of glass-tube about three- quarters of an inch in diameter, and nearly two inches long, open at both ends, and covered one of them with a piece of clean bladder. Into this little vessel I poured some water, in which I had dissolved oi" of its weight of salt; and after placing it upon a shilling with the bladder slightly moistened externally, I bent a wire of zinc, so that while one extremity rested on the shilling, the other might be immersed about an inch in the water. By successive examinations of the external surface of the bladder, I found that even this feeble power occasioned soda to be separated from the water, and to transude through the substance of the bladder. The presence of alkali was discernible by the application of red- dened litmus-paper after two or three minutes, and was generally manifested even by the test of turmeric paper before five minutes had expired. « The efficacy of powers,” continues Woutaston, “ so feeble as are here called into action, tends to confirm the conjecture that similar agents may be instrumental in effecting the various animal secretions which have not yet been otherwise explained.” There is one circumstance connected with Wottasron’s conjecture which must be noticed, viz. the idea that secretion depended upon, or is the effect of a power similar to that which exists in a voltaic circle; but it must be borne in mind that the origin of the power in the voltaic circle was not so completely understood at the time WotLaston published his conjecture as it is at the present day; and, although he himself was an advocate for the opinion that it depended upon chemical action, it nevertheless required the elucidation that it has subse- quently received at Farapay’s hands; the fact being that the chemical action which occurs is the cause of the power, or, in other words, the current is a mere manifestation of the chemical action that is taking place. I shall now pass on to the consideration of the manifestation of current force during secretion in other organs; and first, in the liver. 254 Mr BAXTER, ON ORGANIC POLARITY. Secr. II. On the Manifestation of Current Force during Biliary Secretion. If the platinum electrodes of the galvanometer be inserted one into the gall-bladder, and the other into the hepatic vein, or which will be found better still, in consequence of the blood flowing over the intestines, into the vena cava ascendens in the chest, we then obtain evidence of the manifestation of current force; the electrode in contact with the blood in the vein being positive to the other in contact with the bile in the gall-bladder. The amount of deflection of the needle varies from 5° to 10°. When the electrode, instead of being inserted into the hepatic vein or into the vena cava ascendens, is inserted into the vena porta, the other remaining in the gall-bladder, the former will still indicate a positive state; but the effect upon the needle is not so great. It will not be necessary to detail the results that may be obtained when other cir- cuits are formed, between pieces of liver and ciots of blood, &c. for example, shewing the effects of heterogeneity of the substances in contact with the electrodes, as these can be found in the original papers already alluded to. But the following conclusion may be de- duced: when the electrodes of a galvanometer are brought into contact, one with the bile in the gall-bladder, and the other with the blood in the hepatic vein, or vena cava ascendens, an effect occurs upon the needle, indicating the secreted product (the bile) and the blood to be in opposite electric states. It may be said, and with apparent justice, that if the actions which occur during secretion be similar to those that take place in the exciting cell of a voltaic battery, as was suggested in the previous Section, the electrode in contact with the alkaline bile ought now to indicate a positive state. The force of this objection depends entirely upon the assumption that the bile contains a free alkali. The researches of chemists, and especially Lirsic, have however shewn that with the alkaline bases which exist in the bile, are associated peculiar organic acids, such as the bilic, choleic, &c. As these acid compounds are easily decomposed, we should not be justified in supposing, from finding a number of indestructible basic elements which exist in the ultimate analysis of the bile, that these basic elements therefore existed as such in the com- position of the bile; and although the bile may present an alkaline reaction, this alone would not necessarily indicate the existence of a free alkali. It would appear more reasonable to suppose that these basic elements existed in combination with the destructible organic acids. Similar remarks may undoubtedly be made respecting the composition of the blood, but the chemical evidence in favour of the existence of a free alkali in the blood is far stronger than that for its existence in the bile. The opinion that the fluidity of the blood may be depen- dent upon the alkaline salts has been Jong entertained by physiologists, and would appear to have received strong confirmation from the recent experiments of Dr Richarpson*, to which I may refer. Having so far removed this objection, the same remarks that were made in regard to the secretion that occurs in the intestinal canal, and which I need not recapitulate, may now be * The Cause of the Coagulation of the Blood, Churchill, 1858. Mr BAXTER, ON ORGANIC POLARITY. 255 applied to the formation of the bile. So bere in another class of secretions, additional evidence has been obtained of the manifestation of current force during secretion. Before passing on to other secretions, I shall now notice the fallacy of supposing that the stomach and liver form poles similar to those of a galvanic battery, an idea that has been en- tertained by several individuals. No evidence could be obtained to shew that the stomach forms the positive and the liver the negative electrode of a circuit similar to those of a voltaic circle. It may just as well be supposed that the lungs and the stomach, or the lungs and the kidneys, or the liver and the lungs, and the kidneys and the lungs are similarly related, if we are to be guided by the mere circumstance of their relative connections in regard to the circulation of the blood through these different organs. Each organ, the stomach and liver, would appear to have, however, its own elementary circle, if I may so express it; but no evidence exists to shew that these two organs are so mutually related as to form one circle. There is one fact which is of some interest and deserving of notice, it is this; the blood, from which the biliary secretion is formed, has previously undergone some most important changes during its passage through the coats of the stomach and intestines, and thus an important re- lationship must necessarily exist between these two organs; and the question may naturally arise, Is not the blood during its passage through the coats of the stomach and intestines, and especially by the stomach, thus deprived of most of the elements of its fixed acids, such as the muriatic acid for example, and so far accounting for the small proportion of these elements that are found in the bile? It must be observed, that I am not now supposing that ail the acids found in the stomach mus¢ necessarily come from the blood, for there can be no doubt that some of the acids are formed in that viscus independent of those that are secreted by that organ. But to enter upon this subject would carry us away from our main object, and I shall therefore leave it. Secr. III. On the, Manifestation of Current Force during Urinary Secretion. Upon inserting one of the extremities of the electrodes of the galvanometer into the pelvis of the kidney, and the extremity of the other electrode into the renal vein of the same kidney, an effect upon the needle is produced indicating the electrode in contact with the blood to be positive to the other. A difficulty may sometimes arise in obtaining any effect. The amount of deflection of the needle, when obtained, varies from 3° to 5°. Should we be justified, in this instance, in supposing that the blood is acid to the urine, and not only so, but combines with the urine, in order to account for the effects observed upon the galvanometer, when a more satisfactory explanation can be adduced by regarding the effects as being consequent upon the separation of the acid product from the blood, as already advanced in the previous sections with respect to the other secretions ? The amount. of deviation of the needle being small, may be referred to the same causes as were observed to exist with regard to the acid secretions and fluids in the stomach. The secretion, urine, being acid, counter currents arise and are produced by the reaction of the acid of the urine upon the fluids and substances with which it comes into contact. In judging, therefore, of the effects upon the needle we must take into consideration the acting points in the circuit; there may be at least three acting points in a circuit, viz. at the point of secretion f οὔθ Mr BAXTER, ON ORGANIC POLARITY. and at each of the electrodes. If the direction of the current consequent upon secretion coincide with those that occur at the electrodes, then an increased effect upon the needle is necessarily produced ; but if these currents tend to go in opposite directions, then the result upon the needle will be merely the differential effect. Hence we should be led to very erroneous con- clusions judging merely from the effect upon the needle, either as to the force of the current or its origin. Sufficient evidence has been obtained to warrant the following deduction, viz. that when the electrodes of a galvanometer are brought into contact, one with the urinary secretion and the other with the venous blood from the same part, an effect upon the needle occurs indicating the blood and the urine to be in opposite electric states. It may just be remarked that slight effects may be observed when the electrode is in con- tact with the arterial blood instead of the venous blood the other being in contact with the urine. But no effects are obtained when one electrode is inserted into the vein and the other into the artery of the kidney. Whilst upon the subject of urinary secretion I may allude to a circumstance of some interest. At the time the original experiments were performed it was frequently observed that the blood continued to indicate its positive condition, long after the secreting process could have been going on, which led to the belief that the blood might have the power of retaining its peculiar electrical state. Subsequent experiments have tended to confirm this opinion, but it was never supposed that the secretions could have the power of retaining their peculiar electrical condition, until lately. Reading over some of the Memoirs published at the time of the celebrated controversy between Gatvani and Vora, I was much gratified by accident- ally finding the following interesting document. It is of some value inasmuch as it is a letter written by Vassat1 Eanpt, at that time one of the celebrated professors at Turin, to M. Detamerturi®, then secretary to the Royal Academy of Paris, who requested his opinion “upon galvanism and the origin of Animal Electricity*.’ The position that these two individuals held might be adduced as giving some weight to their authority. Amongst other arguments that Vassart ΕΑΝ ΟῚ brings forward in favour of the existence of Animal Electricity is the following: ‘‘J’ai prouvé ailleurs,” says Vassatt Eanpi, “que les urines donnent une Glectricite négative, et j’ai fait voir plusieurs fois aux D. Gerri, Garorrr et aux éléves de médecine et de chirurgie, que le sang tiré des veines donne dans mon appareil électrométrique (décrit dans le Vol. V° de VAcadémie des Sciences de Turin, Dec. 19, 1790) une électricité positive.” It need scarcely be stated that the galvanometer was not then known, and that the effects observed by Vassat1 Eanpr were those of attraction and repulsion. Although the results obtained by Vassatr Eanp1 may be supposed to be due to other circumstances, such as evaporation or chemical action, than those arising from Animal Electricity, nevertheless, as recorded facts, they are of some value, inasmuch as they tend to establish similar conclusions which have been arrived at by different modes of investigation, and entirely independent of each other. * Journal de Physique, T. xiviit. p. 336, 1799. Germinal an. vii. Lettre de Vassati Eanpr a J. C. DELAMETHRIE Sur le galvanisme et sur Vorigine de Vélectricité animale. Mr BAXTER, ON ORGANIC POLARITY. 257 Sect. IV. On the Manifestation of Current Force during Mammary Secretion. In my original paper only one experiment was recorded as shewing the results that were obtained in the Mammary gland; since then several other opportunities have occurred in which similar results were observed. If we insert the electrodes one into a lactiferous vessel and the other into a vein from the same part, the electrode in contact with the vein is positive to the other, 8° or 10°. Here, in this instance, we get evidence of the secreted product (the milk) and the venous blood being in opposite electric states. It will be now seen that wherever secretion occurs, whether in the stomach and intestines, in the liver, in the kidneys, or in the mammary gland, it will be found that the act itself is not only accompanied with the manifestation of ewrrent force, but that the venous blood is also, in all these instances, in the same state, in a positive electric state. The next question that would naturally arise is the following:—what is the state of the arterial blood ? Although it has been found that the arterial blood indicates a positive state when formed into a circuit with the secreted product, the other necessary element, viz. its electro-negative element, the cation, for example, bas not yet been obtained. Reasoning from analogy, it is in the lungs that a satisfactory explanation on this point must be sought for. Physiologists may not perhaps be disposed to admit that the function of the lungs corresponds to that of a secretory organ; or that the process by which carbonic acid is elimi- nated from the blood corresponds to that by which the acid is eliminated from the stomach ; fortunately a decision on this point will not be necessary, and therefore need not detain us: One circumstance, however, is well known, viz. that carbon, in some form or other, is elimi- nated from the blood during its passage through the lungs; and it may so happen that during the elimination of this carbon, its separation from the venous blood whilst traversing the lungs, that current electricity becomes manifested. Sect. V. On the Manifestation of Current Force during Respiration. When one electrode is brought into contact with the mucous membrane of the bron- chial tubes, and the other inserted into the left ventricle of the heart, the latter electrode is positive to the former, from 2° to δ, When the electrode, instead of being inserted into the left ventricle was inserted into the right ventricle, it still indicated a positive state-—Here then are indications of the arterial blood being positive to the mucous surface of the lungs; how far this state may be due to the separation of the carbon from the venous blood which traversed that organ may be a subject of dispute; the fact, however, is of some importance as indicating the electric condition of the arterial blood. In looking back upon the results that have now been obtained, some surprise may be felt at the circumstance that all these experiments tend to indicate that, during life, the blood, whether venows or arterial, is in a positive electrical state or condition, and that this state or νοι. AR rele 33 258 Mr BAXTER, ON ORGANIC POLARITY. condition is partly produced and maintained by the various secretions that take place in the animal body. How far the fluidity of the blood, and the vitality of the blood, as it is called, are dependent upon this electric state or condition, are questions which must necessarily arise in our minds. The particles of the blood, also, must under these circum- stances exist in a state of self-repulsion; and may not this fact, it may be asked, tend to explain some of the phenomena connected with the circulation of the blood in parts not dependent upon the vis ἃ tergo action of the heart, and also those connected with the coagula- tion of the blood when taken from the living animal? These are questions that will arise ; but I must not wander too far from our present object ; and therefore conclude this Section by stating that a clue has now been obtained to the non-appearance of any effect upon the galvanometer when the two electrodes are inserted into an artery and a vein, a fact previously established by the experiments of Pourrter and Murrer®*. As the blood in the two vessels is in the same electric state, no effect could occur upon the needle; thus proving the fact, well established by Farapay, that in order to obtain currENT Force the cirewit form must be given to the arrangement, i.e. that the electrodes must be brought into contact, or by means of some conducting mass, with the ANION and caTIon originating the power}. Before entering upon the concluding remarks there are one or two points which must be noticed. It may be supposed, Ist, that the effects that have been obtained may arise from thermo-electric actions, since BecquerEtt and Brescuert have ascertained the existence of a difference in temperature between the arterial and venous blood by means of a galvanometer ; endly, that they may also arise from the actions that take place upon the surface of the platinum electrodes. There can be no doubt that a part of the effects may be referred to both of these circumstances, and they must therefore be taken into consideration when judging of the final result upon the needle. As these objections have however been already noticed in one of the original papers§, I cannot do better than refer to the experiments and arguments there brought forward for their refutation. ‘Concluding Remarks. The results recorded in the present and previous papers tend to establish the following conclusion, viz. that the act of secretion in the living animal is accompanied with the mani- festation of CURRENT ForcE; and the phenomena with which this act of secretion appears to be the most intimately related are those that occur in the voltaic circle, as I have endeavoured to point out in the present paper. A difficulty may arise to some minds in perceiving this relation, from the circumstance that in the ordinary voltaic circle metals are employed. If we bear in mind that the metals, although one of them is usually acted upon, serve princi- pally as conductors, and that they are not essential for the development of the power, this difficulty will be easily removed. Now as the manifestation of current force during the actions which occur in the voltaic circle are considered as evidence of polar action, there can * Loc. cit. + Experimental Researches, Vol. 11. p. 51. ΖΦ Loc. cit. Tom. vit. p. 20. § Phil. Trans. 1852, p. 279. Mr BAXTER, ON ORGANIC POLARITY. 259 be no reason why it should not be so considered in regard to organic action, viz. during secretion ; but before we arrive at this conclusion let us compare the phenomena of secretion with another class of facts, viz. with those of osmose. Professor Grauam has communicated a very valuable paper to the Royal Society, entitled On Osmotic Force, which has lately appeared in their Transactions*. In this paper Professor Granam has shewn that osmose is dependent upon chemical action, and not as it has’ been generally supposed, upon capillary attraction. ‘Time will not allow me to enter upon the facts brought forward in support of this opinion, and I must therefore refer to the paper itself, which cannot be too strongly recommended. The conditions under which an osmotic experiment is conducted, viz. the necessity of having two fluids, one on each side of the septum, render it extremely difficult to ascertain by means of the galvanometer the exact mode of action which arises during osmose, so as to com- pare it with that which takes place in the animal body during secretion, in consequence of the reaction of the two fluids upon each other producing their own peculiar effects on the galvanometer ; and the changes upon which osmose depends take place, according to Professor Granam, within the substance of the porous diaphragm, where we cannot apply the elec- trodes of the galvanometer. The fact of osmose depending upon chemical action shews however that the act itself must not be considered as a mere transudation, a mere physical separation, but that it depends upon other important conditions ; and if upon chemical action they are consequently polar in their nature. If this conclusion be arrived at in regard te osmotic phenomena we may with equal propriety consider the phenomena connected with secretion to be at least something more than a mere physical transudation; and as reasons exist for shewing that osmotic phenomena are polar in their nature, why may we not consider the action connected with secretion, and where we can obtain such direct evidence of polar action, as manifested by the galvanometer, to be polar in their nature also? Respecting the chemical character of osmose, and its bearings upon physiology, Professor Grauam adds:—“It may appear to some that the chemical character which has been assigned to osmose takes away from the physiological interest of the subject in so far as the decomposition of the membrane may appear to be incompatible with vital conditions, and osmotic movement confined therefore to dead matter. But such apprehensions are, it is believed, groundless, or at all events premature. All parts of living structures are allowed to be in a state of incessant change—of decomposition and renewal. The decomposition occur- ring in a living membrane, while effecting osmotic propulsion may possibly be of a reparable kind. In other respects chemical osmose appears to be an agency particularly well adapted to take part in the animal economy.” The subject of the present communication has been that of Orcanic Pouarity, and to this it has been my endeavour to confine our attention, and to shew that some of the organic actions which occur in the animal body, viz. secretions, are evidently accompanied with the mani- festation of current force ; a fact which may not be disputed. An endeavour has been made * Phil. Trans, 1884. 33—2 400 Mr BAXTER, ON ORGANIC POLARITY. also to point out with what class of phenomena they appear to be the most nearly allied, viz. those which occur in voltaic decomposition (a conjecture already advanced by Wottasron) ; and as these are considered as polar in their nature we are justified in logically inferring that those which occur in the animal body are likewise polar in their nature; and as chemical force is considered a polar force, so may organic force be viewed in the same light as a polar force also. But the conditions under which polar phenomena are manifested, in the organic, at once stamp them as of a higher order than those which are observed in the inorganic kingdom of nature. Cambridge, Feb. 1858. XIV. A proof of the Existence of a Root in every Algebraic Equation: with an examination and extension of Cauchys Theorem on Imaginary Roots, and Remarks on the Proofs of the existence of Roots given by Argand and by Mourey. By Avaustus De Morean, F.R.AS., of Trinity College, Professor of Mathematics in University College, London. [Read Dec. 7, 1857.] To those teachers who value the logic of mathematics it has always been a subject of regret that the fundamental proposition of the theory of equations—every algebraical equation has as many roots as dimensions, and no more—is either to be taken on trust, or deferred to a late period of the course. Every such proceeding is, in mathematics, a confession of incompetency, either in the state of the subject or in the teacher, This confession I have until now been obliged to make by deferring the proof of the theorem until it can be deduced from Cauchy’s theorem on the limits of imaginary roots, a theorem which incidentally brings out the existence of the roots. Having been recently led to examine the first* of Sturm’s demonstrations of this theorem, in the first volume of Liouville’s Journal, it struck me, from the very fundamental character of this proof, that there must be some equally fundamental demonstration of the existence of the roots, which would be the natural prefix to Sturm’s demonstration. Attentive examination proved my conjecture to be correct; and at the same time I found an addition to Cauchy’s theorem, which makes it include roots derived from the circuit itself, and also roots of the reciprocal of the function in hand. This I shall incorporate with Sturm’s proof in the present paper: joining with it the consideration of Argand’s and Mourey’s proofs, which have points worthy of particular attention. The proof which I prefix to Sturm’s demonstration depends upon a preliminary theorem, which is one of combination and position. It takes no account of the meaning of 0, οὐ, +, —; but only postulates that + and — shall be separated either by 0 or by ». All changes con- sistent with this condition are to be held allowable. Then +0 -+ may become + +: but ἜΟ -- must not become + —-. Either 0 or » may open; that is, 0 may become 0 — 0, or 040, or 0[+0-—2+0-]0 ἅς. Again + may become + 0+ or ++; and soon, And 0 and «© may come together, and either cross each other or recede from each other without crossing ; having, after crossing or recession, either the same sign between them as before, or a different sign. Turorem. In any number of signs, each of which is + or —, interspersed with the signs Ὁ and «, in any manner which satisfies the condition that either 0 or « always comes be- tween + and — and between — and +, let & be the number of occurrences of +0 -ν and / the number of occurrences of -0+. Then it is impossible that k -- ὦ should undergo * I mean the first by Sturm alone: the first in the memoir cited is by Sturm and Liouville jointly. 262 Mr DE MORGAN, ON A PROOF OF THE EXISTENCE OF any alteration, unless by 0 and © coming together, whether with change of place or simple recession. It is supposed that the series both begins and ends with a sign + or —, which remains unaltered ; not with 0 or ὦ. Except appulse of 0 and οὐ. the only other changes are appearance or disappearance of 0 between like signs, appearance or disappearance of οὐ between like signs, opening of 0 or x into 00 or » « with signs between them. A simple induction will shew that, in every case which involves no appulse of 0 and ~, either & and J remain unaltered, or receive the same increment. Thus when + 0+ changes into + +, both are unaltered: as also in — 0 + changed into —0-0+, or —0-— changed into -o[ -o-]o—. But in +0 + changed into + [0 -- 0] +, both & and 7 increase by a unit: in + 0-— changed into +0 [- 0+0 - 0 —, both receive a unit of increase. But when — 0 — is changed into -0[—« +]0-, in which case & aug- ments by a unit, while 7 is unchanged, the change, if continuous, commenced by an appulse of Oand o,asin0Ow 0. Again, when — 0+ ὦ + changes through — 02% + to -- © +04, in which case / loses a unit, there is an appulse of 0 and #. ‘This theorem brings the funda- mental theorem on the roots of equations to rest on what will readily be acknowledged to be its proper foundation, the necessity of 0 or in the transition from positive to negative. Now suppose a line of any sort drawn in a plane, and at each point of it, (@, y), let the sign of a given function of x and y be recorded; with the character of each change, + 0 -, —0+, + ὦ —-, — © +, as the case may be. Every contour, and every portion of a contour, will thus present what we may call a chain of signs, such as +0 —0+ © —0+..., with re- ference to any function of ᾧ and y which may be chosen. If the contour, or part of a contour, change continuously, so as to pass gradually from one form and position to another, changes may occur in the chain; and it is obvious that the change may be so conducted, that not more than one of the signs 0 and « shall be affected at any one moment. If the function examined be Ἢ where P and Q never become infinite for any finite values of # and y, then 0 can only appear when P = 0, and o can only appear when Q = 0, and an appulse of 0 and « can only take place where τ takes the form τ: Next, suppose @z to be a function which never becomes infinite for any finite value of z, and let d(w+y/-1) = P+Q,/-1. We sce then that if & —7 be found to have, on one contour, a value different from what it has on any other contour, a gradual transition from one contour to the other can- not be made without the varying contour passing through points at which P=0, Q=0, or f(z) =0. Such point or points then must exist; or we have the following Tueorem. If f(x +y4/-1) = P+ Qv/-1, and if neither P nor Q can be infinite for any finite values of w and y; if also two contours can be found for which & — J has different values; then such difference of value is proof of the existence of a root or roots which satisfy z= 0. It is supposed that the choice begins and ends with fixed signs. This always takes place when we go round the whole of a closed circuit, from one sign to the same again. But we have also seen that, so long as the initial and terminal signs remain the same, it is impossible A ROOT IN EVERY ALGEBRAIC EQUATION. 263 for a contour which is only part of a circuit to be changed into part of another contour having a different value of & —/, without passing over one or more of what I call radical points. If pz be a rational and integral function, the possibility of assigning closed circuits which have different values of ἢ — J is easily shewn. When an angle gains a revolution by continued increase, the cotangent of that angle passes through two changes of the form + Ο --, and two of the form — ο +, When the gain of a revolution is a balance of increase and diminution, every case of — 0 + which occurs during diminution is accompanied by a case of +0— which occurs during restoration, over and above the two cases of + 0 — which belong to the balance. Bs Consequently, whenever — is the cotangent of an angle which gains a revolution during the progress of (x, y) round a closed circuit, k —7 acquires two units in that revolution, and two units in every such revolution. If the angle change only by increase, we have k = 2, / = 0, for each revolution. Representing a(cosa+sina4/— 1) by a,, &c., and # + y¥4/—1or r (cos θ +sin θ 4/-1) by 1» let p(w + y 4/1) be ar" + bgrg’-! + ... +m,: in which, to avoid a visibly existing root, we suppose that m has value. We see then that Par’ cos(nO + a) + br’! cos[(z — 1ὴ)θ + B] +... + meosp Ὁ ~ ar sin(nd + a) + br*~'sin[ (m -- 1) 0+ β] +... +m sing Tf a closed circuit be taken in which all the values of 7 are infinitely small, we see that Ρ: is either constant, or, where cos or siny vanishes, varies directly or inversely as the cosine or sine of a multiple of @ altered by a constant. In these cases each revolution gives k and ὦ both = 0, or both the same integer: that is, k-12=0. But if throughout the closed circuit r be infinitely great, the value of P: Q is always cot (n@ +a) and & -- ὦ acquires two units for each accession of 27 which n§ + a receives, while @ changes from 0 to 27: that is, k-—1=2n. Hence the proposition that @z always has a root or roots is proved. We then, in the common way, establish the existence of the root-factor, and the number of the roots. Previously to proceeding further, I discuss a point which is of great importance, and bears on many of the proofs of the preceding proposition, Dr Peacock (Zeport on Analysis, p. 305) objects to making interpretation the foundation of important symbolical truths, which, he maintains, should be considered as necessary results of the first principles of algebra, and ought to admit of demonstration by the aid of those’ principles alone. Interpretation is, or at least begins with, the application of meaning of fundamental symbols to the deduction of meaning for compound symbols. It may be applied to throw light on the steps of a demonstration, and in this way it must be applied: without it algebra is a valley of dry bones. It may also be applied to furnish steps of demonstration ; and this sort of application must be sternly resisted: the result is not algebra. But on this point the following distinction suggests itself. Every proposition is true of which the truth can be shewn, Demonstration of the possi- bility of demonstration is itself demonstration; demonstration of the possibility of demonstrat- ing the possibility of demonstration is also demonstration: and so on. Mathematical teaching has used this principle rather too extensively. A proposition proved to be true of commen- surables is allowed to be assumed as to incommensurables, on the feeling that its truth as to 264 Mr DE MORGAN, ON A PROOF OF THE EXISTENCE OF commensurables is proof that a demonstration can be found as to incommensurables. If a step suggested by interpretation, and seen to be a true step by perception of the necessary consequences of interpretation, be allowed to stand part of the proof, without anything further, this question then arises, Can the step of interpretation be supplied by an algebraical substi- tute? If yes, then the substitution ought in strictness to be made, and it must be made on demand: if no, then the proof cannot be called either actually or potentially algebraical. All its geometrical interpretations might very easily be replaced by algebraical ones; not so its arithmetical interpretations. The proof which I have given above is not, in the very strictest sense, algebraical. It hinges on the use of greater and less, when we come to apply the preliminary theorem to @z. Let all the letters denote operations, how are we to prove that X°+ AX + B performed on φῶ is the result of five successive operations of the form X-C? ledge that transformations deduced from quantitative interpretations, upon no assumptions as to the specific magnitude of the quantities, are symbolically valid. I do not believe that any proof * exists except that which is derived from our know- I now proceed to supply the algebraical substitute for a geometrical step which occurs in Sturm’s proof of Cauchy's theorem, and in Mourey’s proof of the fundamental theorem. When a closed circuit is described, say in the positive direction of revolution (that is, in the direction which, on the whole balance of positivet and negative revolution, makes the radius drawn from some one point inside it gain four right angles), then the radius drawn from any one point whatsoever inside the circuit gains four right angles; the radius from any point outside neither gains nor loses, performing as much positive revolution as negative; the radius from any point on the circuit gains two right angles during continuous revolution, and a second pair of right angles per saltwm, in passing through its vanishing position. This is as evident as can be when the figure is looked at. Let one point, within the contour, be taken as the origin: let the radius from this point to (w,y) be 7, and its angle with the axis of 2 be @. Let there be another point within, on, or without, the circuit, at a radius m and angle » with respect to the origin. from the point just named to (a, y) be 8, and its angle σ. Let the radius Remember that 7, m, s, are posi- : rsin?@—msina tive. See les We have then 7 cos@=m cos w+ 8 οοβσ, rsin@=msinn +s sing, tang = : 7 005θ —mcospn Now it is the algebraical property of this last formula, independent of all geometrical interpre- tation to those who algebraize the sine and cosine, that while @ changes from 0 to 27, o gains 27, or gains 7, or gains nothing; g; and never loses. Let ¢ =4+\/: we then deduce r sin (θ -- μὴ) tan Ψ = Ξ------- rcos(@—p)—m * The celebrated proof of Laplace, or rather his improve- ment of the proof given by Foncenex (Lecons de l’ Ecole Nor- male, vol. ii. p. 314), has often been cited as a proof that every equation has roots. The first words of Laplace are ‘ Soient a, b,c, &c. les diverses racines de cette équation...” and the are given symbols. But the assumption of this possibility is a difficulty of the same kind. + The circuit must not be autotomic. dition it makes any undulations. Subject to this con- With respect to an internal point, any point which describes the circuit revolves in one way, proposition proved is that these roots are of the form m+ n/—1. Dr Peacock's form of this proof (Report on Analysis, p- 298), begins by shewing that the possibility of roots stands or falls with the possibility of symbols, all whose symmetrical products positively or negatively, while it is hidden from the internal point by an even number of intervening parts of the circuit, and in the other way, negatively or positively, while it is hidden by an odd number of intervening parts. A ROOT IN EVERY ALGEBRAIC EQUATION. 265 We see that tan y makes the change +0-or —0+ only when sin (θ -- μ) =0, cos(9-p)= +1. If rm when = μ (which answers to taking the second point inside the circuit), then, at Θ =, tan yy goes through —0 +, and at θ =, 4+ π΄ also through But if πὸ Ὁ: that is, tan y recovers its first value at 9 = 2a by gaining a whole revolution. : ς 2 . 0 r =m when @ = μ (which answers to taking a point on the contour) then tan yy passes through — 0 without change of sign, and s sin o, or m/(sin θ -- sin μὴ changes sign without s changing sign: that is, σ᾿ receives, per saltum, an accession or diminution of 7. At @=nu47, tan Wy undergoes the change — 0 + which, not being compensated until 6 =, + 27, shews half a revolution added to Ψ by the time y gains its original value. Appealing to the above as algebraical proof of the requisite property of the circuit, and using the geometrical phrases only as combined abbreviation and elucidation, I shall now pro- fo) δ y ceed to Cauchy’s theorem, which with its extension is as follows. Let pz be any rational function whatsoever, and P(#+y/— 1) being P+ Qr/-1, 7 ἢ τὰς Ἔ be recorded while the point (#,y) describes any closed circuit in the positive (or rather positive-balance) direction of revolution. Let & be the number of + 0 — changes, ὦ the number of —0+ changes. Let m and m’ be the numbers of points within and upon the circuit, at which ᾧ (ὦ +y4/ -- 1) =0. circuit at which ᾧ (w@ Ἐν, - 1) =e. Let p and p’ be the numbers of points within and upon* the Then k—-l=2m+m -- (2p4+p’). Cauchy included only the case in which, by hypothesis, m’ = 0, p = 0, p’ = 0. Let the function be a rational algebraical fraction, in which the roots of the numerator come under a, (cos a, + sina,,/ —1) and of the denominator under ὁ, (cos B, + sin B,4/-1). Let the function be ᾧ (w+ y4/ - 1), Φ +y/ —1 being rcos6 +r sinO@/ — 1, and let r cos -- a, cosa, + (r sin @ —a, sina,) ../ -- ᾿ Ξ 8, (cosa, + sing, γί — 1), r cos @ — b, cos B, + (rsin θ — b, cos B,)../ -- 1 =#, (cos τ᾿ + sin τ, γί -- 1). The function ᾧ (ὦ + y4/ — 1) is therefore a constant multiplied by the following fraction 8785 «τοῦτο (605 σὴ + SING, y/ — 1) (605 σ» + SiN σο γί — 1)... tity «οὐ νος. (Cos τῇ + sin τῇ γί — 1) (605 το + Sin τ΄ 4/ — 1)... * Sturm says, positively, that there can be no theorem when | missed the introduction of m’. Any one who will take up the a root is on the contour, for that different contours containing the same numbers of radical points, may in that case give dif- ferent values of k-J/. But this was said after the first proof, which he and Liouville gave together, and before the second proof, which I am now translating into my own language, as applied to the extended proposition. Had he reconsidered his assertion while employed on the second proof, he could not have VoL. XX. Pari I: point as a question of continuity by the aid of the curves P=0, Q=0, will easily detect the loss of a change of the form +0-, or a gain of —0+, when the circuit passes over an inter- section of the curves P=0, Q=0. In this he will need the following theorem, which is easily proved :—when the circuit passes through an intersection of P=0, Q=0, either both P and Q change sign, or neither. 34 266 Mr DE MORGAN, ON A PROOF OF THE EXISTENCE OF ly Whence = cot (o, + o2 +... — τῇ -- το -- ---) & Now it has been proved, algebraically, that for every one of the radical points, whether of numerator or denominator, within the circuit, o, or 7, gains 27; on the circuit, π᾿ continu- ously, and 7 per saltum without effect upon sign; without the circuit, 0. The theorem is now obvious. As to the excess of & over J, it matters nothing whether we make @ pass from 0 to 27, in any of the angles oj, σῃ» ... Ti, Τὸν --- consecutively, or in all at once. In the first case, σὶ +o. + ... gives to the cotangent 2m +m’ changes of the form +0 — if the circuit be convex, and none of the form -- 0+: while if the circuit be not convex, the changes of the first kind exceed those of the second by 2m+m’. At the same time, -- τι -- τῷ -- ... gives an excess of — 0 + changes over + 0 — changes of 2p +p’. The theorem is universally true for all functions in which a root factor of the first dimen- sion exists for every root. The proof most commonly given (the joint proof of Sturm and Liouville) depends upon the consideration that where two closed circuits having no common area have some portion of boundary circuit in common, the sum of the values of k — 1 for the two separately is the value of & -- for the single circuit made by neglecting the common boundary. And this because the common boundary, being described in different directions in the two circuits, contributes towards ἢ in one circuit what it contributes towards J in the other; and vice versa. Hence any circuit* may be divided into an infinite number of infi- nitely small circuits; and the theorem, being proved true for an infinitely small circuit, is true for the circuit made of the outer line of all the subdivisions. There is no occasion, after what precedes, to shew that if platy -1)=(@t+yV —1-acosa —asinay/-1)"Wl(w+ yr -- 1), where Ψ (a cosa + a sina’/ — 1), does not vanish, an infinitely small contour described about the point (a@cos a,a sina) gives k —1 = 42m or £m, according as the point is within or upon the contour. The theorem fails when the root factor enters with a fractional exponent: unless indeed we propose an extension so vague as a theorem constructed on the trial of all integer powers of @z. Let the function be one in which every root-factor is of the first dimension, subject to the usual definition of equal roots; and let it never become infinite for finite values of # and y. Then the curves P = 0, Q = 0, the intersections of which determine the root-points, are such that two branches, one of each curve, cannot inclose a space. At each root-point, the branches which there intersect, must make known the existence of that root on every circuit which con- tains the point, however large. The four places in which a branch of P = 0 and one of Q =0 meet any circuit, supposed convex, give +0—-, -- © +, τ0 --, — » +, which are just suffi- cient to indicate one root. No second root-point can then be determined by these branches This is not, however, a definition of all curves which cannot inclose space; for P =0 and * Those who remember the treatment of the electric circuit | ber that this is also the way in which an infinitely small cur- by Ampere (I think, but it is long since I read it) will remem- ! rent is integrated into any current whatsoever. A ROOT IN EVERY ALGEBRAIC EQUATION. 267 Q = 0 always intersect orthogonally ; and do not, therefore, contain so much as all pairs of straight lines. There are other conditions of intersection and of sequence on which I do not here enter. I now proceed to give the proof of the fundamental proposition which Argand gave (1815) in the fifth volume of Gergonne’s Annales, p. 204. I repeat this proof here, first to separate it entirely from the interpretation by double algebra which it was Argand’s principal object to illustrate, and which he did illustrate with great effect : secondly, to remark that, in a much more simple form, it is the proof which Cauchy afterwards hit upon, and published, first (1820) in the Journ. de [Ecole Polytech. vol. x1. p. 411, and afterwards (1821) in the Cours d’ Analyse, p. $31, a work, to which, as a student, I was much indebted. Argand’s proof rests upon the easily proved proposition, that r, signifying r cos @ + rsin@4/—1, &e. and p, q, &c. being ascending positive exponents, the length or modulus of a,7," + bgr,? + erg +... may, by taking 7 small enough, be made as nearly equal as we please to that of a,r",, and the angle of the first as nearly equal as we please to that of the second. This, under the inter- pretations of the complete, or double algebra, is instantly perceptible, and the pure algebraical proof is very easy. This being premised, let us take a,r,” + bgrg"~' + ... +k, + l,, which call U(cos Y +sinY\/ -- 1). If it be impossible to take r, so that U = 0, it follows that values of r and θ exist which give for U a value which cannot be lessened. Let m, be this value of 7). and for r, write m, + h,, which, D, being the value of least modulus just mentioned, changes the expression into the form DR te A,h? ae B, μι ΞΕ ΕΝ where, p, g, &c. are ascending positive exponents. Take h so small that the effect produced on A,.h,» by the succeeding terms shall be useless in the following considerations. The first two terms give Dcos Δ τ Dsin A \/- 1 + § Ah? cos (pn +a) + Ah? sin (py +a).,/— 1h. Here 7 is at our pleasure. Assume py +a= A+, the preceding then becomes (D - Ah) (cos A + sind. - 1), which, A and h being positive, as they may be, the angles furnishing negative signs when wanted, has a modulus less than that which cannot be lessened; which is a contradiction. No less acute a person than Servois did not see that this contradiction deduced from the assumption of one of two necessary alternatives, is final in favour of the other. He pleaded to the contradiction that it was not shewn to be large enough; and in so doing he has added one to the many cases which prove that a severer study of pure Jogic would be useful to the mathematicians. He contends that Argand was bound not merely to shew a less than the least, but to shew that this less than the least might be made as near as we please to zero. Argand’s argument was precisely that of Euclid in the proof that pyramids of equal bases and altitudes have equal solidities; the difference is nothing because, whatever else may be named for the difference, it can be shewn to be too large. The minimum modulus must be nothing, because, whatever else may be taken for the least modulus, it can be shewn to be too large. Servois forgot that the opponent who undertook to convince him had allowed him to begin by 34—2 2608 Mr DE MORGAN, ON A PROOF OF THE EXISTENCE OF taking D as small as he should please, before he began to shew that it might have been taken yet smaller. Argand’s proof is quite fundamental, and is the most direct of all. Its so called παϊξτετο, character is nothing but a case of the habit of the mathematicians not to admit the identity of contrapositive forms without proof. To a logician the following forms, ‘Every quantity which is not 0 is not the minimum,’ and ‘the minimum is 0’ are identical, the existence of the terms being known. If Cauchy’s theorem were not to form part of a course, I should recom- mend Argand’s proof; and I should, in any case, insert Argand’s as supplementary to the one I have before given. Argand and Mourey were both in full possession of double algebra up to the interpreta- tion of real exponents inclusive. The manner in which, at what is thereby proved to be the due time, persons of all kinds, unconnected with each other and unknowing of each other’s existence, will take up a subject of speculation, of observation, or of experiment, is becoming better and better known from day to day. Remembering that Mr Airy, more than five-and- twenty years ago, casually told me that he had occupied himself with the interpretation of / - 1 at a very early period.of his studies, I lately begged of him to let me see any notes which he might have made on the subject. The reply was the transmission of a manuscript, drawn up in the form of a paper for a scientific society, dated January 21, 1820, and therefore written in the first three months of the author’s residence at Cambridge. It con- tains, with many examples, a full interpretation of the roots of + 1 and of —1; and commands the full meaning of + and of —, and of x so far as relates to the formation of powers. The idea on which it starts is, like that of Argand, the assumption of proportion, in the case of lines, as involving equal differences of direction, as well as equal quotients of length. Of Argand Mr Airy knew nothing; of Buée as much as this, that he had been told a Frenchman had treated the subject in the Philosophical Transactions. Mourey’s* proof is as follows. It is much defaced in the original by peculiarities of notation: the author had the idea that he was in possession of a new algebra, not the old algebra under extension of interpretation. First, it is shewn that the equation which is expressed in my foregoing notation by I(T — Aq) (7% — Og) --- = My must have a root or roots. The first side of the equation being altered as before, we have $8) «ον $C0S (θ +o; Ἔσο +...) +8in (0+ σι + 02+...) ¥/ — 1}. = πὶ (cosp + sinur/ — 1). As before shewn we know that while @ changes from 0 to 27, no one of the angles σι» 0%... loses value on the whole, while such as gain must increase by 7, or by 27, consequently 0 +6, +o2+... increases by 27 + (0 or some multiple of 7). At some value or values of θ, then, we have cos(@ +o, + .--) =cosp, sin(@+a+...)=sinu. Next, this value of @ being supposed to be determined, we have 18,8)... =7,/ {r° — 2ar cos (8 —a) + a} .a/ ir? -- 2br cos (8 -- B) + }.... * La vraie Théorie des quantités négatives et des quantités prétendues imaginaires. Dédié aux amis de VPévidence. Par C. V. Mourey. Paris, Bachelier, 1828; pp. xii. + 144, 3 plates. A ROOT IN EVERY ALGEBRAIC EQUATION. 269 which vanishes when r = 0, and finally increases without limit with r. At some value or values, then, of 7, we have 78,8)... = m. Consequently, the given equation has one or more roots: that is, every equation of the form ὦ (a — a) (w — δ)... =m has one or more roots. Next, it follows that if every expression of the (m — 1)th degree has — 1 roots, every expression of the nth degree has n roots. First, aa” + ba" +... + kv +l is aw (a +...4 kh) 4+), which, since every expression of the (m — 1)th degree has n — 1 roots, is aw (ὦ — a) (a — 3)... + U, and this, by the preliminary theorem, has one root. Consequently, ax” + ba"! +4... is of the form (ὦ — a) (ax"~' + bu”~* + ...) which again is (ὦ -- αὐ x the product of πὶ —1 such other factors. Whence aw" +... has n roots, if every such expression of one degree lower have 7) -- 1 roots. All the rest follows from the expression of the first degree having one root. ἷ A. DE MORGAN. Untversity CoLtecr, Lonpon, December 18*, 1857. Postscript. I sussotn some brief remarks on a couple of elementary points. 1. L[ find that the following theorem is new to several mathematicians to whom I have proposed it. It may be most briefly expressed as follows: Any two divergent series whatso- ever, of the same character as to signs of the terms, are to one another in the ratio of their last terms. That is, if a+ a,+a,+ ... +a, and &+6,+0+ ... +6, give results which become infinite with m, the limit of the ratio is that of a, to 6,; and this, whatever the signs of a,, ὃν» ἅς. may be, provided only that a, and 6, always have like signs. Thus 14+2+3... and1+3 +54... are infinites in the ratio of 1 to2; and soare1—24+3-... and1—3+5-—...3 and soarel+2-3+4+4+5—64+... and 1+3-—5+7+9-11+4... I speak only of arithmetical summation, without reference to the value of the evolving func- tion, when finite. Apply this to Σ. πῇ and = {(m + 1)"+!—n'*+!t, and we have (1: -- 1), the proposition out of which Cavalieri and his successors produced a limited integral calculus. - 1 Apply it to Στ᾿ and Σ $log (m + 1) — log mt and we may render the connexion of 1 + } + 53. Ἐ- and log m + const. + dn -! +... a part of elementary algebra by easy processes. And simi- larly for log1 +... + log m and n log m — n, and generally for =n and =f?*'pade. 2. I have looked through elementary writings in vain for a classification of the species of spherical triangles, as to character of sides and angles, with respect to the right angle. Ex- cluding the right-angle, the cases which exist are as follows: all cases in which opposite sides and angles are of the same name; and all others in which an odd number of acute sides is * The substance of this paper was read to the Society on the 7th of December, as stated at the commencement. 270 Mr DE MORGAN, ON THE ROOT OF AN ALGEBRAIC EQUATION. joined with an odd number of obtuse angles, except only three acute sides joined with three obtuse angles. This may be remembered by the following mnemonic diagram. 0 1 a 3 Obtuse Sides. Acute Angles. Obtuse Angles. Acute Sides. 0 1 2 3 or two Ws with an inverted NV between them. The numerals, from left to right, describe the numbers of obtuse sides and of obtuse angles; from right to left, acute sides and acute angles ; the lines connecting numerals assert the possibility of the combination. Thus, with two obtuse sides may coexist either one, two, or three obtuse angles; with two acute angles may coexist either one, two, or three acute sides. The base of the triangle being given, and one of the hemispheres formed by its plane, the regions which are the loci of the vertices, for different classes of triangles, may be described as follows. Let the hemisphere be orthographically projected upon the plane of the base, and let orthographic projections be used for originals in description. Let AB be the base when acute, BA’ when obtuse. Let aa’, BB’ be perpendicular to 44’, BB’. Draw an hyperbola AB, 4Ἢ8΄, having aa’, ββ΄, for asymptotes. Then AA’, BB’, aa’, ββ΄, represent secondaries of AB A'B’, and AB, A’B’, represent intersections of the sphere with an hyperbolic cylinder. The hemisphere is divided into ten regions; and each region is the locus of the vertices of one class of triangles for the base 4B, and of one class for the base BA’. A point on the hyper- bola, on either branch, is the vertex of a right-angled triangle, be the base either 4B or BA’. Thus, if the vertex ( be taken within the region BOa, then, the base being AB, all the sides are acute, and the one angle ABC obtuse; or the triangle may be symbolised as 01, no obtuse side and one obtuse angle. But with reference to Bd’, we have 21, two obtuse sides 4.8, A'C, and one obtuse angle A’CB. These symbols are entered in the compartment BOa, and all the other compartments are treated accordingly. A. De M. Dec. 19, 1857. “ar Am ani wile mi : ia a | Ἰ Ὡς ad ile 4 Gee Ξ | ὶ Ai ; ’ 13h ie, 4 Shite df ΝΣ ee a ; i Li age : h Py ὁ. πὰ ee my A alll ol 1] δῇ δ 7 . - = a = “Ὁ Μὰ = i = "( j= 5 ἐν “Ὁ: ᾿ 2 = - ὃ > _ -᾿ “i δ᾽ ΝΕ ---ἕ ΄σ ἜΝ ν 7). ΟῚ >. a ν΄ Ὁ- “6, i, fpe Ὑτυτηδ το τ᾽ ᾿ ‘ ᾿ ἐν ἃ τ) τ ἘΣ ΡΡΕ γ rer με ph Als cpt a py MTC 30 iad feted #159055 {yi 80) : PNG ἃ ΤΠ] τῇ ἘΣ ἐὰν ΓΟ ΤΠ We ον. οὐ σοθῦ > ΠΡ ᾿ τ , ae 7 mine ws — : : POOR PUNCH PO EPS ccice Gh PPS PMAsiaPA. Finest fot spon | Ie, ogre og. «hore op ἀμ ΕΣ are Range] Bay OM τῷ ἰδοῦν. ὯΝ τὐθεν, το hee edie Vabpeni- oo) op steal aR TT ir Oy τ’ --- 0 τ᾽ τοὐπηδιρα ee ᾿ 4 oP ‘SUOTJOVSUBIT, 901 10 Θίπη10 Δ, 5111 ϑιπγαα jo osuedxa oy} soajestuoyy uodn Suryez ut ΑΠΌ 196 oy} 10} ‘ssorg ἀψμιβαθλίιῃ OY} JO SOIGNAG 94} 0} syUSTISpopMouyoR jnjoqwrs 921 Surssordxo jo Ayrungzoddo sty} 5678} ALHINOG AH, ‘sdoymnp aayoadsan way, Jo pata ayy uo hjacyua jsat ysnu youn ‘swadng 70.0.05 91} τὺ paounappo suowmdo pun sjovf finn sof ajqrsuodsas paiapisuoa aq 01 yow 81 hpog Ὁ so hyarw0g aH] ‘LNA WASILYAACV 60F PLE SOE 06é uopuoT ‘abayop tiprssaaug ur soynuayyoyy fo Lossaforg ‘abayog ἤχει, fo “sywuy a ‘Nvoxoyy aq saisnoay Ag ‘wmashy oynuhug ay, fo spnod snor.ma uo pup “Δ ὋΝ Swusrboligy ay UO SRR R Ree eee eee e eee eee eee eee eee eee ee Hee THe teen tees ee eeas aes ene see ΜΟΡΊΟΥ “θο 02) finswauug wa samwueywpy fo sossaforg ‘abayog hynny fo “ὧν ἘΓ ‘NVOUO]Y Gq saisavay fig ‘uoynauasqg fo sioLug fo ἤμοοη, 242 UO Sedge IFO LGESOy WOSSSE Ie SADC τ τ τσ aboyon ἤχει; fo nope “WW ‘ona “fv Ἢ ΛΖ .coeabjy τὰ syuopy nog ,, sry τ poysyqnd pup veg yw AAISAUVIG IPT ἤφ pardoo uoydwasut uyoT poouayy Ὁ UOC Fee eee tere ΞΞ ἤχριοος' qwoorvydosopy 1 abpriquny ay? fo COVEN & “WW ‘sanoy Ἢ Ὃ 4g ‘spoy ousmg wy) pun sumag fo uoyory 242 YO ΝΞ Ξ Ξ“ΞΞ-Ξ-- ΜΟΡΊΟΥ “βοῃοῦ fipsvanug ur sypuaywyy fo sossafong ‘abayjopn Kuo αν wa ‘NvouoW aq saisanay fig ‘suoynpay fo abot ey, uo pun “AT ΟΝ ‘wsrbopjligy ay2 uC settee eee eee een eeeeseeeene golioay LIULOU0LIS FE ΠΕΡῚ χατν 4 Ὃ hg 40077 Ὁ spy uoyonby omigebpy ἤμοαο you, wasoayy, 90} fo foorg v 07 quauajddngy avsres tees sae abpruquoy ‘abayjop Kyun. fo moyag 920) “Gq ‘NosaTvNog ‘M ΡΠ “INaNaDUy pwom ay fo 95) sadorg pun Ubu 32 UO eee eee ΠΣ --"-- ἤχριοος! ay} fo quap asatg-ar4 . sin 80 =0;3 from which cos 80 = — 1, ρὴ = 8. 422 990 6. B. AIRY, ESQ. SUPPLEMENT TO A PROOF OF THE THEOREM, ἃς. If z be a negative quantity = — 2’, then , cos88=+1, ρ τῆ. The combination of the different values of θ which make cos 8θ = + 1, properly carried out, gives Cotes’ Theorem. 27. It will readily be seen that the process here used is perfectly general as regards the order of the equation; the adoption of the sth order having been made merely for convenience of writing. G, B. AIRY. Roya OBseERVATORY, GREENWICH, October 1, 1859. VII. On the Syllogism, No. IV, and on the Logic of Relations. By Avaustus De Morea, F.R.AS. of Trinity College, Professor of Mathematics in University College, London. [Read April 23, 1860.] In my second and third papers on logic (Vol. rx. part 1, Vol. x. part 1,) I insisted on the ordinary syllogism being one case, and one case only, of the composition of relations. In this fourth paper I enter further on the subject of relation, as a branch of logic. Much has been written on relation in all its psychological aspects except the logical one, that is, the analysis of necessary laws of thought connected with the notion of relation. The logician has hitherto carefully excluded from his science the study of relation in general : he places it among those heterogeneous categories which turn the porch of his temple into a magazine of raw material mixed with refuse. Aristotle does not give this part of logic a very hopeful look when (Categories, ch. v, or vii.) he puts forward no better phrase* than πρός τι to denote his abstract idea of relation. And such hope as there is becomes well- nigh extinct when we learn that the rudder is not properly the rudder of the ship, because people do not say (οὐ λέγεται) that the ship is the ship of the rudder. Here, as occasionally elsewhere, Aristotle is rather too much the expositor of common language, too little the expositor of common thought. Surely the question, ‘ What ship does this rudder belong to ”” must sometimes have been heard in an Athenian dockyard: and if this question were not equivalent to ‘ Which is the ship of this rudder?’ in the common idiom, the equivalence ought to have been established by the logician so soon as wanted. Terms may be related, even though they have more meaning than just goes to the relation. A ship is ‘the steered’ and 2 the rudder is ‘the steerer:’ that it happens to a ship to mean more than ‘a thing steered, and to a rudder to mean mo more than ‘the thing which steers,’ is a purely material concomitant of the words. The logicians of our day seem to my mind to combine a want of memory by which * When a noun is thus formed, it is a sign that the mind , except the contemptuous term medley, the corruption of a legal of the language has not possession of the idea. There is a | term hodgepodge, also contemptuous out of law, and the best useful piece of furniture called a what-not, a holder of miscel- | word of all, the only one which perfectly applies, omnium laneous articles: the word is of the same type as πρός τι. We | gatherum, contemptuous and not English. The applier of the are an orderly people, and the notion of unarranged deposit is | term what-not probably was not aware that he had the authority not among those for which we find words of serious approval. | of Aristotle for his mode of proceeding. Dr Roget can produce no nouns which come close to the point 992 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, they do their own literary ancestors less than justice with an assumption by which they take advantage of their own wrong. ‘Their predecessors worked the modern languages into adequate vehicles of scientific thought. They greatly augmented what they found in the Latin of the power of the Greek: and the vernacular idioms, partly by abstraction and partly by imitation, acquired the increased power of the Latin. From the first growth of experimental science down to our own day the logicians have not shewn themselves aware of this: at least they have not known how to use it in efficient defence of ‘the schoolman’ from the sneers of the physical writers. A person who approaches medieval psychology fresh from a long course of thought on exact science, its language, its progress, and its impediments, finds the claim of the scholastic writers presented to him in a strong light. He seeks the old books to learn something about the ‘trammels’—this is, I believe, the proper technical word—in which they bound the human mind: for the human mind, he has been implicitly instructed, is rapid and vigorous in abstract science, if only it take care to follow no leader. But he finds that very much of his own lingual power of expressing abstract thought is due to the action of these schoolmen upon his mother-tongue: he feels that he is at the fountain-head of his own scientific idiom: he learns that those who raised the seed have been ignorant enough to think lightly of those who dug and manured the ground: and he comes to know that language capable of science can only be the result of deep thought upon the mind in relation to words, and words in relation to things. He then remembers the sarcasm* of the spider, and finds it a true description of a needful process : the web first, the fly afterwards. The logician appeals to common thought in proof of his system being an exposition of the necessary laws of thought. In one sense he is right: his system contains the necessary laws of thought; for the actual thought of the lowest type of mankind must be the maximum of the necessary thought: so that, on the Ricardo theory, the logician has created a great deal of rent. But, meaning by a necessary law of thought that mode of action which must guide the thinker who comes up to the point at which the question of law or no law can arise, I affirm that all the difference between Aristotle or Occam and the lowest of the noble savages who ran wild in the woods is only part, and I believe a very small part, of the development of human power. If the logician could leaven his own mind with a full sense of what his foregoers did for thought and for language, a spontaneous x Logicus aranee potest comparari, | reality followers of Galileo and of Newton—of Galileo, the Que subtiles didicit telas operari, predecessor of Bacon in his works, and of Newton, who cannot Que suis visceribus volunt consummari, be proved to have known that there was such a person as Bacon. Est pretium musca—si forte queat laqueari. Again and again has it been asked what discovery has ever When Bacon adopted this sarcasm, he left out the fly, and | been made by that method which Bacon recommended? and propounded the web as the end, not the means: and he has | always without answer. And for this reason, that the mythical been followed by some original writers who have likened the | Bacon cannot be supported by quotations from the Novum schoolman to the spider, which spins all its own nourishment | Organum. It is full time that those who actually read the from its own bowels. great work—for such it really is—which is supposed to have The web which caught the flies at last was a mathematical | taught experimental philosophy her rudiments, should either web: and in time an imitation of the mathematical web was | support the pretensions advanced in its favour, or aid in the Applied to subjects over which the empire of pure calculation | substitution of others of a more correct character. Provided did not extend, Neither the medieval logicians nor the fol- | always that Bacon’s own method—which is very easily pro- lowers of Bacon ever constructed a physical science. Those | pounded—be advanced in Bacon's own words. who delight to call themselves the followers of Bacon are in AND ON THE LOGIC OF RELATIONS. 999 admission would grow that if these same foregoers had worked themselves into the same familiarity with relation in general which they obtained with what I call onymatic relations, and still more if they had cultivated those yet wider fields which lie beyond, the common language would have now possessed facilities on the want of which* he founds his assertion of the sufficiency of the old logic. Though satisfied that the educated world is in advance of the current system of logic, I feel equally sure that a more extensive system would work a still greater progress. The investigation of the subject of relation has kept before my thoughts, and with a de te fabula narratur of a most humiliating character, conclusions which instruction of young minds during more than thirty years has forced upon me. A person accustomed to teach mathematics from the earliest commencement to the highest theories, to pupils wholly unformed in inference upon matter with which they are not familiar, has a perception of the difficulties of the uneducated process of reasoning which few others can arrive at. And that which he cannot help seeing in the efforts of an unformed mind, decided in cha- racter and large in amount, he learns to detect in the more advanced student and in the educated man. At the same time he finds the reason why the deficiency need not be acknowledged, and may even be denied by any one who takes the proper ground, For it is not a deficiency which strikingly manifests itself in habitual faults of commission : habitual faults are only in habitual things. The evil is most patent when new and strange materials are submitted to the mind; and it bears fruit, though of course undetected by positive con- sequences, in the many cases in which want of power is a prohibition. The uncultivated reason proceeds by a process almost} entirely material. Though the necessary law of thought must determine the conclusion of the plough-boy as much as that of Aristotle himself, the plough-boy’s conclusion will only be tolerably sure when the matter of it is such as comes within his usual cognizance. He knows that geese being all birds does not make all birds geese, but mainly because there are ducks, chickens, partridges, &c. A * An existing instrument always appeals to the fact—in | pulchrius est magis eligendum..... Unde licet laici non ha- justice I must say the established fact—that all which ever was done resulted from the use of then existing instruments. In our own day Brown Bess has asked the long range rifle how many battles she won in the Peninsula: to which the rifle has replied by asking Brown Bess how many battles she won forty years before she was invented. + The syllogism is complex, and so is the Act of walking : but in both cases the mind produces the whole without a con- sciousness of parts. Several persons have thought I was car- tying things too far when, in my first paper, I said that a person who calls out John! enunciates two propositions, «« John is the person I want,’ and, ‘* You are John.’ They will probably think that the author cited by Roger Bacon (Opus Tertium, p. 102) also went too far, in the following passage. ‘‘Auctor Perspective ponit exemplum de puero qui cum ei offeruntur duo poma, quorum unum est pulchrius altero, ipse eligit pulchrius, et non nisi quia judicat pulchrius esse melius, et quod est melius est magis eligendum. Ergo de necessitate puer arguit sic apud se: quod est pulchrius est melius, et quod est melius est magis eligendum; ergo pomum beant vocabula logice quibus clerici utuntur, tamen habent suos modos solvendi omne argumentum falsum. Et ideo voca- bula sola logicorum deficiunt laicis, non ipsa scientia logice.”’ It is very difficult to deny that both the premises and the conclusion are truly parts of the boy’s mental act of choice: but quite impossible to admit that they are separate parts. We must distinguish between the compound act of the un- educated thinker, and the analysed compound of the logician ; between the process guided by law, and the cognizance of the law which guides. It is not true that the law by which thought is governed must be part of the thought which is governed: though some writers against logic have spoken as if they would sanction the affirmative. And, similarly, some writers against gravitation are hardly intelligible except when taken as implying that Newton gave the particles of matter some mysterious cognizance of m:r?. It is also clear that the opinion of Roger Bacon’s time tended to the conclusion that logic is a science of invented laws, not an analysis of the actual laws of thinking; the mistake is not yet defunct. 994 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, beginner in geometry*, when asked what follows from ‘Every A is B, answers ‘ Every B is A, of course.’ That is, the necessary laws of thought, except in minds which have examined their tools, are not very sure to work correct conclusions except upon familiar matter. And above all, relation is a difficulty when the related terms are unusual names, even in the most} common cases, As the cultivation of the individual increases, the laws of thought which are of most But difficulty and risk of error make a new appearance with a new subject; and this, in most cases, usual application are applied to familiar matter with tolerable safety. until new subjects are familiar things, unusual matter common, untried nomenclature habitual; that is, until it is a habit to be occupied upon a novelty. It is observed that many persons reason well in some things, and badly in others; and this is attributed to the consequences of employing the mind too much upon one or another subject. But those who know the truth of the preceding remarks will not be to seek for what is often, perhaps most often, the true reason. Waiving all question about common matter being usually the subject of tolerably good inference, about the assertion that logic, though of some use, does not fully repay its labour, and about the observed fact—the like of which is true in regard to all studies— that learners of logic not infrequently reason no better after instruction than before,— waiving these things, not admitting them, I maintain that logic tends to make the power of reason over the unusual and unfamiliar more nearly equal to the power over the usual and familiar than it would otherwise be. The second is increased ; but the first is almost created. An attempt to investigate the forms of thought involved in combination of relations, the results of which are contained in the following pages, has given me personal experience of the truth of the preceding remarks. I have had to work my way through trans- formations as new to my own mind, so far as the separation of form is concerned, as the common moods of syllogism to a beginner in logic. If there be any person who can see at a “glance, and with justifiable confidence, what classes of men, including women, are specified in ‘the non-ancestors of all non-descendants of Ζ, I should not like to submit to his criticism the confusions and blunders through which I arrived at the following results: unless indeed I were able to remind him of some of his own similar experiences. And this could be done with the greatest names in the history of abstract speculation. * He is thrown at once into forms of strict reasoning, with unusual matter on which to employ them. LEither some logic ought to precede Geometry, with familiar instances ; or some acquaintance with figure by measurement ought to precede the reasoning ; or, better than either, both. + Though I take the following only from a newspaper, yet I feel confident it really happened : there is the truth of nature about it, and the enormity of the case is not incredible to those who have taught beginners in reasoning. The scene is a ragged school. Tracuer. Now, boys, Shem, Ham, and Japheth were Noah’s sons; who was the father of Shem, Ham, and Japheth? No answer. T'eEAcHER. Boys, you know Mr Smith, the carpenter, opposite; has he any sons? Boys. Oh! yes, Sir! there’s Bill and Ben. TEACHER. And who is the father of Bill and Ben Smith? Boys. Why, Mr Smith, to be sure. TracHer. Well, then, once more, Shem, Ham, and Japheth were Noah's sons; who was the father of Shem, Ham, and Japheth? A long pause; at last a boy, indignant at what he thought the attempted trick, cried out, It couldn't have been Mr Smith! These boys had never conyerted the relation of father and son, except under the material aid of a common surname: if Shem Arkwright, &c., had been described as the sons of Noah Arkwright, part of the difficulty, not all, would have been removed. AND ON THE LOGIC OF RELATIONS. 335 If Newton were the examiner of my failures, I could recall the occasion on which he lost his own connexion between the inverse square and the ellipse, because his casual diagram put conjugate diameters at right angles to one another, and seduced him into the belief that they were the principal axes. Were it Wallis, I could revive the time when he hesitated at Wil2t= 23/3, sure of the theorem, but doubtful of the validity of the expression, for want of precedent. Were it Leibnitz, I could bring to his memory the co-inventor of the differential calculus, doubting whether to say yes or no to the equation = =d (2) , and working out the decision on paper. And so on. The want of power which most persons feel in the treatment of combined relations, may be well illustrated by cases of the class of relationships which have almost appro- priated the name, those of consanguinity and affinity. Many educated persons, and some acute logicians, would either pause for an unreasonable time, or would not give the right answer, if asked for all the conclusion* that follows about John and Thomas from ‘ William is not John’s father, and Thomas is William’s uncle.’ The only relations admitted into logic, down to the present time, are those which can be signified by is and denied by is not. Allowing to the substantive verb all its range of meaning—and that range is a wide one—and introducing contrary notions, all the relations which were styled onymatic in my last paper, whether arithmetical, mathematical, or metaphysical, are capable of inclusion. All other relation is avoided by the dictum that it shall be of the form of thought to consider the relation and the related pre- dicate as the predicate, and the judgment as a declaration or denial of identity between this and the related subject. Accordingly, all logical relation is affirmed to be reducible to identity, A is A, to non- contradiction, Nothing both A and not-A, and to excluded middle, Everything either A or not-A. These three principles, it is affirmed, dictate all the forms of inference, and evolve all the canons of syllogism, I am not prepared to deny the truth of either of these propositions, at least when A is not self-contradictory, but I cannot see how, alone, they are competent to the functions assigned. I see that they distinguish truth from falsehood: but I do not see that they, again alone, either distinguish or evolve one truth from another. Every trans- gression of these laws is an invalid inference: every valid inference is not a transgression of these laws. But I cannot admit that every thing which is not a transgression of these laws is a valid inference. And I cannot make out how just the only propositions which are true of all things conceivable can be or lead to any distinction between one thing and another. I believe these three principles to be of the soil, and not of the seed, though the seed may * The old riddle-books often propound the following query :—If Dick’s father be Tom’s son, what relation is Dick to Tom? When a boy, I heard the following classical and Protestant version of the puzzle, over which I haye since made grown persons ponder, not always with success. An abbess observed that an elderly nun was often visited by a young gentleman, and asked what relation he was, ‘‘ A very near relation,’’ answered the nun; “his mother was my mother’s Vou. X. Parr II. only child:”” which answer, as was intended, satisfied the ab- bess that the visitor must be within the unprohibited degrees, without giving precise information. When this is proposed, the first answer often is, He was her grandchild; and if the story did not say that the visitor was young, he would sometimes be taken for her grandfather; the matter not preventing, @@~! might as well be mistaken into φ- φτ-ὶ as into φφ. 43 990 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, possess some materials of the soil; of the foundation, not of the building, though the bricks may partake of the nature of the foundation; of the rails, not of the locomotive, though both may have iron in their structure. The canons of ordinary syllogism cannot be established without help from our knowledge of the convertible and transitive character of identification: that is, we must know and use the properties ‘A is B gives B is A’ and ‘A is B and B is C, compounded, give A is C.’ Can these principles be established by concession of ‘A is A, nothing is both A and not-A, and every thing is one or the other’? All my attempts at such establishment end in begging * the question, when closely scrutinised. The logicians do not make their deduction in perfectly precise and formal method, so that a lapse may be clearly pointed out. I suspect that the use of convertibility and transitiveness actually takes place, and must take place, in every attempt to deduce the legitimacy of the two laws, as necessary consequences of the three laws: and if my suspicion be correct, it follows that the two principles must be assumed independently of the three. I cannot argue the question until I find some more precise attempt to maintain the assertion: I suspect that ‘it is as plain as that A is A’ has been confounded with ‘it is true becawse A is A’ In the consideration of the proposition, identification of objects is in truth a relation of concepts. In the ordinary books on logic, the relation before the mind is confusedly mixed up with the judgment, the assertion or denial of the relation. The word is has two different meanings: standing alone, it means identity affirmed; in the phrase is not, it means only identity. I claim to recognise the distinction between relation and judgment, and to assign to each notion its own symbol. Let X and Y be terms, and L a relation in which X may or may not stand to Y, let X.. LY signify, the assertion of the relation, and ΧΟ, LY its denial, This separation of relation and judgment is an important step towards the treat- ment of syllogistic inference as an act of combination of relation; as also towards the knowledge that the ordinary canons of syllogism do actually embrace every case in which one relation only is used, and that relation transitive and convertible. That all analysis of thought should be confined to expression under one class of re- lations is the defence of a system formed under limited views, and a defence which nothing but necessity could have originated. It is the great principle of pebbles invented for justification of arithmeticians who have never got beyond pebbles. Pure arithmetic, dealing with nothing but the notion of number, has all its processes reducible of course to making number more or less. The solution of a cubict equation to 153 figures is within the reach * It is not lawful to employ syllogism in deducing syllo- | characters of the copula, these characters cannot be themselves gism from postulates which are affirmed to necessitate it: for inferred: consequently, unless non-inferentially and immedi- if all syllogism be invalid—and whether or no is the question— ately seen in the three principles, they must be adopted on it may establish i¢self on any basis. The quadrature of the | their own security. The moment this is done, the whole of circle may be deduced from the Habeas Corpus Act by a the common syllogism must be admitted under the extension method which contains only one paralogism. I have heard | to every copula which is both transitive and convertible ; for logic called the science which demonstrates demonstration: | transitiveness and convertibility once separated from the three it only analyses demonstration. So surely as no system of principles of identification, and standing on their own footing, truths can be established upon no truth to begin with, so the restriction of the copula to the identifying verb ‘is,’ no surely can no methods of fransition or inference be estab- | matter how many its senses, is only arbitrary and lawless dis- lished without methods of inference to start with. If then the | tinction. very earliest demand the use of the transitive and convertible + “If the curiosity of any gentleman that has leisure’’ to AND ON THE LOGIC OF RELATIONS. Bor of a calculator who has enough of calculi, life, and patience. And number is defined by the more or less of counting which has taken place in its formation; further counting onwards is the process required in addition; counting backwards is the process required in sub- traction: and to these all other processes can be reduced. The last unit, or item* of numeration, tells the result of all that has been done. Suppose any one to contend that arithmetic should never transcend pure counting, and he would be a faithful imitator of The arithmetician I have supposed should argue from the fundamental character of the counting argument about logic, as not infrequently expressed, and always implicitly maintained. process: he should leave practice and progress out of sight, should refuse to allow the possibility of abstractions which might end in the differential calculus, and should contend for the pure form of arithmetical thought. Every merchant’s clerk would laugh at his book of arithmetic, and would be joined by every speculator on that theory of numbers at which he could never arrive. But our arithmetician should stand firm upon the fact that men naturally count on their fingers. And though those who count on the fingers do not want him, and those who can do better will not have him, he can retire within himself, satisfied that he is the true philo- sopher of arithmetic, and the sole depository of the science. And, all unreasonable as he For it is the truth that 4ll arithmetical result can be obtained by counters: it is mot the truth that all inference can be obtained is, he would be more reasonable than the logician. by ordinary syllogism, in which the terms of the conclusion must be terms of the premises. Tf any one will by such syllogism prove that because every man is an animal, therefore every head of a man is a head of an animal, I shall be ready to—set him another question. When the logician contends that a syllogism which is not onymatic can be reduced to one which is, he always proceeds by a statement of the combination of relations, for his use Halley’s words when inviting to the calculation of the logarithms of all prime numbers under 100,000 to 25 or 30 places of figures, ‘“‘should prompt him to undertake’? to verify this assertion, he ought to find the following as the solu- tion of the celebrated equation 27—2x—5=0. I will not say, with Halley, ‘I can assure him that the facility of this method will invite him thereto.’” # = 309455 14815 42326 59148 23865 40579 30296 38573 06105 62623 91803 04128 52904 53121 89983 48366 71462 67281 77715 77578 60839 52118 90629 63459 84514 03984 20812 82370 08437 22349 91 This result, which will place the power of Horner’s me- thod in its proper light of evidence, was calculated in 1850 by my pupil Mr John Power Hicks, since of Lincoln College, Oxford, and has not been published till now. A hundred places had previously been calculated by another pupil, Mr William Harris Johnston (Mathematician, Vol. 111. p. 289) whose solution was unknown to Mr Hicks. Neither solution was merely numerical exercise; both were performed upon a knowledge of, and by incitement of, the tardiness of mathema- ticians, as well abroad’ as at home, in recognising the true place of Horner’s discovery in fundamental arithmetical ope- ration. ° * In my last paper I criticised the phraseology of logicians when they say that the difference between one and another individual of the same species is mwmerical. An able de- fender referred me to the Greek original of the phrase: in Porphyry, &c., things which, being different, do not differ εἴδει, differ ἀριθμῷ. My thorough conviction that the Greeks never altered the vernacular in scientific terms led me to an examination of the word ἀριθμός, the results of which appear in the Transactions of the Philological Society for 1859. The original meaning of ἀριθμός, never lost, though soon associated with the secondary sense of total, is the item of enumeration, the unit of a collection, which standing alone would be μονάς. Thus Aristotle, (Metaphysics, book xi. or xii.) speaking of the primary meaning, affirms that μονάς and ἀριθμός do not differ in quantity. When an ἀριθμός was spoken of as large, the departure from the original meaning is precisely that which takes place in our own language when a sale is said to be made at a high figure, meaning much money to count. The word swm gives occasion to similar remarks. Summa and sum meant number indicated by the highest wnit of counting: neither had reference to addition more than to subtraction, number to be subtracted being also swm and summa deducibilis, The totum was summa totalis, and sum total still remains in use, sounding like tautology: but the fact is that fotalis, when it dropped off, left its meaning fixed in swmma. The school-word for arithmetic, swmming, is not a derivation from the Jeading rule, addition, but means, or meant, numbering generally. Logicians would speak more to the purpose, in English, if they substituted monadical for numerical: nothing can make a numerical difference to an English ear except a difference of numerical quantity. 43—2 998 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, major, and an assertion that the relations so introduced into his principium exist in the evemplum before him, for his minor. But though this evasion—it is nothing else—is practised, and serves to hide the insufficiency of the onymatic syllogism, it is not distinctly When I first challenged the reduction to an Aris- totelian syllogism of the inference that some must have both coats and waistcoats if most proclaimed, and universally applied. have coats and most have waistcoats, I supposed that among the attempts to answer would be the following:—* Two terms each of which has more than half the extent of a third term are terms which have some common extent; the men who have coats and the men who have waistcoats are two terms each of which has more than half the extent of a third term; therefore the men who have coats and the men who have waistcoats are terms which have some common extent.’ But this was not brought forward: though it had as much right to appear Reid denied that ‘A = B, B=C, therefore A = Οὐ is a (common) syllo- True, says one able expounder, because it is elliptical: true, says another, because it as the following. gism, is material. Both render it into what they call true logical form as follows :——Things equal to the same are things equal to one another; A and B are things equal to the same; therefore A and B are things equal to one another. I pass over the assertion that A = B &c. is an ellipsis of this last, as not worth answer: the imputed material character requires further consideration. When it shall be clearly pointed out, by definite precept and sufficiently copious ex- ample, what the logicians really mean by the distinction of form and matter, I may be able Dr Thomson (Outlines, &e., § 15) remarks that they seldom or never talk much about the distinction without * con- to deal with the question more definitely than I can do at this time. fusion. I can but ask what is that notion of form as opposed to matter on which it can be denied that ‘A =B,B=C, .. A=C’ is as pure a form of thought, apart from matter, as ‘A is B, B is C,.. A is C.’ In both there is matter implied in A, B, C: but in both this matter is vague, all that is definite being the sameness of the matter of A, ἃς. in all places in which the symbol occurs. In both there is a law of thought appealed to on primary subjective testimony of consciousness; ‘equal of equal is equal’ in the first ; ‘identical of identical is identical’ in the second. These two laws are equally necessary, equally self-evident, equally incapable of demonstration out of more simple elements. Does * Because there really is not much to talk about: the sepa- ration is soon conceived, and soon made; and the work begins when, after separation, the analysis of the things separated is attempted. There is much detail in cookery, much in shoe- making, if we start from the raw flesh and the raw hide. The separation of these parts of the animal is easily seen to be wanted and easily made; any very great talk about it can have no effect, unless it be to give a chance of leather steaks and beef shoes. One of the oldest of the schoolmen, John of Sa- lisbury—whose date may be remembered by the record that tacitus, sed merens, continuo se subduxit, when 'Thomas-a- Becket was killed by his side—says nearly as much as need be said, as follows:—‘ At qui lineam, aut superficiem attendit sine corpore, formam utique contemplationis oculo a materia desjungit: cum tamen sine materia forma esse non possit. Non tamen formam sine materia esse abstrahens hic concipit intellectus (compositus enim esset) sed simpliciter alterum sine altero, cum tamen sine altero esse non possit, intuetur. Nec hoc quidem simplicitati ejus prajudicat, sed eo simpli- cior est, quo simpliciora, sine aliorum admixtione, perspicit singulatim. Hoc autem nature rerum non adversatur, que ad sui investigationem hanc potestatem contulit intellectui, ut possit conjuncta disjungere, et desjuncta conjungere.’’ (Meta- logicus, Lib. 11. cap. 20). Add to this illustration from the original meaning of the terms the extension of the words matter and form to any distinction between the quod se habet and the modus se habendi, as also to the distinction of ope- ration and operated on, and the two words may then take leave of each other. But when form and matter are to be adapted to the defence of the existing mode of distinction, it is πὸ wonder if they must be hammered until the anvil is hot. AND ON THE LOGIC OF RELATIONS. 339 the very notion of equation demand the identity of A and A to be conceded? just as much does the very notion of identification demand the equality of A and A to be conceded. We can think of nothing but what has some attributes which have quantity: and the very notion of identity, demanding identity of all attributes, demands equality of quantity in those which have quantity. On what definition, then, of form is ‘equal of equal is equal’ declared material, while ‘identical of identical is identical’ is declared formal ? In choosing the instance of equality, a very near relation of identity, I am rendering but a poor account of my own thesis. I maintain that there is no purely and entirely formal proposition except this: ——‘ There is the probability a that X is in the relation L to YY Accordingly, I hold that the copula is as much materialised, when for L we read identity, as when for L we read grandfather. The mere notion of materiality, like that of quantity (see my last paper), non suscipit magis et minus. And I hold the supreme form of syllo- gism of one middle term to be as follows;—There is the probability a that X is in relation L to Y; there is the probability 6 that Y is in relation M to Z; whence there is the probability a@ that X has been proved in these premises to be in relation L of M to Z. Here is nothing but formal representation, that is, expression of form without particular specification of matter. I now proceed to something of a less controversial character. Any two objects of thought brought together by the mind, and thought together in one act of thought, are in relation. Should any one deny this by producing two notions of which he defies me to state the relation, I tell him that he has stated it himself: he has made me think the notions in the relation of alleged impossibility of relation ; and has made his own objection commit suicide. Two thoughts cannot be brought together in thought except by a thought; which last thought contains their relation, All our prepositions express relation, and indeed all our junctions of words: but the preposition of is the only word of which we can say that it is, or may be made, a part of the expression of every relation; though the same thing may nearly be said of the preposition ¢o. When relation creates a noun substantive, of is unavoidable: if A by its relation to B be Ὁ, it is a C of B. A volume might be written on the idiom of relation: but it would be of the matter, not of the form, of the subject. 1 add a few desultory remarks, because some readers would hardly, from the symbols themselves, form a notion of the wide extent of thought which the symbols embrace. When two notions are components in one compound, as white and ball in the phrase white ball, we have one of the many cases in which the relation is not made prominent, and the compound, as a whole, is the notion on which thought fixes. So little is the relation thought of that its introduction may produce unusual idioms. In speaking of the appurtenance of white to ball, we have the whiteness of the ball, which is idiomatic: but in speaking of the appurtenance of the ball to the white, we have the rotundity of the white, which is not familiar, though intelligible. Here we are sensible of a difficulty which usage puts in the way of logic: language hesitates at realising notions which are not objectively called things. The metaphysical distinction of the ball being a substance, of which the whiteness is an inherent accident, is extralogical: all we have to do with is the junction in one notion of matter, roundness, and whiteness. Whether whiteness and 940 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, rotundity were given to matter, or material and whiteness to rotundity, is of no account: the turner can do only the first, the thinker can do either. The notion of metaphysical or physical order of precedence in the entrance of components, dictates the exclusion of forms of language which are necessary to logical precision. We may think of a horse, and then of the attributes swift or slow: we speak of the speed of the horse, correctly expressing what we have in thought as related by appurtenance to the animal. But we never speak of the horseness of the speed: do we ever think of it? Suppose a horse going a hundred miles an hour : such a thing was never known, Suppose one which goes a million of miles in a second : perhaps this is the first time such a thing was ever heard of. In the first case the speed attributed to the horse is no marvel: in the second case it is not in nature, that we know of. We object to both rates, as predicated of a horse: but to the first rate only as so predicated. That is, it is not the velocity of the horse, but the equinity of the velocity, that strikes us as unprecedented when we speak of a hundred miles an hour: and the logician may use his privilege of making language for every distinction which exists in thought. Relations of appurtenance, and indeed all others, carry with them distinctions of which grammar takes no cognizance: they give time or tense, for example, to nouns. That which hangs in the butcher’s shop under the name of a calf’s head, hangs under that name with perfect propriety : but the noun has a past tense. I am not sure that we should have been so well off as we are if philosophers had invented our language: it may have been that in such a case we should have had less sense and no poetry: but assuredly our nouns would have had moods indicative and potential, as well as tenses, past, present and future. The relation in a compound notion sometimes seeks emergence; and the word 9f demands entrance. When we hear that ‘it was the most bloody battle,” we feel an unfinished sentence: what of? the Peloponnesian war? the Peninsular war? ἕο. If not one of these a separation is wanted which may throw into notice the relation of appurtenance; ‘it was the most bloody of battles.’ Indefinite extension of one component is a bar to the conception of relation, and tends to fix thought upon the whole compound. Thus in si sheep, the relation of sia to sheep is almost dormant, so long as the selective and separative force of six is applied to all possible sheep. Make the collection more definite, and the relation demands expression: six of the sheep, six of his sheep. Not that siv of sheep is unintelligible: and, on the other hand, six his sheep is a form not unknown in old English. Largeness of selection, totality, has the effect of destroying the relating preposition: thus all his sons is as admissible as all of his sons. But let the expression of completeness be retarded ever so little, and the relating preposition demands entrance. We do not say ‘All of men are animals :°’ but we do say, ‘Of men, all are animals.’ The habits of thought of a nation silently accomplish many changes which we call caprices of language. Our modern forms of thought tend to sharpen specification of relation, especially in distinguishing agency from other relations. We no more hear of a person forsaken of his friends; it is now always by. Neither does the active participle bear the expression of relation, except as a vulgarism: squires and hounds are no longer catching of foxes. AND ON THE LOGIC OF RELATIONS. 341 now proceed to consider the formal laws of relation, so far as is necessary for the treatment of the syllogism. Let the names X, Y, Z, be singular: not only will this be sufficient when class is considered as a unit, but it will be easy to extend conclusions to quantified propositions. I do not use the mathematical symbols of functional relation, , Ψ: &e.: there are more reasons than one why mathematical examples are not well suited for illustration. 'The most apposite instances are taken from the relations between human beings: among which the relations which have almost monopolized the name, those of consanguinity and affinity, are conspicuously convenient, as being in daily use. Just as in ordinary logic ewistence is implicitly predicated for all the terms, so in this subject every relation employed will be considered as actually connecting the terms of which it is predicated. Let X.LY signify that X is some one of the objects of thought which stand to Y in the relation L, or is one of the Ls of Y. Let X.LY signify that X is not any one of the Ls of Y. Here X and Y are subject and predicate : these names having reference to the mode of entrance in the relation, not to order of mention. Thus Y is the predicate in LY.X, as well asin X.LY. When the predicate is itself the subject of a relation, there may * be a composition: thus if X..L(MY), if X be one of the Ls of one of the Ms of Y, we may think of X as an ‘L of Μ᾿ of Y, expressed by X.(LM)Y, or simply by X..LMY. A wider treatment of the subject would make it necessary to effect a symbolic distinction between ‘X is not any L of any M of Y’ and ‘X is not any L of some of the Ms of Y. For my present purpose this is not necessary: so that X.LMY may denote the first of the two. Neither do I at present find it necessary to use relations which are aggregates of other relations: as in X..(L,M)Y, X is either one of the Ls of Y or one of the Ms, or both. We cannot proceed further without attention to forms in which wniversal quantity is an inherent part of the compound relation, as belonging to the notion of the relation itself, intelligible in the compound, unintelligible in the separated component. First, let LM’ signifyt an L of every M, LM’X being an individual in the same relation to many. Here the accent is a sign of universal quantity which forms part of the description of the relation: LM’ is not an aggregate of cases of LM, Next let LM signify an L of an M in every way in which it is an L at all: an L of none but Ms. Here the accent is also a sign of universal quantity: and logic seems to dictate to grammar that this should be read ‘an every-L of M.’ The dictation however is of * A mathematician may raise a moment’s question as to | existence. In this particular the compound relation ‘L of whether L and M are properly said to be compounded in the | M’ classes with the compound term ‘ both X and Y.’ sense in which X and Y are said to be compounded in the + Simple as the connexion with the rest of what I now pro- term XY. In the phrase brother of parent, are brother and | ceed to may appear, it was long before the quantified relation parent compounded in the same manner as white and ball in | suggested itself, and until this suggestion arrived, all my efforts the term white ball. I hold the affirmative, so far as concerns | to make a scheme of syllogism were wholly unsuccessful. The the distinction between composition and aggregation : not de- | quantity was in my mind, but not carried to the account of nying the essential distinction between relation and attribute. | relation. Thus LX)) MY, or every L of X is an M of Y, According to the conceptions by which man and drute are | has the notion of universal quantity attached in the common aggregated in animal, while animal and reason are com- | way to LX, not to L: its equivalents X..L-'!.MY, and pounded in man, one primary feature of the distinction is that | Y..M-! ΠΧ, shew X and Y as singular terms, and though an impossible component puts the compound out of existence, | expressing the same ideas of quantity as LX )) MY, throw an impossible aggregant does not put the aggregate out of | the quantity entirely into the description of the relations. 949 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, convenience; not of obligation, as in the case of the double negative. Either some horse The Greek* idiom refused this dilemma. There is no scrape that man does not get into: if we had no other way of knowing or no horse; if not no horse, then some horse. this, we have the assurance of Euripides; but he informs us that there is not no scrape The educated English idiom follows logic, which here Such a phrase as the ‘every uncle of a sailor’ has no meaning except in poetry, where it means the sole uncle. It would be highly convenient if the distinction between LM’ and LM could be made as in ‘L of every Μ᾽ and ‘ every-L of M.’ The symbols L’MX and LM,X, which I shall not need, analogically interpreted would mean ‘every L of an M of X’ and ‘an L of an M of none but X.’ The com- pound symbol 1, Μ'Χ means an L of every M of X and of nothing else; and is really the compound (LM'X) (1, Δ Χ). No further notice will be taken of it. We have thus three symbols of compound relation; LM, an L of an M; LM’, an L of every M; LM, an L of none but Ms. No other compounds will be needed in syllogism, until the premises themselves contain compound relations. that man does not get into. commands, In every case in which there is a first and a second, let the first be minor, the second, Thus if X..LMY, let X and Y be its minor and major terms, and L and M its minor and major relations: if it be the jirs¢ premise of a syllogism let it be the major. minor premise. The converse relation of L, L7', is defined as usual: if X..LY, Y..L7'X: if X be one of the Ls of Y, Y is one of the L~'s of X. And L-'X may be read ‘L-verse of X.’ Those who dislike the mathematical symbol in L~* might write L". This language would be very convenient in mathematics: @~'x might be the ‘ @-verse of xv, read as ‘ @-verse a.’ If X be not any L of Y, X is to Y in some not-L relation: let this contrary relation be signified+ by 1; thus Relations are assumed to exist between any two terms whatsoever. X.LY gives and is given by X..1Y. Contrary relations may be compounded, though con- trary terms cannot: Xx, both X and not-X, is impossible; but Llx, the L of a not-L of X, is conceivable. Thus a man may be the partisan of a non-partisan of X. Contraries of converses are converses: thus not-L and not-L~' are converses. For X..LY and Y..L-'X are identical; whence X..not-LY and Y..(not-L~') X, their simple denials, are identical; whence not-L and not-L~ are converses. * It would be worth the while of some one who has the requisite scholarship to examine the question how far the negatory power of the double negative in Greek determined the course of Aristotle in regard to privative terms. In further reference to the dictating power of logic, I may observe that it does not go far: forms cannot dictate meaning to any but a very small extent. For instance: It is almost universal, but not guite, that transference of not from the copula to the predicate produces no change of meaning. ‘He either will do it, or he will-not do it’? means the same as ‘He either will do it, or he will not-do it;’ and the two of each set are alternatives. But ‘He either can do it, or he cannot do it’ has not identity of meaning with ‘He either can do it or he can not-do it:’ the first pair are repugnant alternatives, the second are not: the same person who can do it, usually can not-do it, or can let it alone, but not always. Again, the junction of nof to the verb usually gives a contrary, or a re- pugnant alternative: he eats or he eats not, he has or he has not, he does or he does not. But we may not say, Either he must, or he must not; these are no necessary alternatives; we can only say, Either he must, or he need not, Either he must not, or he may. Many similar instances might be given. + The affirmative symbol (..) is derived from the junction of the two negatives (.)(.). Analogy would seem to require that the privative relation not-L should be denoted by (.L). Or thus:—Let W denote the affirmation, and V the denial: then XWLY would denote that X isan L of Y, and XVVLY that X is not a not-L of Y. ButI donot at present find advan- tage in a notation which expresses X.. LY and its equivalent X.1Y in one symbol: I may possibly do so at a future time. AND ON THE LOGIC OF RELATIONS. 343 Converses of contraries are contraries: thus L7! and (not-L)~! are contraries. For since X.. LY and X..not-LY are simple denials of each other, so are their converses Y.. L~'X and Y,,(not-L)-'X; whence L~ and (not-L)~! are contraries. The contrary of a converse is the converse of the contrary: not-L~' is (not-L)~'. For X..LY is identical with Y,not-L7!X and with X,(not-L)Y, which is also identical with Y.(not-L)“'X. Hence the term not-L-verse is unambiguous in meaning, though ambiguous in form. If a first relation be contained in a second, then the converse of the first is contained in the converse of the second: but the contrary of the second in the contrary of the /irst. The conversion of a compound relation converts both components, and inverts their order. If X be an L of an M of Y, then an M of Y is an L~ of X, and Y is an Μ΄ of an L7! of X. Or (LM)! is M~'L~*. The mark of inherent quantity is also changed in place. If X be* an L of every M of Y, then Y is an M7! of none but L~'s of X. And if X be an L of none but Ms of Y, then Y is an Μ΄! of every L7! of X. For X..LM’Y is MY ))L7'X or YeMa'L-'X: and X..L MY is L7}X))MY or Y..Mo'Ls" X, When there is a sign of inherent quantity, if each component be changed into its con- trary, and the sign of quantity be shifted from one component to the other, there is no change in the meaning of the symbol. Thus an L of every M is a not-L of none but not-Ms; and vice versa: and an L of none but Ms is a not-L of every not-M ; and vice versa, When a compound has no inherent quantity, the contrary is found by taking the con- trary of either component, and giving inherent quantity to the other, Thus, either L of an M or not-L of every M: either L of an M or L of none but not-Ms. But if there be a sign of inherent quantity in one component, the contrary is taken by dropping that sign, and Thus, either L of every M or not-L of an M; taking the contrary of the other component. either L of none but Ms, or L of a not-M. The following table contains Ὁ all these theorems: | Converse of contrary Combination Conyerse Contrary | GButsary ΘΕ ΟΠ ΣΟ LM ML" 1M’ or Lm ΜΙ or τη Ἐν | LM’ orlm | ML“ or m1" IM Mr" | LMorlm’ Μ’Ὶ," οὐ πα" Lm | myo ee a eee ee * A good instance of the difficulty of abstract propositions: | it is easy enough on a concrete instance. If X be the superior of every ancestor of Y, then Y is the descendant of none but inferiors of X. 2 + The resultant relation in onymatic syllogism is identical with the compound from which it results. Thus (.)).) is (), identically: every complement of a deficient is a partient ; every partient is a complement of a deficient. The contraries then are identical: and this gives the key to the resulting meaning of quantified compound relations: as (()), a genus Voto, Barn iW: of none but species; or (())’, a genus of every species. The complete rule of interpretation of such symbols is as follows. Reject as incapable of meaning all cases in which two uni- versals or two particulars have different middle quantities, or in which a universal and particular have the accent upon the particular. Thus there is no such thing as )) ))', or (.( ()’5 or (( ()': a species of every species of a given genus is non- existing. In all other cases, invert the spicula nearest to the accent, erase the middle spicule, and the result shows the relation identical with the given compound. Thus (.) (.)’ 44 344 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, If a compound relation be contained in another relation, by the nature of the relations and not by casualty of the predicate, the same may be said when either component is con- verted, and the contrary of the other component and of the compound change places. That is if, be Z what it may, every L of M of Zbe an N of Z, say LM))N, then L~'n))m, and nM-'))l. If LM))N, then n))IM’ and nM-'))IM’M-!. But an | of every M of an M-! of Z must be anl of Z: hence nM~'))1. Again, if LM))N, then n)) Lm, whence L-'n))L~Lm. But an L-' of an L of none but ms of Z must be an m of Z; whence L-‘n )) m. I shall call this result theorem K, in remembrance of the office of that letter in Baroko and Bokardo: it is the theorem on which the formation of what I called opponent syllogisms is founded. For example, the combination in one of the mathematical* syllogisms is Every deficient of an external is a coinadequate: external and coinadequate have partient and com- plement for their contraries, and deficient has exient for its converse: hence every exient of a complement is a partient; which is one of the opponent syllogisms of that first given. Identity, in theorem K, does not give identity; as will be observed by watching the demonstration. For an instance, brother of parent is identical with uncle, by mere defi- nition. But non-unele of child is not identical with non-brother: for though every non- uncle of child is non-brother (as by the theorem), yet it is not true that every non-brother is non-uncle of child. If LM be identical with N, meaning that N is an L of an M and of no other signifi- cation, we have LM || N, LMM-!||NM"!, ΤῊ ΜΙ} ΠΝ. Now MM-!X and L71LX are classes which contain X; so that we may affirm L))NM~! and M))L™!N; but not L ΝΜ’! nor M|| L-'N. Having given LM||N, it is natural to ask whether we can deduce identical expressions for L and M: the answer is that no such thing can be done. If by M we mean some one particular M left vague, the form of L can be deduced, as we shall see; but not when N is a name for, and only for, every L of every M. Take, for example, the word uncle: it is identical with brother of parent; either is the other. Can we now construct a definition of brother out of wnele, parent, and their converses. Uncle of child of X is no definition of brother of X; it includes the brothers of the other parent. Unele of every child of X will not do, for a similar reason: if X had as many wives as Solomon, and children by all, nothing in logic excludes the supposition that they were all sisters. In mathematics we have much power of forcing NM~'||L out of LM||N by extension of language: and in a science of truths necessary as to matter it is almost a proof of insuf- ficient grasp when we find either of the forms above unaccompanied by the other. Common gives (..( or ((: a complement of every complement is a * In arithmetical form thus:—Some Ys are not any Xs, genus ; and vice versa. Again, (.) (( gives ).(: a complement | no Y is Z, therefore some things are neither Xs nor Zs. Deny of none but genera is an external, ὥς. Also, (()( gives )(: a | the conclusion, affirm the first premise, and we may deny the genus of none but coinadequates is a coinadequate, &c. A | second, which gives some Ys are not any Xs, everything is defective account of these transformations will be found in my | either X or Z, therefore some Ys are Zs. third paper. AND ON THE LOGIC OF RELATIONS. 345 language has some degree of tendency towards the same sort of enlargement of meanings. As in the very example before us: the brothers of the other parent are called brothers (in law): and under this extension uncle of child is identical with brother: for the word uncle receives similar extensions. A relation is said to be convertible (though it should rather be said that the subject and predicate are convertible) when it is its own converse; when X..LY gives Y..LX, And, L being any relation whatever, LL~! is convertible: but LL~!’ and L,L7! are each the converse of the other. So far as I can see, every convertible relation can be reduced to the form LL-!. If two notions stand in the same relation to one another, they can always, I think, be made to stand in one and the same relation to some third notion. The converse is certainly true, namely, that two notions which stand in one relation to a third, stand in con- vertible relation ‘to each other. But it cannot be proved that if X..LY and Y..LX, then L must be reducible to MM~!, for some meaning or other of M: this is certainly a material proposition. But I can find no case in which material proof fails. Take identity, for ex- ample: it is the very notion of identity between X and Y that X..LL~'Y for every possible relation L in which X can stand to any third notion. Identification of objects of thought by names derives its convertibility from the idea of the names standing in relation of appli- cability to the same object. Identification in thought of unnamed objects can only be con- ceived as convertible by reference, as above, to other notions. Exclude names, and identify X with itself by ‘this is this:’ it would be absurd to repeat the process, and say that there is conversion by reason of the first this of one indication being the second this of the other: such conversion would be only the invention of different names spelt the same way. Among the subjects of a convertible relation must usually come the predicate itself, unless it be forced out by express convention. If all convertible relation can be expressed by ΠῚ, Ὶ this is obviously necessary: for ΠΤ, ΠΧ includes X. Is a man his own brother ? It is commonly not so held: but we cannot make a definition which shall by its own power exclude him, unless under a clause expressly framed for the purpose. Is a brother the son* of the same father and mother with the man himself? Then is he pre-eminently his own brother: for there never lived one of whom we have not more reason to be sure that he was the son of his own father and mother than that any reputed brother had the same parents with him, If we want to exclude him we must stipulate that by brother of X we mean any man who, not being X himself, has the same father and mother. In common language the stipulation is or is not made, according to the casual presence or absence of the necessity for it. Put the question what relation to a man is his brother’s brother, and most persons will answer, His brother: point out that the answer should be, Either his brother or himself, and * When the individual is but one among many, and is | Jn magno quidem periculo versari video existimationem meam, speaking generally of his class, there is an implication that | quia geometris fere omnibus dissentio. Eorum enim qui de others are intended, and the introduction of self produces for | iisdem rebus mecum aliquid ediderunt, aut solus insanio ego a moment that sense of incongruity which, if it could be made | aud solus non insanio ; tertiwm enim non est, nisi (quod dicet to last, would give an air of humour. Thus Hobbes, in a | forte aliquis) insaniamus omnes. Undoubtedly a man is sentence which, altering geometris into Jogicis, might be said | among those who have written on the same subjects with him- ot himself by a person I ought to know, speaks as follows :— | self. 442 346 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, a fair proportion will think that himself was included. I shall hold, for logical purposes, that the predicate is included among its own convertible relatives. A relation is transitive when a relative of a relative is a relative of the same kind; as symbolised in LL )) L, whence LLL )) LL )) L; and so on. A transitive relation has a transitive converse, but not necessarily a transitive contrary: for 1,-1,-} is the converse of LL, so that LL))L gives L-'L-'))L-%. From these, by contraposition, and also by theorem K and its contrapositions, we obtain the following results. L is contained in LL~”, 11", ΓΝ LL LL is contained in L Brsige, inp os Πρ ea al Ueto be L"! Lire Loe | oe ee 1 Fd troche LA eb’ LEY I'L ον 1: I omit demonstration, but to prevent any doubt about correctness of printing I subjoin instances in words: L signifies ancestor and L~! descendant. An ancestor is always an ancestor of all descendants, a non-ancestor of none but non- descendants, a non-descendant of all non-ancestors, and a descendant of none but ancestors. A descendant is always an ancestor of none but descendants, a non-ancestor of all non- descendants, a non-descendant of none but non-ancestors, and a descendant of all ancestors. A non-ancestor is always a non-ancestor of all ancestors, and an ancestor of none but non- ancestors. A non-descendant is a descendant of none but non-descendants, and a non- descendant of all descendants. Among non-ancestors are contained all descendants of non- ancestors, and all non-ancestors of descendants. Among non-descendants are contained all ancestors of non-descendants, and all non-descendants of ancestors. The mathematician forces the predicate itself among its own chain of successive relatives, whether the relation be transitive or not: a, as @°x, appears in the sequence...p ‘a, @-'a, pe, pe, gz,... There isa little tendency towards the same thing in ordinary language, especially when the relation is transitive. Milton, in calling Eve “ the fairest of her daughters,” meaning female descendants in general, allowed ¢°x to be a case of px. Nothing but circumlocution avoids the same thing in our day, and by it language loses much force, or some precision. If we say that Achilles was the strongest of all his enemies, we feel both definite meaning and force: if we say that he was stronger than any one of his enemies, we gain an enfeebling addition of logical accuracy: if that he was stronger than all his enemies, we introduce ambiguity. I now proceed to the syllogism, taking first the case in which the terms are individual notions, units of thought. All syllogisms of second intention, whether mathematical or metaphysical, come under this case; and arithmetical syllogisms are but aggregates of singular syllogisms, each of which also comes under this case. The supreme law of syllogism of three terms, the law which governs every possible case, and to which every variety of expression must be brought before inference can be made, is * this ;—any relation of X to Y compounded with any relation of Y to Z gives a relation of X to Z. This is very nearly the wording of Euclid’s implied definition of compound ratio of magnitudes ;—The ratio of X to Z is compounded of the ratios of X to Y and AND ON THE LOGIC OF RELATIONS. 347 Y to Z. If I had now produced this principle for the first time, and in the present manner, it would surely have been imputed to me that I had made a fanciful definition of syllogism out of a mathematical analogy. But my second paper will bear witness that I enunciated the identity of inference with combination of relation at a time when I had not noted the extreme closeness of the analogy. For when I in that paper remarked that the generality of the notion of composition (of ratio) prevented the Greek geometers from needing to make separate treatment of decomposition, I made no allusion (having in truth none to make) to the analogous point of syllogism. But if I had generalised the mathematical notion, from the Greek, the process would have been both natural and valid, For ratio is no direct trans- lation of λόγος : the Greek* word means communication; and the same turn of thought which made λόγος a technical term of geometry made it stand for any relation in one of its derived meanings; that is, any way in which we talk about one notion in terms of another. Any way of speaking of one notion with respect to a second, joined with a way of speaking of the second notion with respect to a third, must dictate a way of speaking of the first notion with respect to the third. And this is syllogism: it exhibits, in the most general form, the law of thought which connects two notions by their connexions with a third. The character of the connexions belongs as much to the matter of the syllogism as the character of the terms connected. The universal and all containing form of syllogism is seen in the statement of X..LMZ is the necessary consequence of X.. LY and Y..MZ. Whether the compound relation LM be capable of presentation to thought under a form in which the components are lost in the compound—in the same manner, to use Hartley’s simile, as the odours of the separate in- gredients are not separately perceptible in the smell of the mixture—is entirely a question of matter. In the Aristotelian syllogism, figure is a function of the places of the middle term; and its necessity arises from the nature of the proposition being also a function of the places of its terms: we cannot, in that system, say ‘Every X is Y’ without having Y for the pre- dicate. Adopt Hamilton’s expressed quantifications and, as he justly remarks, figure becomes an unessential variation. Introduce the general idea of relation, and figure resumes its importance: but not as connected with the place of the middle term. Whether we say X..LY or LY..X, the figure is the same. Change of figure can be effected only by con- version of relation, The first figure is that of direct transition: X related to Z through X related to Y and Y related to Z. The fourth figure is that of inverted transition: X related to Z through Z related to Y and Y related to X. The second figure is that of reference to (the middle term): X related to Z through X and Z both related to Y. The third figure * Euclid’s definition of radio, most properly when most literally translated, is ‘‘ Communicating instrument is a habi- tude of two magnitudes of the same kind to one another with | that feature of the Greek under which the excessive curve was respect to guantuplicity.”” We talk about magnitude in terms | called excess (hyperbola), the defective curve the defect of magnitude only by how many times one contains the other. | (edlipsé), the irregular angle the irregularity (anomaly), and On the meaning of πηλικότης see the Penny Cyclopedia and | soon. Parabola is another instance, of which elsewhere. the Supplement, articles Ratio. That the communicating instrument was called communication (λόγος) was a case of “948 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, is that of reference from (the middle term): X related to Z through Y related to both X and Z. Before generalisation, whatever may be our preference for the first figure, we hardly feel inclined to admit that inference takes place in no other figure ; that is, demonstrated inference : that premises in the second figure can only yield their result by seeing the first figure through the second. Say that X is Y, Z is not Y: how do we know that X is not Z? If X be Y, it is not anything that Y is not: at this point we are immediately aided by ‘Y is not Z;’ only mediately by ‘Z isnot Y.. The ease * of the transformation prevents our feeling that it takes place. But if we take for our premises X..LY and Z..MY, the necessity of con- version of the major premise, that is, of reduction into the first figure, is sufficiently apparent: we cannot express our inference without it. In my second paper I stated that no inference can be drawn from a negative premise, except by decomposition of a relation, ‘This is perfectly true, so long as contrary relations and inherent quantity are excluded: but not true when they are admitted. The following comparison will illustrate this. Let the premises be X.LY,Z..MY. These premises are identical with X..1Y and Y..M~'Z of which all the inference is X..1M~'Z, or either of its two identicals, X.LM-"Z, X.1m-!Z. Or thus;—X is not any L of Y: of Y we know no more than that it is a certain M-' of Z; and as the M~ in question is vagum—which in English we call certain—all we can say is that X is not any L of all the M-'s of Z. Hence X.LM-"Z is all the inference we can draw. Next, let contraries be forbidden. To deduce the inference, without use of inherent quantity or of contraries, we are compelled to proceed by the old reductio ad impossibile by which Baroko and Bokardo were made to listen to reason: and this equally in all cases which contain one negative premise. Let X.LY and Z..MY give X.NZ: then X..NZ and Z..MY, conjointly, must contradict X,LY; that is, X..NMY must contradict X, LY. That is, one instance of NMY must be identical with one instance of LY, the MY in question being Z. Here N is to be determined by the decomposition of a relation, which is all that need be said until we come to the consideration of the forms of inference. In the mean time I take a concrete instance. Let it be, X is not any uncle of Y, and Z is a parent of Y. The whole inference clearly is that X is not any uncle of a particular child of Z. We also know which child: but as we throw away all reference to the middle term—the question being how much can we know when the middle term is completely eliminated—the inference is that X is not the uncle of all the children of Z. Material+ Nothing is so well adapted to exhibit the simplicity of | correlation of parent and child is not a logical but a material the first figure, as the expression of the four in common lan- | fact. It is undoubtedly material that (parent)-’ is spelt guage as follows :— | c-h-i-l-d: and the pure form of inference, independent of the 1 X..LY, Y..MZ, Xis L of M of Z. meanings of u-n-c-l-e and p-a-r-e-n-t, is that X is not any 2. X..LY, Z..MY, X is L of that of which the M | (uncle) (parent)-'’ of Z, in which all that is meant by uncle is Z. and parent is that they were the symbols we saw in the pre- 3. Y..LX, Y..MZ, X is that of which the Lis the M | mises, all the rest of the force of inference being in the mean- of Z. ing of (...) (...)-!’. Again, it is material that ‘uncle of all 4. Y..LX, Z..MY, X is that of which the L is that of | children’ is either ‘ brother,’ or ‘brother of all wives who which the M is Z. have had children:’ it is material that the universe of our Here ‘that’ and ‘the’ must be read as indefinites. propositions is strictly moral, so that brothers of paramours of + I shall certainly have to meet the old objection that the | X are not included: it is material that we hope there are no AND ON THE LOGIC OF RELATIONS. 349 knowledge, in this instance, converts the conclusion into ‘X is neither the brother of Z, nor the brother of all the wives who have had children by Z.’ Again, let the conclusion be, X is not any N of Z. If then X be N of Z, and Z parent of Y, X must be uncle of Y: that is ‘N of parent’ is ‘uncle,’ and N is to be found by decomposition. | What of parent is uncle ἢ The aggregate of brother and brother-in-law. But to say that X is neither brother nor brother-in-law would be to suppose that Y might be any child of Z: so that all we are to say is, X is neither brother, nor brother-in-law with reference to one particular child. Drop the vestige of the middle term, and we say that X is neither brother nor brother-in-law by every wife: which agrees with the preceding, The mode of decomposition may be thus generalised. Let there be one negative premise, and, L and M being the premising relations, let N be the concluding relation denied. Write down the terms of the negative premise, and between them the remaining term, choosing such order as shall make X precede Z. Combine the relations of the two pairs seen, and the combination must be the direct or converted relation in the negative premise, provided that due attention be paid to the particular character in the affirmative premise. For example, let the negative premise be Z.MY. Take Y, Z, and between them write X, YXZ; in which YX, XZ, are seen, Let the affirmative premise be X. LY: then L-' and N combine to give M~'; or N must be deduced by making ΠΝ identical with M~', a certain L~' being understood. I have had occasion to notice the manner in which, by wilful renunciation of knowledge, the conclusion is made to express not quite all the possible inference. This happens also in the common syllogism. If from ‘Some Xs are not Ys’ and ‘ Everything is either Y or Z,” we deduce ‘Some Xs are Zs’ it is not true that this conclusion embraces the whole knowledge which the premises give. It is known that ‘some Xs’ mean all that are not Ys: the vague quantity is not so vague as it would be if the conclusion were the only thing known. It ought to be noticed that a universal (Hamilton’s definite would be a better word) lurks in the conclusion of every particular syllogism: in the above X()Z would be Xy)) Z, if all that is known were expressed. The particular conclusion of a syllogism is the universal of a narrower name: one premise predicates ewistence for a new and compounded name: the conclusion substitutes that compound in the other premise in a legitimate manner. Reserving the word mood as irrevocably associated with details of quantity and quality combined, let each figure have four phases, determined by the quality only of the premises. The four phases, + meaning affirmative, and — meaning negative, are to be remembered* in the standard order ++ -+ +- -- such persons: it is material that if Z be either Whiston or Dr Primrose, the brother of ‘he wife would be sufficient. All this and more is conceded: matter makes its appearance the moment L and M mean more than ‘certain relations which, and no others, are designated by these letters throughout the syllogism.’ Again, when I am told that Logic does not pro- vide the inference that ‘ Philip was Alexander’s father’ because ‘ Alexander was Philip’s son,’ and that it is our material knowledge of the relation of father and son that enables us to make the inference, I reply that it is certainly material that father and son are related in the manner of L and L-; but that the transition from X.. LY to Y..L-X is a form of thought, and a more general form than any case of conversion admitted by the logician in the common syllogism. It is that which is common to the transitions ‘ X a genus of Y, therefore Y a species of X’, ‘Xa parent of Y, therefore Y a child of X,’ * X an identical of Y, therefore Y an iden- tical of X,’ &c.: and is therefore more abstract than any of them, and equally form without matter to all of them. * Some persons, of whom I am one, but whether it be a gift or a defect I do not know, cannot associate two things with two other things, each with each, merely by conventional 300 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, Let these be the primary phases of the four figures. The order of the phases, in the four figures, is determined by reading from the leading or primary phase, first backwards and then forwards. Thus the phases are as follows: 1 2 3 4 Figure I. ++ πὶ π- -- Π. πε ++ +- -- ΠῚ. +- -+ ++ -- IV. -- -- -+ ++ The following is the table of forms of syllogism, afterwards explained. 1 2 3 4 X--LY Kena” X.-LY IER G Y--MZ Y.-MZ Υ. MZ Υ. ΜΖ Χ..1.ΜΖ X.-IMZ X.-LmZ X. .lmZ I I Χ. IM’Z X. LW’Z X. lm’Z X. Lm’Z X- LmZ Χ. 1mZ X- L,MZ X- 1MZ LM || N NM" ||L | L“N||M | Im] N X- LY biG) O's X.-LY xX. LY Z-.-MY Z.-MY Z. MY No Arma OG X-.IM"Z | X..LM?Z| X.-Lm°Z | X..lm"Z II Il X- ΝΕ X.- IM-"’Z Χ. Im-"Z X, Lm-"Z X-Im"Z | X-LmZ | X. LM@Z| x. 1M2z NM || L LM“ ||N | L-N| M_| mm] Ν Y. {LX ay. ΤΣ Yor oy nx Y- MZ Y-.-MZ YooMe Υ. MZ ΧΙ ΗΖ | X.-PMZ | XTIEMZ | X..Pmz I τ Χ. I'm’Z Χ. 1 ΜΖ] X. MZ | X-L“m’Z i X. L7°MZ| Χ. I-mZ | X. ΙΖ ΧΙ ΜΖ LN || M NMS {τὺ GAMaEN. | λα ΠΝ Y. LX Y.-LX Wie ΤῊΧ Y.-LX Z. MY Z. MY Z.-MY Z--MY XZ |X| eZ | ΣΧ. ΜΟ IV X. L-m"Z X. -m-"Z X.L°M-’Z X. 'M-"Z IV K.1-M-Z| ΧΟΙ ΜΖ, X.1-mZ Χ. 1, Ἰὼ Mm ΠΝ} LN] M2 ΝΜΗ ΙΓ | L2M?]N 1 2 3 4 distribution, unless the association be required every day or every hour. Though I have read music for forty years, I have never known the crotchet rest from the quaver rest, except by context. These two symbols turn their heads one backwards and the other forwards, a difference which bears no imaginable relation to one standing for twice as much time as the other. AND ON THE LOGIC OF RELATIONS. 351 The Roman numerals refer to figure; the Arabic to phase. The first two lines in each compartment contain the premises. The third line contains the conclusion in affirmative form, The fourth and fifth lines In the sixth line N is the concluding relation, derived from reduction into the primary phase of the first figure. show the two forms of negative conclusion, affirmative or negative according as the premises are of similar or different qualities: and the connexion of N with the premising relations is seen, as obtained by simple composition when the premises are of the same quality, and from opponent syllogism, or from the rule above, when the premises are of different quality. When the premises are of one quality, so that N is not to be disengaged by decomposition, it is enough The sixth line is the only one which will need any detail of consideration. that N should be identical with, or should contain, the relation set down opposite to it. Thus in III, 4, the inference is that X is one l~'m of Z, or one N of Z, if only 1-'m be contained in N. The enlargement of course is a weakening of the inference, an addition of scope* and diminution of force. Let the premises, as in II. 1, be X, LY and Z.. MY, L being any one L, and M some one M. The reduction to I. 1 gives X..1Y and Y..M~-1Z, whence X ..1M~!Z is all the conclusion that can be drawn. Of this X .LM~-Z and X.1lm~’Z are equivalents. Again, if the con- clusion be X.NZ, it is clear that X.NZ and Z. MY, M being still some one M, should give X.. LY, and do give X «+ NMY, whence NM and L should be identical. NM||L, upon the condition that M is some one M, left vague, but not any one M, we see that it gives Ν᾿ LM7}). and nothing else: so that N is ΜΠ ΠῚ, where that Μ᾿ is used which is the correlative of If we examine For L is to include in its meaning any N of a certain M, the M in question. but deny LM-". This point is well illustrated by relations in which degree or quantity is conceivable. For example, let X. LY be ‘X is not an external of Y;’ and let Z..MY be " Ζ is a genus of Y’. may be as nearly the whole genus as we please, and even the whole genus itself, the only But in denying this LM~', or rather any L of this vague M-', we do The inference is that X is not an external of all species of Z. Since the species inference is that X is not an external of Z. Again, we ask what of a particular genus is an external, the genus in question being vague. If the particular genus were known, we should say that the required class is not partient of that genus to any extent except some or all of the exience of the genus: but as this exience is quite vague, possibly nothing, we can only say not partient at all or external. In both cases X.{).({Y and Z.. $(({Y gives X.{)-(}Z, or X()Y))Z gives X () Z. I shall remember it in future, having looked it up for the purpose of this note, by seeing that the crotchet rest turns its head forwards, the quaver rest backwards; and assuming that progress is worth twice-as much as retrogradation. In the case before me, the difficulty of attaching —+ and +— to the figures of which they are the primary phases may be lessened to those who are constituted like myself by remembering that the second figure is that of reference ἔθ. and that in—+ we read ¢o the chief sign +, while the third figure is that of re- ference from, and that in +— we read from the sign +. Vou. X. Parr II: * In my last paper, speaking of the world at large as rudely acquainted with the intension of a term under the name of its force, I omitted, by one of those pieces of forgetfulness which are hard to account for, to add that they are also acquainted with ewtension under the name of scope. And many cases occur in which writers choose their terms as if they felt that the greater the scope the less the force, and use them accordingly: but I cannot tind anything like a direct statement of the theorem, though I should suppose it can hardly have been missed by all writers. 45 302 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, In the same way other cases may be treated. But the entrance of contrary relations renders the method of decomposition useless for every purpose except historical com- parison. Except when both premises are negative, the conclusion can always be expressed in terms of the premising relations, without contraries. Thus among the concluding forms of III. 2, we see L~'M’. The following rules may be collected. First, the relations converted in the conclusions belong to those premises which must be converted when reduction is made into the first figure. Secondly, the mark of inherent quantity appears in the ordinary form of conclusion only when the premises are of different qualities. Thirdly, when the conclusion is expressed without contraries, this mark is always attached to the rela- tion of the affirmative premise. These rules would give mechanical canons of inference, if such things were wanted: and it would be well to remember that in the second figure the middle term usually comes second in both premises, and the second premise is converted in reduction into the first figure. I shall now proceed to the consideration of the quantified proposition and its syllogism, presuming the reader to be acquainted with the notation and classifications of my second and third papers. If we take the proposition ‘Every X is an L of one or more Ys’ we may denote it by X_)) LY: and similarly LY )) X may denote ‘ Every L of any Y or Ys is an X. And similarly for the other parts of the notation. I enter on this part of the subject only so far as to illustrate the ancient or Aristotelian syllogism. Though of necessity a part of logic, as involving possible forms and necessary connexions, the quantified syllogism of relation is not of primary importance as an explanation of actual thought: for by the time we arrive at the consideration of relation in general we are clear of all necessity for quantification. And for this reason : quantification itself only expresses a relation. Thus if we say that some Xs are connected with Ys, the relation of the class X to the class Y is that of partial connexion: that some at least, all it may be, are connected, is itself a connexion between the classes. Nevertheless, it may be useful to exhibit the modifying quantification as a component, not as inseparably thought of in the compound; though in this we must confine ourselves to what may be called the Aristotelian branch of the extended subject. If we would enter completely upon quantified forms, we must examine not only the relation and its contrary, but the relation of a term in connexion with the relation of the contrary term. And here we find that all universal connexion ceases. The repugnance of X and not-X or x, which, joined with alternance, is the notion the symbols X and x were invented to express, cannot be predicated of LX and Lx: for Y..LX and Y.Lx may coexist. The complete investigation would require subordinate notions of form, effecting subdivisions of matter. This complete examination would also require the investigation of the manner in which quantity of relation acts upon quantity of term: and this whether the quantity of relation be inherent or not; including an examination of all syllogisms in which inherent quantity of relation appears in the premises. And thus in logic, as in mathematics, the horizon opens with thes height gained: generalisation suggests detail, which again suggests generalisation, and so on ad infinitum. There is no more limit to the formule of thought than to the AND ON THE LOGIC OF RELATIONS. 358 formule of algebra. The logician may, if it please him, reduce all thought to simple assertion.or denial of identification, and the algebraist may define all his science as either vw=y or w=y+ta: one reduction is as true as the other. There is identity or difference in every possible logical judgment: there is equation or inequation in every possible alge- braical judgment. In the Aristotelian syllogism, the premising forms are X )) Y, X)(Y, X(QVY, X((Y; X being the subject and Y the predicate. The forms X((Y and X),) Y cannot appear, unless we add so much of Hamilton’s system as is seen in them. Nor can we avoid doing this here. For conversion from figure to figure is no longer conversion of order of terms. Thus LX )) Y and Y ).( MZ, do not give the first figure, but the third: there being refer- ence from the middle term in both premises; that is, the middle term being the subject of relation in both cases. In all the syllogisms which do not involve the forms (.) and )(, that is, in all which are either Aristotelian or capable of being made so by simple conversion, each premise is a congeries or aggregation of propositions involving individual notions, such as we have hitherto considered. An adequate quantification of the middle term insures the collection of a number of pairs, one out of each premise, in which the same individual from the middle term appears in both the premises: and thus the ordinary laws of dependence upon the quantities of the terms may be established. The whole of the system of relations of quantity remains undisturbed if for the common copula ‘is’ be substituted any other relation: so that the usual laws of quantity may be applied to the table of wmnit-syllogisms given above, precisely as if L and M only meant ‘is. Thus X.LY and Y..MZ giving X..1MZ or X.LM’Z, we find that X).(LY and Y (( MZ gives X).(LM’Z. In the first three figures, the pure Aristotelian modes are derived entirely from the first and third phases, and in no case from the second or fourth. But all the phases of the fourth figure give such syllogisms except the first. Every one of the thirty-two forms of onymatic syllogism may be made to give some conclusion, however the relations may be distributed: but the results are at present of infe- rior interest, for reasons already given. Thus X)) LY, MY) (Z give X)) LY, MY).)z, or X)) LY, Y).)M-'z, or X).)L.M~*z. But direct relation between X and Z is here unattainable, without reference to the matter of L and M. I now proceed one step nearer to the common syllogism, as follows. Let only one relation and its converse appear in the premises; and let this relation be transitive. That is, each relation is either Lor 1,5], and LL is L, L~'L7!is L7!. The most convenient relations from which to form instances will be ‘ ancestor’ and ‘ descendant.’ Every phase of every figure gives its conclusion; but our question will be to determine those cases in which the concluding relation is, or may be, the relation of the premises, L or L“1. We may have a larger conclusion: if so, we throw away a part of it. To illustrate this, let us examine I. 3: let X..LY be the minor premise, and let Y.L~'Z be the major. The full conclusion is X, L,L-!Z. This contains X. L7!Z: for, as before seen, when L is transitive, LL“ contains L~', Thus when X is an ancestor of Y, and Y not a descendant of Z, X is not a descendant of Z. This of course is easy: if X were a descendant of Z, 45—2 954 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, then Y, a descendant of X, would also be a descendant of Z, which he is not. But this is not all the conclusion. The full conclusion is that X is not an ancestor of none but descendants of Z: not Z himself nor herself, not his wife nor her husband, not any descendant of Z, and not the wife nor the husband of any descendant; unless, in the cases where wife or husband is mentioned, there be another marriage and a fruitful one with a non-descendant. Looking through the phases of the figures, and making L the minor relation in all cases, the major relation being L or L~', we have the following Table of cases in which L or L7' is a legitimate conclusion. 1 2 3 4 I Ne ee By Kwek Vii, Y..L°Z Y. L“z — Χ- ΠΣ XK. LZ ἘΠ“ Il πὶ NG ΧΟ, Kaw ΠΝ, Ζ..1,Ὺ ΖΝ — baba PVA X.-LZ ΤΠ ΠῚ Vc Veubx Wak: τ: ΤΖ ὟΣ ΕΖ Vike ἢ Ξ- τ ΠΖ See ee IV Υε ΤΩ͂Ν, Mea Yo ΤΣ 7. LV... Fede 75. ΠΝ] Σ X. LZ X.L-Z ns In rejecting all conclusions which do not contain L or L~', we must not forget that these conclusions exist: I was only, by such rejection, preparing the way for the complete analogues of the common syllogism. For instance X..LY, Y..L~'Z, give X..LL-"Z, not neces- sarily either LZ or L~'Z, though possibly either. But still it 7s a conclusion, and to some persons an important one: for if L mean descendant, and therefore L~' ancestor, then, Z being the Queen, X is entitled to an honorary degree. In my second paper I pointed out the law according to which L and L~" are distributed. The radical forms of the four figures are here ++, —+, + —, — —3 in my former paper, in which, according to usual practice, the major premise was written first, the radical forms were + +, Ἐπ, —+, ——- The rule is, that the radical form does not admit the con- verse relation: but that when one premise differs in quality from the radical form, the con- verse relation is thrown upon the other; when both, upon the conclusion, Agreeing with the logicians that all judgment either identifies or separates two objects of thought, I maintain against them that this great alternative, though a real form inherent in all judgment, does not give the whole basis of the fundamental act of reasoning, or compa- rison of judgments. The old logicians carried their system all the length which its pre- tensions justified: the modern logicians, without abating a jot of the pretension, have tacitly dropped greatly short of the length. The restorer of logical study in England, Archbishop Whately, directs against many of his predecessors the reproach that, strongly as they con- tend for the syllogism containing the whole form of inferential thought, they seem never to use it nor to care about it when they come to their so-called applications of logic. I sup- AND ON THE LOGIC OF RELATIONS, 355 pose that the parties inculpated found that the ordinary syllogism does not very frequently contain the act of reasoning. And in truth, when it does appear, it is usually Barbara or Celarent, that is ‘Species of species is species’? or else ‘Species of external is external: both contained in the principle that the part follows the whole as to inclusion or exclusion, Of the common logical heads, the study of the term and of the proposition, of the ageregates and components of the term, and of the transformations of the proposition, is far more neces- sary, presents points of far more frequent occurrence, and holds out far greater occasion for warning, than the study of the syllogism, when limited to the arithmetical abacus. If the ordinary syllogism deserved the character given of it, a certain chapter in the older books of logic, instead of dropping into desuetude, would have increased in size and importance, with good assurance of addition to both. I mean the chapter De Inventione Medii Termini. This part of the subject was enlarged into many heads by the latest of the older writers: but in those who most resemble the genuine schoolmen, as Sanderson for ex- ample, the pure heading above given is preserved and its subject treated within the limits of the phrase. If all reasoning be reducible to ordinary syllogism, it follows that any asser- tion inferred about two terms must arise from comparison of the two by aid of some middle term, which is therefore to be investigated. Accordingly, a universal negative can only be established by finding out a third, or middle, term, in which one of the terms of the conclu- sion is wholly contained, and from which the other is wholly excluded, So necessarily is this invention of a middle term the act of investigation, if the syllogism, as given, be what it is said to be, that the mode of arrival at the missing element is very properly formalised into memorial verses Fecana, Cageti, &c., which ought to have followed Barbara, Celarent, &c., as practice follows theory. When by the word syllogism we agree to mean a composition of two relations into one, we open the field in such manner that the invention of the middle term, and of the com- ponent relations which give the compound relation of the conclusion, is seen to constitute the act of mind which is always occurring in the efforts of the reasoning power. Was an event the consequence of another? We know that consequence of consequence is consequence, and, X being a suspected consequence of Z, we examine various Ys, and try if for any one of them we can establish that X is a consequence of Y and Y of Z. We do not consciously refer our search after relation to the notion of relation, nor our act of composition to the notion of composition ; so that our descriptions of mental processes, when exhibited in tech- nical terms, are as strange as our daily syntax when explained in phrases of grammar to an uneducated but tolerably correct speaker. The person X, did he commit the act Z? Non-possession of motive is, taken alone, probable innocence: non-production of motive is probable non-possession. We try for a motive Y, to which X is related by possession, and Y to Z by sufficiency. Here are the premises—X is the possessor of the motive Y; Y is a sufficient motive to commit Z; therefore X is the possessor of a sufficient motive to com- mit Z; and this compound relation is extensively contained in—or intensively contains—the relation of ‘sufficiently in connexion with the action to give the evidence of actual commission a claim to consideration under ordinary notions of probability.’ A very complicated concluding relation: but very familiar in action both to judge, counsel, and jury. But all this is not 900 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, ordinary syllogism. The old logicians were right in attaching importance to the invention of the middle term: but their right notion was deprived of efficient action by their determining that no connexion worthy of a logician’s attention could exist between terms except is or ᾿ ἧς not. Any two notions whatsoever may happen to possess relation to each other in the mind. Choose two notions at hazard: the chances are small that they are related by inclusion or exclusion, total or partial, in any manner worth consideration: but these chances are multiplied a thousand fold if we turn our thoughts to the likelihood of their existing in some other relation. Indeed, some relation must exist between any two things, over and above the relations, usually well settled, of identity or difference. When we examine any book of ordinary reasoning, we find that the onymatic syllogism is not very frequent, the combination of relations much more frequent, and the introduction of composition of terms and transformation of propositions by far the most frequent of all. Syllogisms are rather chapters than sentences, in many cases. When the acts of inference follow one another very quickly, or the reasoning is very consecutive, people begin to cry mathematics. I have read and heard the statement that Fearne’s celebrated work on con- tingent remainders is algebra: it is no more algebra than a remainder-man is e—y; but the reasoning, if I may speak from a very old recollection of a few chapters, is remarkably sustained and connected. Chillingworth is a writer who delights in the technical exhibition of a syllogism, when he gets one: but the instances exhibited do not come very thickly. Nothing that I know of can be written add in syllogism, except mathematics: and this merely because, out of mathematics, nearly all the writing is spent in loading the syllogism, and very little in firing it. It has sometimes been made a reproach to logic that the mathematicians, who reason more consecutively than any others when about their mathematics, do not regard the syllogism with respect in theory, and disdain it in practice. I shall proceed to examine how this matter stands. First, as to the merest technical exhibition of the syllogism, it is, or should be, evident to all parties that such display of form is no more necessary to a proficient than the spelling of every word as he reads it. Those who cannot exhibit their inferences syllogistically need to learn; but those who can do not need to practise: which is exactly what may be said of spelling. When I wrote this last word, I was quite unconscious of s-p-e-l-l-i-n-g: no per- ceptibly separate acts of my mind dictated the writing down of the separate letters. This is all that need be said in answer to those who despise the analysis which is good for the learner, because the logician himself ends, in practice, by using the composite process with which the learner began. There is a useful but very limited field of exercise for the syllogism in geometry. There is hardly an instance, over and above the elimination of B from A=B, B=C, which is not an overt use of the principium et ewemplum: whenever P is true, Q is true; in this case P is true; therefore in this case Q is true. The reduction into the pure technical form, except in a few instances at the commencement, would be useless. The attempt of Herlinus and Dasypodius, of which Mr Mansel (Appendix to Aldrich, note L) has reprinted one pro- AND ON THE LOGIC OF RELATIONS. 357 position, is of use to the learner when carried to the extent to which he and Mr John Mill have carried it, that is, the exhibition of one proposition, to be repeated a few times for prac- tice, But there is a far better logical exercise in Euclid. This great leader has, equally with Aristotle, a style of his own, and one full of its own technicalities: but utterly divested of any the smallest distinction between form and matter. This is most fortunate for that student for whom a further guide is provided: the book before him is raw material on which the exercise of thought about form and matter can be far more profitably carried on than it could have been if Euclid had made the distinction, And this especially on two points. First, a geometrical proposition may either be a purely formal consequence of those which precede, or it may require (as most do) a further infusion of geometrical matter. When Euclid has proved that a non-central point inside a circle, or outside, is πο a point to which three equal lines can be drawn, he holds himself mot to have completed the proof that a point to which three equal lines can be drawn is the centre. But his demonstration is nothing except his often repeated transition from one to the other of the contrapositive forms of a universal affirmative proposition. It is not in his system to establish a purely logical inference once for all: accordingly, ‘not-X is always not-Y’ is converted into “Ὑ is always X’ by one and the same train of thought whenever it is wanted. ‘That the common end of three equal lines is the centre follows equally from the non-centre not being such common end, whether or no the reasoner can say what a circle is, or a centre, or a common end. Again, this same want of admission of what logicians call contraposition gives rise to the majority of the ex absurdo demonstrations: in fact the reductio ad absurdum is usually nothing more than the mode of making the passage from the direct to the contrapositive form. When (in I. 6) it is to be shown that equal angles give equal sides, what Euclid really shows—that is, the geometrical matter of his proposition—is that unequal sides give unequal angles. His unequal sides immediately produce unequal areas with a pair of sides equal, each to each: whence, by I. 4 contrapositively taken, the included angles are unequal. All that is ex absurdo serves only to show that ‘unequal sides give unequal angles’ is identical with ‘equal angles give equal sides,’ and to admit of the direct, instead of the contrapositive, form of I. 4. From such instances, and many others, I derive my now long fixed opinion that geometry is of little, though some, account for technical exercise in the syllogism; of more for exercise in the transformations of the proposition ; of most of all, and of very much, for exercise in the separation of form and matter. It says but little for the truth of the views taken of logic that this science and geometry lived so long in the same family—the old school of arts—without any attention being paid to the bearing of the first upon the second. But it is to be remembered—to say nothing of Euclid being κύριος στοιχειωτής and above criticism—that the form of contraposition, though known and duly registered, was, by reason of the neglect of contrary or privative terms, very little used or thought about: and also that the distinction of form and matter was never completely envisaged, though influential. That the logicians—and it must be remembered that Jogicus meant student or graduate in arts, in all its intension—prone as they were to syllogise, never threw the propositions of Euclid into technical form, must be taken as a point 908 Mr DE MORGAN, ON THE SYLLOGISM, No. IV, ἃς. in their favour. Perhaps it would have been asserted as a matter of course that they did so, omission of all mention being taken as equally a matter of course, if the publication of Herlinus and Dasypodius, the only one of its kind, had not come in as the exception which proves the rule. It is to algebra that we must look for the most habitual use of logical forms. Not that onymatic relations are found in frequent occurrence: but so soon as the syllogism is con- sidered under the aspect of combination of relations, it becomes clear that there is more of syllogism, and more of its variety, in algebra than in any other subject whatever, though the matter of the relations—pure quantity—is itself of small variety. And here the general idea of relation emerges, and for the first time in the history of knowledge, the notions of relation and relation of relation are symbolised. And here again is seen the scale of gradations of form, the manner in which what is difference of form at one step of the ascent, is difference of matter at the next. But the relation of algebra to the higher developments of logic is a subject of far too great extent to be treated here. It will hereafter be acknowledged that, though the geometer did not think it necessary to throw his ever recurring principium et exemplum into an imitation of “Omnis homo est animal, Sortes est homo, &c.,”’ yet the algebraist was living in the higher atmosphere of syllogism, the unceasing composition of relation, before it was admitted that such an atmosphere existed. I expect agreement in what I have said neither from the logicians nor from the algebraists: but, for reasons given in my last paper, I do not submit myself to either class. Not that I by any means take it for granted that all those who have cultivated both sciences will agree with me. When two countries are first brought by the navigators into com- munication with each other, it is found that there are two kinds of perfect agreement, and one case of nothing but discordance. All the inhabitants of each of the countries are quite at one in believing a huge heap of mythical notions about the other. “At first, the only persons who though similarly cireumstanced nevertheless tell different stories are the very mariners who have passed from one land to the other. This will go on for a time, and for a time only: multi pertransibunt, et augebitur scientia. A. DE MORGAN. University Cottecre, Lonpon, November 12, 1859. APPENDIX. ON SYLLOGISMS OF TRANSPOSED QUANTITY. In my Formal Logic, and in my recently published Syllabus of a proposed System of Logic, I gave instances of the syllogism of transposed quantity: that is, the syllogism in which the whole quantity of one concluding term, or of its contrary, is applied in a premise to the other concluding term, or to its contrary. As in the following:—Some Xs are not Ys; jor every X there is a Y which is Z: from which it follows, to those who can see it, that some Zs (the some of the first premise) are not Xs. Such syllogisms occur in thought and in discussion. It also happens that the premises and conclusion are stated independently, and their connexion not seen, It may also happen that the premises are stated simultaneously with the contrary of the conclusion. The fol- lowing sentences, though they will not pass current in a paper on logic which produces them as an example of fallacy, would be very likely to slip through without detection, as part of an ordinary page of writing :— To say nothing of those who achieved success by effort, there were not wanting others of whom it may rather be said that the end gained them than that they gained the end: for they made no attempt whatever. But for every one who was more fortunate than he deserved to be, as well as for every one who used his best exertions, one at least might be pointed out who aban- doned the trial before the result was known. And yet, so strangely are the rewards of persever- ance distributed in this world, there was not one of these fainthearted men but was as successful as any one of those who held on to the last. Might not many educated logicians pass this over, supposing it presented without warning, as containing nothing but what might be true, without seeing that, except under forced interpretation, it combines in one the assertions that αἰΐ are and that some are not ? The syllogism of transposed quantity is essentially a*case of the numerically definite syllogism, though the number of instances is in every case of the indefinite, or rather unspe- cified, character of the algebraical letter: and the same may be said of every onymatic syllo- gism. Those who have commented upon the arithmetical syllogism have for the most part missed this point: they have not seen that the numerical definiteness of the premises is the definiteness of general, not of particular, symbols. That is, they have not caught the dis- tinction between the form and the matter of arithmetical definition. The following slight account of the numerical syllogism will be sufficient for the present purpose. Let us understand by mXY that m or more Xs are Ys; and by m:XY that m or fewer Xsare Ys. ‘Then by mXY we also mean, if # and y be the whole numbers of Xs and Ys in the universe, both (ὦ —m):Xy and (y—m):xY. Let τὸ be the number of in- stances in the universe; v, y, , the numbers in X, Y, Z; and a’, ψ΄, κ΄, the numbers in y, z Then mXY is (ew —m):Xy, or (y —w +m) xy, or (mp+u-—aw—y)xy. And mXy is (γι τὸ -- ἃ τ τὸ - ἡ) xY, or (m+y-—2)xY. 45—5 *356 Mr DE MORGAN, ON SYLLOGISMS OF TRANSPOSED QUANTITY. Let mXY and nYZ coexist: we infer (m+n—y) XZ, or (m+n+uU-—H@-—y—2Z) xz. Let mXy and nYZ coexist: these are mXy and (n+u—-—y-2x)yz;3 from which we infer (m+ n+u-—y—x-y) Xz, or (m+n -- Κ) Xz and (m+n — 2) χΖ. Call the number of instances the (logical) extent of the term or proposition. Then it appears that when the two premises have the middle term Y in both, or y in both, the two forms of conclusion take from the premises, the one both terms direct, the other both terms contraverted: but when the middle term enters in both forms, Y and y, the two forms of conclusion take each one term direct from the premises, and one term by contraversion. In the first, the coeficients of extent in the forms of conclusion are the united extent of the pre- mises diminished by that of the middle term, and the united extent of the premises and of the universal diminished by the united extent of the three terms. In the second, the coefh- cients of extent are both described by the united extent of the premises diminished by the extent of the contraverted term. We can now deduce either the ordinary syllogisms or those of transposed quantity, be- longing to any one case of the numerical syllogism. Let the premises be mXY, nyz, so that (m +m — 4) xz, (m+n—- 7) XZ, are the forms of the conclusion. From m= ἃ, we deduce nxz and (v+n-—3x) XZ; of the second of which we can say nothing without further knowledge of the relations of extent. On the meaning and character of the second form I may refer to my Formal Logic. From wXY and nyz we have then nxz; that is, using the notation of my second and third papers, from X))Y and Y)(Z we deduce X)(Z. Similarly, from τὺ τε ἃ, n= 2’ we show that X)) Y))Z gives X )) Z: from "ἡ = κ΄ alone, we deduce that X () Y )) Z gives X () Z. To find such syllogisms of transposed quantity as this form gives, let m = ὦ: then mXY and azyz give mxz. ‘That is from ‘Some Xs are Ys, and for every X there is something neither Y nor Z’ we deduce ‘Some things are neither Xs nor Zs.’ When one term imparts its quantity to another, let the imparting term have a symbol placed above its spicula of quantity, and the receiving term below. Thus what we have just arrived at is that X()Y)(Z gives X)(Z. It must be specially observed that the term which imparts is always particular: thus when we see X () Y) (Z, in which Z is universal and z particular, the meaning of X () Y is ‘ For every z there is an X which is Y.’ It is also to be remem- bered that in the formation of the symbel of conclusion the spicula of the imparting term is always to be inverted: thus X()Y)(Z does not give X ((Z, but X)(Z. When aterm takes the whole quantity of a term external to its proposition, it will be convenient still to call the proposition wniversal, and, for distinction, externally universal. The ordinary universal may be called internal, When a conclusion is spoken of as uni- versal, it is meant as being internally universal. The circumstances under which two premises have a valid conclusion are precisely those of the ordinary syllogism. Two universals, either or both of which are externally so, give a conclusion, universal or particular according as the middle term is of unlike or like quan- tities in the two premises. A universal and a particular with the middle term of unlike Mr DE MORGAN, ON SYLLOGISMS OF TRANSPOSED QUANTITY. *357 quantities give a particular conclusion. And, as before noted, the symbol of the conclusion is derived as in the ordinary syllogism, with this difference, that the quantity of each imparting term must be changed in forming the conclusion. Let external and internal universality be denoted by Y and U; and sameness and oppo- sition of the quantities of the middle terms by S and D. The ordinary syllogism is valid under the conditions expressed in UDU, USU, UDP, PDU: and each of these forms has eight varieties. The syllogism of transposed quantity is valid under any one of the forms YDU, UDY, YDY, YSU, USY, YSY, YDP, PDY: and each of these forms has also eight varieties. There are therefore 64 varieties of transposed syllogism; twice as many as there are of ordinary syllogism. In the forms YDU and UDY, or whenever an external and internal universal give . a universal, the internal universal is really a simple identity or contrariety of its terms. Thus in ())) which gives ((, or ‘For every Z an X is Y, every Y is Z, give every Z is X,’ the first premise tells us that there cannot be more Zs than Ys, and the second that there cannot be more Ys than Zs. Hence the Ys and Zs exist in equal numbers; that is, since every Y is Z, Y and Z are identical. But in ())-(, or X()Y).(Z, which is XQY))z, Y and z are identical by the same reasoning, or Y and Z are contraries. Σ When two external universals are conjoined, the concluding terms, or one concluding term and the contrary of the other, must be arithmetically of the same number of instances, though not necessarily identical and, it may be, even externals of each other. Thus in OO or in ‘For every Z an X is Y, and for every Xa Y is Z’ we see that there cannot be more Zs than Xs, nor more Xs than Zs. Hence Z and X are of the same numbers of instances. Again in ONC or ‘for every z an X is Y, and for every X there is something neither Y nor Z’ we see that there cannot be more zs than Xs, nor more Xs than zs. Hence X and z are of the same number of instances, though the conclusion X))Z shows that no two instances, one of each, are identical. The vague quantity of a particular conclusion is the vague quantity of the particular premise, when there is such a thing: when there is no such thing, it is either the quantity of the middle term, or of its contrary, whichever is universal in both premises. The following is the list of varieties of syllogism :— UDY YDU YDY YSY ( )) Peet με Pe O@ne () GO MOC ο. EC (-) CC, ©! )((κ( ὡ Oo (ὦ ( ( CG): το (( © DV χοῦ) eC C1 CDG (γ(ζ »») Ὁ) or eC I COCO) Gc Que@)> ©) oad ( ( od ys Gee Ye *358 ΜῈ MORGAN, ON SYLLOGISMS OF TRANSPOSED QUANTITY. PDY USY YDP YSU τ τ | Cares OC aC) Ὁ) τς CC | J) oir | )-):CC)-) NOCERCe | )) CC ΡΟ 70 Mee) |) CY OC) Cee.) νος DIO) |): | mE |.) Con OO) μα αν δ λ οὐχ Coie ΠΟ ΟΣ τ Ss: ΠΟ [τ:-} We have here given the symbols of the premises, followed by the symbol of the conclusion: from which the syllogisms may be read at length. Thus the last syllogism - under YDY is as follows ;—For every z there is an X which is not Y; for every X there is a Y which is not Z: whence every X is Z. The several cases in UDY and YDU are inverted readings each of the other: those in YDY and YSY are essentially different. The cases in USY and YSU are but strengthened forms of those in PDY and YDP; the particular in the second being converted into an internal universal which contains it by an alteration in the quantity of the middle term, without any accession to the conclusion. The forms in YDY are derived from those in PDY and from those in YDP by strengthening—or at least rendering less vague,—the particular proposition by giving it that quantity which makes it an external universal : and the conclusion is thereby strengthened into a universal. The following comparison will illustrate and extend the preceding remarks. In ordinary syllogism, the existence of valid inference depends upon the presence of U and D; that is, either U must be present twice; or U once, and D. In UDU, the most powerful of valid forms, the inference is U: in USU, PDU, UDP, it is only P. In USP; PSU, PDPVESE, there is no inference: of these the first three may be said to be one remove: from inference, and the fourth two removes. All this may be said of the cases in which the external universal is allowed to enter. Thus PSP, now two removes from inference, is made valid by such accessions as make it PDY, YDP, or YSY: each accession being either a change from S to D, or alteration of P into Y. Of all the transposed syllogisms, one half, being all in the first compartment, are either identical with, or contained in, syllogisms of the ordinary kind. Of those in UDY, the one marked (QO is contained in, and contains, ||)) ; and ).(() is similarly identical with |-|)). Of those in YDU, ())) is identical with ((||: and so on. Of those in YDY, ())( is contained in, but does not contain, )))); )-)C is contained in (-))); and so on. Of those in ViSi¥s 00 is contained in ))((; and soon. Overlooking strengthened forms, it thus appears that the really new cases are all contained in PDY and YDP, of which either is but an inverted reading of the other. A. DE MORGAN. University Conmiecr, Lonpon, March 15, 1860. VIII. On the Motion of Beams and thin Elastic Rods. By J. Ἡ. Rours, M.A. Fellow of the Cambridge Philosophical Society. [Read April 23, 1860.) Dunrine a residence in Switzerland my attention was occasionally directed to certain speci- mens of its ancient national weapon—the steel cross-bow, which I had seen exposed for sale in the shops of dealers in curiosities, and the idea struck me that some such instrument might be serviceable in experiments on the resistance of the air to projectiles of different forms and specific gravities; besides I was curious to compare the efficiency of the “ Arbalete” with that of our English long-bow. Accordingly I had no less than three arbaletes made for me in succession by a very skilful Swiss armurier™, the third arbalete stronger than the second, and the second stronger than the first, but I could not succeed in obtaining anything like the velocity I had anticipated; in fact, 200 feet a second seemed about the superior limit, a velocity far too low for my purposes. This induced me to investigate ab initio by the aid of analysis the motion of a vibrating bow. In order to simplify the subject, I began by considering the motion of an uniform thin elastic lamina depressed at one end, and then suddenly released; the partial differential equation which I obtained in this case is the same as that of Poisson, and of which he has given a solution in the shape of a definite integral, which however does not seem easily available for the particular calculation I was engaged in. The differential equation for the motion of a common long-bow or steel-bow is of course dif- ferent; it depends on the law of the thickness of the spring, which is much stouter in the centre than at the ends; but as the simpler equation for the motion of an uniform rod admits of such exact and pleasing integrations, and is so suggestive of the general phenomena of motion of vibrating rods, uniform or not, I attacked that in the first instance. A highly interesting problem closely connected with that I have more particularly considered, is the determination of the law of vibration of a Railway Girder under the action of a passing load; this when the load may be considered as collected at one point, and the mass of the girder is neglected, in comparison with that of the load, leads to an equation derived from simple considerations, but of which the numerical integration or tabulation, when the pres- sure of the load is not considered as approximately constant, is so extremely difficult, that I should not have ventured to attempt it, even if it had not been, as it has, accomplished by Professor Stokes, who, in conjunction with Professor Willis, has well nigh exhausted the * M. Tschiimy of Moudon. Vou. X; Parr 1]. 46 900 Mr ROHRS, ON THE MOTION OF BEAMS subject of vibrating railway girders. There is one case, however, and that the most common one, in which the deflection of the girder is very slight, and the pressure of the load so nearly constant as to be capable of being so assumed, at least for a first approximation; in this case—whether the mass of the girder be neglected or not—the problem admits of an easy solution by the aid of Fourier’s Functions; and the central deflection can be exhibited in the form of a rapidly convergent series, and in a shape very convenient for discussion. Pro- fessor Stokes has also considered this case, but he has solved it by a quite different method from the one I have employed. And now to proceed to the analysis. Let AO, Οἱ Δ΄ be a section of a slender elastic prismatic quadrangular rod made by the plane of the paper, which is supposed parallel to either of the narrow sides of the rod. Let the length of the rod = a, beatels : thicknesss.icsscce = ἐδ ἐπ πο DECAUGH oo. τον = Ts P’P, P,'P, an element length = ds, OX tangent at O the axis of a, OP = 5, y the ordinate at P. Then the element P,'P’ is kept at rest by the tensions and thrusts arising from the elasticity of the rod and by reactions R, R + oR acting along PP’, PP, We shall assume that the elements of the rod are incapable of sliding over each other, and that the thickness remains uniform, so that all the laminz of the rod will always have the same centre of curvature at the same time. We shall also suppose the rod to be so slightly disturbed that the longitudi- nal motion of its elements may be neglected; consequently the neutral axis will be in the middle of the rod. Also if we suppose m to be the mass of an unit’s length and breadth of the rod, and Y a pressure arising from other than molecular forces extending over an unit’s length and breadth of the rod at point (y, ), we shall have As a Roe +8 (= a) εἰ ( 8. 5. ea (eee - ; Le ας nos 12 dy dR 1 SS ss SII. dha dima where Cis a constant, depending on the elasticity. AND THIN ELASTIC RODS. 361 3 ch a’ {dy : : If now = be very great, and ἢ very small, h? ΖΡ A may be neglected in comparison 9 with the other terms of the equation in which it stands, and we have finally, If the rod had not been of constant thickness but of constant breadth, we should have had an equation of the form αἱ & dy ——t ΞΞ -- 2 Ξ See m ΖΕ yp (5) α ae (ps) ast LOU First, for the motion of an uniform girder under the action of a passing load. Let the load be supposed to occupy a small length 28 of the girder; 23 will be supposed ultimately to vanish. Let Q be the pressure distributed over the space 28, supposed uniformly so. Let υ be the velocity of the moving load, supposed uniform, along the girder. Then if a be the length of the girder, Y will be made up of two parts mg and Y’, where Y’ will = 0 till s = vt, if the load be supposed entering on the girder at time ¢ = 0. Y’ will = a from 8 = vt, till 8 = vt + 28, and Y’ will = 0 from thence to the end s=a. : : : 7s Now expanding Y’ in a series of sines of —— between s = 0 and s = a, we have a ΥΞΣ E {sin τι sin 2 | τὶ ein =) Tm a 2 . anret , ns => Ε sin ~—— sin =) a a a in the limit when 6 = 0. oe ch* Hence writing 6? for — , 3 Ty τ: ὧν 2(*2 . wnt . ans m — ESS sin sin ) + mq. dt? ds a a a 7 Let now y = y’ + γ΄ where y” is the part due solely to the statical deflection of the girder by its own weight. - ταὐν" Then 0- -- ὃ ἅτε + mg; dé det a PE ane oe, 2p? Mg 2 tik τὸ ἘΠῚ ἘΞ Q (: ποῦ =) 46—2 902 Mr ROHRS, ON THE MOTION OF BEAMS dy Ἂ : P β ΝΟΥ͂ τς » (or dropping the accent and assuming y for the dynamical deflection) = a ΓΝ τ and aa are both = 0 when 8 =0 and s =a, because the beam is at rest at those points, and the radius of curvature is infinite there, 4: a ae (ς:2 -ο, 3= 0, and S=G@,' y and ἘῸΝ ano at those points ; ae there except for one instant). Therefore expanding y in a series of the form = (P. sin ™) ; a we have dP, aint 2 . πρυΐ mm —— = L pp 4 See : dt’ at a a an bt . πηυΐ Let P,, = A, sin —= / ma? where 4, and B, are constants to be determined. If now the breadth of the girder be assumed =1, and W be its weight, δ΄ the central deflection for the weight W, a) Wane b* 3884. mag =W; Wa’ = , 384 ὃ 5 5 2Q δ @ "aint W5 ΠΕΣ a 3840 ag Again, because the girder has no initial velocity, 22 arid πηῦυ A, +—B,=0; a Wie n 5g 3 2 x 480 a 48 Wo’ arin! Se Se 3) 2 ton ag Q if δ be the deflection due to a weight Q at centre of the girder. AND THIN ELASTIC RODS. 363 : a : Ist. We observe from (2) that if »=0, and vt = >? the displacement at the centre by our series is 183 (5, + jt be.) = 3, οἱ, as it ought to be, and the series is so convergent that taking only the first term we get a good approximation. 2ndly. The statical value of B, is not much increased; as an extreme case, let δ-1, a=420, v= 60, v 1 δ΄ ΞῚ, ===, ἴον 20 3840" 5 B, = 3 fe “i; ee 384 Ἢ wn? | an? -- eS 5 x 32 49 or the denominator is diminished in the ratio of 12 Tn Ὁ ane - ΞΕ: 225 Now as the smallest value of 7’n? is 10 nearly, this is about 201 : 200 nearly. oe But 4, = — —— Tan 1 A,= = = B, = —°07B, nearly. ἐπ If a = 460, δ΄ = Ὁ = 44, Dian A, will = - :042B, nearly ; this is the same result almost exactly as was obtained by Professor Stokes in this example (the Britannia Bridge). 3rdly. We observe from (2) that if 3 be very small, B, varies little from its statical : ui ΝᾺ : : value provided v be not very great; but if ( be very considerable, then the dynamical © a part of the denominator in B, may rise to importance. To prove the relations assumed between δ΄, 6, and 6%, we proceed thus, Let Q be the diet at the middle of the girder, R, R’ the reactions at the ends of the girder. Then R= α δε 364 Mr ROHRS, ON THE MOTION OF BEAMS a : Also between s = 0, and s= >> we have for the equation to the curve dy W a Q aie α oo ae (Saale Ἐπεὶ τὴν ya Wa? 5 Qa’ whence eae πὶ =o +06. δ᾽ 384 τὸ 480" ἣ If we had not assumed the pressure Q constant, we should have had to determine the motion of the girder ay’ dy 2Q_ . anvt . ms =F > sin sin Ξροροδόδοσοσ τ σασϑας ee dt* ds'* ᾧ α Ὰ α α () m ---Ξ τ πέρ --- Q. qe eS Now y=y' τ γ΄, and y” = φ (9), where ᾧ (5) is the statical form of the girder at rest, and s= vt; dy’ ne ' ᾿ ἣν ὩΣ +m —. Φ (vt) = Mg — .........:..6:....(6).ὅ The statical value of y’ is given by the equation ov Q » 2 b*y - τέ -- οὐ" τα Γι πο - =} ς Hence between 1 and 2, eliminating Q if it were possible, we might solve the problem. An easier example where the pressure varies, is the motion of a beam suddenly loaded in the middle and allowed to sink. Deflection of a girder suddenly loaded in the middle. Let M’ be the mass of the load imposed, M the mass of the girder, Q the pressure at time ὁ exerted by the load on the girder. Then supposing the load collected at one point we shall have where it must be observed that y is the part of the ordinate due only to the weight in the middle, so that if y’ be the value of the ordinate when no weight is imposed on the girder, y + γ΄ is the complete ordinate. AND THIN ELASTIC RODS. 365 . Ns Let y= (P, sin) δ a dy &@P, @P, Q Th δ ' “ἢ ae de "mM a 9 Ξ (Stopping at the term P;, which will render the results sufficiently approximate.) Hence substituting for Q, and writing M for ma, and putting M +2M ᾿ ΠΡ kala hls @P, ΦΡ, ar! “dP ἀρ - @P, ΦΡ, v8int ᾿ Maar Σ᾿ digs ες whence P, and P; can be found when ὁ is given. Let ὃ be the statical central deflection due to the mass M’, then μὲς ae If we assume 6 = 3, which makes M = 8M", we shall find 4 P, + 240P, = — 1°940 (1 — cos 7: x 50.) nearly, 4 P, - = = -οὗ (1 — cos 7. at) nearly ; the greatest depression is 1°80 nearly. If we assume e = 1, or the mass of the girder indefinitely small, in proportion to that of the load, the greatest depression = 20 within two places of decimals = 20 (= ms “) , πὶ 81 π΄ a very close approximation to its true value 20 in that case. To determine the motion of an elastic rod, fixed at one end and free at the other. This problem is much more difficult than that of the vibration of a rod, of which the two ends are at rest and the intermediate parts only in a state of vibration. I tried it by many methods, but returned to the one I first thought of, as after all the easiest and best for numerical calculation. I may observe by way of preliminary, that Fourier’s series do not apply to this case, on account of the values of the derived functions of f(s), the equation of 366 Mr ROHRS, ON THE MOTION OF BEAMS the curve being unknown at the limits s= 0, s =. The plan I have adopted is as follows. I take a particular solution of the differential equation ἂν pty ae” ag? where mb” = δ᾽, involving exponentials and sines of s with the argument p, and then obtain an equation with an infinite number of real roots to determine p, from the conditions of the problem, viz. that ay: dy must=0, s=a, and y and dy must be = 0 whens=0. We Ss ds ds have in this way an infinite number of solutions y = D, f(s, ¢), where D, is an arbitrary constant; consequently, putting t= 0, we can take any number m of such solutions, and determine the m constants by the condition that y= = }D,, f(s, 0)} may coincide in m points with the bent spring at rest; it will be found that, taking only the first three values of D,,, we shall have a very close approximation. Let then y = cos p°b't (A, sin ps + B, cos ps + Cie” + D,e~**) + sin ρ᾽ δ (A, sin ps + B, cos ps + C, οἵδ + D,e~™). This, it will be observed, satisfies the equation d' Ξε = — 6" = , where δ = mb”. 5 Also the factor of sin δ᾽ ὁ will be zero, according to the condition that the rod starts from a position of rest. Hence, by the conditions ἀν dy dy = —=0 = a, — =U, =U, de? ds® ) S205 eel a andy=0, s=0 we have B,+C,+ D,=9, Ay +C,- D, = 0, - B, cos pa — A, sin pa + C,e?" + D,e“™ = 0, B, sin pa — A, cos pa + ΟἹ εἴα — D, ec?" = 9; —2 whence cos pa = Pape C cos pa + sin pa + | ; ’ τὰ e4 + cos pa —sinpa }” ἥ εἴα — 2 sin pa -- “ἡ i= 1 - 3 e?* + cos pa — sin pa ePt +. e-P* + 9 cos pa B,= - D,( ali e?* + cos?*— sin pa AND THIN ELASTIC RODS. 367 Assuming for the vibration of the rod y == {(4, sinp,s + B, cos p,s + Ce? + D,e-) cos ρὲ bt, where 4,,, &c. are the n™ particular values of the general 4,, &c., we find the above ex- pression yet further reduced, and A,= D, {1 +e-™*(- 1)", C, B D, Yes [ἘΞ era ? n — D,, {1 + οι (= 1)"*22, These expressions are only useful in computing the first one or at most two values of C., B,,, and A,,, as after that e~?»” may fairly be assumed = 0, Perhaps it might be possible to develope directly any f(s) in a series of terms similar to those we have found as particular solutions of the equation, but in any case it would be labour thrown away, as it is not necessary to consider more than the first two values of pa, and, by equating the differential coefficients in the statical curve, 30 Loge v= ; (as - a} where s = 0, we shall obtain an approximation quite close enough. Of course the most simple and natural method of solving any partial differential equation between two variables s, t, would be to obtain a particular solution Yn, = ΤΣ (Pr$)s φ (,). where p,, is one of an infinite number of roots of the equation y (h) = 0. Then, if we could develope any other given function of s, say Ψ (8) in a series of the form =.{4,(fp,s)}, we could always determine D,, directly, supposing we knew the value of y whent=0. But such expansions are not always possible, or at all events practicable. 2 Roots of cos pa = — a ς- τα" By trial and error, we find 15 , p,@ = 1.875 = Ἢ nearly (measure of 107°, 27’), e? © = 6.521), e-Pi4 = 153, cos ῥ᾽ - —.3, βίη ρα = .954, 1 P2@ = 4.69, cosp.a = - πε A, = «847 D;, Vou. X. Parr II. 47 368 Mr ROHRS, ON THE MOTION OF BEAMS 15 8 15 8 and y =¢,D, (.847 sin ps — 1.153 cos p,s +.158e8 *+e ὃ 4%) : 4.695 -4.695 + t, D,(1.01 sin p.s — .99cospys -- (165. “+e % ) . ors 1 eit SRE 57 τς}. (sin + ὁτὲ 7 — cose) a os 718 po 2s Bia 718 + t, D, OV Se τοῦ τς ~ em ? t,, t,, &c. being abbreviations for the circular functions of ¢ they represent, + &e. + &e. + &c. If now we determine D,, D,, D;, D, by the conditions that dy dy dy ds ἀπὸ νἀ ds®’"_—s ds” shall be the same in the statical curve S20 fe. & y= 55 (as -ἢ, =z @ and in the dynamical curve when ¢ = 0, and at the origin s = 0, the 1°, 4", 5‘, 8", and οἷν differential coefficients will also be equal to each other and to zero in the two curves at the origin, so that the contact there will be of the ninth order. We shall find it unnecessary to determine D,; and D,, they are so small as to be fairly omittable from any but an exceedingly close approximation. Hence we shall have 8.1D, + 4410. + 123.D; + 949 ἢ), = 36, 10.92 D, + 208 D, + 968 D; + 2662 D, = 36, 50.2D, + 10,630 D, + 234,700 D, + 1,771,000 D, = 0, 71.03 D, + 49,910 D, + 1,843,000 D; + 19,490,000 D, = 0; whence D, = .426, D, = -.016, D, = .000288, and may be omitted, Deine ya, t= 0 = 2.307 D, — 2D, + 4505, &e. = .998 nearly. Here where the error is greatest, it is scarcely perceptible. , The greatest value of = is 3.5 εξ é nearly. a AND THIN ELASTIC RODS. 369 Had we supposed the curve to have retained always its statical form, the greatest value ἘΠῚ of = would have been only /8 ae dy. : : : . ἜΣ a will also, during the motion, attain to more than its primitive value. Ss 2, d If pibt = 77, oA will equal -- $55 nearly, This will be nearly the numerically maxi- a dy : Bete mum value of = without regard to sign, and hence we see that a bent rod within the δ᾽ ἂν 8.ὃ breaking limit at the centre may be broken by the rebound after it is set free, as Ξ ae PG only at starting, and an addition of ‘th to the strain might determine fracture. ΕΣ The following is a table of values of Ξ = 0) for two successive values of ¢. dy 36 If t= 0, ds? = a 5 ; , d’ ὃ prb't = 14°. 22", p,b't = 90°, a = 8.38. 0 , ὃ woe = 28°, 45 1, ees = 180, oo = 3.84 —. a ὃ wee = 43°. 7H, e+ = 270, one = 2.46 — rent ὃ .-. = 57". 802, ς-- = 860, ve = 1.437. ? ὃ τις = 71°. 523, eos = 450, noe SOG .65 Hr=86>, 15, «oo = 540°, cco SS «18ὃ soe = 100°, 88' τς = 630°, Cd τς 3.50 ee = 201°, wa = TT, oo =e ον αΞ That the maximum value of 2 is »/ gee on the hypothesis of the deflected rod preserving, a during the motion, the statical form due to the amount of displacement at its free end, may be thus shewn. Aj —2 370 Mr ROHRS, ON THE MOTION OF BEAMS Let, as before, m be the mass of an unit's length and breadth of rod, # the reaction upwards at point s=0. Then remembering the value of R, we have ad’y κεν m | — ds=mb” —., és dt ds’ Now if δ΄ be the extreme deflection at time ἔ, 3 s E : y= ~ = (as* - =) according to hypothesis ; ae ee. \ da sb’? whence ia v= - a 5 Ἢ and δ΄ -- ὃ οο -- -ἰ, a whence 6 is greatest deflection, and the maximum value of di. . ,-8 — is &—. dt 1s oO / ae To determine the velocity of an arrow discharged from a bow. Let ABC be the bow, B the centre, APC the cord, which is supposed perfectly flexible, and always stretched between the points AC and the arrow at P. If the bow be much B A c P thicker at the middle than at the ends, which is usually the case, the amount of displacement of the centre of the cord P will be much more than twice that of the ends A, C. In a bow with which I experimented the displacement was very nearly four times that of the ends, and it will be assumed* that this ratio is constant during the motion. Let then E be the initial place of P before the cord is displaced, AC perpendicular to BP, DE=a, EP= 4c. Let the depression of the centre of gravity of AB = ex, where e is a small fraction, e may be taken about .2 or .3 at the outside; .2 is, I believe, very near it in the bow I * This assumption is only an approximation, for if E | of the second and higher orders ; and the ratio in the text is, I coincides with B, the limiting ratio of DP : DE is 2 : V3. | think, sufficiently near the truth to be adopted as the mean of But in a practical furmula, regard must be had to quantities the varying ratio of PE : DE. AND THIN ELASTIC RODS. 371 employed. Of course when the displacement of the bow is considerable e is not constant, but it is nearly so. The depression of the centre of gravity of AP is 3a, and EP = 4ω. Hence if R be the upward reaction at B, m, the mass of AB, m, of AP, and m, of half the arrow, and if 9} be the force applied at P to stretch the bow, and R be assumed to vary as a, and ὃ be the extreme value of HD when the bow is fully bent, before the string is released, we shall have by first principles d*x Fa me + 2m, + 4m,;) —,=- ---- ae © 3 ts ) dt? ὃ Hence if V be the final maximum velocity of the arrow py πεν A + AM, Ex. The unit of weight being 1 ounce, the accelerating force of gravity 32.2 feet per second, to find the velocity when F' = 200 x 160.9, gme = 5.5 gm, = 1, em, =.5 and ὃ =1, 200 + 4 + 32.2 yng 4/200 4 4 88. 5.5+4+ 2.542 = 200 nearly. The greatest velocity I obtained was, I believe, about 215 feet, or, at the outside, 220 feet a second with a bow, each leg of which was 20 inches long; the stock in which it was set 2 inches broad; the breadth of the bow 1 inch, its thickness near the stock x inch, and 4 inch at the ends; the initial value of BE was 4 inches, and of PE 12 inches; ὃ = 8 inches nearly. The steel was of the best “* St Etienne,” forged and worked with great care, A bow made of English steel, exactly similar, broke after using it some time, so I presume the steel is strained nearly to its breaking point, and 2 = 200 lb. probably. I may observe by the way that the strength of steel varies greatly, and that the best fine-grained steel is not nearly so well adapted for springs as a tougher and more irony steel. The velocities of the arrows I measured by means of sights; the depression for known short distances, combined with the range at 45°, enabled me very closely to calculate the velocity. The range at 45° gave the resistance of the air, which was sufficient to reduce in some instances the velocity 10 feet a second at 40 yards: A rough attempt at a Balistic Pendulum, which I constructed, invariably gave the velocity too little, sometimes by as much as 20 feet a second. Let F = 80, or 97 = 160, ὃ = 4 foot, and the bow be of wood. An ancient English archer’s bow probably ‘* drew” as much as 1601b., as even amateur archers use bows up to 100 lb. strength. 372 Mr ROHRS, ON THE MOTION OF BEAMS If the denominator of the formula = 5, which would be somewhat near its value, the whole string weighing about 10z, and the arrow 1 ounce, V = 250 nearly. I suppose that 160 lb. would be, however, quite the limit of the strength of an archer’s bow under any cir- cumstances. V probably never exceeded 300 feet a second, and when we consider that a rifle-ball has an initial velocity of 1800 feet a second, it is easy to imagine how inefficient in comparison with our present weapons must have been those of our ancestors. The wonderful ranges said to have been attained by Robin Hood and William Tell are no doubt mythical. A curious fact concerning the possible amount of the velocity of an arrow is suggested by the formula we have last found, viz. that, supposing their material to be the same, any two bows of similar figure will impart the same maximum velocity, provided that the arrows and cords are also similar and proportional, that is to say, provided that their masses vary as the cube of the linear scale to which the bows are constructed. For, according to the received law of the strength of beams, &c. if F be the maximum strain a bow can support, & the breadth of the bow, and h its thickness at centre, and a its length, fy 2 Fa poles a “. Fa « kh? « the cube of the scale, and m,, m,, m; vary also as the cube of the scale; Fa ; “. ————————_ 15 constant. me+3m.,+ 4m; If we had assumed R = F'sin — = 26’ Y would have equalled 5 em, + qm + 4m; This assumption for the value of R, which is quite empirical, seems to agree more closely with the results of experiment, than the first assumed value, especially when the cord and arrow are light; and when in consequence the curve assumed by the bow in motion deviates more from its statical form than when the cord and arrow are heavier. h If we assume the law of thickness of a bow to be Z =———,, where A is the thickness 8 1+ — a in the middle, or when s = 0, Z the thickness at any distance 8 from the middle, y will to first approximation, and e = 29 nearly. But a formula based on a first approximation only will not nearly express the real velocity when a bow is bent to the degree to which it is in practice. AND THIN ELASTIC RODS. 373 I will conclude this paper with a short resumé of some of the more interesting results of experiment as to the range &c. of the bolts, and the general power of the steel cross-bow. First, the “ bolts” I used were of two kinds, either capped with iron cones, or with blunt leaden heads; the iron cones were about 41 inches long, and three-fourths of an inch broad at the base, the shaft was cylindrical, about 6 inches long, made of deal, and scooped out in such a manner as to leave three edges between the two ends. he iron-coned bolts pene- trated from about an inch and a third to an inch and a half into sound deal planking. The weight of the bolts varied from about an ounce to an ounce and three quarters, and their flight was remarkably true. The greatest range of the blunt-headed bolts was about 240 yards at 45°, the iron-pointed ones would probably go 20 or 30 yards further; but as the only available ground I could find for my experiments as to range, was a portion of a public road I had measured, I thought it imprudent to launch pointed missiles upon it, at a distance such that persons might be passing, and yet not be visible to me. With a weaker bow the difference was 15 yards in the range between the two kinds of bolts; the iron-headed bolt had a range of 150 yards, and the blunt-headed bolt only 135; the experiment in this case was attended with no danger, as the road was bounded by open fields on both sides up to that distance*. As far as I could judge by mere inspection, I should think the strongest cross-bow with which I experimented was at least as efficient a weapon as any I have seen among the numerous collection of ancient arbaletes preserved in the Musée d’Artillerie at Paris, and yet the velocity of a rifle-ball is more than 8 times the velocity of a bolt discharged from so powerful a bow as the one I possess. The value of & the coefficient of resistance was obtained from the equation ay —— ΖΞ ὶ νεῖ 653} dx* "εἶ COS a where for s I wrote ( 1 } 15 1 Ὸ @x—., cosa 28 When a = 45°, this empirical approximation for s gives results of considerable accuracy; the approximation being much closer than can be obtained by taking several terms of the series for y, developed as in the books by Maclaurin’s Theorem—at least it does so for the particular 5 1 values of @ and & which I had to deal with. For bolts of from 1 to 13 oz., & varied from τὸς 1 1 2 δ to —— and ——, when the bolts were not conical-headed. To determine the value of the 1100 1200 other coefficients of higher powers of v in the expression for the resistance, no velocity less than that imparted by gunpowder is sufficient. In all cases & was found to be much greater than its theoretical value. J. H. ROHRS. * The ranges of these two bows were afterwards increased to 260 yards and 180 yards respectively, for blunt-headed bolts, by using cords lighter than in the first set of experiments. IX. On a Metrical Latin Inscription copied by Mr BuaKEstEy at Cirta and published in his ‘Four Months in Algeria? By H. A. J. Munro, M.A. Fellow of Trinity College. [Read February 13, 1860.] ‘OnE of the most remarkable objects of antiquity which has been brought to light is a tomb of imposing dimensions on the south-west side of the city...On the fourth side [of the lower tomb] three sarcophagi are still lying. A fourth was taken from one of the niches, and on it is an extremely curious inscription, remarkable both for its portentous latinity and the blunders of the stonecutter in executing it. It is the epitaph of a Cirta banker who lived to the age of more than a hundred years ete. I give the inscription exactly as it appears on the stone without any division of the words. There are eight unequal lines and two or three gaps :— HICEGOQVITACEOVERSIBVSMEA ** TADEMONSTROLVCEMCLARAFRVI TVSETTEMPORASYV MMA PRAECILIVSCIRTENSILA REARGENTARI AMLXIBVIARTEMTYDESIN MEMIRAFVILSEMPERETVERITASOMNISOM NISBVSCOMMVNISEGOCVINONMISERTVSVBIQVERISVSIVX VRIASEMPERFRVITVSCVN CARISAXICISTALEM POSTOBITVMDOMINAEVALERIAENONINVENIPVDICAEVITAMCVMPOTVI GRATAMHABVICVNCONIVGESANCTAMNATALESHONESTEMEOSCENTYV MCELEBRAVIFELICES ATVENITPOSTREMADIESVTSPIRITVSINANIAMEMPRARELIQVATTITTYLOSQV OSLEGISVIV VSMEE MORTIPARAVIVTVO'V'EQREVNAMNO A MEDESERVITIPSASEQVIMINITALESEIICV OSEXORECTOVENITAE The old gentleman probably intended to write: Hic ego qui taceo versibus mea fata de- monstro, lucem claram fruitus et tempora summa. Pracilius, Cirtensi Lare, argentariam exhibui artem. Fides in me mira fuit semper et veritas omnis omnibus communis. Ego cui non misertus ubique Ὁ Risus, luxuriam semper fruitus cum caris amicis, talem post obitum Dominz» Valeriz non inveni. Pudice vitam cum potui gratam habui cum conjuge sancté. Natales honeste meos centum celebravi felices. At venit postrema dies ut spiritus mania membra relinquat. ‘Titulos quos legis, vivus mez morti paravi ut voluit Fortuna. Nunquam me deseruit ipsa. Sequimini tales: hine vos exspecto. Venite.’ BuLAKESLEY’S Four Months in Algeria, p. 283. I subjoin_ at once the above inscription arranged in verses. The nature of these singular verses it is the purpose of the following paper to elucidate. LATIN INSCRIPTION AT CIRTA. 375 Hic égo quitéceo vérsibus méa viti déménstro licém clara frtitis ettémpora stimma. Praécil(iu)s Cirténsi laré irgéntar(ia)m Exibui Artem. fydés inmé mira fiit sémper &tvéritis 6mnis. émnibis comminis ég0 cui ndnmisértus ubique Ὁ ristis, liixiir(ia) sémper friiitus cuncdris amicis, talém pdst(dbit)tim d(jmin)ae Valér(iae) ndninvéni pudicae. . are Εν ν᾿ - Ω vitam cim pétui gratam hab(ui) ciincénjuge sdinctam. SS) DY τα AOD OW Coe oe ae natales honést® m(éo)s céntiim c(e)brévi félices. 10. dt v(@nit) podstréma d(fés) utsp(irit)us inénia mémpra relinquat. 11. titulds quoslégis vivus méé mérti paravi, 12. utvdluit fortina: niinquam médéséruit {psa. 13. séquimini tales: hinc vés expéctd. venitae. v. 12. Perhaps utvolui: fortuna nimnoén, ete. Wuen I read Mr Blakesley’s book last autumn, this inscription at once attracted my attention. On examining it I saw, as indeed its author tells us, that it was verse, and verse of some importance as a landmark in the history of the Latin language. Not long before that time I had been reading the two poems of Commodian, an early African Bishop, of whom I will presently say more. They, as well as our inscription, are composed in what is intended to be hexameter verse, verse that is to say written by men of some education, who lived however at a time when that most extraordinary change had already taken place in Latin, and probably also in Greek. I allude to the loss of quantity which was the very bone and sinew of the old language, and to the consequent revolution in the nature of the accent which then degenerated and hardened into a mere stress, resembling the Italian or German or English accent. Of course in the schools of Italy, Gaul and Spain the know- ledge of the old quantity was maintained, just as it is in England at Eton or Cambridge ; but the poems of Ausonius and Claudian are in all essential points as artificial an imitation of Virgil or Horace, as the Musae Etonenses or the Arundines Cami. As prosody therefore and the writing of nonsense or sense verses appear unfortunately to have been quite neglected in the schools of Africa, a worthy Bishop or rich banker, like Commodian or Praecilius, read Virgil by accent alone, and in attempting to imitate him set to work in much the same way as a modern Roman or Englishman would do, who had made himself in other respects a good Latin scholar, without having learned the rules of Prosody: rules which swineherds in the Vou. X. Part II. 48 970 Mr MUNRO, ON A METRICAL days of Homer and ploughmen in those of Plautus had imbibed with their mother’s milk and could discriminate with the nicest precision. As soon as I had seen it too, I looked for an acrostich. The habit of writing acrostichs is very ancient in some kinds of Latin poems. Cicero in the de divin. 11, 54, tells us that the Sibylline verses and some of the poems of Ennius were so composed. Commodian’s longer poem, the Znstructiones, containing more than 1200 verses, forms eighty sections, each of which is an acrostich, and denotes its title by its initial letters. The last section of all, read backwards, gives Commodianus Mendicus Christi. In the second line of our inscription we find Lue. Praecilius plainly enough, and the initial letters of the last ten compose the word Fortwnatus. Perhaps the initial letters of the first three verses H. L. C. may stand for hoe loco cubat, or conditus est, or hwne locum consecravit, or hune lapidem condidit : these or similar expression being common enough in epitaphs. It is no wonder then that, cramped by the requirements of metre and the necessities of the acrostich, the style is some- what stiff and crabbed. Yet the Latin, making the due allowances, is not bad or ungram- matical, and is very superior to many inscriptions of a late date. Indeed it is very much better than Commodian’s, and gives in my opinion a far correcter representation of this kind of verse. Of the two poems of Commodian the one I have just mentioned has often been printed, but always after one very corrupt manuscript, and is therefore in many parts mutilated and imperfect. The other poem was first published a few years ago by Dom Pitra in the first volume of his Spicilegium Solismense, and is still more corrupt than the former. For this, as well as other reasons, our inscription is a more trustworthy represen- tation of this style of verse. Commodian is supposed by Cave and Dodwell, whose opinion has been generally acquiesced in, to have written about a.p. 270. Dom Pitra in his introduction to Com- modian’s second poem places him as early as 250, Clinton in his Fasti Romani, Vol. 2, p- 450, puts him more than a century later, for the following reasons: 1. Jerome who wrote in 392 makes no mention of him in his catalogue. 9, Gennadius who wrote in 493 places him after Evagrius who lived in 388, and after Prudentius who lived in 400. 8, Gennadius observes that he followed Lactantius, and Lactantius lived in the reign of Constantine. The first two reasons seem to me of no weight. Jerome passed over many more important writers; and the work of Gennadius, Presbyter of Marseilles, was intended as a mere sup- plement to Jerome; so that Commodian would have a place in the one list, because he was excluded from the other. Gennadius observes, so far as I can see, no chronological order whatsoever. Audentius, a Spanish Bishop, who comes immediately before Commodian in the list, is placed by Cave, I know not how rightly, in the year 960. Honorius merely repeats Jerome and Gennadius. The third argument would have more weight, if we suppose that Gennadius wrote with accurate knowledge of those times. But proud of his own Gallic culture, he speaks of Commodian as a worthy man, but talks contemptuously of his ‘quasi versus’; and says ‘ Tertullianum et Lactantium et Papiam auctores secutus’, ‘he followed the doctrines of Tertullian etc.’; meaning merely, I presume, that there was a resemblance between Commodian and these fathers. Now Tertullian he certainly did follow; but no two styles can be more different than those of Lactantius and Commodian. I cannot there- LATIN INSCRIPTION AT CIRTA. 377 fore think that this vague expression of Gennadius is sufficient to outweigh the strong internal evidence that Commodian lived in the days of persecution, at the very latest in the beginning of the fourth century. I should be inclined to infer that our inscription was of about the same date. Praecilius speaks of his Cirtensian home. Now Cirta, the old capital of Numidia, was very flourishing in the third century. During the civil wars of the fourth century waged by Constantine and his rivals it fell into entire decay, and was rebuilt by him under its present name of Constantina. If Praecilius had written after these events, he would perhaps have given the city its new name; and besides this a wealthy banker of all men would have been least likely to have enjoyed the uninterrupted peace of mind and outward prosperity of which he speaks so feelingly. To Mr Blakesley’s copy, followed by his explanation, I have appended my own arrange- ment of the verses with the accents, and the quantity marked where it differs from the true prosody. Of course Praecilius himself did not know what the quantity was. His verses are a mere reproduction of his own idea of what those of Virgil were, read by him according to accent. But this shall presently be explained at greater length. I wish first to say a few words about the Latin accent generally ; next to shew that before the third century Latin verses of every kind, popular as well as learned, were written by quantity alone; that on the different kinds of metre accent had no direct influence at all; that however sometimes consciously, sometimes unconsciously, certain poets sought sometimes a coincidence, sometimes on the other hand a contradiction between the ictus metricus of the verse and the accent ; that in the course of the third century by some extraordinary degeneracy of the language, accent began entirely to supersede quantity which practically became a dead thing and was kept up only by artificial training, and that this led necessarily to the destruction of the old language and to the formation of its daughters the modern Romance languages; that nearly about the same time the same strange change came over the Greek and occasioned its total disorganisation, and that it was owing to the utter effeteness of the learned at Con- stantinople and the absence of national life in the people, that the Romaic could never extricate itself like the Romance languages, but always had and still has to struggle with a dead, spurious, abortive Hellenic. Having touched on these topics as briefly as possible, I will conclude with a special comment on each line of our Inscription. The rules of the Latin accent may be told in a few words. Like the Greek, it had no relation to quantity or the length of the syllable, but was a mere raising or sharpening of the tone of voice at the syllable on which it was placed. As in Greek too, there was both a circumflex and an acute; every independent word had one of these two accents. All the unaccentuated syllables were supposed to have the grave accent. Whether the rules of the Greek and Latin accent were ever different from what we know them to have been in histo- rical times, more resembling for instance that of their common sister the Sanscrit, I shall not stop to enquire. Within the records of history the two had this in common, that the accent could never go farther back than the third syllable from the end of the word. It is an instructive fact, that Cicero, who knew only his own language and Greek, in the Orator, 18, declares it to be inconceivable that this should not be so. “ Nature herself’, he says, ‘has so 48—e2 378 Mr MUNRO, ON A METRICAL modulated the speech of man, as to place on every word one acute tone, and not more than one, and that one not beyond the third syllable from the end.’ In modern times many have found it impossible to conceive what he thinks it impossible not to conceive. Such crea- tures of habit are we. As to the limits within which the accent might range, the two lan- guages are agreed; as to the place it might have within these limits they differ greatly. In words of more than one syllable, with few and peculiar exceptions, the Latin accent was never on the last syllable. In this respect it departed widely from the other Greek dialects, but agreed curiously with the Aeolic, with branches of which dialect in Italy the Latins were so long in contact. But in another and even more important point the Latin was in direct opposition to the Aeolic, as well as all other Greek dialects. In Greek the length of the last syllable limited the range of the accent; the length of the penultimate made no difference whatsoever. In Latin polysyllables the length of the last syllable was quite unimportant; the length of the penultimate absolutely determined the place of the accent. If it is long, the accent must be on it, if short, it cannot be on it. To give a few examples. Monosyllables in which the vowel was long by nature, were circumflexed; as sé, ros, mos, pons, mons, res, os (oris), est (‘eats’). Those in which the vowel was by nature short, were oxyton; as mél, cor, vir, mors, nua, os (ossis), est (‘is’). Dissyllables, the penultimate of which was short or only long by position, were paroxy- ton; as arma, virum, vénit, deos, esse (‘to be’), essent, lectus (‘bed’). Those in which the penultimate was long by nature, if the final syllable was also long either by nature or position, took the same accent; as érzs, fato, Romae, celant. But if the last syllable was short, the penultimate was circumflexed; primus, vénit, iram, musa, lectus (particip.). Polysyllables, if the penultimate was long either by nature or position, had the accent on that syllable; and whether that accent was a circumflex or acute, was determined by the same rule as in dissyllables: regina, adire, pietate took the circumflex; inférret, Albani, labores the acute. If the penultimate was short, all polysyllables, whatever the quantity of the antepenultimate or of the last syllable, were proparoxyton ; as IJtdliam, profugus, litora, caelestibus, asperrimi. The following are exceptions to these general laws. The enclitics que, ve, ce, ne attract the accent to the syllable immediately preceding, whether long or short: armaque, as well as armisque ; illave, istéce, sictne. When ce and ne suffer apocope, the accent is then on the last syllable: illite, adhie, istéc, audin, vidén, tantén, erudelin. In a few other cases too the accent is on the last syllable, as in nostrds (‘of our country’), vestras, ceujds, Antids. The atonics as they are called, that is words so closely joined with another that they become as it were a part of it and lose their own accent, are much more numerous in Latin than in Greek; comprising all the prepositions, many conjunctions, and the relative, not the interrogative, gui, quae, quod. Particles too are often joined enclitically to the word pre- ceding them, Quintilian quotes from the first line of the Aeneid quipriémus abéris, where both qui and ab are atonic, that is to say really form but one word with primus and oris respec- tively. An ancient Latin seems to have been able by the sense alone to distinguish in justo from injisto; or praeter méssd from praetermissa, even dissyllabic prepositions being atonic. Of circum litora Quintilian says that some grammarians taught that circum, like the Greek LATIN INSCRIPTION AT CIRTA. 379 dissyllabie prepositions, had an accent on the last syllable. But his ear, he says, could detect no trace of one. Yet many of the later grammarians appear to have held this theory, so fre- quent are their allusions to it. We may safely infer from inscriptions, the oldest manu- scripts, the ancient Grammarians and other sources of information that there were hundreds of cases in which writers felt themselves at liberty to unite two or more words into one or to keep them separate. Qui cumque or quicumque, ubi ewmque or ubiewmque, magno opere or magnopere, ni mirum or nimirum are a few instances out of many. Some other exceptions to these general laws will be noticed in the course of this paper. It appears from what has been said that we English in reading Latin place the accent generally, but by no means always, on the proper syllable. But then we have entirely changed its nature, making it a mere stress, instead of a simple raising of the tone without any lengthening of the quantity. And Praecilius and his contemporaries already did the same. From them and their still more degraded descendants the Italians and other western nations inherited this debased accent which had overthrown and usurped the rights of quantity. In the second line of the Aeneid we read Italiam fato préfugus with the accent on the right syllable; but on the same principle we ought to say, and Praecilius indeed and the Romans for centuries after him did say, Lavinaque. We flatter ourselves that we thus preserve the quantity ; but that is a mere delusion. It we feel by a mere mental process. Whether we pronounce préfugus or profigus, quantity is equally violated. In the same way we read Greek with this debased Latin accent, and fancy that we preserve the quantity while sacri- ficing the accent. The modern Greeks read old Greek with the ancient Greek accent debased in the same way into a mere stress. We think them, they think us in the wrong; and in different ways we are both equally in the wrong. Μήνιν αείδε θέα in an English or Italian and μήνιν ἄειδε θεά in a modern Greek mouth are equally remote from the accent and quantity given to the words by Iiomer or Demosthenes. The thing is so manifest, it would be a waste of words to prove that while Greek was a living tongue, metre was determined by quantity alone, and that accent had no influence on it direct or indirect. In Homer or any other poet verses may be found with identically the same cadence, flow and structure, in one of which the accent shall in every foot agree; in another shall in every foot disagree; in a third shall sometimes agree, sometimes disagree with the metrical ictus. But in prose as well as verse quantity was of far more importance than accent. This is attested by every technical writer on the subject, from Aristotle down- wards. In the third book of his Rhetoric he gives elaborate directions about the rhythms suitable for the different styles of prose, whether it be an iambic, trochaic, dactylic or paeonic rhythm ; but says not one word of the accent. With Dionysius too accent was quite subordinate. The due proportion and due admixture of long and short syllables were all- important. Nearly the same may be said of Latin. Their poetry from the most ancient recorded times was purely quantitative; the old Saturnian verses quite as much so as the Aeneid. And _ in prose too quantity was far the more important element. Cicero and Quintilian attest this as decidedly as Aristotle or Dionysius. The notion of an old lingua rustica in which the people 980 Mr MUNRO, ON A METRICAL composed accentual verses in contradistinction to the quantitative poetry of the learned, is a delusion, a chimera, borrowed not from the fresh youth of the language, but from its anile decrepitude. That in one sense there was a people’s language, that peer and peasant did not speak precisely alike, is a truism. But in that sense the language of Cicero’s orations is different from that of his letters, and both from that of Plautus. There was not even a lingua rustica to the same extent that there must have been in Greece, when Attic became pre- dominant and the other dialects sank into patois; or that now prevails in England where among many different dialects one has been for centuries the universal language of literature and refinement. As in the present day the ploughmen and herdboys of the Alban and Tusculan hills, the head-quarters of the old Latin race, speak the pure lingua Toscana with the pure bocca Romana, so in old times the whole ‘ Latinum nomen’ spoke the Latin unde- filed of Plautus and Terence and Cicero and Caesar. In historical times the closely allied Umbrian and Oscan and Sabellian always remained distinct languages, and never degenerated into mere patois of the Latin, An accentual verse without quantity could have had no mean- ing to an old Latin ear; for the accent was no stress. Ennius did much for the artificial Roman verse; but that he invented quantity is as true as that Dante invented the Italian language. We still possess many fragments of Livius Andronicus who represented his first play before Ennius was born. I believe indeed that accent had a greater, I will not say direct, but indirect influence on the verses of Lucretius and Virgil than on those of Livius Andronicus and Naevius. ; While the language was uncorrupted, the accent had no power, no tendency to lengthen a syllable. To give a single illustration of this: The highest authorities declare that in the whole of the old dramatic poetry there is no instance of a short vowel being lengthened before a mute and liquid; thus patres, patribus, patrius, lacrimae, agros, indugredi, have the accentuated syllable always and necessarily short. The learned poets in imitation of the Greeks allowed these syllables to be common ; and they used indifferently tenébrae or ténebrae, latebrae or latébrae, changing the accent with the quantity. Nay Ovid even ventures, though only once, to write mune similis voluert, nunc vera volécris in the same line. Most languages when allowed their free development have shewn a tendency towards con- traction. This was seen for instance in the passing of Ionic into Attic. It was eminently characteristic however of the Latin. The author of the Varronianus well observes ‘that one could not better describe the genius of the Latin language than by defining it as a language which is always yearning after contraction. The various modes in which this tendency developed itself may be seen in that and other learned works. When we first become histori- cally acquainted with the Latin Language in the oldest extant inscriptions, this tendency, especially in regard to the suppression of final letters and syllables, had been carried to such an extent as to endanger the conjugations, declensions, and consequently the syntax, nay the very existence of the language. Thus we find dedro, for dederwnt: first the final ¢, then the n having fallen away. Nay Mommsen, one of the highest authorities on such a subject, has jately proved the existence of deda (for 8rd pers. plur. perf. ind.) ; that is to say dedanti, the same form as the Greek πεφύκαντι, had become successively dedant, dedan, deda. Then as to the declensions, we find many instances in the oldest inscriptions where the final s or m LATIN INSCRIPTION AT CIRTA. 381 has been suppressed; so that Cornelio stands equally for nominative, dative, accusative and ablative. But probably on the whole the changes which had taken place up to this time were beneficial. As we know it, the Latin compared with the Greek labours with an undue propor- tion of long vowels and accumulations of consonants. And had its forms been stereotyped by a learned literature much sooner than they were, rhythm would have been almost swamped under the dead weight of ponderous long syllables. Musa the nominative would have been as long as the ablative. When the language then had probably reached the proper stage of development, perhaps because it had done so, there arose a succession of great and brilliant writers, Naevius, Plautus, Ennius, Terence, Pacuvius and others; who fixed the grammar and prosody of the language, and made it what it was and is, one of the master languages of the world. But these writers, proceeding all of them of course on the basis of quantity, the only one which could have had any meaning to them or their hearers, fixed this quantity in certain cases, according to the style of verse they were writing, on different principles. Ennius, in introducing from the Greeks the learned hexameter, observed stricter rules of prosody than he did in his tragedies and satires, and than did his predecessors or contemporaries Naevius, Plautus and others. Of course the Greek and Latin poets alike, in order to have a definite metre, are obliged to divide syllables into long and short, and to say that all long and all short shall be of the same value respectively, and that every long syllable shall be twice the length of every short. Yet all long syllables and all short syllables are not in reality of exactly the same lengths respectively. There are also many doubtful syllables which may at pleasure be either long or short, When then a syllable had become decidedly and indisputably short, as the final e in bene and male, though originally long, Ennius in his hexameters determined it should be short; but he would not suffer the e in probe to be so. Thus also he allowed dederunt and dedere to remain side by side, though the final syllable of darent was made irrevocably long. He wrote at pleasure magnus or magnu, but he in no case would permit the last consonant in pater or datur to be neglected. His rules, with only slight modifications, were observed through the whole flourishing period of Latin literature and gave to the learned poetry a finish and precision which it could not otherwise have had.’ And to attain this end he sacrificed much. For a large proportion of the noblest words and forms in the language were thus altogether excluded from the hexameter: all the innumerable cases for example where a short vowel came between two long ones. Ennius on the other hand as tragedian and satirist, Naevius, Plautus and others constructed their verses on the same essential principles of prosody, but gave a far wider latitude to doubtful syllables, Thus not only were bene, male short, but probe might be, though it was not necessarily so. Again pater, datur, darent, and hundreds of similar forms might have their full metrical value, or the final consonants might be slurred over and neglected, as in scripsere for scripserunt. We must not suppose however for a moment that pater could be a monosyllable, a sound impossible for an old Roman tongue. The French pére, like mére and frére arose in a widely different way. Even in the middle of many common words position might be neglected, and voluptatem might have the second syllable short, although it as often has its full metrical value. So in many prepositions, conjunctions and adverbs, ad, in, enim, quidem etc,, the last consonant might at pleasure be suppressed or not; and in hundreds of words like domi, manu, sequi, 982 Mr MUNRO, ON A METRICAL the last vowel might be either long or short. Again meus, twus, boves, and many other words might be either dissyllables or monosyllables. But I cannot dwell longer on this wide question which has been so fully developed by Ritschl, the highest authority on the subject. In one of the last numbers of the Rhenish Museum that scholar gives some hexameters written according to the rules of the dramatic poets probably between 600 and 650 v. c., and interesting in many respects. ‘They generally go by the name of the Praenestinae sortes. Here are one or two of them: Non sumus mendacis, quas dixti; consulis stulte. This verse might have been written by Ennius or Lucretius who ends a line with penden- tibus structas. Conrigi vix tandem, quod curvom est, factum crede. Here the i in conrigi is short, as it might be in Plautus. Yet the principle of quantity is not departed from, any more than it is by Virgil or Horace, when they use mihi or τοδὲ long or short at pleasure. Qur petis postempus consilium: quod rogas non est. Quod petis is simple enough; consilium has the quantity given to it once by Horace; rogas with the last syllable short is found in Plautus and Terence, and is no more a vio- lation of quantity than amat, the last syllable of which was originally as long as amas ; and to Plautus and Ennius was still common, long or short indifferently. Here is one more instance : Est equos perpulcer, sed tu vehi non potes istoc, which admits of just the same explanation. I have dwelt thus long on this part of my subject, in order to protest against the absurdity of supposing that quantity was any less the principle of the old, than of the Augustan Latin poetry, and of imagining that the accent, then a mere heightening of the intonation, could have determined its laws. But in genuine Latin verse was there any coincidence, or any contradiction, intentional or unintentional between the accent and the metrical ictus or arsis, as it is called, of the verse ὃ Three of the very highest authorities on such a question, Bentley, Hermann and Ritschl, have all asserted that the old dramatic poets intentionally sought an agreement between accent and ictus in their iambic and trochaic verses, especially in the middle, the most important part of the verse; while the learned Augustan poets aimed at nothing of the kind. This assertion with respect to the dramatic writers has recently been denied and in great measure explained away; and it seems clear that those scholars to some extent mixed up their feeling of the English or German accent or stress with their conception of the Latin accent. But I must say a few words on this subject, as I wish to shew that the influence of the accent is on the contrary more perceptible in the Augustan and later poets, than in the earlier; as indeed I should a priori have rather expected, considering the way in which it finally superseded and extinguished the old quantity. The nature of the Latin accent must always be remembered ; that which in contrast to the Greek Quintilian complains of, its stiffness and monotony (rigor et similitudo); the fact that almost every word in the language was barytone, and that, when the penultimate was LATIN INSCRIPTION AT CIRTA. 383 long, the accent was almost invariably upon it. This alone would often according to the nature of the verse cause either an agreement or disagreement between ictus and accent. Thus in the old Saturnian verse it is difficult to avoid a frequent coincidence between the two, at the end of the first half, and throughout the whole of the second half of the verse, But this coincidence certainly was not sought. Take the often quoted line, as simple a form as you can have of the verse, Dabunt malum Metelli—Naevio poetae. In the two first feet ictus and accent disagree; in the next from the nature of the Latin accent they agree. ‘Take again this line from the tomb of the Scipios, Cosol Cesor Aidilis—qui fuit apud vos. Here, as gui and apud are atonic, it happens that five times accent and ictus disagree, and only once coincide. And so in many others of the best known verses, especially in those of the great master of the Saturnian metre, Nevius, the poet would appear almost unconsciously to have striven against the coincidence of the two. Immortales mortales—si foret fas flere Flerent divae Camenae—Naevium poetam. In the first of these verses, since δὲ is probably atonic, we have four contradictions, only two agreements between ictus and accent. Yet had the words been thus arranged: A£dilis Consul Censor etc. and Mortales immortales etc., coincidences would have been much more numerous. Let us now examine the hexameter and iambic. With that unerring instinct which never failed them the old Greeks at a particular stage in the development of their language invented the heroic hexameter, the noblest and most perfect metre of the noblest and most perfect of languages. In that verse, for some reason or other which every one can feel, but I for one cannot explain, the caesura was the central force which bound the two parts together, gave to them all their beauty and significance, and allowed an almost infinite variety of rhythm; by the judicious application of which poems of any length might be constructed without their ever palling or wearying the reader. Without this caesura the verse would be an inorganical unrhythmical mass. As the Janguage changed its forms, the different dialects developed different forms of verse, all exquisite in their kind. In Athens the drama occupied the place that the old epic had filled in Ionia: and as suitable alike to it and the dialect in which it was written, the iambic senarius was happily selected as the principal metre. In this verse too the caesura is the central force which gives it a variety of cadence, almost rival- ling the heroic, and rendering it equally suitable for long poems. On the whole therefore, though it is inferior in sweetness to some of the lyric metres, it may be looked upon as only second to the hexameter. Considering the nature of the Greek accent, any influence of it upon these or other Greek metres is quite out of the question. It is only an Eustathius, living when the language was prostrate, who could suggest that the second syllable of Αἰόλου, which he met with in his Homer, was long on account of its accent, never asking himself, why he did not find Αἰόλῳ the dative so used, and ignorant that Homer really gave the form Aid\oFo, another form of Αἰόλοιο ; and that ἀνεψιοῦ is used with the same quantity. From the Greeks the Latins borrowed these two metres, and feeling that the right obser- Vou. X. Parr IL 49 984 Mr MUNRO, ON A METRICAL vance of the caesura was all-important, they on the whole applied it even more strictly than their masters. The ordinary caesura therefore falling in the middle of the third foot, it has been argued, in opposition to Bentley’s and Ritschl’s notion of an intentional coincidence be- tween ictus and accent in that part of the iambic senarius and trochaic septenarius, that from the nature of the Latin accent this could not fail to be generally the case, and that if you read Aristophanes or Euripides with the Latin accent you will find it to apply to them as much as to Plautus or Terence; and they at all events intended no coincidence between their own ictus and the Latin accent. Take the fifth line of the Mercator of Plautus: Graece haee vocatur Emporos Philemonis. From the nature of the accent in vocatur Emporos, it corresponds with the ictus. Yet though Ritschl and Bentley have pushed their idea of an intended coincidence much too far, from a somewhat mistaken notion perhaps of the true nature of the ancient accent, I cannot help seeing even in Plautus and Terence an unwillingness, though probably only half conscious unwillingness, to allow in certain cases ictus and accent to be in violent opposition. Take the next line to what I have just quoted, Eadem Latine Mercator Macci Titi, where in the word mercdtor accent and ictus are in direct contradiction to each other. Such verses as these occur not unfrequently in Plautus, and though I think they are rarer in Terence, we meet with them occasionally in him also. Now when we reflect that a spondee occurs as frequently in the fourth as in any other foot of the verse; and yet that we find perhaps twenty instances where accent and ictus are in opposition in the fifth foot, as in the first verse of this play, Duas res simul nunc agere decrétumst mihi, for one instance similar to that just quoted, Eadem Latine Mercator Macci Titi, it would seem clear that this latter rhythm was intentionally avoided by Plautus and Terence, and that the accent alone can explain why this was done. I am likewise led to this conclusion by what I am now going to shew, that this connexion between ictus and accent gradually established itself much more firmly in times when quantity was yet in possession of all its rights, and probably contributed much to the eventual supplanting of quantity by accent and the consequent destruction of the language. In the exquisite pure iambic odes of Catullus ictus and accent must from the necessity of the case coincide in the middle of the verse. At the beginning and end he probably neither sought nor avoided such coincidence and wrote with equal satisfaction Senet quiete seque dedicat tibi and Gemelle Castor et gemelle Castoris and Quis hoc potest videre, quis potest pati. In the first of these verses accent and ictus disagree in the first and last foot; in the second LATIN INSCRIPTION AT ΟἸΒΤΑ. 385 they agree throughout ; in the third they disagree in almost as many places as they well could in this kind of verse. Yet led by his own delicate instinct he makes them coincide in far the greater number of lines in the ode from which the last verse is quoted : Mamurram habere quod comata Gallia Habebat ante et ultima Britannia. Et ille nune superbus et superfluens Perambulabit omnium cubilia Ut albulus columbus aut Adoneus. Eone nomine, imperator unice, Fuisti in ultima occidentis insula, and so on. Decimus Laberius, the famous writer of mimes in the time of Augustus, entirely I believe avoids in his extant fragments such verses as the sixth line of the Mercator of Plautus quoted above, though he rather seeks than avoids such a cadence as this, Non me flexibilem concuryasti ut carperes, Read coneiirvas and observe the change of rhythm with the change of accent. This increasing tendency (for of such tendency I feel no doubt) to make accent and ictus agree would be most likely to be perceived in verses written to please the popular ear. Dom Pitra in his valuable preface to the poem of Commodian (p. xxiv.) speaks of his verse as written in rhythm; then quotes Bede’s definition of rhythm, ‘ verborum modulata com- positio, non ratione metrica, sed numero syllabarum ad judicium aurium examinata, ut sunt carmina vulgarium poetarum’; and then gives as a good example of this rhythm the celebrated scomma, sung by Caesar’s soldiers during his triumph in the usual scoffing style employed to avert the envy of the Gods: Gallias Caesar subegit, Nicomedes Caesarem, Ecce Caesar nunc triumphat qui subegit Gallias; Nicomedes non triumphat qui subegit Caesarem. He then observes that such like plebeian verses without metre were even more usual among the Greeks than the Romans. In all this he is strangely mistaken. Bede who wrote centuries after the downfal of quantity, means by his rhythm the accentual Church hymns, such as those attributed to St Ambrose whom he quotes. In classical times of course rhythm both with Greek and Latin writers meant simply the several proportions and arrange- ments of long and short syllables; definite sections of which formed the several metres dactylic, iambic, etc.; and has nothing in the world to do with accent. Caesar’s veterans were incapable of perpetrating a false quantity. Their verses are in as strict accordance with the laws of prosody as the Aeneid. Yet in every instance with a single exception in the first line accent and ictus are in agreement. In this specimen we have trochees in all the odd places; but from other examples of the same kind we know that, as in the comic metres, every foot except the last might be a spondee, Observe this other scomma sung by Caesar’s soldiers: Urbani, servate uxores moechum calvyum adducimus. Here accent and ictus are in opposition in the first two feet, but in the middle and end are 49—2 986 Mr MUNRO, ON A METRICAL quite in accordance; and we feel that this ought to be so. We might read for instance the second of the verses above quoted, thus: Ecce Caesar nunc triumphat qui devicit Gallias. But we feel that Ecce Caesar nunc triumphat devicit qui Gallias would be inadmissible, though that would give precisely the same rhythm for the fourth foot which we had in the verse before quoted from Plautus : Eadem haec Latine Mercator Macci Titi. Such progress in popular poetry had the desire of agreement between accent and ictus already made. In the song of Galba’s soldiers a century later, Disce, miles, militare ; Galbast non Gaetulicus, quantity is accurately observed, but accent agrees with ictus in every place, and we feel that such a rhythm as Disce, miles, militare ; Gaetulus non nunc adest would not have been tolerated. That the popular taste for this agreement between accent and ictus was already very decided, may I think be inferred no less certainly from its ostentatious avoidance by a learned writer of iambics. I allude to the tragedies of Seneca. He is most strict in his observance of the regular caesura; and this, as he always has iambi in the even places, necessitates an agreement between accent and ictus in this place. If then he concluded the verse with the same kind of fall as Nicomedes Caesarem, the writer must have felt that he would be conforming himself to the vulgar taste, and therefore in the fifth foot of his verse which, if not always, is almost always, a spondee or anapaest, he contrives that ictus and accent shall be nearly always in violent opposition. The Hercules Furens thus opens: Soror Tonantis, hoc enim solum mihi Nomen relictum est, semper alienum Jovem Ac templa summi vidua deserui aetheris, Locumque, caelo pulsa, paelicibus dedi, ete. In the eleventh line we first come to an apparent exception, but only an apparent one which really proves the rule. Passim vagantes exerunt Atlantides, where Atlantides is a Greek word and accentuated on the penultimate; and we know from Quintilian (and the unanimous statements of the later Grammarians confirm what he says) that in the time of Seneca the Romans, when they adopted Greek words, always gave them the Greek accent, though Quintilian adds that he remembers when a youth that the most learned old folks pronounced such words, Afrews for instance, with the Latin accent. Indeed in looking through a good deal of Seneca, I have been surprised to find how many of the apparent exceptions consist of such Greek words. At y. 495 of the same play, Umbrae Creontis et penates Labdaci, LATIN INSCRIPTION AT CIRTA. 387 I came to the first instance of a cadence like Nicomedes Caesarem; and here too Labdacus is a Greek word. Now such cadences must from the nature of the Latin language have pre- sented themselves to him in almost every line, had he not purposely avoided them. When he ends a verse with a word like aetheris, he keeps ictus and accent separate by most violent elisions. Deserut aetheris, promissa occupet, imperia eacipit, devicti intuens, all occur in the first few lines of one play. This is the more striking, since in the other parts of the verse he but rarely elides long vowels. Very striking too it is, when we think of the following fact. The older poets were free in their elision of long vowels; and Virgil produces many of his most exquisite effects of harmony by its judicious employment. But when we examine Ovid’s Metamorphoses, we find that he confines such elisions within very narrow limits, and so does Seneca’s contemporary Lucan ; and the philosopher Seneca (and I see no reason why he and the Tragedian should not be one and the same person) in the few hexameters which he scatters through his prose works entirely abstains from the elision of long vowels. I cannot help comparing the mode in which Seneca sought to avoid in his iambics the favourite popular movements, with the course pursued by two greater poets than himself. Euripides adopted in his later plays a style entirely different from that of his earlier, seeking no doubt by a freer use of trisyllabic feet and a less ornate diction to approach nearer to the style of conversation of the educated, and avoid the cadences loved by the vulgar. Aristotle approves of this in the third book of his Rhetoric, and says that the uneducated only prefer the more highly coloured poetical language. No less remarkable is the con- trast between the unbroken flow of Shakespeare’s earlier versification in which the sense generally terminates with the verse, and the broken style of his latest versification in which the line perpetually ends on a weak monosyllable, such as and, if, etc, Thus in the Tempest we have verses like the following: Had I been any god of power, I would Have sunk the sea within the earth, or e’er It should the good ship so have swallowed, and The fraughting souls within her. Thy father was the Duke of Milan, and A prince of power. Thy mother was a piece of virtue, and She said, thou wast my daughter, and thy father Was Duke of Milan. Euripides, Seneca, Shakespeare, all alike sought in different ways, suitably to the genius of their different languages, to avoid the monotony of movement, dear to the vulgar, not unwelcome perhaps to the educated ear. We may derive similar lessons from the history of the Latin hexameter. In it, as I have said, the caesura in the middle of the verse is the central force which binds its two halves into one organical whole; without which it would be no verse at all. Now as the ictus metricus or arsis of the dactyl is on the first syllable, while in the iambic it is on the last, we have the opposite result in the hexameter to what we found to be the case in the iambic. In the iambic ictus and accent are generally in agreement in the 988 Mr MUNRO, ON A METRICAL neighbourhood of the caesura; in the hexameter they are for the most part in oppo- sition. Arma virumque cano Trojae qui primus ab oris Italiam fato profugus Lavinaque venit. These two verses are on the whole very different in their movement; yet in the middle of both alike ictus and accent are in opposition, owing to the nature of the Latin accent This opposition was of course quite as unintentional, as the general coincidence in the case of the iambic. Both kinds of verse were adaptations from Greek models, and there any inten- tional agreement must from the nature of the case have been out of the question. Μῆνιν ἄειδε θεά has precisely the same rhythmical movement as arma virimque cano; and there accent and ictus coincide at the caesura. Nay it has been shewn above that words like illic, illine, tantén, talén are always accentuated on the last syllable. Now in Virgil and other poets we frequently find verses thus commencing: Expediat ? tanton placuit, Arte morer ? talin possum, Nune hue nune illic, Nune hine nune illine; and these verses have the same rhythmical effect as I¢aliam fato, etc.; and yet have accent and ictus in agreement, not opposition, at the caesura. There is another peculiarity to be noted in the structure of the hexameter. Even in Homer, although he has many other varieties, the most common cadences at the end of a verse are either such as ἄλγε ἔθηκε, τεῦχε κύνεσσιν, ξυνέηκε μάχεσθαι, or else ἐτελείετο βουλή. ἥνδανε θυμῷ. In the Latin hexameter, at least in the poems of its great master Virgil and his successors, cadences similar to those just quoted are almost universal, gut primus ab oris, Lavinaque venit, conderet urbem. Now in Greek such cadences are of course totally independent of accent. ἄλγε᾽ ἔθηκεν, Φοῖβος ᾿Απόλλων, αὐτὰρ ᾿Οδυσσεύς, Παλλὰς ᾿Αθήνη, δῖος ᾿Αχιλλεύς have all different accents, sometimes agreeing with, sometimes opposed to the metrical ictus ; and yet we feel and see and know that the rhythmical movement is in all the same. The case is very different in Latin. From the nature of its accent ictus and accent must generally, not by any means always, be in agreement, when the verse terminates in the manner mentioned. But this coincidence was of course merely accidental, for the accent did not determine the choice of such cadences, but merely a judicious imitation of Greek models. Indeed Virgil excludes carefully such terminations to a verse as vis animdi, saécla animan- twm, common in Lucretius and others, where accent is just as much in agreement with ictus, as in primus ab oris, moenia Romae. Rhythm, not accent, determined his practice. All the great masters then of the elevated heroic have with fine tact, the reasons for which we can feel, if we cannot explain, given to the end this free open fall in opposition to the involution of rhythm which the caesura occasions in the middle of the verse ; avoiding unless for special effects such terminations as ilicibus sus, procumbit humi bos, per inceptos hymenaeos. And here we come to a phaenomenon similar to what we have already encountered more than once. In the oldest specimen of what may be called popular hexameters extant, the Praenestinae sortes, some of which we quoted above, this regular fall of the end of the verse had not yet so fully established itself, and out of the small number of verses, the exceptions to this cadence are very large: quod régas non ést, where quod and non have probably no accent, but join on to the following words; tempus abit jam ; vehi non potes istoc. We find also id sequz LATIN INSCRIPTION AT CIRTA. 389 satiust, fit nisi caveas: where there is a resolution of the arsis in the first syllable of the last foot. Ceciderunt occurs as a molossus in the middle of another verse. In the Titulus Mummianus, another very old specimen of hexameters, three out of six verses have not the usual cadence of later times; and we meet with one resolution of the arsis facilia for the dactyl of the fifth foot. In Virgil, to take the most perfect master, the caesura of the verse occasions generally a contradiction, the conclusion an agreement between accent and ictus. The other feet of equally harmonious verses may have them either altogether agreeing or altogether disagreeing. ‘Arma virtimque céno Tréjae qui primus ab Gris ‘Ttaliam fato profugus Lavinaque vénit. In the first of these verses we have this agreement in four out of six feet; and had he written qui Tréjae, as Lucretius or Catullus would probably have done, there would have been this coincidence in five out of six places. In the second verse we find a disagreement five out of six times. Yet the two verses are equally good. Nay we find in the best Latin poets many lines where accent and ictus agree throughout, as in the following from Virgil : Pallida, dis invisa supérque immane baréthrum. Non potuisse tuaque animam hanc effundere dextra. Hune congressus et hunc, illum eminus, eminus ambo. Esto nunc sol testis et haec mihi terra vocanti. Dé quod vis et mé victusque volensque remitto. In Catullus we meet with Omnia sunt deserta, ostentant omnia letum. In Lucretius are hundreds of verses like the following : Quanam sit ratione atque alte terminus haerens. Impia te rationis inire elementa viamque. Crescit barba pilique per omnia membra per artus. Then with regard to disagreements between accent and ictus, we have just seen that the second verse of the Aineid, a very excellent one, exhibits five such. So does the thir- teenth verse: Carthaégo Itéliam contra Tiberinaque longe Litora. And had he chosen to write Carthigo Itéliam lénge Tiberinaque contra Litora, since the preposition before its noun has no accent, or, if it has one, has it on the last syllable, there would not have been a single agreement in the whole verse between accent and ictus. Or take this other verse, Aurinci misére patres Sidicinaque juxta Aéquora. Here we have no coincidence in the whole verse, as jyuata is unaccentuated, except in the third foot where there is at once agreement and opposition. Again in the following verse of Lucretius, Ille leonis obesset et horrens Arcadius sus, 390 Mr MUNRO, ON A METRICAL we have agreement throughout the first four places; disagreement in both the last. It may be said that Arcadius sus is an unusual cadence; so it is, but certainly not on account of the accents. Such cadences are comparatively unfrequent in Homer also. They are avoided by Virgil, except when he wishes to produce some particular effect. But as we have already said, he eschews still more for reasons already hinted at such termina- tions of a verse as vis animai, saecla animantum, which are very common in Homer; and where accent and ictus coincide in Latin. He makes a striking exception to this rule in the case of Greek words, in grateful recognition probably of Homer’s movement: and delights in such cadences as luctu miscere hymenaeos, molli fultus hyacintho, neque Aoniae Aganippe. Once or twice indeed, in acknowledgement of his obligations to Lucretius, he ends a verse with a cadence like this, magnam cui mentem animumque. But as a general rule rhythms like these are much more carefully avoided by Virgil, than others in which accent and ictus are opposed. The natura tua vi, fortis equi vis, et horrens Ar- cadius sus of Lucretius may easily be paralleled by the other’s legitque virum vir, et odora canum vis, sub ilicibus sus, and the like. His motives for so doing can hardly be doubt- ful: accent had nothing to do with the matter in either case. He avoided the former kind of movement as weak and unimpressive, except in the case of Greek words; the latter he often purposely sought in order to produce some peculiar effect, It was clearly too for the sake of the rhythmical movement, not the unusual accents, that he so often in- dulges in hypermetral cadences, like robora totasque, itpsique mepotésque, and the like; and that we sometimes find in him such verses as Quam pius Aeneas et quam magni Phryges et quam. If he really ended two lines in the Georgics with arbutws hérrida and vivaque sulpura, the last foot with its accent on the first syllable is much more harsh than in the other kind of hypermetral lines. Take this other verse of Lucretius, Proxima fért humanum in péctus templique méntis. Here again we have agreement in the first four places, and disagreement in the fifth ; and had Lucretius seen fit to write, as surely he might have done, so far as rhythm is concerned, templaque circum Mentis; there would have been agreement neither in the fifth nor sixth foot. Indeed there are hundreds of excellent and regularly constructed verses in Virgil and the other poets where we have this contradiction between accent and ictus either in both the last places or in one or other of them. I will not needlessly cite many instances, but what can be finer than the following verses from the first Georgic, perhaps the most consummate model of rhythm in the whole of Latin poetry ? Spicea jam campis cum messis inhorruit, et cum Frumenta in viridi stipula lactentia turgent. At Boreae de parte trucis cum fulminat, et cim Eurique Zephyrique tonat domus. In these two examples ictus and accent are in violent contradiction in the sixth, per- haps the most important part of the verse. Then again there are many scores of lines in Virgil, the fifth foot of which is formed in some such way as this Lavindque, where the LATIN INSCRIPTION AT CIRTA. 391 accent is in equally violent opposition to the ictus. Let us take this one other illus- tration. We know from abundant testimony that déinde, périnde, préinde, évinde were accentuated in the manner indicated. Servius among others notices this fact in his com- ment on Aeneid vi, 743, Quisque suos patimur manes; éxinde per 4mplum. The accents of éwvinde per dmplum exactly correspond to those of mélti comitantum; and yet how different the rhythmical effect of these two endings. Lucretius again terminates a verse, ΥἹ, 1017, with wnde vacefit. Now we have the most conclusive evidence that vacefit, and all cognate words, tepefit etc., were accentuated on the last syllable. Yet I believe that to Lucretius the movement of these words was the same as Virgil’s wnde La- tinum: to Servius or Priscian it was doubtless otherwise. As the caesura is of vital importance in the hexameter, and the metrical beat of the dactylic rhythm is on the first syllable of the foot, and the Latin accent is such as we have described it, it is perfectly true that in general those verses will be smoothest and easiest in their movement, in the first three or four feet of which ictus and accent are opposed ; the most impetuous and violent those in which there is the greatest amount of agreement in the first four or all the six places. In the iambic and trochaic for cognate reasons we found the contrary to be the fact; the metrical beat of the iambus falling on the second syllable, and the caesura of the senarius occurring, as in the hexameter, in the middle of a foot; the metrical beat of the trochee falling on the first syllable, and the caesura of the trochaic tetrameter always coming at the end of a foot. That the rhythmical movement however, and not the accent, is the occasion of this, may be shewn from many considerations, and also by this fact which should never be forgotten, that the Latin hexameter is entirely borrowed from Homer and Homer’s Greek imitators, and any notion of accent having the least influence on his rhythm is belied by every line in the Iliad and Odyssey. Many verses of Lucretius in which accent and ictus have exactly the same relation to one another which they have in many most easy-flowing verses, are more violent and unusual in their rhythmical effect than any of the verses quoted above in which ictus and accent coincide throughout. Out of hundreds of examples take these: Et membratim vitalem deperdere sensum. Quidve tripectora tergemini vis Geryonai. Quidve superbia spurcitia ac petulantia? quantas. Take again the following; Séd béna magnique pairs servabat foedera caste. Here the accents are arranged exactly as in a verse of this kind: Séd yéterum béna pars servabat εἴς, Yet how different is the metrical effect of the two. The following line of Virgil is quite unexceptionable : Thesatiros ignétum argénti pondus et auri. Substitute férri for argénti. The accents remain identically the same; yet instead of a Vou. X, Parr II, 50 992 Mr MUNRO, ON A METRICAL verse you get an inert unrhythmical mass. Again owing to certain exceptions to the general rules of Latin accentuation we find verses in which accent and ictus coincide throughout, and yet the rhythmical movement is smooth and easy, as in this of Virgil : Sanguine adhiic campique ingentes ossibus albent, and the following from Lucretius: Nec potuisset adhic perducere saecla propago. Nune huc nunc illtic in cunctas undique partis. Nunc hine nunc illine abrupti nubibus ignes. Others too without such exceptional accents are simple enough in their rhythmical move- ment; as these of Virgil, Funera nec cum se sub leges pacis iniquae. Omnia jam vulgata. Quis aut Eurysthea durum, and this of Tibullus (Lygdamus), Non ego firmus in hoc, non haec patientia nostra, and these two consecutive verses of Lucretius, Tam manet haec et tam nativo corpore constant, Quam genus omne quod hic generatim rebus abundat. What shall we say of the following excellent verse of Virgil, Quid loquor? αὖτ᾽ ubi sim? quaé méntem insénia miitat ? in which accent and ictus agree throughout, and at the same time also disagree in the first three places? Of this from Lucretius, Cum metus aut dolor est et cum jam gaudia gliscunt, in which accent and ictus agree throughout, and at the same time disagree in the first, second and fourth places; and the third foot is made up of the enclitic est and the atonic or proclitic et? In this case however I will not vouch for the fact that cum, aut, cum, jam had all dis- tinct accents: I believe they had to Lucretius and Cicero, not to Servius and Priscian, The whole history of the language proves that atonics went on increasing in number, until they had reached quite an inordinate amount at the time when Latin was passing into its Romance daughters. This would seem to be the main cause of the total disappearance of so many of the most serviceable Latin particles from these dialects. This simultaneous coincidence and contradiction between the two would seem indeed to be a strong ground for assuming that the former has no direct influence whatever on the rhythm. Movements like Quantus Athos aut quantus Eryx, Arma viri ferte arma vocat must of course occur perpetually. Sometimes indeed the poet will by peculiarity of rhythm designedly produce a peculiar effect, and accent and ictus will agree in all places as in these verses of Virgil : Saucius ora ruitque implorans nomine Turnum, Impius haec tam culta novalia miles habebit. But this agreement is surely accidental. LATIN INSCRIPTION AT CIRTA. 393 Take again this line of Lucretius, Prava cubantia préna supina atque 4bsona técta. Here unquestionably sound is meant to echo sense; and the rhythm appears to be modelled on Homer’s πολλὰ δ᾽ ἄναντα κάταντα πάραντα τε δόχμιά τ᾽ ἦλθον, where in the first four feet oddly enough accent and ictus are in flagrant contradiction. Again if the verse of Lucretius be read in the following manner: although I do not mean to say that he intended it to be so read : Prava—cubantia prona—supina atque absona, the rhythm is by no means unpleasing g, not nearly so much so as that of many verses where the coincidence in question does not exist. Or substitute this verse, Procumbéntia semisupina atque absona tecta. In this case the rhythmical movement is much more disagreeable, yet coincidence is less com- plete between ictus and accent. Again the many Latin words which have no accent, and the necessarily frequent occurrence of whole feet formed out of the unaccentuated parts of accen- tuated words would afford a strong argument that accent has no direct influence upon rhythm ; for Cicero and other ancients lay it down as contrary to the very nature of things for one word to have more than one accent. Rhythm we have now seen was in Latin as in Greek quite independent of accent which had no direct influence on it whatsoever. But as quantity on which it rested was divided into various portions by caesura, pause and due arrangement of words, it well might be that in consequence of the limited range of the Latin accent it might gradually obtain a certain indi- rect influence over some parts of the hexameter, as of the iambic or trochaic: habit being all-powerful in this as in more important matters. I wish therefore now to shew that there was this tendency, a feeling in favour of an association of accent and ictus, and in particular cases a studied endeavour to avoid such. Lucretius obeys of course the genius of the hexameter in his management of the caesura. But his favourite movement at the end of the verse is to have not only the two, but the three last feet arranged in such a manner as to produce in general a coincidence between accent and ictus. Take the first forty-three verses of his poem, a highly elaborated passage, and you will find more than half the number to have cadences like these, quae terras frugiferentis, not terras quae; exortum lumina solis, tibi suavis daedala tellus, not suavis tibi; tibi rident aequora ponti, diffuso lumine caelum, genitabilis aura favoni, and so on. This produces a grand and stately, but somewhat monotonous effect. Catullus carries this peculiarity even farther than Lucretius, and with his usual grace; but the result is the same. Virgil and his followers, and before him the author of the Dirae whose two short poems are chiefly noticeable, because they seem to have been to some extent taken by Virgil as a model, manifestly wish to avoid as a rule this cloying monotony. Virgil says Trojae qui primus, not qui Trojae; labentem caelo quae ducitis annum, not quae caelo labentem. Not but that he employs this cadence, and frequently too, to produce a solemn and majestic effect. We have not to read far in the Aneid to find Albanique patres atque altae moenia Romae, Tantae molis erat Romanam condere gentem, Illum expirantem transfixo pector 50—2 994 Mr MUNRO, ON A METRICAL flammas. But he felt with his unerring tact that the inordinate employment of this cadence necessarily occasioned monotony ; and he gained ease and variety with the sacrifice perhaps of some grandeur. In a speech of Jupiter to Mercury in the fourth Aeneid there are many con- secutive lines twice repeated with this movement ; but the result is to my ear unsatisfactory, stiff not stately. I will now refer to a studied pursuit of such a rhythmical movement as produced a general contradiction between accent and ictus, a stronger proof perhaps of the increasing power of the former, than a studied agreement. Horace, wishing in his satires to produce verses Sermoni propiora, nearer to the style of ordinary conversation among the polite and educated, and hating the ‘profanum vulgus’, must have clearly felt, as Seneca did, that the rhythm which produced that almost unvarying coincidence between ictus and accent now prevailing in the last two places of the hexameter, occasioned, where the verse was not very elevated, a vulgar monotony pleasing to the common ear, like the chants of Caesar’s soldiers. While therefore in the first four feet he allows his rhythm to proceed much in the same way as that of other poets, he has in the last two places, one or both, made accent and ictus to disagree in a proportion extraordinarily great, if he be compared with his con- temporaries or successors, even those in his own line, Persius and Juvenal. The first two satires will give I believe more than forty illustrations of what I mean; and the result thereby produced is certainly very striking and, as he meant it to be, unpoetical. If time allowed, I might illustrate my views of the increasing influence of the accent by various peculiarities in his odes also. I will mention but one which I have carefully noted. In his earlier alcaic odes he not unfrequently has an iambus for the first foot in any of the first three lines of the stanza. The first book contains, if I have counted right, thirteen instances of an iambus so placed. Of these thirteen instances, six have the cadence vides ut alta, fri paratus, where the accent is on the short syllable of vides, frui, etc. In the second book out of eight cases only one eéhors gigantum has this cadence. In the third, out of seven instances not one has that cadence. In the fourth, in which he generally observes more stringent rules, there is no instance of an iambus whatever. This can hardly be accidental. As Horace disliked generally the short syllable at the beginning, the accent must have brought it out in stronger relief, and have induced him to avoid the conflict between it and the short syllable. In his sapphics the poet, in striking contrast to his mistress Sappho, never has a trochee for the second foot. Catullus however in his two short sapphic odes, which seem to some extent to have been followed by Horace as his model, has three instances of a trochee in that place. In all the short syllable is unaccentuated Seu Sacas sagittiferosque Parthos, Pauca nuntiate meae puellae, Otium, Catulle, tibi molestum est; and yet in the sapphics of Catullus as of Horace the fourth syllable of the verse is commonly accentuated. Of course to Sappho a short syllable in this place was just as acceptable as a long, under any conditions whatever. But the magnificent freedom with which she wielded this noble measure, was quite unattainable by Horace, or even by Catullus. Similarly the first and fifth syllables of the first three lines of the alcaic stanza of Alcaeus were indifferently short or long. I have thus endeavoured to shew that already in the Augustan age accent exercised a certain LATIN INSCRIPTION AT CIRTA. 395 though quite subordinate and indirect influence on Latin versification. Quantity was as yet altogether intact, and in full possession of all its rights; and the accent was as yet no stress, but a mere heightening of the intonation. Quantity was still in full force in the early half of the second century, as we know from the poets of that period and such critics as A. Gellius. After that time there is a great break in the extant Latin literature; and during the century that followed the language must have grievously degenerated. In the third century quantity was far other than it had been. When the boys of Rome salute Aurelian in his triumph, their verse is no more like that of Caesar’s veterans. Unus homo mille mille mille decollavimus, Tantum vini hibet nemo quantum fudit sanguinis, are virtual accentual verses. In better times the accent had no power to prevent the accentuated i of fierem and fieri, which once had been long and in the time of Plautus was yet common, from becoming necessarily short in the time of Virgil; while the unac- centuated ὁ in fiébam and fiebimus was still long. But now the accent has become a stress, and can render a short syllable long. The passage has been made from the ancient to the new. Of course for some centuries after in learned schools the knowledge of the old quantity was kept up by artificial means. But we can see from the greatest grammarians, Servius, Priscian, etc., that it was acquired, as we acquire it; was no longer a living reality; and that a writer when left to his own resources wrote like Praecilius or Commodian. We see too that, with the exception of Claudian and one or two other happy imitators, the artificial verse was less poetical, less vivid than the accentual popular songs and Church hymns, which by degrees more and more confirmed themselves in a total rejection of quantity and a full acceptance of the power of the accent, now become purely a stress like our own or the Italian. Rhyme was soon added; until we come at length to the Dies irae, Stabat mater and to the poems of Mapes, many of them beautiful enough in their simplicity. ‘These are really the same rhythms, as the song of the Roman boys in Aurelian’s time. A large part of their impressiveness is owing to the trochaic rhythm which suits admirably the accentual unquantitative Latin. The other accentual imitations of old metres, such as the many written in mimicry of the Asclepiad Maecenas atavis, are for the most part far less successful; as the writers were unable to distinguish between this and a dactylic rhythm. To make the subject at all complete, it ought to be shewn as could easily be done, that about the same time or soon after the same strange change came over the Greek language. It likewise completely lost its quantity. A very few words on this head must suffice. Why it was that in the third century such a complete revolution occurred in the speech and the whole life of the old classical peoples, I cannot tell. Ancient things then seemed to be passing away. Almost continual wars, pestilences and famines oppressed the human race; and when at the end of that century some vigorous rulers appeared for a while to uphold and restore the perishing empire, the new order of things was far other than the old. The modern world had already begun. It seems to be with languages as with other things: when they cease to grow, they begin to decay; and after the period of the Attic orators the Greek underwent a rapid 896 Mr MUNRO, ON A ΜΕΤΒΙΟΑΙ, ι degradation. After that time poetry, and prose when it has the least merit, are merely imitative. Yet for centuries the prosody of the language continued safe. The first symptom of decay, and a very noticeable one it is, with which I am acquainted, is afforded by the choliambies or scazons of Babrius who appears to have flourished not later than the beginning of the third century: Bentley calls him the last of the good writers. The most marked feature of that verse is the concluding spondee. Now Babrius is not content, as Hipponax and all the older writers of it were, with the simple quantity; but the first syllable of every concluding spondee has an accent acute or circumflex. That this could be accident is of course out of the question in several thousand lines. There are a good many corrupt verses, and when Lachmann published his edition, he, strange to say for a man so singularly observant of such points, did not perceive this peculiarity ; and among the verses emended by him and some others of the leading scholars of Germany, a large proportion, as might be expected, neglect this law: which makes its constant observance by Babrius the more striking. To Hipponax this would have had no meaning. The fact that this concluding spondee could not trust to quantity alone, but required the support of the accent, shews that the latter had then begun to be a stress; and that the noblest language for form and structure which the world has ever seen, was already stricken with a mortal malady. After this period decay advanced with rapid strides; Greek or rather Hellenic soon ceased to be a spoken, a living tongue; certainly as soon as the seventh century, probably long before, the distinction between long and short syllables had been entirely lost. Yet the effete Con- stantinopolitans still clung with tenacious pedantry to the galvanised corpse of the old Greek, and would not allow the Romaic to develop itself freely, as the Romance tongues were doing. As for verse, they had recourse to some of the basest expedients that have ever perhaps been devised. For a long time they measured verse by the eye; said ἡ, w, and the diphthongs shall be long, because the ancients said they were; ¢, ὁ shall be short, and the other vowels long or short at discretion, Finally after struggling for centuries against it, they were obliged to let accent have its rights and exercise the power it had acquired in their spurious Hellenic as well as in the living Romaic. They adopted universally the old comic tetrameter catalectic, written of course accentually, the accent making every alternate syllable long as well as its own syllable, and all monosyllables being indifferent. ‘Qs ἥδομαι καὶ τέρπομαι Kat βούλομαι χορεῦσαι is a good model, as it so chances that the accents of this line correspond to the quantity. Had this not been so they would have had no idea of its rhythm. Thus if the accents of a tragic tetrameter catalectic suited, it might be turned into a good accentual iambic tetrameter, as for instance Ὦ βαθυζώνων ἄνασσα Περσίδων ὑπερτάτη, thereby completely reversing the movement of the metre. Nay the majesty of Homer was not safe, if these conditions were fulfilled by any of his verses: if they had fifteen syllables, if there was a break after the eighth syllable, and if with all this the accents suited. We need not look far in the Iliad to find the following : ἀλλ᾽ ἕνεκ᾽ ἀρητῆρος ov ἠτίμησ᾽ ᾽Αγαμέμνων. ἢ κεν γηθήσαι Πρίαμος Πριάμοιό τε παῖδες. LATIN INSCRIPTION AT CIRTA. 397 The golden harp of Apollo transmuted into a vile droning hurdy-gurdy! A modern Greek gives to these verses the identical rhythm of A captain bold of Halifax who lived in country quarters: as well as to this Ithyphallic, ᾽ iva} “ \ ~ ΄ ’ Ov βέβηλος, ὦ τελεταὶ τοῦ νέον Διονύσου. The writer of a well-meant book on Greek pronunciation, a member of this university, finding this quoted by Dionysius, has committed the enormous blunder of supposing that Dionysius is talking of accentual verse which to him was a nonentity ; and of asserting that the people of old Greece employed them, because they were unable to appreciate quantity. When that verse was written, the meanest peasant had as perfect a knowledge of quantity as Plato. But the Hellenes and Philhellenes of to-day tell us in vain that they speak and write the language of Xenophon. You might as well take the language of Dante and Ariosto, had Dante and Ariosto never lived; mix it up with the Latin of the schoolmen and canonists of the middle ages, add some half-understood purple patches from Cicero and Virgil, and say, Here you have the language of Caesar, Cicero and Virgil. 9 κεν γηθήσαι Πρίαμος! In spite of all passionate protestations to the contrary, Italian has retained far more of the old Latin than genuine Romaic has of the old Greek ; and for this reason among others that Greek is a much more copious language than Latin, Romaic a much poorer one than Italian, The latter has preserved much more of the old vocabulary and the old pronunciation; has even changed in much fewer cases the place of the old accent: the point on which the modern Hellenes most boast of their close adherence to antiquity. In sober truth the debased Latin accent may be said to have created the Italian and the other Romanic tongues. Siede la terra dove nata fui represents exactly the pronunciation and accentuation of Sedet illa terra de-ubi nata fui in the sixth or seventh century. The Hellenic of Tzetzes, Tricoupi or the Vretannikés Astir is as much a dead language as the Latin of Dante or Petrarch, Bentley or Lach- mann. : After this lengthened introduction I will now make a more minute dissection of our epitaph. It is, as I have said, decidedly a purer and a better specimen of accentual verse than the corrupt poems of Commodian; and far more complete than the many later inscriptions to be found on tombs and other monuments, where the writers seldom break so entirely with quantity as Praecilius does. The key to the right understanding of these and similar verses is to remember that Praecilius in studying his Virgil read him by accent and not by quantity, for which he had no natural feeling whatever, and which neither his nurse nor his schoolmaster had ever taught him artificially. What struck him in every line of Virgil was first the caesura, the keystone of the whole; or rather that which he took to be the caesura; a point on which he often differed from Virgil; and secondly, owing to the peculiar nature of the Latin accent and the usual cadence of the Virgilian hexameter the dactylie fall of the end of the verse whether read accentually or according to quantity. Of the portion pre- ceding the caesura he had a far less distinct conception. ‘Arma virtéimque had to him the 398 Mr MUNRO, ON A METRICAL regular dactylic cadence, because accent and quantity are here in agreement; Italiam fato was quite another thing. Cano, Tréjae, primus, quéque had all the same quantity to him> and therefore the same force in verse; just as they would have to an Englishman know- ing the language but ignorant of its prosody: so that it quite depended on the general structure of the verse whether they should be long or short. ‘The same is to be said of préfugus, litora, siperum, cénderet etc. Again Italiam fato had to him precisely the same rhythm as his own Sequimini tales, or Praefatio néstra with which Commodian opens one of his poems. Virgil could commence a verse with Arcébat lénge; why should he not do the same with Cirténsi lare? Where they came conveniently to hand, he seems to have preferred dactylic openings, speaking of course accentually; but finding as many of Virgil’s lines without this movement as with it, he did not trouble himself to avoid a different rhythm when it suggested itself to him. Another leading peculiarity of his versification should be noticed: he did not acknow- ledge the synaloepha, and in reading Virgil never elided a vowel. Of course Virgil did not altogether suppress the elided vowel; that would have ruined his harmony ; he allowed the one to run into the other and produce a composite sound. This absence of ‘elision is characteristic of all the later accentual poems, church hymns and such like, in striking contrast by the way to the frequency with which it is employed in Italian poetry. Prae- cilius accordingly must have recited many of Virgil’s verses with a singular kind of trochaic jumping cadence which has had a powerful influence on the structure of his own poetry. He must have read Litora multtm 7llé et terris jactatus et alto, Trojand a sanguine duct, Spretaequé injuria formae, Teucroriim dvertere regem, etc. He had no feeling for such lines as Aggeribus socer Alpinis atque arce Monoeci Descendens gener adversis instructus Eois, read as Virgil read them. He preferred Descendens génér adversis tnstructus Eois, which sounded as gratefully to his ear as his own Cirtensi lar@, argéntariam éxrbut artem. And similar conceptions, I presume, of the harmony of Virgil and Homer will be entertained by the youth of England, when the advancing intelligence of the age shall have completely sacrificed the ornamental to the useful and proscribed at Eton and Cam- bridge the practice of writing Greek and Latin verses. Praecilius could have known no distinction between the circumflex and the acute; both must have been to him one and the same stress. For obvious reasons however I have printed his words with the accents which Cicero and Virgil would have employed ; and, in order to prevent confusion, I have not for instance given to invéni the circumflex which of course it would have had, if the final 7 had become short in classical times. I have already given the epitaph as copied by Mr Blakesley, subjoining first his version and then my own arrangement of it into verses, in which the faults of orthography com- mitted by the stonecutter are corrected; but those are left which I conceive to be due to Praecilius himself, as being characteristic of the Latin pronunciation of his time. It will be seen for the reasons already so often alluded to that the harmony of the verses such as it is depends mainly on the caesura in the middle and the accentual dactylic cadence at the conclusion of the verse. 1. Taceo, if not a mistake for the usual jaceo, is a play on it to contrast with versibus LATIN INSCRIPTION AT CIRTA. 399 silent myself, I speak in my verses. If mea vita is right, it proves, as does clara in v. 2, luxuria in v. 6, and sanctam for sancta in v. 8, that the final m, as Mr Blakesley observes, was now mute: and this is confirmed by thousands of late inscriptions. In the best classical times it had, as is well known, a dull half-suppressed sound which was often represented in writing by a half letter. In the oldest inscriptions it is frequently omitted, before the date when the poet Attius, a great grammatical reformer, fixed its place in the orthography of the language; thus rendering at least one essential service to the grammar of his language, on the declensions and conjugations of which the loss of the final m would have had the most disastrous consequences, as is well proved by the transition of Latin into the Romance tongues. The early loss of this final letter con- tributed much to the rapid decay of the Umbrian, as we know from many existing monuments. In demonstro the m was almost or quite mute, but the ὁ was proportionately lengthened. We know on the authority of Cicero and others that this was generally the case when m preceded s or f. Hence thensaurus is the genuine Latin form of the Greek Oycavpos. The unaccentuated de was thereby rendered prebably shorter than it would otherwise have been to Praecilius. Commodian opens his Instructiones with this line Praefatio nostra viam erranti demonstrat. The later poets, even those who profess to observe quantity, perpetually shorten this de in composition, Even so good a grammarian as Servius, who lived of course when quantity had to be acquired by artificial rules, tells us that in a word like amicus you know by its accent the second syllable to be long, but must learn the quantity of the first arte, that is by your gradus. Now as Praecilius had no gradus, he took the liberty of making the first syllable of demonstro short at the conclusion of his verse, and the first syllable of honeste long in the first part of verse 9. He knew no difference in quantity between demonstro and recondo, honeste and venisse, the accent in all cases determining only the length of the penultimate. In the same way vérsibus sounded to him the same as préfugus, titulos in v. 11 the same as litora. His first line had for him precisely the same cadence as Hic ego qui taceo, numeris mea fata recondo would have had. In v. 2 the last two syllables of Praecilivs coalescing probably rendered the accentuated i peculiarly long, and the prae proportionately short, though even to writers who profess quantity in the fourth and following centuries this prae was essentially a short syllable. Even so early as the first century ae came to be used to denote the short open e in words like praemo etc. V. 3 was no less rhythmical to Praecilius than Emollit mores didicisse fideliter artes. The first syllable of /are with its clear liquid sound was perhaps more distinctly long to him as to a modern Italian, than the @ in arma or fato. Compare the Italian pro- nunciation of mare, mano, rosa οἷοι; and fydes in v. 4 and the Italian fede. In the ac- centual church hymns, attributed to St Ambrose by Bede and others, quantity is observed with more or less care; yet we find such a verse as Qui éras ante saecula. The e of eras was as long as the @ in dare. The frequency with which Commodian uses words like duce, quoque, neque, homo and such like for spondees or trochees is very striking, The last syllable was of course quite as indifferent to him and Praecilius as to a modern Vou. X. Part ILI. 51 400 Mr MUNRO, ON A METRICAL Italian. In an old inscription in Gruter occurs this line Hune quoqué tristes veniunt et laeti recedunt. Péter became quite as long as méter, frater. Compare the Italian padre, madre, frate, and the French pére, mére, frere. The o in popolo from pdpulus is perhaps longer than in pidppo from populus. The strongly accentuated a@ in argentariam made the first two and the last syllables so much the shorter; the same may be said of the ὁ in exibui. Even professed Grammarians like Priscian and his contemporaries, when they are expounding the rhythms of prose sentences, often pronounce the accentuated ὁ of words like exhibui, hospitibus, perspicere to be long. ‘The same may be said of the w in luwuria of v. 6. In y. 5 the unaccentuated mon was naturally short to Praecilius especially after the caesura, when the movement of the verse suggested it, just as in the noninveni of verse 7. The movement of v. 6 was to Praecilius identical with Arma virumque cano placida composta quiete. The sound of n or m before ὁ in ecuncaris was we know something between an n and g. So was that of n in anquiro or angelus; of ry in ἄγγελος. Again the first clause of v. 7 has the rhythm of Arma virumque cano, Praecilius’ favourite trochaic canter. The unaccentuated post, like other prepositions, was closely connected with the noun it governed and formed indeed one word with it. Compare the postempus, pomeridianus, etc. of the old writers, of Cicero and Virgil. Obitus was pro- nounced obtus; domnus, domna were early corruptions. Compare dompna, donna, dame. The editor of the new poem of Commodian calls himself on the title page Domnus Pitra. In all periods of the language this tendency to contraction was very strong. With obtus compare doctus, raptus οἵοις with domnae lamna for lammina, autumnus, vertumnus and fifty similar words. The quantity of Valériae in this verse appears at first sight the most difficult point to explain in the whole epitaph. The last two syllables are of course con- tracted into one as in argentariam, luxuriam. The accent therefore of Valériae is espe- cially emphatic. I offer the following explanation. Gellius tells us that it was an exceptional peculiarity of the Latin language to accentuate the short penultimate of the genitive and vocative in words like Valerius, Vergilius, and to pronounce Valéri, Vergili . The singular pertinacity with which Servius and other grammarians point out this fact, proves it to have been something very unusual. Thus Praecilius would have read in Horace Contra Laevinum Valéri genus, unde Superbus; and this and similar verses might well have been impressed on the rich banker’s mind, if his wife Valeria was used to recount the ancient glories of her name. ‘The whole verse therefore had to him the same flow as Arma virumque cano Valéri nec amare pudice. Of non inveni I have already spoken. In v. 8 habui is a dissyllable like the Italian οὐδὲ, and the whole verse had the cadence Vitam cum potui memor ire per omnia saecla. In v. 9 the metrical value given to the first syllable of honeste and of felices is solely due to their position in the verse. Even before quantity was tgtally lost, there was a strong tendency to shorten the final e in adverbs, as had been done from the earliest times for bene and male. But Praecilius would have given himself little concern about such matters. To him almost every final syllable was as essentially short, that is to say as unaccentuated, as in modern Italian. Meos is a monosyllable, as it so often is in Ennius, LATIN INSCRIPTION AT CIRTA. 401 Plautus and Terence. Praecilius too probably linked it on to centum. In celebravi the first two unaccentuated syllables were slurred over as one. Compare Ut si perseveraveris, the beginning of one of Commodian’s verses. Many other illustrations might be given. The flow of this verse might be represented by the following fictitious one Natales venisse per arma sacrare fideles, the writer’s favourite trochaic amble. V. 10 at first sight would appear to have eight feet; but venit is a monosyllable, as we find in the classical fert, vult; and in late inscriptions fect for fecit, vivt for viwit and such like. Perhaps Praecilius could only thus distinguish the present vénit from the perfect vénit. Dies is a monosyllable. This word probably soon became shortened in familiar speech and unable to support an independent existence; and so made way for the jornus of middle Latin, and the giorno, jour etc. of the Romance languages. Commodian uses diem for one short syllable and mediws for a pyrthie, and frequently Zabolus for diabolus. Probably he and Praecilius pronounced dies zes. We know from Servius that the d of medius was universally a sibilant in his day. Ut was quite atonic and therefore absorbed in the strongly accentuated spiritus, contracted by its accent into a dissyllable. Compare the spirto of Italian poetry and the French esprit. Their want of accent will perhaps explain the curious fact that so many of the most serviceable Latin particles have like ut disappeared from the Romance tongues, and been replaced by the awkward perche’s perciocche’s cependant’s and the like. The line will therefore have this cadence Nune it amica sed altus inania membra relinquat, the loved dactylico-trochaic run again. Compare with this and the preceding verse such lines of Commodian as the following: Componi|tur alia | novitas caelli terraeque perennis. V. 11 presents no difficulty. Z%tulos might be a dactyl to Praccilius as well as litora, conderet, moenia. With legis compare lare; its position in the line makes it long. Mee is for meae, as venitae on the other hand for venite, and the final ae is as short as the 7 in mihi or the @ in mea. This tendency to abbreviate final syllables was strong in all periods of the language, even the most classical. The ablative morté was once as long as the dative morti, the nominative musa as the ablative musa. V. 12, Deséruit has naturally the same quantity as ewibui, and the me is probably atonic, and attached to the verb like the mi, ti, si, me te, se’s of the Romance lan- guages. In ν. 13 Sequimini tales sounded to Praecilius exactly like Jtéliam fato, Praeterea supplex, or the Praefatio nostra of Commodian. Its position in the verse determined the quantity of sequimint. The end which I have proposed to myself in this paper, has been to shew by a real visible example the essential difference between the old classical languages with their fully developed quantity moving in harmonious combination with the light musical accents; and their debased and degraded state, when they had forfeited the first and had transformed the second into a stiff monotonous stress; a stress inherited by ourselves, and the other chief European nations, so that it is now difficult for us without much thought to bring the reality fully before our minds and persuade ourselves that the capacity of a language for 51—2 402 Mr MUNRO, ON A METRICAL LATIN INSCRIPTION AT CIRTA. that rich variety of beautiful rhythm has passed irrecoverably away. Only a few weeks ago I read a pamphlet by a noble lord, in which he proposes to restore to our language the prosody of the ancients by the help of the two universities who under the sanction of a royal commission shall appoint syndicates composed either of resident or non-resident members, who shall authoritatively determine what syllables shall be short or long or common. Alas! when the world was younger, the cowherds and milkmaids of Ariana executed that task with a marvellous precision, and constructed glorious forms of language, to be afterwards developed into Indian Vedas and Greek Iliads and Latin Afneids. But that faculty has long been lost; and neither noble lords, nor royal commissions, nor universities, no nor syndicates,, resident or non-resident, can now bring back, Quod fugiens semel hora vevit. But what this university can do, and long has done, is to encourage and enforce a study of that ancient prosody, without the knowledge of which not only the poetry of Homer, Sophocles and Virgil, but in an almost equal measure the prose of Plato and Demosthenes and Cicero and Livy would be robbed of all its power and beauty. APPENDIX WueEn this paper was prepared in the early part of last year, I was not aware that the inscription had been copied by any one except Mr Blakesley. I afterwards found that it was inserted in the collection of Algerian inscriptions published by the French govern- ment; where references are given to various French Journals, none of which I have seen. My paper, as the reader will perceive, was more adapted for oral delivery than for the press ; and soon after it had been read, I received the second volume of Corrsen’s elaborate work on the Aussprache Vokalismus und Betonung der Lateinischen Sprache: in which volume the subject of Latin accentuation is treated at very great length. For these and other reasons I had quite given up the thought of publication, when my attention was lately called to an article in Fraser’s Magazine of last month written by a most able and accomplished critic who signs himself J.S. It is headed ‘Arnold on translating Homer.’ Its main purpose is to prove that the movement of the best English hexameters which the writer has seen is so very unlike the movement of any Greek or Latin hexameters, that the thing is an absurdity and a translation of Homer in such a metre altogether out of the question. With much of his elegant criticism every reader will agree. Some of his principal positions however are so contrary to all that I have attempted to prove above and appear to me to be so wide of the mark, that I have been mainly induced to print my paper in order to make public this difference of opinion. The evidence to my mind is so overwhelming, I hold it to be an axiom that the old Greeks and Romans had an instinctive feeling for and knowledge of quantity; that upon this instinct depended the whole force and meaning of their rhythmical measured verse ; that their accent resembled our accent only in name, in reality was essentially different ; that the internal decay of those languages occasioned the ruin of quantity, that consequently the accent, before an intonation, now became a mere stress like the Italian, Romaic, English or German accent; and that if the English hexameter has been or ever is to be successful, that success has been or is to be attained by following out the analogy of other modern metres and making accent replace the ancient rhythmical beat. The critic in question looks on all this as a mere delusion; maintains that Virgil’s accent was the same s our accent; that, though we cannot tell what Homer’s accent was to Homer, to us it is the same as Virgil’s, that is to say as our own; and that in English quantity is as distinguishable as in Latin or Greek by any ear that will attend to it. Then after defining the English hexameter accurately enough; and also the Virgilian so far as quantity is concerned, he goes on to say with regard to the latter, that quantity is not the only condition of the metre. ‘The accent also must be distributed according 404 APPENDIX. to certain laws. Of the six long syllables [forming the six metrical beats] the two last must be accented. Of the remaining four, any one, two, or three may be accented. All four must mot. Subject to these conditions, the accent may be placed any where, and the rhythmical effect depends mainly upon the management of it.’ He adds in a note: ‘ The rule with regard to caesura is, I believe, involved in the rule for the accent.’ But on this last point he does not explain himself any farther. I have already discussed this question so fully, that I will only here repeat that accent has nothing to do with the Virgilian hexameter. Its rhythm depends entirely on caesura, pause and a due arrangement of words. Accent may agree or disagree with the metrical beat throughout. Surely too the Virgilian is constructed on the model of the Homeric verse, and with it accent could have had nothing to do. But no. ‘ All that we know of the Greek pronunciation, is that the rule of accentuation was in Quin- tilian’s time different from the Latin. What it was in Homer’s time, Quintilian himself probably did not know.’ Quintilian knew this rule as well as Homer, and so do I know it; and so does J. S.; otherwise what is the meaning of those marks which he so carefully places over all the Greek words quoted by him? This casual, and, because casual, most important remark of Quintilian I shall presently say more of. Meanwhile let us concede for the moment that by a miraculous anachronism Homer read his verses with the Latin accent. Yet surely it was with Virgil’s, not with our or Dante’s Latin accent. However ‘in Homer we do find now and then a line which reads like an English hexameter—viz. a line in which all the six long syllables are accented, as αὖτις ἔπειτα πέδονδε etc.’ Such lines are rare even in Homer, as any one may satisfy himself if he will read a few pages of the Iliad.” To this I would reply: 1, Our English reading of Homer and Virgil has in itself no meaning. All the won- drous harmony we feel is derived from the mental process by which we superinduce our acquired knowledge of the quantity and rhythm, 2. Verses like those just mentioned, instead of being rare, are among the very commonest types of Homeric rhythm. There must be in Homer thousands of verses like Τὸν δ᾽ ἠμείβετ᾽ ἔπειτα ποδάρκης tos ᾿Αχιλλεύς, or Ὡς Edhar οὐδ᾽ ἀπίθησε Γερήνιος Ἱτπότα Νέστωρ. 1 have counted sixteen or seventeen of them between vv. 78 and 178 of the first book of the Iliad. If the same proportion holds throughont, there must be as many as four thousand in the Hiad and Odyssey together. But I confess that to me this obtruding of the Anglo-Latin accent on Homer seems almost an absurdity. And where I would ask is this said Anglo-Latin accent in words like Πηληϊάδεω or tempestatimque ? Does it outrage Cicero’s ‘ Nature of things’ and occur more than once in the same word ἢ Let us now recur to the pregnant passage of Quintilian. What he says is simply this. Even the most learned old people in his youth pronounced all Greek words with the Latin accent, as ’Atrews. In his time, and ever after as we know from abundant testimony, Greek words, provided they retained their Greek form, retained their Greek accent, as Atreiés, aér, aethér (Ὁ Atrets, etc.); Atréa, aéra; but deris, Achilles. Quintilian and his contemporaries gave a pedantic preference to everything Greek, even in points where their own language had the advantage. They naturally therefore liked the more varied and flexible Greek accent. But everything proves that this change in the place of the accent made no real difference to their ears in the metrical movement of the verse. Let us write down these two verses from the fourth Georgic with Virgil’s accentuation : APPENDIX. 405 Altéque Pangaéa et Rhési Mavértia téllus Atque Gétae atque Hébrus et Actias Orithyia. Quintilian pronounced Pangaea, Actids, Orithyia; and doubtless this accentuation in his opinion gave to the words a certain additional volatile grace. But the rhythmical movement was to him precisely what it had been to Virgil. Virgil again said Harpyfa Celaéno, Orphei Calliopéa, Chariybdis, Eurysthea, Daréta, Théseus, Caénea, etc. etc.; Quintilian Harpyia Celaend (? Celaend), Orphet Callidpea, Charybdis, Eurysthéa, Dareta, and so on. Nero, or whoever the poet was, who is satirised by Persius, in ‘closing his verse’ with Neréa delphin (? delphin) luxuriated no doubt in the Greek intonation with which he trilled forth these words. But in all these cases the rhythm remained unchanged by change of accent. The accent to them, whether Greek or Latin, was only a heightening of the tone with which the syllable where it fell was pronounced. In the time of Priscian and Servius a great change had already taken place. Greek words were still pronounced with the Greek accent; but both the Greek and Latin accent had changed their nature. Quantity had perished; and was only to be acquired by artificial training. What was casually noticed by Quintilian, was to them a matter of vital importance. In his comment on Georgie 1, 59 Servius says, ‘Sane Epiros Graece profertur, unde etiam e habet accentum. Nam si Latinum esset, pi haberet, quia longa est.’ Again and again does he notice this Greek accent. To his ear the accentuated syllable was long, every other short. By study alone he learned the real quantity. We know, he says, the ὁ in amicus to be long, because it has the accent; the quantity of the a we know only arte. He therefore pronounced Epiros, just as a modern Greek does ἤπεῖρος ; he knew only by his art that the ὁ was long. On the other hand he said Epirus; he knew by his art alone that e was long. So also he pronounced Pangaéa, Actids, Orithyta, Charybdis, and the like. ‘To the ear of Virgil or Quintilian altaque was as perfect a dactyl as ardua; éwinde had the same quantity as evire. To the ear of Priscian or Servius altéque was an amphibrachys, éainde a dactyl. The rules of prosody alone taught them otherwise. Now when I think of all this; when I read the hexameters of a Praecilius, or the Political verses of a Tzetzes, or the drama of a modern Athenian, it seems to me almost preposterous to main- tain that quantity exists even potentially in any modern language with which I am acquainted. When I was in Athens a few months ago, I met with tragedies which looked to the eye like the tragedies of Sophocles. The words were apparently ancient Greek; the metre was the senarius scanned according to accent. Reading them produced in me a strange dream-like sensation. The a of σπλάγχνον was long because it had the accent, so was the α in ἄειδε for the very same reason. The a in σπλαγχνεύσας was short, because it was unaccentuated ; so was the a in ἀείδει. The was short in opyyos, long in opiyya. To me it is the same with English. Neither my ear nor my reason recognises any real distinction of quantity except that which is produced by accentuated and unaccentuated syllables. To say ‘ Rapidly is a word to which we find no parallel in Latin; the first short but accented, the second long but unaccented, the third short ;’ or, ‘Sweetly cometh slumber, closing th’ oerwearied eyelid is a correct Virgilian hexameter ; Sweetly falleth slumber, closing the wearied eyelid, 406 APPENDIX. contains two shocking false quantities,’ conveys to my mind no intelligible idea) To me rapidly is an accentual dactyl, cometh and falleth alike accentual trochees; and nothing more; although I am of course aware that two or more consonants take longer time in enunciating than one. The argument of quantity is a mere paralogism arising from our misreading Virgil. ‘ Céntemplate,’ says Rogers, ‘is bad enough; but balcony makes me sick.’ Let us adopt Rogers’ pronunciation and construct an Anglo-Virgilian verse : Comfortably the world from a high balcony contémplate, Read now balcony and contemplate, and we get assuredly ‘ two shocking false quantities.’ OF course I feel puzzled when I find so accomplished a critic holding such contrary opinions. He utterly repudiates accentual hexameters. Then after constructing several verses in what he calls Virgilian measure, he adds that to him the effect of such metre is not bad. And indeed it he goes back to the sixteenth century, he may find many zealous allies both in England and in France. Even so great a master of harmonious verse as Spenser was at one time enamoured of them. In a letter to his friend Gabriel Harvey he informs him that Mr Sidney and Mr Dyer ‘have proclaimed in their apetw παγῳ a general surceasing and silence of bald rhymers...By authority of their whole senate they have prescribed certain rules and laws of quantities of English syllables for English verse, having had thereof already great practice and almost drawn me into their faction. In another letter he goes farther. He likes Harvey’s hexameters so well that he also ‘ enures his penne sometime in that kinde.’ Thus for instance: See ye the blindefoulded pretie god, that feathered archer? Wote ye why his mother with a veale hath covered his face? ‘Do we not all recognise at once the movement of our new friend ?’ Verses so modulate, so tuned, so varied in accent, Rich with unexpected changes, ete. But there were difficulties in the way; ‘the chiefest hardness is in the accent...as in carpenter the middle syllable being used short in speech, when it should be read long in verse, seemeth like a lame gosling’...Yet ‘why, a God’s name, may not we, as the Greeks, have the kingdom of our own language and measure our accents by the sound, referring the quantity to the verse. I would heartily wish you would either send me the rules or principles of art which you observe in quantities, or else follow those which Mr Sidney gave me, being the very same which Mr Drant devised, but enlarged with Mr Sidney’s own judgement and augmented with my observations, that we might both agree and accord in one, lest we overthrow one another and be overthrown of the rest.’ Think of the author of the Faerie Queene talking in this style! Imagine Demodocus writing to his friend Phemius in Ithaca, and telling him to send the rules which he observed in quantity, or else to accept those which Orpheus invented, Musaeus enlarged and he himself further improved; that they might not overthrow one another, and be discovered to be ‘impostrous pretenders to knowledge’ of quantity by those long-eared Achaians who had up to this time listened with rapture to their songs; but who might at length find out APPENDIX. 407 that ἄειδε with the accent on the first and the second long was only a ‘lame gosling.’ Was it in this way that Homer’s verse was invented and handed down to him? Luckily the stiffnecked Muse was too strong for Spenser’s logic, as she had been for Dante’s, when he wished to discard the vulgar jargon for the sounding heroic of Virgil. Italian too much resembles Latin not to have always entertained a pious horrour of so ghastly a parody on its poor dead mother. In France such hexameters were once common enough; and at first sight it might appear that quantity was more possible in French than in most modern languages. The accent such as it is is very variable and is rather a heightening of the voice than an emphatic stress. The Latin accent, having become all-powerful by the destruction of quantity, must have displayed especial energy in creating the langue d’oil; so that after performing such feats as gathering up semetipsissimum into méme, it would seem to have perished by its own intensity. The following is not a bad specimen of French quantity : Rien ne me plait sinon de te chanter et servir et orner; Rien ne te plait, mon bien, rien ne te plait que ma mort. Plus je requiers et plus je me tiens seur d’estre refus¢, Et ce refus pourtant point ne me semble refus. The clear precision of the French intellect however soon recognised the truth and repudiated all such pedantic frivolities, But the main object of the writer I am criticising is to put to shame the accentual English hexameter. He quotes from Virgil Incipe, parve puer, risu cognoscere matrem, and the following verses, and then triumphantly asks, ‘Can anybody produce me an English hexameter resembling, in the succession of sounds, any one of these three lines? I think not. But if I shift the accents a little and write, Incipe, parve puércule, risu noscere matrem. Matri longa tulérunt séx fastidia menses: Incipe, parve puércule, fac ridere parentes,— do we not all recognise at once the movement of our new friend ? Why dost thou prophesy so my death to me, Xanthus? It needs not, &c.’ It is not I maintain the shifting of the accents, but the abolition of the caesura that changes each of these verses into two lumbering unrhythmical masses. Some of the most harmo- nious Latin verses have ictus and accent agreeing throughout. The most unrhythmical un-Virgilian verses in Ennius, Lucretius and the like, as I have remarked above, are not those where the accent is so arranged, but where it is distributed in such a manner that according to the laws laid down by J. S. we ought to have good symmetrical hexameters. Nay by slightly changing the verses just quoted I will make them quite rhythmical again, and yet the accents shall be precisely the same: Incipe, parve, vidén, sine risu noscere matrem., Matri semper abhine per séx fastidia menses: Effice, parve, vidén, sine té ridere parentes. The accents are identical in both sets; for we know that viden and abhinc are oxyton, Vou. X. Parr Il. 52 408 APPENDIX. and sine and per atonic. If J. S, replies that his ear only acknowledges viden and abhine; let him for once give these words the accent which he gives to the while and awhile. Or let us apply his reasoning to cognate cases. What I would ask are the usual English metres but accentual adaptations of quantitative Latin measures, iambic, trochaic and the like? What is the English ten-syllable line of Shakespeare and Milton? Has not accent here replaced the Latin metrical beat, and is not caesura, essential to Seneca’s verse, altogether unnecessary? or if it be said you cannot fairly compare a verse of five feet with one of six, let us take the present French Alexandrine, This may not be an attractive measure to an English ear. But we cannot deny that it has been brought to its present perfection by the labour and genius of centuries; and that it gives entire satisfaction to a nation exquisitely alive to beauty and precision of form. Now in it there must be no caesura; the sixth syllable cannot be the middle, must be the end of a word. Suppose I quote the opening lines of the C&dipus Tyrannus, ὦ τέκνα, Καδμου τοῦ πάλαι νέα τροφή, τίνας ποθ᾽ ἕδρας τάσδε μοι θοάζετε ἱκτηρίοις κλάδοισιν ἐξεστεμμένοι, and reading them with the Anglo-Latin accent exclaim: ‘Can anybody produce me a French Alexandrine resembling in the succession of sounds any one of these three lines? I think not. But if I shift the accents a little and write, ὦ τοῦ πάλαι ἹΚαδμου νέα τροφη, τέκνα, ἕδρας τίνας ταύτας ἐμοὶ θοάζετε, ἱκτηρίοις τούτοις κλάδοις ἐστεμμένοι, do we not recognise at once the movement of our old friend, with whom we are all so painfully familiar ? Je chante ce héros qui regna sur la France, ete.’ I neither defend nor attack the English or German hexameter. No lengthened com- position in either language, not even ‘Hermann und Dorothea,’ gives me full satisfaction. The monotony is too killing, But then what a dull heavy lumbering verse our English ten-syllable line was in the first half of the 16th century! What a glorious measure it soon became in the hands of Spenser, Shakespeare, Milton, and Dryden! Yet the five accents form the basis of all their infinite diversity of movement. With this analogy before my mind I can conceive it, though I do not know it, to be possible that in the hands of genius the English hexameter might be rendered even more majestic and sonorous than the iambic; might come in time to have somewhat of the same relation to it that the hexameter of Homer has to the senarius of Sophocles. However that may be, I feel convinced that six accentuated syllables must take the place of the six rhythmical beats, though the skilful and varied arrangement of some of these may give scope to great di- versity of movement, just as the accent of our iambic is shifted about in certain places with such success by Shakespeare or Milton. Quantity must be utterly discarded; and longer or shorter unaccentuated syllables can have no meaning, except so far as they may be made to produce sweeter or harsher sounds in the hands of the master. Trinity CoLLEcE. July, 1861. X. On the Theory of Errors of Observation. By Auaustus Dr Moraay, F.R.AS. of Trinity College, Professor of Mathematics in University College, London. [Read Nov. 11, 1861.] Tuts paper is an attempt to simplify the mathematical treatment of the subject, mixed with a statement of the grounds on which, in my belief, it ought to rest. I touch only those heads on which I have something to say as to one or other of these points: and I make no remark on preceding writers, except so far as may be inferred from the following preliminary observations. In this subject I fancy I have always seen a mixture of modes of thought and modes of treatment which makes it a difficult speculation, though easy of application in practice. Whether this or that be psychological postulate, result of experience, or deduction from one or the other, is often of harder determination than it ought to be: a difficulty sometimes arising from, or augmented by, the very circumstance on which facility of application depends. The peculiar pliability of the function e *, which serves our turn be*the law of facility of error what it may, is so dexterously used that we hardly know how much of any result is independent of it. This function makes its appearance only as a mathematical instrument. Had any other instrument been of more convenient use, it seems as if our results would have had another expression, I shall succeed in shewing that there is no possible choice in this matter, by introducing e~™ into the representation of results obtained without any reference to it, expressed or implied. The theory of probabilities professes to give the way in which belief in elements should affect belief in combinations. The word probability has two different senses, the collision of which is a grand source of confusion: it is used to refer both to the state of the mind, and to the external dispositions which are to regulate the long run of events: to our strength of prediction, and also to the capacity of circumstances to fulfil our prediction. I shall use the words probability and facility, as follows. Head and tail are to our minds of equal proba- bility, so long as we know nothing to distinguish them: but to say they are of equal facility is to make an assertion involving points of symmetry, density, surface, &c. as regards the coin, and we know not what about the habits of the person who is to make the tosses. Ac- cordingly, our theory does not, as many suppose, arrogate to itself a predictive character: it does not prophesy that in six millions of throws with a die, something near to one million will be aces. All it does is to justify to the mind the following alternative, Either very near to one million of aces, or determinate presumption, depending upon the amount of departure, against equal facility in the different faces. Such equality of facility is as likely as a pencil line or a perfectly rigid bar, When we talk of actually applying our theory to observations, 52—2 410 Mr DE MORGAN, ON THE THEORY OF we mean that we carry with us into the field of practice a true knowledge of equal facility of positive and negative error, as to what its effects will be. What use we shall be able to make of this knowledge experience alone can tell us: theory has nothing to do with the answer to this question. Every part of exact science has a defined foundation, upon which it is the condition of science that the superstructure shall entirely rest. The theory of probabilities postulates for its foundation ἃ +6 equally probable—or to our minds similarly situated—cases, of which a favour one event, and ὁ the alternative. Assent cannot be claimed to any fraction as express- ing a probability, unless this derivation of its terms can be substantiated. Nevertheless, as in other branches of science, we find in ourselves a certain amount of rude, but not very inaccu- rate, knowledge of those details which it is our business to deduce from first principles. Geometry, for instance, does not give us more confidence in the proportion of the diagonals of squares to their sides than we began with. But the mischief of natural knowledge is this: with full confidence in a great deal of truth, we have also full confidence in a great deal of falsehood. Many persons begin by believing that doubling the side of a square doubles the area as well as the diagonal. And if we be liable to such mistake in judging of space, a matter in which our most unbiassed thoughts and our keenest perceptions keep watch upon one another, we must needs be in still greater danger in a subject of comparatively rough and infrequent experiment, in which the instruments of the mind have been trained under bias both of prejudice and self-love. The greatest stumblingblocks lie in the way when the argu- ment is from the finite to the infinite, or from the infinite to the finite. I shall take an example of each, From a sack of white and black beans, mixed and shaken, we take out a score, and find 13 white and 7 black. We naturally conclude that the sorts are in the proportion of 13 to 7, or not far from it: and also that we can have no reason to declare against that proportion on one side rather than the other, Is it not just as likely beforehand that the selected portion should belie the general average by excess as by defect? Before this is granted, we are tempted to recal the story of a Cambridge professor whom some living persons remember, who is said to have sturdily refused to concede that the whole is greater than its part until he saw what use his opponent would make of the concession. Let a person be required to stake upon his own statement of the proportion in such manner that the nearer he is to the truth the more he is to receive. He will do wisely to name 13 to 7. What odds then shall he offer that the next bean drawn is white? Surely, it will be answered, 13 to 7: nevertheless, this answer is wrong; he ought to offer 14 to 8. Next, let 4 and B be arranged in every possible order, in infinite sequence, but so that in the long run A shall occur five times as often as B: that is, let the unending succession AAAAAB, AAAAAB, ἕο. be made to take every possible variety of arrangement. Let an arrangement be drawn at hazard; what is the probability that its first letter shall be A. Any one can see, if he take the point of view, that we merely ask, on the supposition that in the long run A occurs five times as often as B, what is the chance of drawing 4 at the first trial. And the true answer is, five to one in favour of 4. But how are we to meet the fol- lowing reasoning? Let every one of the arrangements be made to have a duplicate; no two of the original arrangements being entirely alike. Of each related pair let one be headed ERRORS OF OBSERVATION. 411 with a new 4 and the other with a new B: it is now an even chance for 4. But have we not simply restored the original state of things? The addition of one more A or B does not alter the ratio of As to Bs in an infinite number. What was the original collection of arrangements except A followed by every possible arrangement and B followed by every possible arrangement? How then are there more ways of beginning with 4 than of begin- ning with B? No beginner can answer this sophism: no proficient can make sure of having avoided the like, if he should take an assumption about the long rwn, or derived from the long run, until he has obtained verification from fundamental principles. The science is essentially enumerative of equally probable cases, and draws all its con- clusions from distribution of these cases under heads, and subsequent enumeration of the numbers under the several heads. The cases may be infinitely many, and it may require all the power of algebraic development or of the integral calculus to present the results of the enumerations: but this does not affect the truth of my assertion, though it places an array of symbolic reasonings between the beginner and clear perception of the fundamental method. In the subject of this paper there has always been a leaning towards the assumption of some complex results upon native evidence ; especially on points connected with the average: and the probable whole has not infrequently been assumed to be a congeries of the most probable parts. ‘This turns out, on proper examination, to be true in some very marked cases: and the conclusion is made welcome for its own sake, as well as for the letters of introduction which sound demonstration furnishes. But in the theory of probabilities, no less than in the conduct of life, if we open our houses to strangers upon the strength of pleasant looks and plausible stories, we shall certainly be swindled at last. That a probable whole must be composed entirely of probable parts is a fallacy of almost universal sway: it resembles the mistake made by Frankenstein, who constructed every limb and feature of his man upon the most approved model of separate beauty, and produced the ugliest monster that ever was seen. Stories which are throughout of the highest probability may be true; there are such truths: but those who note actual occurrences see that very complex wholes without impro- bable parts are extremely rare. The common mind weighs the probable against the particular improbable which the evidence seems to favour; it always forgets that in ἃ priori reasoning, it is the probable against one or other of all the improbables. I now proceed to the statement of my own views : If w be a quantity which may take various values, x, #,, 73,... ἴῃ number; we have λ΄ 'Σω, Χ τ Σω", &c. for the average value, average square, &c. If y, z,... be other quantities, having pu, »,... values severally, it is clear that average product and product of averages are convertible terms, if combination of values may take place in any manner. Thus in S2°yz", we have Apv terms, which are the terms of the product =w°.dy.=z4: and division by Nuv shews that S:a?yct= Σ: αὐ. S:y . S:s4, where 2:a° means 2w*: d, the average 2. Let us now examine the average of all the values of (ἢ +y+2+...)*, which are \up... in number. Take a product of the type a y}z%, in which a+ B+ =k: and let Awv...=N. For given values of a, ὃ, ὁ, this term (with a, y, 3) occurs in every value of (w +...)* in which ἂς ἜΜ +2, is seen: that is, in N:)\yv of the terms: and the same of every term of exponents 412 Mr DE MORGAN, ON THE THEORY OF a, 3, y, whatever be its letters or suffixes. Hence, P being the coefficient of expansion of wy? 2%, we find that Σὰ" γῇ 2’,—where & refers to all terms of the type, from all the values of (ὦ +...)*—has the coefficient ΡΝ: λμν. Divide both sides by N, and we see that the multinomial theorem holds of averages, That is to say, if we expand (ὦ + y+ +...)*, and for each power of a, y, &c. write its average, we have the average value of (v+y+2%...)*. This theorem lies hid in many cases of multiple integration, Any number of values of a letter may be equal, so that by different sets of equal values, forming parts of the whole set, any probabilities of occurrence of any amount of value may be represented. If any letter have balanced values, that is, if — a occur as often as +a, a being any one of the values, it is obvious that all the average odd powers of that letter vanish, and all the average products into which any odd power enters. Thus if all the letters, or all but one, be balanced, we see that the average square of the sum is the sum of the average squares. Let the quantities x, y, *,... increase without limit in number; but to avoid the prolixity of the language of limits, let us say that the number is infinite; and let the number be o. As to the several values of the letters, those of any one may be finite or infinite in number ; the results will be in no way affected by the transition from one supposition to the other. First, let all the values of all the letters be positive, A term having h letters, with assigned exponents—of which the sum must be &—-appears in each value of (v + ...)* in as many ways as there are some sort of mutations of h out of ¢: and this number is of the order σ΄. Accordingly, σ being infinite, we need only retain the terms in which h is greatest; and the same after substitution of averages. Now h is greatest, and is =, when each letter enters only in the first power: and the multinomial coefficient is then 1.2.3...4. Hence the average k' power of (w+...) is 1.2...% times the sum of all the products of the form Σ τα =:y=:x... with & letters in each product. But, by the same reasoning, this is all that need be retained of (Z:w+Z:y+...)% Hence the following theorem:—AIl values being positive, and the number of letters which take value being infinite, the average of the ith powers of the sum of values is the 4" power of the sum of average values. By way of verification, let each of the o letters be either 0 or 1: and let n, represent the number of combinations of 7 out of σι The sum of the &™ powers of all values of x+y +... is Οὐ +1, 1° + 2,2" +... +0,0%, which is the operation (1+ £)’ performed upon 0*; where Em* =(m +1)". This is the operation (2 + A)”; and A*O* vanishing when n> k, the highest term is k,27-* A*o*, which,—since o is infinite, and A*0" = 1.2...4,—is o*27-". Divide by 2’, the number of values, and the average k'" power of the sum of values is ($0), or 5+4+...¢ terms)’, or the k power of the sum of the averages. Another simple application of this theorem, easily verified by the integral calculus, is as b follows. If da.dx, p(a+dzx).dx, &c., the elements of if pud«x, be multiplied together [1 b k & and k, the sum of all the products is (/ pode) 71.2...4. α ERRORS OF OBSERVATION. 413 Secondly, let all the letters be of balanced values. Every collection of terms of one type out of Σ (ὦ +...) now absolutely vanishes, if any one or more exponents be odd: and every collection out of =(w+...)* now vanishes, if k be odd. The term of most letters in =(v+...)* is that in which each exponent is 2, of all types that do not ageregately vanish. The number of letters is Δ, and the multinomial coefficient is 182 ΠΟΤ (Lie 2) OF 1.3...2k —1 x 1.2...k, Hence the average 2k" power of the sum of values, which I shall denote by A,,, is 1.8...9}} —1.1.2...4 multiplied by the sum of all terms of the form D:a?.S:y?... with & letters in each product. But 1.2.3...k multiplied by this product is all that is to be retained of (Σ: ὡ + D:y°+...)" or ΤΣ: (ὦ Ἐν Ἑ ---) {* or 45, Hence the following theorem :—If all the values of each letter be balanced, and the number of letters which take value be infinite, then A,, being the average 2k" power of all the values, we have Aj, Ξ NS ὌΠ 1 As a verification, let each of the o quantities take the values +1 and —1. The sum of the 2k" powers of the values is o +1, (σ — 2)*+2,(¢ -- 47" +... +6,(—o)", which is the operation (£ + ΑΓ performed upon 0, This is {2 + A? (1 + A)-1}70%, and its highest term is that which has A**(1 + A)-*, of the terms of which only ΔΙ ΟἿ has value. The term to be retained is therefore k,27-*A*0™, or, o being infinite, o*2°-"2.3...2k:2.3...% or 1.8. ὅ...9 1} --᾿ ο΄ σ΄. Dividing by 2’, the number of values, and remembering that 9 the average square for each letter is 1, we see the verification of the theorem. We may now adjust our supposition to the problems of our subject by supposing that each letter has an infinitely great number of continuous values, those infinitely near to v entering proportionally to the element of an integral, @vdv, so that the average k'" power of values of this letter is [pr.v'de, taken from one extreme, as — FE, to the other, + E. If the extreme of integration give [φυω-- 1, then, all the original values being equally likely, gvdv represents the probability of a value taken at hazard lying between v and v + dv. Let dv be called the modulus of facility of the value in question: I shall assume that [φυ v*dv, taken between extremes, is a finite quantity for every positive and integer value +a of k. The usual limits are — co and + co: I shall denote f by f*. Should finite limits ever be in question, we must deduce out of our forms the consequences of supposing gua discontinuous function of the form (-- οὐ )0(— £) dv (+ £)0(+ @), where + are the limits of error, 5 \ \ Since φῶ is an even function, f pe .a**'dv=0, And from this, and if pe . «dx being always finite, it readily follows that pa .«” vanishes when w = + ©, for all values of m and n, 0 included. Remembering Ἵ gaudex = 1, the following results are easily obtained, m, a, k, being positive integers. 414 Mr DE MORGAN, ON THE THEORY OF ‘ \ Ἵ pra .a"~*da = 0, [ pa .a"de ΞΕῚ.2.3...9πὶ: ‘ A fh pra ada = (2k +1) (2k + 2)... (2k + 2m) i: pr. a* de. Dismissing for a while the idea of the number of quantities being infinite, I now ask what is the law of facility of value which gives 4,,=1.3.5...2—1.A%, for all integer values of & We know one such law, which has the modulus ,/e.e~™:4/7: this, c being (24.)~', satisfies all the conditions. If we could determine a function V, for which if Vode =0 from k=0 upwards, for all integer values, we might add this function to the modulus already obtained. We might almost deduce, ἃ priori, that though such functions could be found if we could dispense with \ the jirst condition if Vdx = 0, the necessity of this condition is an insuperable barrier. This discussion will, however, be rendered unnecessary by the following mode of proceeding. If to the quantity whose modulus is in question we add a constant, the character of that modulus is unaltered: @a being the modulus, all we have to say is that @adwx now represents the probability of the value lying between const.+# and const.+«+dzx. If we add a quantity of variable value &, of indefinite modulus ya, we may, at the close of our investigation, so change ψῶ that every case of a Wa «dx shall diminish without limit, from £=1 upwards. We suppose Ww to be an even function, The modulus of the sum, & + 2, as we shall pre- sently see, is f ψᾷ -- x) pxdx, which, multiplied by dg, represents the probability of the sum lying between g and gq +dq. Expanding Ψᾷῳ -- 2) by Taylor’s theorem, and paying attention to preceding conditions, we have for this modulus, ὅσ being (24,)~}}, "9 y” vi Wq + c ) 3 u φ' Ww = Daa an Geta Isat vee) dv Ea efor ἐψ (ἃ Ὁ α) Ἐἐψ ἃ -- α)) de= LAE fe“ fy τῷ ψα - Φ} αι τε. .95.4 Expand ετ τ in powers of z, make the multiplication, and we have as many terms as there \ are cases of " V2.0 dx. The integrations already described shew that the only terms independent of ψα are those which arise from the cases of ἢ ΝΜ; πρὴγ ὁ «(ΞΡ ΕΘ a a ΘᾺ) ᾿ (- 1) [ν «: a*dx, which is ἘΣ Nes Bee 2. 8.0K 3.5 8:8 ERRORS OF OBSERVATION. 415 ‘ ‘ and that all the other terms give the forms (series) x ff vw . ada, (series) x i ya.atdx, &e. Hence, when these last integrals are diminished without limit, we obtain ,/c ες: / 7 as the final modulus. There is a want about the preceding investigation, and also about that of Laplace, which has never been complained of, and for a sufficient reason. It requires very high principle to scrutinise the accounts of a debtor who is always making mistakes in our favour, and always accompanies his statement by a cheque. All the methods in which e~** is employed give much better approximations than could have been expected from the demonstration, for even rather small values of σ. It would have been no matter of surprise, judging by the rejections of the process, if every decimal place of correct result had demanded a cipher in the numerical value of o. Nevertheless, we get three places when we hardly want the second, and do not deserve the first. The reason must be sought at the beginning of the process: or rather presumption of the fact; for I can give no account of the matter which sufficiently explains the phenomenon, If we take any even function gx which gives f prde =1, and if fh pu .x°dx = A,, we have, far more closely than we could expect, “: 245 2 foe dx = τ e~ “dt. 0 0 Tv Let all values be supposed equally probable, the most extreme case of a theory of errors. This supposes pu such a discontinuous function as 1 1 (- ojo (-=,) a (+=) 0 (Ἐ 9), and gives A, = (12a*)~!, and 242 = wo Let a=1. Then # =°01 gives ‘02 = ‘027; and #= 1 gives 1-- ὁ. Now try the case of uniformly descending facility, the limits being — 1 and +1. This gives (-©)0(-1)1+a#(0)1-@(41)0(+0); also A,.= 1, and a 2 N3-@ | 2 —|—)}] = —— sli Cmte ( 4 Vf 7 0 Here #=°01 gives 02 =-018; ‘05 gives 10 =°10; Ἵ gives ‘2 -- "198: “5 gives "76 = ‘78; and 1 gives 1 = ‘99. x 1 No. (+0234...) " If we assume [ pu dx = — ἡ εταἱ; 0 ν΄ πνο cives ‘ and if [ pv.akde = A,: the first approximation, in which 6, ἅς. are rejected, g ὁ = (24,)~'; the second gives Aye 5A, — ἡ (254, — TA4) ree (1 -- 9.4.) ρον ἘΠ Vor, Χ. Parr 1]. 53 416 Mr DE MORGAN, ON THE THEORY OF I shall now enter upon the consideration of the subject from its first principles, making no use of what has preceded, but treating the observations made as finite in number. When we have a number of discordant values from which to choose or construct a result, without any other knowledge than that of such similarity of circumstance of the different values as renders it impossible to prefer one to another, we naturally substitute the average for the true result unknown, upon a number of associations which are all covered by the phrase that this average is given by the observations, one with another. That the proper result should lie deep among the observations seems inevitable; and it is therefore some* kind of average, according to the general notion of the word. The mathema- tician, seeing that the balanced character of error makes it more likely that the sum of errors should be zero than anything else, however little likely in itself, sometimes claims to equate the sum of the errors to zero, and thence to deduce the most probable result; and thus he arrives at the average as the best substitute for the truth. But why does he give an exclu- sive attention to the sum of the jirs¢t powers, when zero is also the most probable value of the sum of the third powers, or of any odd powers? This question cannot be answered. If small discordance only were supposed, the first power might have an easily understood claim of preference: easily understood, not so easily admitted in full. But the argument just stated, so far as it is valid, applies equally to all amounts of discordance. The truth is that the average may stand upon a much stronger base of speculation than is usually given, namely, as that from which we have no reason for departure one way rather than the other. It is not merely the mean value of all the given values: it is also the mean supposition of all possible suppositions as to the mode of obtaining value. This may be shown in the following way. A single observation is, before others are made, the most probable truth. If the second observation agree with the first, the common value is the most probable truth: and so on, so long as the observations show no discordance. If then @ (a, a2, ...) be the most probable result of the discordant observations a,, a,, &c., the function @ is subject to the condition (a, a, ...) = a. Now in every case in which this condition must be satisfied, whatever the intent of the process may be, it may be shown that the average Sa:s is the most probable * A corndealer might dissent: for to him the average ought The farmers have a right to the word average: for its to be something near the highest, if not above it. The harvest | origin is certainly agricultural. -dveria, havings, or posses- is, at the time of reaping, never better than an average, rarely | sions, was a word applied to things in a lump: thus averia so good. ‘If the fine weather should last three weeks longer, | ponderis—whence averdepois—refers to the whole mass of we may expect an average yield,” is the strongest admission ever made in the corn market. It is not until the next crop is so far advanced as to admit of gloomy prediction that we hear of ‘*the abundant harvest with which Providence blessed the fields last year:” and this only as a covert hint not to expect the same again. All words are subject to strange mutations when they come into connexion with prices: as the daily ac- counts of the markets show. “1 did not,’’ said the farmer, ἐς σοί as much as I expected for those calves; and I never thought 1 should.” To ‘*expect” is to ** demand.” goods sold by common weight, as opposed to the selected arti- cles subject to troy weight. Averia, alone, meant precisely what a farmer now calls stock, that is, all the animals which a farm feeds. Afterwards the word was applied only to horses used in farm labour. Averagiwn was labour of the farm horses, &c. to which the lord was entitled, and for which the composition was the averpenny. Averagium also is of very old use in the sense of the loss of part of the cargo by sea or land thrown over the whole: and probably this is the use by which the word was brought into common life. ERRORS OF OBSERVATION. 417 result, so long as we have nothing by which we can compare the goodness of two different forms of @. If we want to know the most wnsafe value, the one to be avoided above all others, we easily detect @ (a, a, ...) = a, and we shall conclude that the average of discordant attempts at this determination is the most probable greatest falsehood. If jurisprudence could establish the principle that the corporate guilt of a conspiracy is that of the conspirator, when all the conspirators are equally guilty; then jurisprudence would also be compelled to take the average guilt of variously guilty conspirators as the corporate guilt of the com- bination. In the function @ (a;, a2, ...@,), symmetrical with respect to its subjects, make ἅ, ΞΞ ΤῸ +e, ας ΞΕ ΕΞ δὲ. &e., E being taken at pleasure. Expanding by Taylor’s theorem, and remembering ᾧ % E, ...) = E, we have for the function E + Pde + Qde? + Ree + SSe? + Tde’e + USeee +... where by Seee, for instance, we mean the sum of all terms of the form e,e,e,, where a, ὃ; c, are different suffixes. Now e,=e,=... =e must reduce this identically to E +e; whence, s being the number of values given, we have s-1 s-1s-2 R=0, S + Te Ξ U=0, &e. 2 2 29 ~ 8-} Ps=1 2 at Q, R, S, &c. depending on T and s only. Now E + Pe, or E + 8. ἶΣ6 is (a, + a+...) 25. Hence ᾧ (αι, ας; ... α,) is s-12a augmented by terms of which we have no knowledge what- ever, either as to sign or value, and no means of getting any: we are therefore wholly without reason for supposing that the value of (a,,...) lies on one side of the average rather than on the other, and must take this average as the most probable value ἃ priori. The average, then, is the most probable result solong as we know nothing of the law of facility of error: but this is only so long as the observations are either not made, or not dis- closed. So soon as we see the second observation, we have some information about the law of error: not much, but some. The second blow begins a fray; the second instance begins an induction; the second observation begins a law of error. If you love life, says poor Richard, don’t waste time; for time is the stuff that life is made of: if you value the results of the theory of probabilities, don’t throw away presumptions, for presumptions are the stuff that results are made of. All information must be used up: and in known obser- vations there must be information. Nevertheless, we have arrived at this result. When the observations present little dis- cordance, and differ by terms of the first order, the average can only differ from the most probable result by quantities of the second order at lowest. We may even predict that in all cases the terms of the second order must disappear; or Q=0, R=0. For if & be the absolute truth, and therefore e,, 65» &c. the real errors, we must suppose that a change of sign 53—2 418 Mr DE MORGAN, ON THE THEORY OF in all the errors would change the side of the truth on which the most probable result lies. That is, we must suppose that the coefficients of all sums of the second, fourth, &c. orders vanish. This will presently be confirmed. To deduce a law of error from observation with theoretical strictness, we should require to know, first, the truth, secondly, the individual results of an infinite number, co, of obser- vations. If ταν represent the number of the errors which lie between v and v + dv, it is then required that we should find what function of y is r+o. This function is the modulus of facility. But it will be foreseen that a preferable plan would be to determine 4,,, the average 2k power of an error, in terms of k, and then to investigate the form of @ which \ satisfies the equation [2 .v "dv = A,, for all values of k, A, being unity. I now ask what supposition we can admit as to the values of 4,,. That 4,, should be finite for all values of & is obviously indispensable: no law of error which allows large errors to occur so frequently that the average tenth power, for instance, of an error, increases without limit with the number of observations, is worth consideration for comparison with our experience. Thus it would be absurd to contemplate any result derived from the modulus 1 : πᾷ + αὖ, in which even the average error, independently of sign, is infinite. Further, we cannot doubt that all observations are made under laws which, if the units of measurement be sufficiently great, must give 4,, diminishing without limit as & increases without limit. For the errors will then be always fractional parts of a unit, and 4,, must diminish without limit as k increases. The final modulus, ,/c.e~” : 4/2; does not satisfy this condition, Be the unit of measurement what it may, 4,, increases without limit with &. The transit observer has learnt to use and to be satisfied with a modulus which asserts and takes into theory the pos- sibility of an error of a century at a single wire. Reckoning in seconds, let ὁ = 25, which gives a probable error of little less than 0.1 on each wire, and may nearly represent a tolerable observer. When k is great, A., is nearly 4/2.(&:ce)*. The average 100th power is about eighty-four thousand millions of millions, as great as it would have been if every error had been but little less than 1°,5. That is, errors of more than 1°,5 occur often enough to compensate those which are less, in the summation of 100th powers. Now,—not speak- ing of errors of clock-reading which, though errors, are self-detecting, and are corrected, and are truly no more connected with the errors we are speaking of than those which arise from setting the transit for a wrong star—we know that 1°,5 of actual error of observation is never made. The defence of our modulus lies in its sufficiency so far as A, and A, are con- cerned, beyond which we have no occasion to use it. Since [ pv.a®da is to be finite for all values of k, it is clear that gx must be of the transcendental character: and since @a must be even, e~* would seem at once to be the form on which we must depend. This function made its appearance as the means of expressing results connected with high numbers, in the hands of De Moivre, in the second edition (1738) ERRORS OF OBSERVATION. 419 of his Doctrine of Chances. 'Two extracts will show how nearly his forms approached to those of Laplace. (p. 236.) “1 also found that the Logarithm of the Ratio which the middle Term of a high Power has to any Term distant from it by an interval denoted by J, would be denoted by , a very near approximation, (supposing m = 4m) by the Quantities m+1l—4xlog.m +1-1 +m—1+4xlog.m—1+1-—2m x log.m + eee (p. 242.) ‘If, in an infinite Power, any Term be distant from the Greatest by the Interval 1, then the Hyperbolic Logarithm of the Ratio which that Term bears to the Greatest will be a+b) 2abn expressed by the Fraction — x ll; provided the Ratio of ὦ to m be not a finite Ratio, but such a one as may be conceived between any given number p and απ, so that J be expressible by p n/n in which case the two terms L and R {equidistant from the greatest] will be equal.’ Let the modulus of facility be assumed to be pe = ie «εἰ (p+qa? + rat+ sa°......). Tv Let it be found that a few at least of the observed averages 4,, A,, &c. diminish rapidly. Let (2c)"! =: then from | pe.a%da = Ay, we find (A, being unity) pt gh + 3. rh? + Spits tt sesces = I, p+ 3qh + 3.5.7? + Bole aie ae goose ΞΤ πὰ 8Ρ τ 3.5qh + 8: 5. 1 ὙΠ τ 13. 70.808 δι a τς. Ξ AE 5.» 5 ΡΠ 915. OTe 1S om eee si: eee = Agha ": and so on. If q, 7, &c. be small compared with p, we may make successive approximations, of which two will be sufficient, seeing that practice is well satisfied with the results of one. But practice does not know that one reason of her contentment with the first approximation is the practical accordance of the second with the first, in everything but value of constants. This circumstance tends to lessen our surprise at that pliability of the function e~™ which has been illustrated. For first approximation we have p=1, p=A,, or (00) Ξ 4,. And we have f/c.e™ : 4/a for the modulus. This is the well-known case: 6 is the weight of the observation ; and the probable error—critical error would be a better name—is °476936 : ν΄“. The second approximation is obtained from pt+tgqh=1, p+s3sqh=Ah', 8 - 1δη], τε Ajh-*, 8h -- A, A,—h ahi a wenn or 3h? — 64A,h + A, = 0, p= 420 Mr DE MORGAN, ON THE THEORY OF Here h has two values, which become imaginary when 34,°— A, is negative. We may reject this supposition : for when the first approximation is absolutely true, we have 34,?—4A,=0, and we may presume that, in any law we shall have to represent, large errors are more infrequent than in the first modulus, so that 4,, when second approximation is necessary, loses more than A,. The two values of ἡ are greater and less than A,, so that the values of g have different signs, with values of p greater and less than unity. To determine which sign q should take, observe that 15p — 4, will, from the third equation, take a different sign from q; so that 45 15 = τὸ - A,h7! — A,h-* 2 will have a different sign from g. This quantity vanishes in the first approximation, and will, as above explained, become positive in any law we may have to represent. Let A, = 34, -- a), a being positive. This gives 2+ 3a —a 1 h=(1 +a) 4,, P= araee 9 Gea) ass And a= /(342 — A,) : 4.4/3, by the magnitude of which we judge of the necessity for a second approximation. : δ - : Our modulus is now nye .e (p + ga*), and for the chance of an error lying between π πη 2 he if ede — 4 me-om', T (Yo 20 remembering that p+qh=1. Now q: 2c being small, and m rather small in all cases in — mand +m we have which e~° is not, Taylor’s theorem shows that the preceding is very close to q 1-2 mae (1-5, spend sey uvsilgese® J T ν΄π 0 0 ὌΠ acs CR a nearly. Hence the probable error is 3 476936 ν.34.. (: + 3 «ἢ > which is that of the first approximation increased in the proportion of 8 to 8 + 3a*, or 844." to 114. - A, This change is small if the first approximation be good; for then A, = 8344." nearly: but if this be not the case, the alteration of the probable error is of importance. The reader will observe that since ᾧ is negative, our second approximation involves the supposition that, when ὦ is great enough, the modulus is negative, which is incapable of inter- ERRORS OF OBSERVATION. 421 But the I find that if we make the hypothesis that pretation, and will remain so until we have discovered” the plusquam impossibile. numerical effect is too small to require attention. the error must lie between + 6, with a modulus of the form Ae~“” (οἷ — x’), the results, e being greater than unity, or not less, and ¢ as large as it commonly must be, do not differ by anything at all appreciable from those of our second approximation. I now proceed to consider the mode of deducing probable results, Let there be a number of functions of the quantities to be determined—say a, y, x,—and of observed constants subject to error. Let P,, P,, &c. be functions of x, y, z, and the constants, which would severally vanish if true values of variables and constants were used. Hence all value is error in’ Py, P,, &e: likely, that is, let nothing whatever be known to the contrary; and let @P,, be the modulus Let it be known that positive and negative values of P,, P,, &c. are equally of P,, with reference to the constants; that is to say, fixed values of.z, y, x, being used with observed values of the constants, the chance of the nth function lying between P, and ἜΤ ΕΒ Ol. ab,» Τὺ then lying between 2 and aw + da, &c., we know that the probability of this combination, after the observations, is Eng OP, pP....dudydz dP, dP,... divided by the complete integral of this differential. oP, φΦῬ,... χξηζ a maximum: and if all values of z, y, , be ἃ priori equally probable, in which case &, η, ζ, are constants, PP, @P,... is to be a maximum, If @,P, be A,e—**?-, then This appears to me to be the only way in which Any other method, however valuable as an illustration, would never have been allowed to impose Edw, ndy, (dz, be the probabilities, ἃ priori, of the variables The most probable conjunction of antecedents is therefore that which makes WiP,+ WeP.+... is to be a minimum. probable value can be deduced from the acknowledged foundations of the theory. a result contradictory of any result of this method. Accordingly, and treating of methods as demonstrations only, without reference to accessory value, I am much inclined to speak of all other methods as the slandered Caliph is said to have spoken of the Alexandrian books: “ΤΕ in the Koran, useless; if not, pernicious: destroy them.” In practice WP will always be a function of even form, and rapid convergence. We. have then to make a minimum of 2(W"0. P*) + 54, S(y"0. P') +... in which y/"0 is always small compared with Ψ. We begin by making ΣΟ. P?) a minimum; or, P, being dP:dzx, we solve the equations =o. PP.) = 0, ἕο. For a second approximation, sub- stitute the values of w, &c. thus obtained in 1 Σ(ΨΓΌ, P*P,), &c. and solve Σ(ΨΌ. PP,) + {the value obtained for 1 3(/"0. P°P,)} = 0, &e. We shall probably always be compelled to estimate errors by taking the most probable value of each variable as the reputed truth, and taking the reputed errors derived from these values as the real errors committed. It may be worth while, nevertheless, to show the effect * The negative probability may no doubt be an index of | the removal from possibility of the circumstances, or of the | alteration of data which must take place before possibility be- gins. But I have not yet seen a problem in which such inter- pretation was worth looking for. I have, however, stumbled upon the necessity of interpretation at the other end of the scale: as in a problem in which the chance of an event hap- pening turns out to be 24; meaning that under the given hypothesis the event must happen twice, with an even chance of happening a third time, 422 Mr DE MORGAN, ON THE THEORY OF produced upon our probable results by the supposition of an extraneous and more accurate mode of estimating errors. For this purpose it will be sufficient to take the simple average of observations of one quantity. In this case the functions are a,— @#, a.—@,...a,—a@: and the first approximation, obtained by making ={wW’0(a—.2)°} a minimum, gives for 2 the weighted average X(W/"0.a)+XW"0. Let 7 be a value which we see reason to prefer to this average for determination of errors, so that a, = 7'+e,, &c.: whence we get w= T+ Σ(ΨΌ. ὁ) (ΣΨ΄ Ὁ)“ ". Take ΣΥΨ' Ὁ (α -- 2)}, or ΣΥΨΌ (Τ' -- x + e)*}, and substitute for 7' -- ὡ. We get for the next equation of approximation Σ(ΨΌ. α — ὦ) +1 V=0, where ie ΠΣ ΚΟ δ) if Z(Ww’0.e)\? μ [2000 .e)\3 γα (30.6) - 8 3(y"0. δ) π᾿ +33 (Ψ'Ό. Ὁ) -΄᾿᾽ = SG foes When we can only obtain reputed errors, we have Σ(ψ'Ό. 6) -- 0, and T=a2. The first correction of the average is Ty". e) .- ΣΨ 0.4) 1 τ ὍΣΟΝ ΣΨΌΌ af I have verified the value of V by application of both Taylor’s theorem and the rever- sion of series to Sy,'(y - 6) =0. Supposing the observations made under one law of facility, and turning back to our second approximation, in which the variable part of ψὼ is - οὐ + log (p+ qx), we find for the correction of the average the average cube of the reputed errors multiplied by (g : p)?+ec. This is a very small quantity, having the sign of the average cube: whence we infer, as we might have expected, that a positive average cube of error indicates a presumption that the average of observations is less than the truth; and vice versa. The only law of facility under which the average is necessarily the most probable result is that in which S\//(a—a)=0 gives Σεί(α — a) = 0 independently of the values of letters and of the number of the functions. Let .’« = y(cu), and we see that Syx = 0 and =x=0 must be true together for all values of a: and we know that y# must be an odd function. Hence Syv + xv = 0 gives Σὰ + v = 0, or Lya + y(-=z) = 0, or χίξω) = Tye, which admits of no solution except yw = av. Hence we deduce ,/ee~™ : ὑπ᾿ as the only modulus which absolutely gives the average as in all cases the most probable result, The greatest mathematical difficulty of the subject, the connexion of the sum of an unlimited number of errors with its modulus, may ‘receive the following illustration of its demonstration, though of a character requiring much consideration, and at first of a repulsive aspect. It must be premised that an observation, as yet so called, need not be simply a result of perception, but may combine both thought and sense. It is enough that, being subject to error, positive and negative errors should be equally likely. Thus the average of a number of similarly situated observations may itself be considered as an observation, in anything hitherto laid down, ERRORS OF OBSERVATION, 423 Further, it appears that the average of an unlimited number of observations, under one law of facility, must be the most probable value; must be in fact the true value, For if + be the true value, the average of o observations is + +a 7'Ze; and στ΄ and Se both vanish. This can also be proved from the development of Σψία -- ῳ) -- ὁ. And next, we know that any observations which, independently of their value and number, give the average, weighted or not, as the most probable value, must be made under the modulus 4/c. e~":4/7. Now let there be any number of sets we please, each of an infinite number of observa- tions, o,, o2, &c. in number. Let each of the averages, M,, M,, ἕο. be held an ob- servation, We know that the average of the whole is the most probable result, namely, =(¢M) : Xo, independently of the number of sets of observations; consequently, the modu- lus of each is of the form asserted; but each is the average of an infinite number of observations, This argument is subject to the difficulty that ,, M,, &c. are equal, being each of them the truth in question. But if, instead of supposing the observations infinite in number, they were to be taken as only very great, and the several parts of the reasoning asserted approximately, instead of absolutely, the whole would become a demonstration of that kind which, though far from satisfactory, is cogent enough to throw doubt upon any contradictory conclusion, however arrived at, until absolute fallacy is detected. And this will never be done} for all the steps are substantially true, though requiring the introduction of limits for their explanation. I shall not enter upon the special points connected with the method of least squares, in the common case in which the functions P,, P,, &c. are of the form av +by +... To this form all cases will be reduced in practice: for when we deal with (a, y, #) we generally know approximate values of v7, y,%. If #=a2,+&, &c., where &, &c. are small, our function takes the form A + B & + &c., powers and products being rejected as inconsiderable. And &, &c. become the subjects of discussion. The probable—or critical—error depends more upon the universal use of the final modulus, 4/ce-™ : 4/z, than any other part of the subject. No attempt has been made to halve any curve of error except the final curve. The modulus being a, an even function, and 2 [pede = 1, the probable error is, approximately, 0 oh a a (0h ncn Ra gp share ὀλέϑα 4Φ 6.4. * 12,49" 10 18.47" 5 280 where ᾧ, f,, $,,» &c. are the values of px, φ΄ “, pw, Kc. when x= 0. The all important theorem that the square of the probable error of the sum is the sum of the squares of the probable errors of the aggregants, is entirely the property of the final modulus. We see that it is not lost in the second approximation: but it would not remain true in the third. The mode of assigning the quantity of the probable error is the most unsafe part of the first approximation; that is, of the simple use of the final modulus. It may be wrong, as the second approximation shows, in any proportion between that of 8 to 8 and that of 8 to 11. When any use of caution is made of the magnitude of the probable error, it would be Mor. X> Parr I ; 54 424 Mr DE MORGAN, ON THE THEORY OF advisable to increase it by one quarter before using it, if the number of observations be not very great, I shall conclude this paper by some consideration of a point which is not connected with my present subject more than with other parts of the theory, but which requires notice, were it only for the confusion of language which has often prevailed in connexion with it. Geometers have long abandoned the notion of indivisibles, in which area is a congeries of an infinite number of lines; and length of points. We may imagine a square with every line parallel to two of the sides drawn in it. The logician must say that the square is made up of an unlimited number of equal parallels: the mathematician must refuse the assertion, in every sense the admission of which would compel him to add all these equal lengths into an area. The mind is reconciled to the refusal partly by the attention being necessarily directed to the consideration of length as a magnitude per se, and of area as another and essentially different kind of magnitude. But when we come to the conception which our minds must entertain of probability, we find that the indivisibles exist, without any distinct notion of descent from one species of magnitude to another species. Suppose the square to be a target, one point of which must be hit by the head of an arrow which ends in a mathematical point : such an arrow exists in thought as much as a geometrical line. ‘That any one should name the parallel which will be struck is ineredible; that any assigned parallel should be the one struck is not incredible; for it is not impossible. What then is the probability of striking a given parallel? It is certainly not an assignable magnitude: it is certainly not even an infinitely small quantity comparable to certainty in the sense in which daw is comparable to a. It is smaller than (dx)", however great 72 may be, the side of the square being unity; and, so far as we can make a symbol for it, that symbol must be (dz)”. But it is not 0, according to usual interpretation; for only the impossible has the probability represented by 0. It is that indivisible of probability which a line is of an area. Difficulties of this kind actually present themselves in problems, and are often made to lead to a process which is quite unintelligible except as derived from an admission of indivi- sibles. A function such as (2, y, 2) is found to be as the probability that certain variables shall have exactly the values w, y, x: it is required to ascertain the probability that the variables shall lie between given limits, and instantly (a, y, x) dwdydz is put down for integration. But if x be a function of w and y, then φίω, y, s) dady is made to appear. I believe that the suppressed process is as in the following reasoning, which I take to be perfectly legitimate. Let there be a line of a length a, from which a point is to be taken at hazard, and let the probabilities of that point being at distances x and y from the commencement be in the proportion of gx to dy; required the probability that the point shall define a distance between p and g. Any infinitely small distance is made up of an infinite number of points, the number being proportional to the length. Let a be the number of points in dw: the probability of the point selected being in dx is maP, where P is between ge and d(x + dz): 7 a say this is map(w +@dz), (@<1). This is subject to the condition ( πιαφίω + Oda) = 1. ο ERRORS OF OBSERVATION. 425 But ma may be written as dw; and the probability of the point defining a distance between w and w + dw is φ(ῳ + Odx)ndw divided by i p(w + Odx)ndx; whence, by principles a common to all questions of integration, we deduce pu dx divided by yi peda, Let the pro- blem be proposed as I have stated it, and I will defy any one to produce a solution without either the distinct recognition of indivisibles which I have made, or an assumption which hides it, something which “ of course we may suppose.” In elementary writing the difficulty can often be avoided, and not merely evaded; especially the difficulty of the introduction of the differentials, and of the management of the differential of that quantity which is a function of the others. If yy and @w be the moduli of y and x, the modulus of x+y is [ ¥q-2) φῶ ἀφ, which, multiplied by dq, represents the probability that « + y shall lie between q and ¢+dq. This may be obtained as follows. That the variables shall lie between 2 +dwv and y+ dy has the probability gx Wy dw dy; and notions of integration with which we are perfectly familiar, and chiefly by \ q-2 geometrical application, give J (oe de f Wy ay} for the probability that #+y shall lie p-@ between p andg. If fry = ny, this is i ἵνα -α) — ψι - o\pede, The probability that # + y shall lie between g and q + dq is the differential of this with respect to g or [vq —x)pudx x dg. In the same manner we obtain the following form, which is more convenient in some respects than that commonly given. If @,«, be the modulus of #,, the probability that #, +...+ a, shall lie between g and q + dq is dq ios sealed PCG, = ®5_1) Ps-1(%s-1 = @,=5) cecces p(«, => 4.) PX dx,_,... eee dx, dx. If those notions, sound or unsound, clear or confused, on which a point has been connected with a line, and a line with an area, as its indivisible, were carried into the consideration of d magnitude in general, then = would be called the indivisible of y. This would more than halve the number of letters in differential coefficient; but, independently of substantial objections to the introduction of the notion into elementary writing, a greater abbreviation is wanted. The length of the most common words is a serious obstacle, especially in teaching : and no body of educated men ever had the sense of the people at large, quem penes arbitrium merely because they choose that it shall be so, Popular usage will in course of time cut down the excellent words—excellent, because they say exactly what they mean—numerator 54—2 426 Mr DE MORGAN, ON THE THEORY OF and denominator into numer and denomer, which the arithmeticians dare not do. That nothing shorter than ‘the differential coefficient of y with respect to a,’ sixteen syllables of sound and forty-three letters of writing, can be found to express the ultimate element of the differential calculus, is a misfortune and a discredit. And more especially when it is remem- bered that this conglomerate of letters does not express the modern meaning of the symbol. This meaning is ‘the limiting ratio of the increment of y to the increment of ὦ ς᾽ and, when first introduced, must be preceded by explanations which would allow ‘the limiting ratio of Ay to Aw’ to be sufficient. It is a grand absurdity that the common mame of the most common symbol, the least amount of phraseology which gives a complete designation, should be longer than its definition need be, The reform which I should propose, if it were possible to create a discussion, would consist in expressing dy : dw as ‘ the rate of y to a’ and.‘ the ὦ rate of y,’ in abbreviation of ‘the ratio of the rate of variation of y to that of 2” This is a most useful notion, and gives all the simplification of expression which can be imagined to be practicable. A. DE MORGAN, University CoLLEGE, Lonpon, July 31, 1861. ADDITION. In the last sentence a nomenclature is recommended which is simply jluvional. It is very much to be regretted that the notion of fluxions disappeared with the notation. Though satisfied that the doctrine of limits must be the basis of sound demonstration, I advocate the early introduction and use both of the infinitesimal and of the fluxional principles in aid of conception: and I observe that the fluxional principle begins to gain some currency in works published on the continent. It is not correct to make Newton the first proposer of the notion of magnitude as generated by flux: the intension and remission of the schoolmen were really positive and negative fluxions. I had made up my mind that Newton was more conversant with the schoolmen than is supposed, long before it was made known that the very scholastic Logic of Sanderson was a study of his early youth. It is impossible here to give any sufficient account of the old doctrine: I will content myself with one quotation. Nicolas D’Oresme (Horem, Oresmius) who died Bishop of Lisieux in 1382, wrote a tract De Latitudinibus Formarum, which was printed in 1482, 1486, 1515, and perhaps oftener. Though consisting of definitions and statements, without any calculus, there is in the work a certain prelibation of co-ordinates. The latitude being constant, we have a rectangle: the variation, therefore, of the latitude makes the difficulty of finding areas. Among other statements, we find that in every semicircle the intension-of the breadth (which is nothing but dy: dz positive) begins from the utmost ERRORS OF OBSERVATION. 427 degree of velocity and terminates at the utmost degree of tardity in the middle of the are. The remission (dy: dw negative) begins from the same middle point with the utmost degree of tardity, and terminates with the highest degree of velocity. But lest any body should babble about this, utmost velocity is understood in respect of any other which is not of the same kind of figure, for it is not denied that one semicircle begins with a greater velocity than another. By how much greater the semicircle, by so much greater the initial velocity* and the final tardity. Here is a clear idea of fluxional velocity, and even of infinites and zeros in other ratios than that of equality. Probably more infor- mation may be found in two manuscript works of Oresmius mentioned as existing by Fabricius (Bibl. Lat.) with the titles De uniformitate et difformitate intentionum and De proportione velocitatum in motibus. I give this account as tending to shew that the fluxional principle is not the comparatively recent introduction of one mind, but the common property of an old and wide school of thinkers. Marcu 17, 1862. * In quolibet circulo incipit intensio latitudinis a summo | non enim nego quin unus semicirculus incipiat a majori veloci- gradu velocitatis : et terminatur ad summum gradum latitudi- | tate quam alius. nam quanto semicirculus est major tanto nis tarditatis scilicet in medio puncto arcus. Remissio vero | incipit a majori velocitate intensio latitudinis sue et terminatur que incipit ab eodem medio incipit a summo gradu tarditatis | ad majorem tarditatem et e converso de remissione. (From the et terminatur ad summum gradum velocitatis patet in figura | reprint in Tannstetter’s collection of five tracts, Vienne, 1818 ; c.d. Verumtamen ne possit aliquis garrulare intelligo summam | the quotation begins on the verso of ὦ iii.) velocitatem respectu alicujus alterius quod non est talis figure: ΧΙ. On the Syllogism, No. V. and on various points of the Onymatic System. By Avucustus DE Morea, F.R.A.S., of Trinity College, Professor of Mathematics in University College, London. [Read May 4, 1863.] Tuts paper contains the following points:—1. A criticism of Hamilton’s system, as further explained in his posthumous work. 2. An explanation of the character of the system of Aristotle and his followers, which I affirm to have been exemplar. 3. The mis- conception of the character of this system by recent writers. 4. Enforcement of the right of both correlatives in any pair, and of all in any set, to equal fulness of treatment. 5. Appli- cation to the distinction of affirmation and mon-affirmation; syllogism of indecision. 6. De- duction of the eight onymatic forms from purely omymatic meaning; alleged demonstra- tion of the necessity and completeness of these forms. 7. Restrictive propositions, their affirmation and denial introduced in every view except the purely onymatic view, whenever complete treatment of all correlatives is allowed. 8. Completion of the exemplar system. 9. Extended comparison of the onymatic relations. 10. System of primary and secondary relations by copula of identification. 11. The same when the copula is any one of the simple onymatic relations. 12. The full system at which the Hamiltonian quantification aims. 13. The logical basis of extension and comprehension, [14. Addition on a recent phase of the controversy. December, 1862. | The Society is by this time aware that any introduction of philosophy proper is also the introduction of controversy ; which, though not necessarily personal in the modern sense, must be ad hominem in the old sense. Such dispute is now as nearly as possible excluded from mathematics and experimental physics: but it was not so of old. There was a time when the investigator in either was nearer to the foundation, and had more to do with the subject from the psychological point of view. It was then that Newton called philosophy—meaning physics,—a litigious lady, and said a man might as well be engaged in lawsuits as have to do with her. But though Newton and others—and Newton above all—have tamed this shrew in her dealings with mathematics and physics, she keeps her character as to all subjects in which first principles must still be probed, and questions of boundary must still be fought. And logic is a subject in which little more than a commencement of. either has been made. I have good hope that in this paper my personal part of a long discussion will come to an end. Since July, 1860, when Hamilton’s Lectures on Logic were ably published Mr DE MORGAN, ON THE SYLLOGISM, No. V. 429 by Professors Mansel and Veitch, with various additions from the scattered papers of the author, I have had what till then I never had, the means of knowing with precision what the system is which has led me into fourteen years of controversial thought and writing: how this occurred will presently appear. The whole dispute differs from many others by an inversion of character. It often happens that a contest of principles degenerates into a duel: but that which I speak of took its rise in personal accusation, and was gradually refined and sublimated into a legitimate war of systems; perhaps because there was a mathematician on one side of it. Logic affirmed that Mathematics could not understand her principles, far less extend their development: that Mathematics was a cracker of shells who could not even so get at the kernels; having no more merit than belongs to those who walk straight in a ditch. Mathematics replied, in effect, that she could understand and would cultivate the field of Logic: that Philosophy, which confessedly could not bite the kernel, had settled nothing ; while she had at least cracked the shell: and that, while she herself could either find or cut straight ditches, the only ditch in which Philosophy and Psychology had allowed Logic to walk—that into which the blind lead the blind—was one in which, for good reason, there was no walking straight at all. Such a suit is not abated by the death of one of the original parties: for it concerns undying things, and the undying part of persons. Hamilton’s mode of controversy was conspicuously ad hominem: the adverse mind was his field of psychological observation: his ways and means lay very much in inference from his opponent’s alleged errors to his opponent’s intellectual organization. Though I need not follow the example all lengths, its existence will allow me a liberty of nearer approach than I should of my own mere motion have taken; and this liberty may be used with advantage to the subject. Character and motive being left untouched, I hardly see how such freedom can be entirely avoided: anatomists may fight a theory upon a third body; but psychologists are compelled to make some dissection of each other. In order that I might finally dispose of Hamilton’s system of enunciation and of syllogism, I found it expedient to challenge contradiction of a very curious assertion by appeal to a literary journal. It was necessary that I should somewhere state, and prove from the posthumous papers, that my distinguished opponent, the great logical teacher of his day, had actually laid down, as valid syllogisms, forms of argument which were mere paralogisms. I could not expect permission to originate such an assertion in these Trans- actions: though the Society may disavow all sanction of the facts and opinions to which it gives currency, leaving the responsibility on their authors, yet there are extremes both of fact and of opinion which the disavowal will not reach. Nor could I forget that my opponent had taken pains to put tlre Society and the University itself into the position of parties to the discussion, so far as it lay with him to compel their appearance. He had in fact refused, not merely to the Society, but to the whole academic body, from the Chancellor downwards, all escape from responsibility for my logical’ heresies. I there- 1 At the beginning of the article in the Discussions, pre- bridge, then, either is the knowledge of Logic,—even of “ Logic sently noted, Hamilton says,—‘‘ If, as has been said, Mr De | not its own,”—in that seminary now absolutely null, or I am Morgan’s Memoir may represent the Transactions, the Trans- | publicly found ignorant of the very alphabet of the science actions the Society, and the Society the University of Cam- | I profess. The alternative I am unable to disown; the deci- 450 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND fore addressed two letters to the Atheneum journal (published July 13 and August 17, 1861) in which I exposed a curious apparent blunder relative to Aristotelian logic, and also one of the paralogisms above alluded to. I gave two months for denial, with notice that, if I fully expected that editors, or pupils, or other disciples, would dispute my conclusions: but (published Nov. 2 and Dec. 28, 1861), in which I pointed out how hurry, illness, and halting between two no denial were attempted, I should myself point out grounds of ewtenuation. nothing whatever appeared. I accordingly wrote two other letters systems, deprived the errors, which I then assumed to be undeniable, of the very gross Further [Having waited more than a year, and illiterate character which would have attached to them had they been deliberate. account of them will appear in the proper places. I again called the attention of the logical world to a point which nothing but testimony could settle, naming those from among whom I expected an answer. ‘This plan was successful; and an account of the defence made will be found in an Addition, I have allowed the body of the paper to stand as it was written. December, 1862. ] Rik prospectus of the intended New Analytic issued in 1846, with Requirements for a prize II. My correspondence with Hamilton in 1846-7, printed by him in 1847, with notes and additions, Ill. Analytic, the prize Essay of 1846, published in 1850, with additions, including a note by My knowledge of Hamilton’s system is derived from the following sources. Essay; both reprinted, the first with some omissions, in the Lectures on Logic. as part of our personal controversy. Mr T. Spencer Baynes’s Essay on the new Hamilton himself. IV. A review of Hamilton, myself, and others, afterwards acknowledged by Mr Mansel, in the North British Review of May, 1851. V. A letter of Hamilton, dated August 7, 1850, forthwith published in the Atheneum, and reprinted in the Discussions. VI. An article’ in the Discussions (1852) inserted probably at the last moment, under sion I care not to avoid; and the discussion, I hope, may have its uses.” The last words of this paper are—“So much for Mathematical Logic; so much for Cambridge Philosophy.” I neither claim to represent the University, nor do I admit the alternative on which Hamilton risked himself: but, when I found I must prove that my opponent, though well knowing his alphabet, spelt new words incorrectly, I thought it right that the Society and the University should not give that faint appearance of sanction which this publication in- volves, until denial from Hamilton’s followers had been chal- lenged. * This hurried article, as I shall sometimes call it, contains those quantities which are one and the same quantity, but of which the greater the one the less the other, and the apparent assertion that ‘‘some at least” is ‘* possibly none.” 11 seems to show excitement: its sarcastic photographs have stronger lights than those of preceding writings. As Hamilton had had the last word in the Atheneum journal eighteen months before, this is presumption of some new call to action. I surmise, from a stray sentence at the end, that the writer was roused, when his work was all but published, by information of the effect which my objections had produced south of the Tweed. After comparing a mathematician to an owl by day- light and a dram-drinker, Hamilton says,—‘‘ For a time, I admit, Toby Philpot may be the Champion of England.” Those who examine the whole discussion in time to come, will note the manner in which his instinct made him feel that ma- thematics would destroy his fabric, unless he could first destroy mathematics. Every proof of hurry in this article is an act of charity: the following is worth notice. Hamilton knew well (rx. i. 43) that in respect of “‘irrefragable certainty”’...... ‘Logic and Mathematics stand alone among the sciences, and their pecu- liar certainty flows from the same source.’’ He knew just as well that the contest between him and me turned wholly on the “‘ forms of intelligence’’—the necessary laws of thought— of an exact science. How then came he to object to me that a mathematician in ‘‘contingent matter” is like an owl by daylight ? How came he thus to assert, by implication, that he and 1 had been arguing contingent matter? How came he to imply that the logical half of exact science is contingent? Except under this implication his assertion, supposing it true, would not help him. The answer is that he was in a great hurry, and pelted the mathematician with whatever came up- permost. There is also hurry in Hamilton's appeal for help to War- burton’s remark, that in his time the oldest mathematician in England was the worst reasoner in it. The person alluded to was Whiston: and no man of letters, writing very deliberately, would have taken Warburton as sufficient authority against ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 431 asterisked paging (621*—652*) and not altered —except as to paging—in the second edition. To get the page of the second edition, add 55 to that of the first. VII, Various editions of Bishop Thomson’s Outlines, beginning with the second in 1849. III. The late Professor Spalding’s Introduction to Logical Science, 1857. IX. The Appendixes, passim, to Hamilton’s Lectures on Logic, 1860. If there be any other writings’ which treat Hamilton’s system at all extensively I am not acquainted with them. I shall quote these works by the numbers prefixed. Some preliminary remarks are wanted upon the quantifying words some and any. ‘The word any is affirmed by Hamilton (in V.) to be exclusively adapted to negatives. This cannot mean that any is unfit to be used in an affirmative: surely any one knows better than that. What is meant must be that no other word suits a negative, universally expressed, except any. I reply that all our quantifying words, though tolerably precise in aftirmatives, are ambiguous in negatives. ‘He has got some apples’ is very clear: ask the meaning of ‘he has not got some apples’ in a company of educated men, and the apples will be those of discord. Some will think that he may have one apple; some that he has no apple at all; some that he has not got some particular apples or species of apples. Say ‘he has not got all apples,’ and some will take him as not possessing all the apples in existence, while others will understand that he has other fruit besides apples. ‘ An apple’ and ‘the apple’ are perfectly clear: but ‘he has not got any apple’ is not free from occasional ambiguity. The word any, when used in a negative, may have either a universal or a particular meaning: it may either stand for any whatsoever, or for a certain or uncertain one or more. It has been said that a healthy person who cannot eat any wholesome food does not deserve to have any food to eat. The first any is particular; it applies, inter alios, to a person who refuses cold mutton, though ready for any other digestible: the second any is universal, and excludes all victuals whatsoever. A person who has just dined heartily need not take any food (universal): a convalescent ought not to take any food (particular; beef tea, but not pickled salmon). Some will perhaps make it depend upon the verb used; they will see the universal in ‘eed not take any food’, and the particular in ‘ought not to take any food’. Some will make it a question of emphasis, laying stress on any, when the word is particular: but the ambiguity is there, let the grammarian and rhetorician treat it as they will. A logician may, if he please, postulate that any shall always have the universal sense in technical enunciation: Hamilton did not do so, but implicitly maintained that any is always universal. Accordingly, he asserted that ‘No Χ is Y’ is properly expressed by ‘Any X is not any Y.” But though ‘No fish is fish’ be certainly false, ‘ Any fish is not any fish’ is false or true, according as the second any is universal or particular, Choose Whiston on the point in question. The two are now chiefly remembered by their several paradoxes: Warburton, by his maintenance of the absence of the doctrine of a future state from a permanent national religion being, per se, proof of Divine support; Whiston, by his acceptance of the Apostoli- cal Constitutions as genuine and authoritative. Whiston seems to have reasoned well enough from his wrong estimate of cer- tain writings: Warburton defended his peculiar thesis with great ‘‘ingenuity”’, say his admirers; but the word is one which admirers often substitute for ‘sophistry.’ ‘There is ον. Σ. Parnes | enough here to show that the condemnation of Whiston’s rea- soning upon the authority of Warburton, as a well-adjudged case, is probably nothing but hurry. ' There is an elementary work which is unfortunately spoiled by a misapprehension of the meaning of one of the forms of enunciation. But it will be a book of true method of inference to all who read the forms in the exemplar system of my second paper. The author’s mistake consists in making ‘Some X is not some Y’ the simple contradiction of ‘ All X | is all Y.’ δῦ 432 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND what fish you please, it is not any fish: turbot is not trout. This is a slight error, easily prevented by a postulate. The idioms of logical. quantity have had very little consideration given to them. The word some, although it may have points on which logicians divide, has one case of sub- division upon which all logicians unite against the world at large. The distinction is that of certain some, and some or other: the first has an unknown definiteness, the second is truly indefinite. People in general incline to the unknown definite: the logician demands the true indefinite, but can in many cases follow the usual tendency. Whenever one term of a pro- position is a definite, known or unknown, the some of the other term is the unknown defi- nite. As in ‘All men are [certain some] animals’, or as in ‘ The men he spoke of were [certain some of those who were] here yesterday’. But when one term is truly indefinite, then certain some is not admissible in the other. Thus ‘any men are [certain some] animals’ is not true when ‘any’ implies unlimited selection out of ‘all’, ‘This is most obvious in the unusual exemplar forms: thus ‘some animal is any man’ would reduce mankind to an indi- vidual if ‘ certain’ some were intended. In some subsequent parts of this paper the reader must watch himself on this point. Logie may take liberties with language for the expression of thought: but she must not declare her alterations to be actual parts of speech. I fully understand and agree to the assertion that complete quantification may be made to allow simple conversion ; that ‘ some X is not any Y’ may infer ‘Any Y is not some X’. Nevertheless, this cannot be admitted if subjection and predication remain notions attached to the subject and predicate: for pre- dication is posterior to subjection; the subject comes jirst into thought, and the question of predication follows. For instance, ‘some man is not any animal’ is a falsehood: designate the man, and a search through the animals will find him. But ‘any animal is not some man’ is true: choose any animal, man or not man, and we can then show some man which he is not. In order to make this last proposition as false as its converse, the right of prece- dence must pass over to the second side with the term which originally had it. Of “ any animal’, first chosen, ‘some man’ may be denied: of ‘some man’, first chosen, ‘any animal’ cannot be denied. The same thing in every case in which some comes into contact with all a@ any. Hamilton saw this, and it made him insist upon enunciation being pure equation or non-equation of subject and predicate, meaning identification or differentiation of simul- taneously entering terms. But Hamilton had the faculty of fastening upon his whole species any use of language into which he had drilled himself. Thus (IX. ii. 294) he says—‘* Why, for example, may I say, as I think,—Some animal is not any man; and yet not say, con- vertibly, as I still think, Any man is not some animal? For this no reason, beyond the caprice of logicians, and the elisions of common language, can be assigned.” If he should think it, he may say it: but in common language, and this with no elisions, ‘Any man is not some animal’ does not contradict ‘Every animal is man’, as he intends it should. For though every animal had been man, yet any man would not have been some animal. Com- mon language makes subjects of terms and then predicates of them. The word some has three distinct uses. First, as mon-partitive; here it is only mot-none, some-at-least, some-may-be-all. And this is the old sense of the logicians. Secondly, as ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 433 singly partitive; some-not-all, some-at-most, but without any assertion or denial about the As Hamilton says, this’ some is “both affirmative and negative”, meaning that it makes any proposition rest. Thirdly, as doubly partitive; some-at-most, and the rest the other way. which contains it both affirm and deny. I cannot find any notice of the distinction between non-partition and single partition. The logicians are much given to halt between the two. Perhaps they would defend their course, as follows:—When we say ‘‘ Some are”, meaning ‘“ Some-not-all are”, we say nothing at all about the rest. Perhaps also “‘ the other some are”. In what way then, do we differ from those who say “ Some-perhaps-all are”. Not at all, I reply, in the expressed “some” of particular subjects: but much in the implied “some” of particular predicates, in which “some not all” is of double partition by necessary inference. For example, say ‘ a// men are some animals’ ; some-perhaps-all and some-not-all no longer give equivalents. If it be some- not-all, even though nothing were intended about the rest, the exhaustion of man contained in all forces double partition: the rest of the animals are not men, Hamilton confronted mon-partition, under the name of indefinite definitude, with double partition, under the name of definite indeyinitude: the second phrase is defensible; the first is Of single partition he takes no distinctive notice whatever. false contrast. That his par- tition is really double cannot be doubted. His “some” is “ both affirmative and negative” ; he represents ‘Some X is not any Y’ as inconsistent with ‘No X is Y°. This must mean that ‘Some X is not any Y”’ tells us that the remaining X is Y. There is however no need Had Hamilton advocated single partition, all his syllogisms would have been valid; and my chal- to enlarge upon this: the diagrams and explanations (VI. 631*, 632*) are sufficient. lenge would have had a host of respondents. One opponent nearly committed himself: he imagined that Hamilton really did adopt single partition ; but he found out his mistake in time. The system which was to form the base—or one of the bases, for Hamilton permitted the old system to exist alongside of his own—of the New Analytic consisted in applying the quantifiers ‘some’ and ‘all’—in negatives ‘any’ universally taken—in every way to both subject and predicate: the word ‘some’ was intended to be doubly partitive, affirmation or denial of it was intended to convey denial or affirmation as to that ‘other some’ which in 1 The common usage of mankind inclines to partition: even the affirmation of none is, whenever it can be so made, affirmation of some of the alternative kind. No person pays any respect to the doctrine that from negative premises nothing can follow: the negatives have theirimplied affirmatives. This happens from the earliest childhood: for example, Jack Sprat could eat no fat, His wife could eat no lean; And so, betwixt them both, They licked the platter clean. How this arose we learn from the second verse, long lost to the nursery, but recovered by Mr Halliwell. For (i.e, as implied) Jack ate all the lean, And Joan ate all the fat, The bone they picked quite clean, And gave it to the cat. The same ambiguity accompanies the mention of definite number. Thus ‘four of them’ may be any four, or certain four. The context only can decide. When we are told that a man had his horse out four times in one day, we must know what we are talking about before we can tell whether it was one or other horse, or one particular horse: and the same if the phrase were ‘one of his horses.’ If the statement be made in proof of his almost living on horseback, we shall certainly sup- pose various horses: if of his want of consideration for his poor beasts, we see that one only is meant. Here the difficulty is real; raise it upon a matter in which context is not required, and we pass into the region of jokes; as in the case of the man who is reported to have been reduced to despair of compliance with the prescription by seeing on the apothecary’s label, ‘ Two of the pills to be taken three times a day.” 55—2 494 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND common language is ‘all the rest’, The hypothesis was truly and consistently applied to every form of enunciation except one: and in that one, by a curious forgetfulness, the second side of the double partition remained unnoticed. According to Hamilton, ‘Some X is not some Y’ quadrates (VI. 632*) with all the other forms, is useful only to divide a class, and (IX. ii. 283) is consistent with all the other negatives; which is true of non-partition or single partition; but is false of double partition. It is a singular commentary on Hamil- ton’s assertion of his system as actually in thought that his ‘Some X is mot some Y’, sys- tematically interpreted, is an equivalent of the Aristotelian ‘Some X is some Y’ being the simple contradiction of ‘ Any X is not any Y. This I must prove at length. Remembering that all Hamilton’s propositions are simply convertible, and that his ‘ some’ is both affirmative and negative, we see in ‘Some X is not some Y” that all the other * Some X is some Y’, that ‘Some Y is not some X’, and that all the other ‘some Y is some X’. Now all these four assertions are true when X and Y are! equivalents, when X is part of Y, when Y is part of X, and when X and Y have each part, and part only, in common with the other. wholly external each to the other, and then false: it is therefore the simple contradiction of Consequently ‘Some X is not some Y’ is true except only when X and Y are ‘Any X is not any Y’, and consequently the equivalent of the usual ‘Some X is some Y’. It would have caused but little alteration in the details of my criticism if this oversight had not been made. I now proceed to write down the forms of Hamilton’s system, on all the three suppositions: the doubly partitive case being closely taken from himself (VI. 631*, 632*) in what appears to be his latest exposition. Hamilton’s forms. Expressed in Aristotelian forms, Affirmatives. All X is all Y when doubly partitive. Every X is Y Every Y is X Every X is Y Some Y is not X when singly partitive. Every X is Y Every X is Y Every Y is X Every Y is X ὶ } Every X is Y ents x is Ye when non partitive. Toto-total* Toto-partial All X is some Y Some Y is not X Parti-total Some X is all Y Every Y is X Ὶ Every Y is X. Some X is not yf Some X is Y Every Y is X Some X is not Y Some X is Y Some X is not Y Some Y is not X Parti-partial Some X is some Y Some X is Y. 1 Take notice that the mere application of this ‘some’ de- nies that its term is singular. 3 This proposition was objected to by me as being only a compound of the toto-partial and the parti-total: this was when I supposed the partition to be wholly vague. Mr Mansel (tv. 116) declared ‘‘all X is all Y” to be a simple act of thought; and Hamilton (v1. 633*) supports this view. I now quote Hamilton’s cooler thoughts, written without an opponent in the field (1x. ii. 292). ‘ For example; if 1 think that the notion ¢riang/e contains the notion ¢ri/atera/, and again that the notion ¢frilateral contains the notion triangle; in other words if I think that each of these is inclusively and exclu- sively for perhaps includedly should have been invented] applicable to the other; I formally say, and, if I speak as I think, must say—all triangle is all trilateral.” This is all I want: here is one proposition compounded of two. Hamilton remarks that when I declare this last to be compound (v1. 633*) I do not attempt to explain how a// should be compound and some simple. I never said this, nor thought it: what I said was that the proposition X||Y is a compound of two pro- ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. Hamilton’s forms. Expressed in Aristotelian forms, Negatives. when doubly partitive. Any X is not any Y No X is Y when singly partitive. No X is Y when non partitive. Toto-total No X is Y. Toto-partial! Any X is not some Y Either toto-partial or Some Y isnot X Some Y is not X. parti-partial affirma- tive: any affirmative which contains Some Y is not X. Parti-total! Some X isnot any Y Either parti-total or Some X isnot Y Some X isnot Y. parti-partial affirma- tive: any affirmative which contains Some X is not Y. Parti-partial Some X is not some Y Cannot be false except when X and Y are singular and identical, [Should have ‘anything but a toto- been total negative’] In the hurried article (VI. 635*) we are informed in the text that the Aristotelian ‘ some’ is ‘possibly none’; and, in a note, that the Aristotelian ‘ not-some’ does not definitely exclude ‘none’. I suppose that if there be a point in which all preceding logicians agree, it is that not-some is none, and not-none is some. But I do not wish to give further attention to this extraordinary product of haste: I pass on to its source. When Hamilton combines some-at-least and some-at-most in one word, some; not-all and mnof-none are then of course constituents of the meaning of one and the same proposition. The ordinary logician, if he should choose to take ‘some-at-most, possibly none’ into his system,—as from Hamilton’s words I suspect some must have done—will see two new particulars emerge, equivalents of the old ones, but not identical with them. For ‘ some-at-least-possibly all X is Y’ is con- vertible with, ‘ Some-at-most-possibly-none X is not Y’: and ‘ Some-at-least-possibly-all X is not Y’ is convertible with ‘ Some-at-most-possibly-none X is Y’. If equivalence be for a moment confounded with identity, a person already accustomed to mot-none and nof-all in one proposition, might shape his language to the supposition that the logicians who use none limitation at least, itis a true canon. For another instance, I may cite my.own form (-). It is a remarkable instance of the positions X))Y and X((Y; true when both are true; false when either is false. It is important to note that twe wholes may compound into a third, without the parts of the two com- pounding into the parts of the third: I never said that αἱϊ is compound of somes: but only that a proposition having two alls is compounded of two propositions having each one some, 1 That disjunctively joined affirmatives should have the logical import of negatives, seems at first sight absurd: but other instances of it may be found; and I suspect that, under want of perception of analogies which characterises early spe- culation on all subjects—and which 1 look at with profit and amusement in my own earlier papers, nothing doubting that I shall in time do the same with this one—that Hamilton, who (vi. 650*) sneers at my disjunctively affirmative form of the negative (-), had not long before (v1. 632*) given two of his own negatives the same kind of form. 436 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND and all in two equivalent propositions, none in one and all in the other, use them both in one and the same proposition. I have pointed out, in the fourth of the letters alluded to, how an insufficient summary (IX. ii. 281) probably led Hamilton into the erroneous language of the hurried article (VI. 635*): it is hardly worth repeating here. I proceed to the further consideration of the system before us. I shall apply my own notation to Hamilton’s forms: thus X (-( Y will designate ‘Some X is not any Y’. It will be seen that, in the doubly partitive system, no one proposition simply contradicts another: though )-( and (-) would have done it if (.) had been truly brought under definition. I shall take for granted that when any premises are given, every conclusion which those premises can yield must be drawn, I do not mean that in the common syllogism I must be noted as a conclusion whenever A is so: because I can be inferred from A. I mean that every possible conclusion must be stated, either immediately or mediately. I will grant to the framer of a system the right to be governed by the hypotheses on which he sets out, in the acceptance or rejection of any premises. But, should he accept a certain pair of premises, I will not grant him the right to stifle a part of the conclusion because he has no form in his system by which to express it: he ought to invent the form. Against any one who demands such a right I quote Hamilton, who insists upon it that language is to be found for all that is in thought: and I aver that when premises are put into the head, all the conclusion is in thought to all who can master it. There are two ways of offending against the reasonable principle stated above. First, by curtailing the conclusion to as much as can be expressed in the system. Secondly, by excluding combinations of premises because they have no conclusion except what cannot be expressed in the system, and for no other reason. Both these faults are committed: to which must be added the still greater fault of conclusions which do not follow from the premises. The canon of validity laid down is that one premise must be affirmative (or both); and that one middle term must be universal (or both). I take this from the earlier writings, and by induction from the latest list of syllogisms: I shall not stop to consider the general canon (IX. ii. 285). It will be remembered that by affirmative and negative Hamilton refers to his own division, to his affirmatives which (all but one) contain negations, and to those negatives which are but disjunctively joined affirmatives. Speaking his lan- guage, and especially remembering that all his propositions are simply convertible, I affirm that both articles of his canon of validity are erroneous. As follows: 1. Both premises may be negative. Let us try )-():). If ‘Any X is not any Y’ and ‘Any Y is not some Z’, it follows that ‘Some Z is not any Y’, and the remaining Z is Y, and therefore not X. Consequently, we have a right to the Aristotelian conclusion, ‘Some Z is not X’. 2. Both middle terms may be particular. Let ustry )) (). If ‘All X be some Y’ and ‘Some Y is some Z’, whence ‘ Some Z (the rest) is not any Y’, it follows that all this remaining Z is not any X. Hence we have the Aristotelian conclusion, ‘Some Z is not X”. Here we see pairs of premises yielding conclusions from which we are debarred, because those conclusions are not such as require the doubly partitive ‘some’ to express them. I now pass on to the syllogisms which are allowed admission (IX. ii, 287). ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 437 Hamilton arranges these under twelve heads. Each head has three syllogisms: one with both premises affirmative ; two others formed by making one premise negative without altera- tion of quantities. Thus )()) is accompanied by )-() ) and )()-), as follows : )()) All X is all Y and All Y is some Z. ):()) Any X is not any Y and All Y is some Z. )()-) All X is all Y and Any Y is not some Z. The negative syllogisms take the number of the positive one from which they are derived, with the letters a and b. The canon of inference may be stated as follows :—-When one premise is ) ( the form of the other is that of the conclusion: in every other case the erasure of the two middle spicule shows the form of the conclusion. I now make a table, adding a word of necessary remark in certain cases, a b I MG) ἢ Oi My γί ἧ II CO) False (.) Incomplete (():) Incomplete 1Π )Q)) bd )) False GJ) ᾿ IV K.Gal ᾿ (ἀν ᾿ CC) False Vv GGG ἢ MGC f γι (ί ὦ VI DUC ᾿ didy) G ἐν ὙΠ Η VIE. ((Ὁ False ((() ἀποοιρίοις ((Ὁ) : VIII ())) False Quy t ))) Incomplete IX Cae) o CQ) False )(() * Xx ())¢ “f Clipe ν (γὴν False XI (((( t (((( False GEEG False XI }}})} ; BD) False »}}»} False Of 36 syllogisms, 21 have no error either of commission or omission: which arises as follows. Those marked (*), 15 in number, are safe because they contain ) (, the sign of equivalence. Let the other signs have any degree of absurdity, or even of contradiction, any one of them joined with ) ( only means that one of the terms is to be extracted, and an equivalent inserted in its place: consequently X ) ( Y (-( Z, for example, must give X (-(Z, let (-( mean what it may. Two others, marked (}), contain and conclude with the vague form (-), which ‘‘ quadrates with all the rest”; and their principle is that some (when singly partitive) of the part is an equivalent of some of the whole. Remember that Hamilton did not intrude double partition into the meaning of (-). Four more, marked (}), involve ‘ some” only in one term of a universal affirmative, in which double partition is of the same effect as single. All the rest—being precisely all those which give working effect to the peculiar differentie of Hamilton’s system,—are either false or incomplete: eleven false, four incom- plete. [I proved this in detail, in due compliment to the reputation of the proposer: but I omit! the proofs, because I find that the point is not to be contested. December 1862. | 1 In the Atheneum journal I took for my instance a case of | all lawyer; any lawyer is not any stone; therefore some man IV. 6, which I called the Gorgon syllogism. ‘‘Some man is | (i.e. lawyer) is not any stone (i.e. all the rest are stone).”” 498 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND I now ask what is the real basis of this system? It is formed on what I call the pepper- box plan; all and some are shaken out upon subjects and predicates in every possible way. I am a decided advocate for this process, as a preliminary mode of collecting materials: and I have now before me 512 modes of enunciation—and this only an instalment—obtained by using the pepperbox with some of the pairs of correlative notions which are scattered through the systems. It would have been well for logic if Aristotle had followed this plan. But it is an error to assume that because certain junctions of correlative concepts give an incomplete system, therefore the introduction of all the remaining junctions must complete that system. Any person who makes this supposition may become liable to the remark made by Hamilton upon Aristotle—and which 1 now make upon himself—that he commenced his synthesis before he had completed his analysis. As soon as the distribution of ‘all’ and ‘some’ had been made, and also introduction of the partitive sense of ‘some’, very slight attention would have shown that the enunciative forms present an imperfect system of the kind which I called complex in my Formal Logic, and ¢erminally precise in my third paper. Contrary or privative terms being refused admis- sion, it would have been seen that there are jive terminally precise relations; or rather, three terminally precise, and two of which one terminal ambiguity is due to the refusal of priva- tive terms, which refusal prevents statement of the relation in which one name stands to the contrary of another. On the principle—which I will not argue further, for with great per- sonal respect for its deniers, I tell them their denial is absurd—that no system of enuncia- tion can be admitted to the name until it is as powerful at denial as at assertion, and at asser- tion as at denial, five contradictions ought to have been introduced. The conjunctive proposi- tions should have brought in their disjunctive denials; and the whole would then have stood as follows. I use both Hamilton’s language and my own; but the symbols are now to express Aristotelian forms. 1. All X is some Y : X toto-partially inclusive of Y : X a sub-identical of Y : X )e) Y, conjoined of X))Y and X)-)Y. The contradiction is ‘Either X(.(Y or X((Y’, which I denote by X (,( Y. 2, All X is all Y : X toto-totally inclusive of Y : X an identical of Y : x ΠΥ, con- joined of X )) Y and X((Y. The contradiction is ‘ Either X (-( Y or X)-) Y’, denoted by Ἐκ), CY. 3. Some X is all Y : X parti-totally inclusive of Y : X a superidentical of Y : X (°( Y, conjoined of X((Y and X(-(Y. The contradiction is ‘ Either )-) Y or X)) Y’, denoted by X),) Y. 4, Any X is not any Y : X toto-totally exclusive of Y : X an external of Y, X)-(Y. The contradiction is X () Y, which, as explained, should have been the partipartial negation, ‘Some X is not some Y’ of Hamilton’s system. {Mr Baynes (Nov. 22, 1862) cheerfully accepts this syllogism | have meant, not what J suppose him to have let pass. But under the name I have given it, declares it valid, and will | what Mr Baynes takes for Hamilton’s meaning needs no de- defend it if it be ‘seriously assailed.” This is hasty writing: | fence; what I suppose him to have passed cannot be seriously he means that he will defend what he supposes Hamilton to | assailed. December, 1862.] ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 439 5. Some X is some Y : X partipartially inclusive of Y : X a complew particular (For- mal Logic, p. 66) of Y : or X (-()-) Y, conjoined of X (-(Y, X() Y, X).) Y. diction is ‘Either X)) Y, or X).(Y, or X((Y’, denoted by X)); ((Y. The contra- These enunciations constitute the system! at which Hamilton was aiming, but which permutations of ‘some’ and ‘all’ did not and could not reach. I do not think it worth while to set out all the syllogistic forms: these are best obtained by resolution into simple pairs of premises. I shall presently have occasion to exhibit a more perfect completion, I now proceed to inquire how this system was received in the time preceding the publication of Hamilton’s Lectures. The day will come when, but for such hints as I now give and the explanations which they will directly or indirectly produce, an in- quirer into the early history of the expressed quantification of the predicate would be in serious difficulty. From 1847 to 1860 he will trace a stream of eulogy and controversy, of which Hamilton’s quantification is the subject: but not a direct word does either advocate or opponent let fall about this quantification containing a very striking departure from Aristotle and his followers. Hamilton himself gives no information until 1852, when he announces his plan in terms which, to the inquirer I am supposing, will appear as clear as any terms could be: but still neither friend nor foe seems to know more about it than before. It is not until after 1860, when those remains were published which had for eight years been known by nearly sufficient extract, that all Hamilton’s admirers are suddenly and publicly challenged to show that his real* system does not lead to mere paralogism: What does all this mean? Is it reserve? which not one of them undertakes to do, Is it misapprehension ? Previously to 1852, Hamilton did not indicate intention to depart from Aristotle in the meaning of the quantifying designations. In his Prospectus (1846) he announces that he is to put the key-stone on the Aristotelic arch: not a hint is given that the but- tresses are to be changed. In his correspondence with me, not a word of so much as Ὁ Tt is clear that Hamilton never examined the syllogism upon the doubly partitive hypothesis. ΤῸ my mind by far the most probable hypothesis is that, after the attack of illness which he never wholly recovered, he really believed that he had examined the syllogism: a sudden interruption of this kind often has strange effects in the way of confusion between what had been done and what was to be done. This supposi- tion receives some confirmation from the note at the end of the table of propositional relations—‘ The preceding table may not be quite accurate in details’ (1x. ii. 284, vr. 637*). Such a memorandum in a private paper is for personal use: it was copied into the hurried article (v1.), which means that no deli- berate examination had taken place up to 1852, even of the table of propositional forms. Now it is clear that a minute verification of the cases of syllogism must have either ended in, or been preceded by, such examination of the table of enuncia- tions as would have led to the erasure of the note. 3 There is another point, which I cannot decide. Hamilton taught his own system publicly from 1840 downwards. What use of ‘some’ did he adopt? Neither he himself, nor Mr Baynes in his New Analytic, nor Bishop Thomson in his Wop OS, Ieee 10 Outlines, nor Mr Mansel in his edition of the Lectures on Logic, give any information on the point. Iput the question in my letters to the Atheneum, but no reply was made. I cannot bring myself to think that my acute opponent actually taught, year after year, a system of syllogism containing a cluster of paralogisms. I lean strongly to the supposition that he retained the Aristotelian sense, or made no further departure than the singly partitive meaning: but if this be the fact, what hinders those who can from establishing it? I repeat the ques- tion again, and I trust that, if the point can be cleared up, those who have the means will not allow me to be the only person who shows interest in Hamilton’s literary fame. For the honour of Scotland, a land noted for the logical turn of its sons, the question should be settled. Should judgment at last go by default, the decision must be that for sixteen years un- detected paralogisms formed a third part of the system of syl- logism taught in the University of Edinburgh as the ‘ key- stone of the Aristotelic arch.” [Mr Baynes replied satisfacto- rily on this point, as will appear in the Addition. I leave this note as showing what I thought on the subject when this paper was communicated. December, 1862.] 56 440 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND allusion. In my Formal Logie (1847) I published my suspicions of what the system was, in which I made it clear that I supposed the non-partitive quantity to be the one adopted. This was soon followed by Bishop Thomson’s second edition (1849) and by Mr Baynes’s Essay (1850), the first containing information communicated by Hamilton himself, the second But, though both writers drop a sentence or two which seem to hint that their own system is the singly a student’s account crowned—and augmented—by Hamilton himself. 8 Ny partitive—most writers, as already noticed, occasionally use at Jeast a singly partitive phraseology in their preliminaries—not a doubly partitive syllable escapes from either. In my second paper (1850) I made it still more apparent that I attributed only the non-partitive sense. Hamilton made an indignant remonstrance (V.) against the use of ‘*all” which I attributed to him: but not a word about “some”: [it turns out (see Addition) that he took me to be using ‘some’ in his own new sense, which first appeared in print with his criticism on my supposed objections to it. He had forgotten his own previous silence. | His editors (IX. ii.) say that his notation had a uniform’ import from 1839-40 onwards, Mr Mansel (IV. 113, 116) gives evidence (1851) of having on his mind the impression that Hamilton differs from Aristotle: but not a syllable is there in his article from which we can infer more than single partition, or at most the double partition which single partition forces out of the universal affirmative. I feel justified in so much use of our private correspondence as to state that he has informed me that all his sources were in print. He makes no allusion to Hamilton’s pamphlet (II.), probably because he did not, any more than myself, gain any knowledge of the system from this source. I rest perfectly satisfied, until contradicted, that Mr Mansel had no complete idea of the double partition, nor of its consequences. As one editor, indeed, he has given me and others the means of arriving at knowledge of the whole case: but both editors, in their short preface, imply a caveat against being supposed to agree with their principal in all points, Mr Mansel’s article is a valuable repertory” of the non-mathematical logician’s objections to the results 1 They say that “this” (p. 278) was his uniform import. By the preceding sentence it appears that “this” is ‘‘ the meaning which the author attached to them [the symbols] on the new doctrine.”” These symbols, therefore, never had more than one meaning; but they certainly were doubly partitive at Jast; therefore they were doubly partitive throughout. But the diagrams on which (and on their explanation) the note is made do not agree with the later diagrams (v1. 632*): the partial negatives, for example, are not disjoined affirmatives; and the whole gives more than a suspicion of the singly parti- tive sense. I hope that the second edition will be more precise on this point. In the text I give nothing but facts. My own belief is that Hamilton neither publicly taught, nor privately communicated to any of those who have since acknowledged communication, any thing beyond the singly partitive system. If, as his editors seem to suppose,—and not against any presumption which I can bring forward—his double partition was elabor- ated by 1846, I feel almost sure that he intentionally reserved it. He had a perfect right to do so; the same right which Titus Oates’s fox had to carry a stone over the brook to see if the ice would bear, before he attempted to carry over the goose. But such reserve always brings perplexity into history : Hamil- ton has made it easier to cook his goose than to write its biography. The following gives a strong suspicion—even more—of reserve in 1850, abandoned in 1852. Jn (v.), he says, “ The language I use is that of the logicians; only the quantity of the predicate, contained in thought, is overtly expressed...... οἷς In the reprint of this letter (v1. 0205) he adds to the words **some is not’’, the following in brackets—** [Some is should, however, have been held its direct and natural result; for, as we shall see, two particulars in the affirmative and negative forms, ought to infer each other. Compare p. 635*,sq.]”” [This makes the forgetfulness above noted very strange. | 2 I quote at length the chief point of reference :—‘‘ Before quitting this part of our subject, we will describe the principle of Mr De Morgan’s complex syllogism, as that part of his system which comes in some degree into rivalry with the quantified predicate of Sir W. Hamilton, which we are about to examine. When we say that the latter accomplishes all the ends attained by Mr De Morgan, with a vast superiority in clearness and simplicity as well as in accuracy of thinking, we have said all that is necessary in the way of criticism. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 44] of mathematical habit: and I confidently predict that it will often be cited as such when the number of those who stow both logic and mathematics in one head shall be greater than it now is. So far I have not produced a single hint. of double partition. When I examined the late Professor Spalding’s work (1857) I could not trace a phrase which was not perfectly reconcileable with the Aristotelian sense, or at most with single partition. On re-examining my copy for the purposes of the present paper, I found inserted a number of the Edinburgh Weekly Review (July 18, 1857) containing an account of Mr Spalding’s work, and citing' him as among the chief objectors to Hamilton’s junction of “some at least” and ‘‘ some at most.” Lastly, I mention myself, who might have been expected to have read the whole riddle at once in the publication (VI.) of 1852. But I, at that time, had good reason to feel estopped, as the lawyers say, from all interpretation of Hamilton’s meaning: the reason is described in the Appendix to my third paper. I found that, in spite of the most distinct assertions, as well on the part of Hamilton as of his expositors, that ‘all’ is the exponent of universal quantity, I was wholly in the wrong for not divining that ‘any’ must be used in every® negative proposition (V. passim). The sarcastic pictures caused the article (VI.) to be to me, so long as its author lived, a joke and nothing else; I mean that whenever T sat down to read in earnest I was always captured by the fun. it serious examination, my disinclination to interpret was augmented. And when, at last, I gave When I saw that Mr De Morgan refuses to quantify [ Mr Mansel means partitive- dy] the predicate ina single affirmative proposition. Accordingly, the universal affirmative, all X is Y, may form part of two complex propositions, either ‘all X is Y, and all Y is X’, or ©All X is Y, and some Y isnot X’. Hence a syllogism in Barbara which, in Sir W. Hamilton’s system, would be ex- pressed in the form ‘ All X is some Y, all Y is some Z, there- | fore all X is some Z’, becomes in Mr De Morgan’s hands the following complex reasoning [a hasty word; eapression must be meant: for Hamilton’s syllogism contains all this reasoning ; and this by the partitive force of ‘some’ ]. All X is Y, and some Y is not X. All Y is Z, and some Z is not Y. Therefore, All X is Z, and some Z is not X. The reader who is desirous of further details must seek them in Mr De Morgan’s own work. Those who will take the trouble of comparing his fourth and fifth chapters with the system we are about to describe, will, we are convinced, dis- cover abundant grounds to justify our preference for the latter. We have followed Mr De Morgan through a tedious journey, during which we have more than once had occasion to express our respect for his talents, and our regret at their perversion. We take leave of him in the words of an eminent logician and mathematician :—‘ Enimyero que confuse tantum coynoscun- tur, ea sepius confunduntur, ut adeo casus similes videantur que sunt dissimiles, et secundum ideam confusam qui agit, facile omittit quibus vel maxime fuerat opus. Atque ideo logica naturali instructus in applicatione sepissime aberrat. Exemplo nobis sunt illi qui, in mathesi cum laude versati, methodum mathematicam extra eandem perperam applicant, etsi sibi rem acu tangere videantur.’ (Wolf, Philosophia Ratio- nalis, Proleg. §19).’’ I suspect that the text of the last two sentences is corrupt: and I propose conjectural emendations. Remember that all that relates to quantity is mathematical ; for naturali read sine mathematica, for cum laude read minus, for extra eandem read inscitia naturali. 1 “ Unless indeed objection be taken, as is done by some of them [his disciples], and particularly by Professor Spald- ing, to Sir William’s employment of both the alternative meanings of the word ‘some’, as " some at least’ and ‘some at most’. There seems good reason for suspecting that the ac- ceptance of the latter interpretation would again open the door [how is this possible?] to extralogical considerations.” I again examined the work: and again without success. I then remembered that Mr Spalding himself had sent me this review, as written by a friend of his own: and I suppose his friend had mixed up reminiscences of conversation with those of the printed pages. I conclude that Mr Spalding did object to the doubly partitive system, but showed his objection only by suppres- sion, 3 Hamilton’s editors have judiciously ignored the whole controversy. But on one point they have made an indirect reference, seemingly intended to intimate that any one who Jays down ‘all’ as the symbol of universal quantity, does in fact lay down ‘any’ as its substitute in negatives. ‘They say— “The comma (,) denotes some; the colon (:) a//”’; which is all that is given in explanation of the symbols of quantity. They then say—‘‘ Thus;— C:—, A is read all Cis some A. C:+:D is read, No Cis any D.” (1x. ii. 277, 278.) The word thus implies exemplification. To read ‘No C is any D’ may be permitted: but he who ¢hwses this reading of ‘C:+:D’ upon “the colon denotes a//”’ reminds me of my old French master, an unhorsed hussar of 1815, who gravely taught that ‘‘ All the French words are derived from the Latin: thus ‘Seigneur’, which is ‘ Lord,’ comes from the Latin ‘ Dominus’.” 56—2 442 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND ‘some at least’ was ‘ possibly none’, it presented itself as on the cards that ‘some at most’ might be ‘ possibly all’, and the system in some unfathomable way Aristotelian. It must be noted that a person might take “some at most” to be singly partitive, by supposing that the limitation “at most * refers to what are spoken of: thus ‘some at most are...” might be read as ‘‘ we speak of some at most, be the rest what they may ; His ‘some’ is laid down as both affirmative and negative; his ‘some are’ is declared inconsistent 2 of these we say they are...” But Hamilton takes pains to explain his meaning. with ‘all are’, and his ‘some are not’ with ‘none are’, ἃς, [I have insisted on this, being in doubt whether it might not be denied : but I believe it is admitted. There is, however, a mode of speaking which may lead to error. It is said that Hamilton gives two systems, the “some at least” of the older logicians, and his own “ some at most ”: and the headings of his own table (VI. 637*) adopt this distinction. But it must be remembered that the ‘ some’ of the table is always ‘not-none’; so that his new system is that of ‘some at least and at most. In no other way could IFI, or ‘Some—is some— be a combination of my 0 }) ¢G as (VI. 632* diagram d) it certainly is. December, 1862. | I now come to the consideration of the genuine Aristotelian system: I mean the system which was sketched out by Aristotle and held its ground down to the end of the seventeenth century. When (1847) I began this long discussion I knew Aristotle only, or almost entirely, as a collection of books of reference. Now and then it became necessary to decide for myself which of two contradicting statements about an opinion of his was true: so soon as one or both were rejected, my business with the Organon was settled for the time. In all cases of agree- ment I took it for granted that the leader was correctly followed. This assumption lasted until I was shaken by the translation of ἀριθμὸς καὶ λόγος into number and speech’ which I exposed in my second paper. Being thus led to suspect that the mathematician Aristotle had been but loosely read, and shamelessly interpolated, by unmathematical followers, I paid more attention to his text. I took for my principle of interpretation that he meant what he said; and truly he is a writer who deserves this compliment. And I found that, though the great bulk of his ancient followers are faithful translators, our modern logicians, though nominally his adherents, have drifted into a system of quantification of their own, and have towed his name after them. When I discussed Hamilton’s system in my second paper, imagining it to be non-partitive in quantity, after pointing out that two of its propositions were without contradiction in the system, I noticed that very slight change would produce perfect logical consistency. This change was nothing but the substitution of any for all, in affirmatives as well as negatives. I proposed, though this is not absolutely required, that the implicit singularity should become explicit, as in ‘any one’ and ‘some one.’ This gives to six of the eight 1 Plato, in the Phedrus, says that Τοῦτον δὲ τὸν Ocv0 | Metaphysics: he will not allow it to be a quantity; and he πρῶτον ἀριθμόν Te καὶ λογισμὸν εὑρεῖν : but whether Thoth is held to have invented speech I cannot say. Conic sections are for mathematicians only, or it might have been that Apollonius would have passed for the first inventor of curtailment and exaggeration. Smiglecius (Disp. 9, qu. 6) remarks that Aris- totle does not count speech as quantity in the fifth book of the says that Aristotle made it quantity in the Categories only as “vulgarem ea de re opinionem secutus.’ But when or how the world at large joined number and speech as cognate quantities he does not state: nor how a writer must be held to have con- cealed his own opinion from deference in an example freely chosen by himself, where another would have done as well. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 445 propositions the meaning which they have in the common system; and makes them, as usual, three pairs of contradictories. The remaining pair ‘any one X is any one Y’ and ‘ some one X is not some one Y’, are also contradictory: the first giving X and Y as singular and identical, This system I called exemplar: its form is that of enunciation by selected example, the unlimited right of selection being expressed by any, the possibly limited right by some. This mode of expression stands opposed to the cumular form in common use: ‘All X is some Y” being the cumular of * Any one X is some one Y’; the aggregate of all its cases. Hamilton’s criticism on all this can be seen in V. and VI. At this moment I am concerned with only one sentence of it (V.): he is persuaded, he says, (VI. 6275), that my ‘ ‘Table of Exemplars’ stands alone,,.in the history of science”: he also (VI. 648*) calls it a * still- born monstrosity.’ 1 dispute his judgment in all that relates to quantification; I do not dispute his learning: I therefore quote these words as a strong testimony to my originality; and I highly value its definite character. But it only applies to half of the system: the remaining half does not stand alone in history as part of my paper. TI assert that the system of Aristotle and his followers consists of four EXEMPLAR propositions, with unquantified predicates, 1 therefore maintain that the exemplar system which I gave in 1850, as a reduction to logical consistency of Hamilton’s system, is a true! extension and step towards completion of the old system. This assertion is mere statement of a fact, and a very simple one. Do the old logicians use the singular, or do they use the plural? Do they say ‘Every, each, any,—man’, or do they say ‘ All men’? Do they say ‘Some man’, or ‘Some men’? If the first, they are exemplar, they speak by selection of example: for Every, Each, Any (with singular noun), are Every one, Each one, Any one; and Some, with a singular noun, is Some one. The modern logician says ‘All man’: he speaks of the extent of the genws ‘man’ as divisible into species: he means that the collection of individuals “ All men’—all that exist, or all that can be imagined to exist, according to the universe he is in for the time—is divi- sible into smaller collections. My assertion is that ¢ All man is animal’, thus understood, is a glaringly wrong translation of the ‘ Omnis homo est animal’ used by his foregoers, and of the πᾶς ἄνθρωπος ζῷον of the leader. Our English word αἰ], when singular, refers only to some whole divisible into parts: and ‘ a// man is animal’, before the phrase undergoes logical technicalization, is false, for it means that man is animal in legs and arms, body and soul. But the Latin omnis means each, every, any one, as in sine omni periclo, omnis parturit arbos. The whole divisible into parts is totus, totus ager, tota mens, totus in illis. And totus may naturally” replace omnis, and does: while omnis does sometimes replace totus. Thus we have omnis insula for the whole island, omnis sanguis for the whole blood. But omnis in the singular may collect the individwal from its parts, never the class; when it is all, it is all the individual, not the collected species. In Greek πᾶς is both ὅλος and ἕκαστος : but ras ἄνθρωπος is each or every man, not the whole man. Should a point be raised upon any 1 Of this, as my second paper will show, I had not the 3 In French the transposition is permanent, as in fout: the least idea when I first gave it: in my mind the exemplar sys- | language derives no word from omnis. In Italian, ¢utto and tem was a derivation, by correction, from that of Hamilton, | ogni (singular) still translate ¢ofus and omnis. which certainly suggested it. 444 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND ambiguity, and advanced in opposition to the preceding, I suppose it will be conceded that the particular quantity, if free from the same ambiguity, will settle the matter. A logician strong in ancient association naturally tends towards the Latin singular: modern habits tend towards consolidation of plurality of individuals into pieces of extent. Hamilton compromises as follows (VI. 636*). He wants to translate ‘*Some dogs do not bark” in fully quantified form. He does not say Quidam canes sunt nulla latrantia: this would offend the logical ear. Neither does he say Quidam canis est nullum latrans: this would be purely exemplar. He does say Quoddam caninum est nullum latrans: he speaks singularly of an indejinite section of dog-nature, and so conciliates the ancient exemplarity of phrase and the modern cumularity of thought. I take, beginning with Aristotle, a score or more of logicians of all ages, and of every kind of note: I choose them merely because I happen to have access to them at the time of writing. I go direct to the places in which the technical propositional forms are laid down, and to the chapters on conversion. Some writers vary their phrases a little as they get deep into their subjects: but we know that they would all desire that their systems should be described by what they lay down in their fundamental explanations. Hamilton (VI. 626*) has collected a large number of quantifying words both in Greek and Latin; and might have got more: but it would have been difficult to have found any early writer who heaped his defining chapters with all this variety, or with any noteworthy amount of it. Aristotle (Analyt. Pr. cap. 1, &c.) defines the universal as that which belongs to every-one or to no-one, τὸ παντὶ ἢ μηδενὶ : the particular as that which belongs to some-one, or not to some-one, or not to every-one, πὸ τινὶ ἢ μὴ τινὶ ἢ μὴ παντὶ. Instead of a long quota- tion from cap. 2, on conversion, I pick out all the quantitatives as they stand: they are pyde- μία, οὐδὲν, πᾶσα, τι, TIS, τι, τινὶ, τινὶ, μηδενὶ, οὐδενὶ, τινὶ, μηδενὶ, τι, παντὶ. τινὶ, μηδενὶ, οὐδενὶ, παντὶ, τινὶ, τινὶ, μηδενὶ, οὐδενὶ, τινὶ, τινὶ, παντὶ, παντὶ: not a plural among them. If all this be not exemplar, it must be because Aristotle said one and meant many. But so (by inference) does every person who says ‘Any one man is some one animal’ he means to speak of ald men, and he does it. So that in what sense soever Aristotle is not exemplar, the exemplar system itself is not exemplar. Some will say that Aristotle only distributes: then the exemplar system distributes; and that in modern use does not. Again, a person using cumular language would say that a universal negative is upset not only by predication of all, but of some: he would never say that ‘none are’ is contradicted by ‘all are’ and also by ‘some one is’; he would certainly find intermediate room for the indefinite plural some. Now Aristotle (Anal. Pr. cap. 26) says that the universal negative is destroyed if the predicate be affirmable of πᾶς or some one, εἰ παντὶ καὶ εἰ Twi: this must be every-one or some one. He had previously said that the universal affirmative is upset if the predicate can be said to belong to no one or not to some one; καὶ yap qv (ηδενὶ καὶ Av revi μὴ ὑπάρχῃ, ἀνήρηται [τὸ καθόλου κατηγορικὸν]. Hamilton, in various’ places, appends to the word all the parenthesis ‘* [or every], thus 1 In one place (1X. ii. 303) there is a boldness of assertion | sameness of a// and every which, but for repeated illustration, which may be quoted as showing that genuine feeling of the | my readers would hardly believe to have existed. Alexander ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 445 making it appear that in his mind ‘ all man is all animal” and “ Every man is every animal” are precisely the same English, and require precisely the same comment. In one place (IX, ii. 300) he translates Aristotle thus:—“ For all or every [πᾶς] does not indicate...”. Here, as elsewhere, he distinctly proclaims that he sees no difference between our English all and every in the two forms. But ‘all man’ has parts, which are species of the genus man: “every man’ has no parts, but makes assertions about the individuals of every species. I repeat that the modern logician has accustomed himself to the identification of two distinct things: he sees distribution in the cumular, and cumulation in the distributive, until the two readings are no longer distinct in his mind. He would speak of a country in which there are no single adults of either sex, as one in which all the Jacks are married to all the Joans: and, though not without ambiguity, he would be understood by mathematicians and other unlogi- cal persons after a moment’s thought. But he would also crave permission to say that every Jack is married to every Joan; which, to all but those whose English has been spoiled by modern logical technology, would enunciate the maximum of polygamy. The expositors and translators, from Boethius to Thomas Taylor, B. St Hilaire, and O. F. Owen, give correct literal translation. I find exemplar language, to the exclusion of cumular, in Paulus Venetus, the Cologne regents, Isenach, Pacius, Burgersdicius, Keckermann, Crackanthorpe, Sanderson, Aldrich, &c. On the other hand, Molinzus, Wallis, Wendelinus, and the Port-Royal, out of about forty systems which I have examined, give more or less into plural forms. hat most rigid* disciplinarian, Crackanthorpe, collected quantitative terms in profusion, and would have admitted a plural or two if such a thing had been canonical. His universal signs are omnis, quilibet, quicunque, quandocunque, nullus, nemo, nunquam: his particulars are aliquis, alius, unicus, alter, nonnullus. One of his singular terms is omnis quando collective sumitur non distributive: that is to say, the cumular is with him referred to the non-distributive” singular. He describes particular quantity as ‘ individuwm incertum et vagum,’ But the strongest testimony to the preponderance of exemplar expression is indirectly given by Hamilton himself, who says (IX. ii. 296) that the objection to “all man is all risible” because each man would then be all the class risible, “‘is only respectable by authority, through the great, the all but unexclusive, number of its allegers”. Now the original is Aphrodisiensis is quoted as saying that it is “impossible that ; certainly interpreted Aristotle in Hamilton’s sense. If (1x. all man should be all animal, as that all man should be all risible.’’ Restoring every for all, the Greek will be seen to mean that whether the terms A and B be coextensive or no, ‘Every A is every B’ is impossible; for that, without any question about the mater of the terms, each individual would be many individuals. Hamilton, with his eyes quite shut to this point, will have him speak of material impossibility, which is clear in the assertion that ‘all man is all animal.’ But seeing that there is no material impossibility in ¢ All man is all risi- ble’—which was believed to be true—he mends the text, and will have Alexander to declare this proposition only useless. Hamilton’s quotation accordingly runs thus:—‘ For it is am- possible that all man should be all animal, as [useless to say, (ἄχρηστον elrew must have dropt out)} that ald man is all risible.”’ Boethius (1x. ii. 308; Padtrolog. xiv. col. 323) has ii. 301—315) omnis be translated by every throughout, it will be seen that the Greek commentators take Aristotle in the sense I contend for, and that there is diversity among the others. 1 Hamilton generally calls him ‘Oxford Crackanthorpe’. He was for some five years fellow of a college, but his Univer- sity sympathies could not have been marked : * Puritan Crack- anthorpe’ would have been a better name; Anthony Wood would have protested against the other epithet. His book on Logic, written probably about 1600, was first published in 1622. 2 This sentence, the quotation from Pacius presently given, and other things, lead me to suspect that my word exemplar is a synonyme of the word distributive, in its old sense. 440 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND “omnis homo est omne risibile”: that is, the logicians are, almost to a man, exemplar. Let who will believe that they nearly all refused a form tantamount to ‘the class A is the class B’, because they thought that each individual in A would thereby be pronounced to be all the class B. They meant the proposition in the sense of its correct translation, ‘ Every man is every risible’, at which they laughed because, in Latin, as in English, the form implies that every separate man is every risible animal. I cannot here properly give the volume of proof which it is easy to collect of the old mode of enunciation being what I call exemplar. What I have given will be sufficient for unbiassed minds, so soon as it shall appear that no equal force of citation is to be produced on the other side. I affirm then that the exemplar table which I gave in 1850 is the Aristo- telian system, fully quantified, and made as complete in its forms as it can be so long as privative terms are excluded. But it must be remarked,— First, that the system is not originally derived from distributions of quantification and search after their meanings. The leading idea is that of assertion or denial of class being contained in class, and of class being excluded from class. Indications of this origin are not wanting. Particular negation is very frequently enunciated by ov πᾶς ἔστι; that is, by denial of total inclusion or agreement: the greatest interest in ‘some are not’ is seen in ‘not every one is.’ If quantification had been a leading idea in the mind of Aristotle, he would not have been unable to use the pepper-box: but to him' the signs of quantity were but incidents of expression. Secondly, when a term was a genus, the ewempla were species taken individually, not ultimate individuals. Thus when the quantified term was ommne animal, the hic, iste, ille, &e. of the distribution would be homo, bos, asinus, &c.: when omnis homo, if homo were injima species, the details would be Plato, Socrates, &c. Hamilton disputes the rational existence of ‘ Any one X is any one Y’, and affirms (VI. 628*) that ‘any’ and ‘any one’ necessarily imply that there are more. This is not true: we have but a strong presumption of more. My critic had arrived at a conviction that some ought to be doubly partitive: but this was his own exclusive possession, The examination of his argument will show that any has no difficulty about it except what applies equally to all. When it is clearly understood that par¢ is that which may be the whole—that is, when partition is formally excluded—it will then be seen that if there be that which is any part of Y, there can be but one part of Y, the smallest part is the whole, the whole is an individual. Any does not mecessarily imply more than one: speaking of existence at this moment, any Queen of England is any Queen of Scotland: every Queen of England that can be found is all the Queens of Scotland there are; it would be treason to deny it. The following addition to my statement as to quantification appears to me so evidently the true reading of the ancients, that I see no means of proving it to any one who, having 1 My belief is that, in the mind of Aristotle, the four | retained, that the proposition was universal or particular, not forms were merely intended to signify, in common language, | the subject. The departure from principle, which gradually the affirmation and denial of total inclusion, and the af- | clouded the theory, was the expression of denial of totality by firmation and denial of total exclusion. The entrance of | a destructive example: as denial of ‘X wholly in Y’ by ‘cer- quantifying adjectives or pronouns was only a non-essential | tain X notin Y’. But there is a vestige of creation in ob πᾶς incident of common language. Hence the old notion, so long | ἔστι; as mentioned in the text. OO Δνδπαᾷθαςυιι.. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 447 examined Aristotle, &c., entertains any doubt about it. Predication, the assertion made in an affirmative proposition, is not identification, far less equation, of subject and predicate, but simply declares the predicate notion to be true of the subject. ‘This predicate has an adjective force, is rather an attribute of the subject—a frequent name in later times—than another and containing swperject, and the proposition is very close to the character which in my third paper I have called physical. The predicate is applied in its totality to the individual of the subject class; and is distributed over as many individuals as the proposition speaks of, one by one, being given whole to each one of them. Thus ‘Every man is mortal’ says of each man all that it says of every other. No such image ever presented itself to the ancients as a notion which, instead of being applied whole, is itself cut piece- meal and assigned, bit by bit, to the bits of the subject. Such imagination is possible, because it is actual. We have before us (VI. 643*) the assertion that our attribute of mortality is divisible: that when we sum up men, we also sum up their mortalities: that Newton has this mortality, Leibnitz that, &c. But none of this is in the old notion of predication; and my present controversy is with those who arraign Aristotle and his followers at the bar of this principle, and declare that a plea to the jurisdiction must be overruled. The Hamiltonians, and many others, read their great exemplar—as I may call Aristotle —in cumular sense, until they have lost! the perception of τις, omnis, quidam, being indivi- duals: so that when a table of forms is presented in which singularity is enforced by the word one, enormous learning declares that it stands alone in history. A plain statement will show that the declarant read history through coloured glass. Aristotle (De Interpr. cap. vii.) denies quantity to a predicate: he says that no affirmative could then be true—ovdenla yap κατάφασις ἀληθῆς ἔσται. And he* instances πᾶς ἄνθρωπος πᾶν ζῷον. Wholly exemplar in his enunciation, quite ignorant that πᾶς ἄνθρωπος meant all man,—the whole eatent of the term man,—he said a plain Greek thing in a plain Greek way to Greeks who knew Greek. He said it is false—formally false, apart from the matter— that every man is every living being, meaning that then Socrates would be every living being, so would Plato, &. When he affirmed a certain quantification to be always false, he meant false in quantity. And he was perfectly right: for there never was man who was more than one living being, The proposition ‘Every X is every Y’ makes singular terms both of X and Y. 1 There is much interesting discussion in Mr Spalding’s | is in the third edition (p. 57) of Mr Mansel’s edition of Ald- Introduction to Logical Science (1857), but one single sen- | rich (1856). Here οὐ πᾶς, when a substantive is put on, is tence curiously instances the want of power to see the singular | translated not all men: another instance of the obliteration of which marks the modern logical mind. It will clearly appear | the distinction between every man and all men. that Mr Spalding was a man of extensive reading and acute 2 I noted in a former paper that the ordinary practice of perception. He says (p. 63), that the logical some is always | translating {wov into animal has led to the Tepresentation that indeterminate, some or other, not certain definite objects : “it | Aristotle ranked the immortal gods under animals. I have is always aliqui, never guidam.”” This is perfectly true; even | since found that Francis Patricius, when collecting his proofs the collector Crackanthorpe does not admit guidam. But it | that Plato was more orthodox (in the Christian sense) than should have been ‘‘it is always aliguis, never quidam,” quidam | Aristotle, cites this supposed opinion as one of his proofs. being singular. I hope none of the Greek Fathers have been belied in the The only remark on thesubject which I know of, published | same way. since Hamilton’s denial of the existence of the exemplar form, Vout. X. Part II. 57 448 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND Now Hamilton, who could not read in any other than the cumular sense, and who was possessed of the quantified predicate not merely as that which could be, and ought to be, but as that which is and must be,—asserted (IX. ii. 263), positively for this occasion only, that his great leader' talked “nonsense.” He misconceived the nature of the falsehood imputed to a universal predicate: he thought that Aristotle’s objection to ‘every man is He charges the founder of logic (sie motus Ulysses?) with rejecting a logical form on the ground every living being’ arose out of horses and dogs, rats and mice, &c., not being men. of certain matter making it false. To the last he could not see that the Aristotelian proposi- tion attributes the whole predicate to every example of the subject: to the last he fixes on Aristotle the ¢ αἰ man is all animal’, of the modern school, the erroneous translation of ‘ Omnis homo est omne animal.’ And he totally omits to notice Aristotle’s assertion that not one pro- position of the form ‘ Every A is every B’ can be true: from which, on his plan of interpreta- tion, he ought to have accused his master of denying the existence of co-extensive terms. Mr Baynes’s work (1II.) gives links which had long been dropped in the history of this discussion: and its author is a decisive instance of the manner in which Hamilton’s teaching made cumular quantity the only one known in the history of logic, and the only one which can result from scientific analysis. The ambiguity which misled Hamilton seems to have come into general discussion by the sixteenth century: for by that time, taking the common belief that only man can laugh, the disputants had completely substituted ‘Omnis homo est omne risibile for Aristotle's instance, as placing the true issue in clearer light. They then asserted, in plain and rational terms, that every man is not every laugher, for each man is only one. Mr Baynes calls this “the inconceivably inconsequent ground that if all man is all risible, then necessarily each man is all risible.” Here omnis homo is translated all man, and made to mean all men. Mr Baynes proceeds thus ‘“...to take a parallel example (one, some making up twenty. It may be so understood by postu- lation: but it never has been. But though the charge against Aristotle is a mere miscon- ception of his meaning, Hamilton fell into the very error of which he accused his leader, namely, that of rejecting a form because certain matter falsifies it. He is speaking (v1. 627*) of the use of any in affirmatives. “ Now, let us try ‘any’ as an affirmative:—‘ Any triangle is any trilateral.’ This is simple nonsense: for we should thus confound every triangle with every other, pronouncing them all to be identical. Nor, in fact, does Mr De Morgan attempt this. He wisely omits the form. But what an omission!’’ I pass over the last assertion with the observation that the very first proposition in the table here criticised is ““ Any one X is any one Y”’: 1 «The whole doctrine of the non-quantitication of the predicate is only another example of the passive sequacity of the logicians. They follow obediently in the footsteps of their great master......He prohibits once and again the annexation of the universal predesignation to the predicate. For why, he says, such predesignation would render the proposition absurd; giving as his only example and proof of this, the judgment,— All man is all animal...... Yet this nonsense (be it spoken with all reverence of the Stagirite) has imposed the precept on the systems of Logic down to the present day” (1X. ii. 263). Again (rx. ii. 296), Hamilton declares that ‘a general rule or postulate of logic is—That in the same logical unity (propo- sition or syllogism), the same term or quantification should not be changed in import.” Hence he infers that if in ‘ All man is all risible”’ the first all be distributive, so is the second. Hamilton may lay down this postulate for himself and those who like it: but there never was such a postulate in logic. On the contrary, the universal practice, down to our own time, implies that in‘ Every man is ——’, all that follows the word is is predicated of each man. If we say ‘some men are twenty men’ the obviousfalsehood drives us to metaphor: some men have each the power of twenty. But Hamilton would have it that it has been postulated that this proposition means a literal truth; i.e. this man is one, that man another, &c.—a certain these words are followed by ‘‘ giving there is but one X and one Y, and X is Y.’’ What I have here to do with is Hamil- ton’s distinct rejection of the form because it is false as applied to plural notions. It és false that ‘*any triangle is any trilate- ral’: he who makes this assertion *‘ confounds every triangle with every other’’; that is, asserts the existence of only one triangle. And it is false that ‘‘ every animal is a man”: but this does not compel the rejection of ‘* Every X is Y”’ from the forms of enunciation, on the ground of the instance declar- ing horses and dogs to be men. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 449 however, which they do not take) that if twelve inches are one foot, then necessarily each individual inch is also one foot.” The example is clearly not a parallel, but it well illustrates the divergence. The parallelism of the twelve inches and the all man is so perfect, that they make the same angle—to a tenth of a second—with omnis homo. The reader who consults the whole of the rare and interesting matter which Mr Baynes’ has produced will see much force in the following argument :—If the ancients had cumular meaning, and wrote as they did, they took inconceivably inconsequent grounds, and ought to have discovered that an inch is a foot. But it is very unlikely that so large a collection of acute thinkers should make puerile mistakes, and fail to see that a foot no longer than an inch follows from their principles, if it really did so. Therefore it is very unlikely that the ancients, writing as they did, had cumular meaning. This preliminary argument could not be weighed in time still recent, because the possibi- lity of anything except the extended term was not in thought. Some readers will perhaps never have seen, until they meet with it here, the assertion that when Aristotle and his long train of followers shaped enunciation in the singular number, it was because they were thinking as they spoke. Those readers may have learnt to see nothing but cumular in the exemplar form, nothing but plural in the singular; and they may attribute this lapse of vision to Aristotle. If so, I remind them that the singular and plural could hardly have been so easily confounded in a language which had the dual interposed, It is worth? a thought whether 1 Mr Baynes is particularly worthy of citation on this sub- | gatur has quoque enunciationes esse falsas, in quibus attribu- ject because, quite fresh from the teaching of his distinguished | tum, quod reciprocatur, adnexam habet particulam omnis, notare guide, he threw himself into the history of quantification, and | oportet, hane particulam omnis, habere vim quam in scholis made very valuable researches. It is curious to see how he | vocant distributivam; ut omnis homo, proinde valeat atque qui- speaks of cumular quantification, as both a logical and a ver- | libet homo, vel singuli homines ; et similiter omne animal, idem nacular necessity. Speaking of the exemplar grounds of ob- | valet, quod singula animalia, vel unumquodque animal seu jection to ‘* Every man is every risible,” which to him could | quodlibet animal. Quapropter si veré diceretur, ‘omne animal be nothing but “all man is all risible,” he says (p.93),‘‘When | est omne sensu preditum’ etiam homo esset omne sensu pre- we consider these grounds, and remember the real ability of the | ditum; nam qui dixit omne animal, non exclusit omnem ho- men by whom they were successively urged, we cannot but be | minem, homo igitur esset quodlibet sensu preditum: proinde struck with a wonder amounting to marvel, that they could | hac ratione fieret, ut homo esset equus, et bos, quandoquidem remain satisfied with them, and that a truth so obvious on its | equus et bos sunt sensu predita.’’ (Pacius in Aristot. de In- first enunciation, so imperative on its fuller exposition, should | ¢erpr.cap. vir.) The reader will see how clearly Pacius has have been so uniformly and so long thus rejected.” Neverthe- | laid down the difference between exemplar and cumular, and less, the clear exposition of Pacius, so deservedly high among | how distinctly he has stated that the exemplar, not the cumular, the expositors of Aristotle, would have stopped the wonder of | is the ancient reading. any one who knew the exemplar sense: though somewhat long, | Should a teacher be so accustomed to read exemplar enun- I repeat it from Mr Baynes, ‘‘ Hunc errorem ut Aristoteles | ciation in cumular sense that the first time an exemplar table tollat, ostendit universalem notam nunguam posse adjungi | is explicitly presented he declares it to stand alone in the attributo; quia tunc omnis affirmatio falsa esset. Quod de- | history of science, that teacher and his pupils may well regard clarat exemplo hujus enunciationis ‘omnis homo est omne | the above with ‘“‘a wonder amounting to marvel’. The issue animal,’ que sine dubio falsa est: nam si homo esset omne | isa simple one. Aristotle, Pacius, &c. say—We enunciate in animal, esset etiam asinus et bos. Sed notare hic oportet, alia | the exemplar form of thought: the moderns reply—You do no attributa latius patere quam subjecta, alia vero reciprocari cum | such thing; orif you do, we have lost the power of seeing the subjectis......Ubi igitur attributum latius patet, ut in exemplo | distinction, whence there is no difference. Aristotelis, res dubitatione caret: certum enim est, affirmatio- ® Were it only because it has hardly been thought of. In nem esse falsam, nec posse dici, ‘omnem hominem esse omne | English the confusion of singular and plural has occurred. animal’, Sed merito dubitatur de attributis, que reciprocan- | The proverb says, ‘‘One’s none; two’s some”. That one tur cum subjectis, veluti si quis dicat, ‘omne animal est omne | should be none (ne one, not one) defies etymology: but as not sensu preditum’, et ‘omnis homo est omne aptum ad riden- | one, and on/y one, both deny some with its ordinary plural im- dum’: nam hic absurditas illa non qué apparet, ut in illa | plication, and some and none pass for alternatives in life as well enunciatione, ‘omnis homo est omne animal’. Sed ut intelli- | as in logic, the way in which the confusion arises is seen. 57—2 450 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND the possible ambiguity arising out of the dual and plural did or did not dictate adherence to the singular: the plural must have frequently taken in the dual, frequently not: that is, the plural must have frequently meant two or more, frequently more than two. If this be the explanation, it does not alter the fact. And to the fact must be joined the utter extinction of the exemplar form in the minds of modern logicians: an extinction so thorough that such a chart of logical history as Hamilton’s mind had not that course laid down, even as a possibility. It was easier to him to imagine Aristotle talking such “nonsense” as that a form must be rejected because it was not true of all matter, than meaning his singular number to speak of one: and a table of exemplar forms published in 1850, appeared to him to stand alone in history. The exemplar form of statement is that both of geometry and of algebra. A proposition in Euclid assumes some one case of satisfaction of hypothesis, and the demonstration lies in the perception of the receiving mind that nothing in the reasoning is adverse to the implied assertion that this some one may be any one. But this form of exact science is more pointedly exemplar in phraseology than the system of Aristotle: its distributor is quilibet, not omnis. We have an exemplar ladder in English: its steps are any one, each one, every one (quilibet, unusquisque, omnis). The third is certainly not all; for it is but one : but it is more truly the grammatical singular of all than either of the others, near as they are. It would be difficult to describe the differences of meaning: that there are differences will appear by our being able to make sentences in which all three shall occur, without power of transposition, For ex- ample :—“If you feel able to cope with any one, try each one, and so you will master every one”—the order cannot here be altered. The first, any one, has a purer unitarian character than the others: the second and third are more nearly transposible. Without further inquiry, any one is the most proper for strict exemplar use, as being applicable in negative predicates. The form ‘any one X is any one Υ 15. much wanted in geometry. In my last paper I pointed out that many indirect demonstrations are only refusals of the knowledge of contraposition ; others, far less excusable, arise from refusal of the right to convert ‘any one X is any one Y.. When X and Y are of the same number of instances, the propositions ‘Every X is Y’ and ‘ Every Y is X’ are equivalent: which is most evident, if there be any gradations of evidence, when X and Y are singular, Consequently, if there be but one X and one Y, and if the X be the Y, it need not be proved that the Y is the X. Suppose that a person, holding himself to have shown that Junius was an individual, and knowing that Philip Francis was an individual, and that Francis was Junius, were to proceed as follows to prove that Junius was Francis :—If not, let Junius be X, another than Francis: then be- cause Francis is Junius, and Junius is X, it follows that Francis is X; that is, Francis is another than Francis, which is absurd. So it is, and so are you too, would be the answer of common sense to the proposer of such a proof: is the principle of difference so much clearer than that of identity, that any one has a right to suppose the sameness of ‘X is ¥Y° and‘Y is X’ to want corroboration by help of X being no other than itself? But this is done in geometry. Not to insist on antiquity, let us take Legendre, a professed amender of Euclid: he knows that through a point can be drawn but one perpendicular to apiane, and one parallel to a line; yet ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 451 (Book V. Prop. 7) having proved that the parallel to a certain line is the perpendicular, And this is supposed to be the climax of rigour; proof by syllogism that if the sole A be the sole B, the he gives further proof, of the kind' above, that the perpendicular is the parallel. sole B is the sole A. But where would syllogism have been, if this had not been true ? It must not be forgotten, in defence of Euclid, and of geometry without logic, that the above procedure may give evidence. When thought has not been analysed, and those who teach are determined that it shall not be analysed, Euclid presents the perfection of the way of doing without. But the time must come when his rich mass of raw thought shall be the material of exercise for logical analysis; when it shall be employed to place the forms of thought in their due order of sequence; when it shall be the ground on which it shall be learnt that the conversion of identity by help of syllogism is reasoning in a circle. I shall proceed to connect the exemplar form with others: but there are several points which it will first be desirable to notice. By a restrictive proposition I mean one which, of its own nature, imposes some absolute condition, positive or negative, upon the quantity of one or both of its terms, or of one or both of the contraries of its terms. I say absolute condition: not relative, as in ‘ All X is Y’, which demands that the Ys shall at least equal the Xs in number. The only such propositions yet met with are ‘Some X is not some Y’, which requires that, when identical, X and Y are not singular: and ‘Any X is any Y’ which imposes on X and Y both singu- larity and identity. But besides singular identity, we shall find ourselves, so soon as we begin to carry every mode of enunciation into every case, obliged to recognise penultimate identity, in which the contraries of our two terms are singular and identical; also singular and penultimate identity, in which both’ our terms and their contraries are singular and iden- tical; and singular and penultimate contrariety, in which two singular terms are each iden- tical with the contrary of the other. the score, as we shall see. The laws of thought will produce these forms* by 1 We may laugh at the geometer establishing by syllogism the conversion of identity, but such is the force of habit that the logician may be a geometer without carrying away into logic the illustrations which lie nearest the surface. My op- ponent, Mr Mansel—out of formal logic—is a mathematician, and applies psychological thought to first principles. In formal logic he argues in favour of * All A is all B” being a simple proposition, in opposition (1v. 116) to my assertion that it is complex; and Hamilton quotes his argument with approbation. Mr Mansel says “1 cannot assert ‘all A is B and all Bis A’ without having thought of A and B as coextensive, i.e. without haying made the judgment ‘all A is all B’.” Euclid (1. 4), the universe being triangle, proves that “all isosceles is isogo- nal’’, and then (1. 6), proves that ‘‘all isogonal is isusceles”; and then, and not till then, does his reader become aware that “all isosceles is all isogonal.’’ Both the components are in thought before the compound. Geometry is the richest field of coextensive notions: it swarms with instances of coextension gained by synthesis of counter-inclusions. I admit that a com- pound cannot be decomposed except by those who have got it to decompose: but, on the other hand, those who have hold of the components may put them together. In the dining-room pudding may be treated as compound of flour and plums: but if before that, in the kitchen, flour and plums had not been treated as components of pudding, the dining-room process would have been Barmecide theory. 2 Tam duly sensible of the figure which a universe of two instances will cut: but I may say on my own behalf, that though I shook it out of the pepperbox, I did not put it in. The laws of thought, which did put it in, are solely responsible for this contempt of established authority. Nor can I even claim the invention of the mode of shaking which brought it out. Hamilton had used the method, and produced, if not singular identity, at least its denial: this was the first of the class of restrictives. I think that here, as elsewhere, it will be found that one instance is but ill understood until more arrive. 8 By introducing ‘‘ some X is not some Y ”’, the denial of a restrictive, Hamilton, when non-partitively interpreted, has given a conclusion to two invalid forms (.( )) and (()-). It will presently be pointed out that every one of the thirty-two invalid forms gives a conclusion, the denial of a restrictive. 452 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND The time is coming when no one of two correlatives will be introduced without as full Logie abounds in pairs! of which both must enter thought Of converse relations, and of contrary (or contradictory) relations, we generally see one embodied, an introduction of the other. together, but of which one only has been allowed to become prominent in language. while the other is but as a shadow. Part and whole give a marked instance: our language is familiar with a whole of several parts, but hardly knows such a phrase as ‘a part of several wholes’. How loosely the subject of correlation is considered may be seen in the case of assertion and denial. In logical writings these are—I do not say defined, but—treated as alter- natives. In the wide world it is generally assumed that all which a person cannot assert he can and will deny: let any one hesitate at affirming, and four out of five of his hearers will report him as having contradicted; and the four will be precisely those who see no use in logic. The books on logic so far favour this inaccuracy that they take no notice of any inter- mediate’ between affirmation and negation, ‘he following brief summary will show how easily a sufficient notation of syllogism will enable us to collect all cases of what I shall call indecision. I mean inferential indecision; in which inability to affirm or deny a conclusion is a necessary consequence of inability to affirm or deny a premise. When two premises, A and B, give a conclusion ©, it follows from the usual law of opponent reduction, as I call it, that the assertion of either premise, with hesitation at denial of the other, is equal hesitation at denial of the conclusion, For one premise, with denial of the conclusion, is denial of the other premise. Hence any hesitation at affirmation of the contrary of the other premise, is equal hesitation at affirmation of the contrary of the con- clusion, ‘That is to say, there are syllogisms in which assertion and non-denial give non- denial; there are others in which assertion and non-assertion give non-assertion: of four possible forms these are the most systematic ; each form including the other three. The syllogisms of undecided denial, in which assertion and non-denial give non-denial, are precisely those in which assertion and assertion give assertion. ‘Thus )))-) gives )-); or X))Y):)Z gives X)-)Z. Assert either X)) Y or Y)-) Z, and refuse to deny the other, and we must refuse to deny X)-)Z. This gives rise to two forms of the other kind. Assert X)) Y, and refuse to affirm Y ((Z, or assert Y)-)Z, and refuse to affirm X(.(Y; in either case we must refuse to affirm X ((Ζ. 1 Many common words, when they represent material ob- jects, have meaning of which relation to other objects is an essential part; whence arises some confusion, An island is land surrounded by water: is the surrounding water a part of the island? Yes, for no water, no island: no, for if you walk into the water, you quit the island. The ambiguity is easily explained in this case: there is the object named, and the rela- tion by which it is named: the object does not extend into the water, but the drojts of the notion do, perhaps as far as those of the crown. Again, whatis a box? Is it a space bounded by an envelope of wood, or is it the envelope itself? Not the first, for we certainly move a box from town to town, which no one can do toa bitof space, And yet, when I asked a little girl what would happen if the nails used in fixing a card of address were too Jong, she answered that they would “ get indo the bow, and spoil the things.”’ We get over these ambiguities in common life; but they are sore puzzles in philosophy. 2 “But negation and affirmation must be contradictorily opposed ; as Aristotle has expressed it,—‘ Between affirmation and negation there is no mean,’”” (Hamilton, vi. 636"). True enough so far as this, that of affirmation and negation one must be true and the other false; but not true of enunciation. I may not know which is true and which is false; I may have the courage to avow it, and to follow Hamilton's principle of finding language for all that is in thought. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 453 In syllogisms of undecided assertion, in which assertion and non-assertion give non-asser- tion, the law of validity is as follows, When one proposition only is particular, that parti- cular must be the undecided assertion. Every form is valid in which a universal and a par- ticular occur: but when both are universal, or both particular, the middle term must be balanced, that is, of the same quantity in both. The symbol of the conclusion is derived as in the ordinary syllogism, with this exception, that the spicula which we are to obtain from the decided proposition must be inverted. Thus, denoting want of power to assert by ~ affixed, we may shew that ))()~ gives ()”, )))-)” gives (), Οὐ (( gives ΟἿ, (-)°C) gives ((", )(")-) gives )-(", &. But ))))” and )-))-)” give no conclusion. For example :—‘‘ We can hardly undertake to say that all men are responsible for the effects of their actions, independently of motive: for there are men who are really incapable of any consecutive tracing of consequences, a thing we must hesitate to affirm of beings whose responsibility is for consequences”, This form is (-()-)”, giving ))~; as follows: (-( Some men are not capable of tracing consequences. ):)” We will not affirm that there are beings responsible for consequences who are inca- pable of tracing consequences. ))° Therefore we will not affirm that all men are responsible for consequences. For will not, we may read’ must not, cannot, ought not, need not, &c., provided only that we make the conclusion follow the premise; all that is wanted is mon-affirmation, be the restraining cause what it may. The forms of indecision are precisely those in which affirm- ation and denial give denial: but the mere presentation of indecision would have been a valuable addition to the logic of the middle ages. Here there was nothing but sharp asser- tion and denial: and theology, the science in which the word dogmatism got its evil sense, was made to look even more positive than she really was. Forbearance is not categorical ; and the syllogism of charity is the syllogism of indecision. The portion of all possible thought within which our concepts are and are? to be 1 The terms of relation can be applied: and it will be good exercise to learn to see the combinations. If we call ‘that which we cannot affirm to be a species’ an unaffirmed species, we may read as follows. In X))¥((“Z, or X((¥Z, we see that a species of an unaffirmed genus of Z is itself an unaffirmed genus of Z. In ))-((, giving ))Y, we see that an unaffirmed species of a genus is an unaffirmed species. In (( )-)¥, giving )-)¥, we see that the genus of an unaffirmed deticient is itself an unaffirmed deficient. In )(¥)-(, giving )-)’, we see that the unaffirmed coinadequate of an external is an unaftirmed deficient. 2 Falling asleep while I was considering how to answer this objection—that a definite universe is material—in the most elementary form, I found Logicus, Mathematicus, and Neuter, in the middle of an argument upon the very point. L. In ** All X is Y” we have a pure form of thought, divested of matter: we see how we think, independently of what. N. It’s not true, though. M. He does not mean that whenever he says X he says Y. L. By no means: X and Y are names; and my proposition asserts that whatever I may name X, I may name Y. N. Why, so may I, or so may any man; but — L. Nay! I meant with truth, according to received meanings : X and Y are representations of concepts, and the concept X is asserted as what ought never to be in thought without the con- cept Y. M. But concepts are matter of thought, are they not ? L, Yes: but X and Y are but concepts as concepts, recognised as different concepts by difference of symbol, stated to be thought as included and including by the proposition. M. But if your form contain concepts as concepts, and if concept be matter, surely your form contains matter as matter. N. You wont get out of that, I see, let concept be which it will, Greek or Hebrew ; it may be one or the other forme. L. You con- firm me entirely in what I was going to say, that the goodness of formal inference may be perceived independently of the meaning of the terms; concept is to you as would be X or Y. M. Then my remark is admitted to be just? L. Certainly : matter as matter is present in every enunciation; but the per- ception of the formal force of a proposition is independent of the material differences between the different matters which it contains or might contain. ΜΙ, That is to say, you treat con- cept as algebra treats number? L. Precisely : logic preceded algebra in the use of general terms. M. But algebra never 454 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND contained is the universe. When that universe is in any way divided into two parts, the name by which the individuals in one part are distinguished from those in the other is a term. ΑἸ] terms are names; but some names are not terms. When animal is the universe, hairy is a term, a divider of the universe: mineral is not a term, but a vacuous name; sentient, sensu preditus, is not a term, but an omnitenent name; mineral and sentient equally fail to divide the universe, the first by non-continence, the second by non-exclusion. These contraries, the vacuous and the omnitenent, must stand or fall together. When we speak of terms only, we see as clearly that contrary terms have no term which is a common whole as that they have no common part; for nothing less than the universe contains both: no term contains both. To what I have said (in former papers) on the exclusion of omnitenent' names, I add that, even in the prevailing system, the predicate of a negative must not be of universal extent, for then some of the subject would be shut out of the universe in which it is to be: and that if the predicate of an affirmative be a universal, the proposition asserts no more than is held to be asserted of the subject by its mere presence. In order fairly to put the exemplar and cumular forms into connexion, it is necessary to examine them with the fullest introduction of both sides of every correlation which makes any appearance at all. Until lately I have never felt assured that they were not two different systems, presenting points of agreement. But before making the investigation, it may be shown that neither one system nor the other can claim to dictate the precise forms of enunciation. That claim is made by another system, more fundamental than either; and is made demonstratively. The logicians have admitted only one idea of relation: the connexion between terms as terms: I call the system thus produced by the name of onymatic. They make what ap- pears to me a confusion between the term and the objects of thought which it represents: they identify terms which are not identical as terms, whenever they can identify the objects represented. Now two terms, as terms, whatever may be the case in etymology, cannot have any relation to each other in logic except what they gain by their relations to things signified or excluded. cable. And the only relation of a term to a thing is that of applicable or not appli- And a term, as a term, has its contrary: a term? without a contrary is no term. 1 This universe is sometimes all that exists objectively, and sometimes all that can exist in thought. If there be any one who demands yet more, and wants room for that which cannot talks about a pure form of numbering from which matter of number is excluded. With us numbers lie hid in sealed pack- ets, marked outside with letters: but they ave numbers, whether before or after assignment or discovery of their values ; differ- ences of value exist or may exist, though ignored as to amount so long as only the consequences of difference as difference are in question. L. It is, I dare say, not quite correct to affirm that the form of the proposition is void of matter: we introduce different matters, leaving the differences unsymbolised, except as differences. But for this, the form should rather be “‘ Every is » than “ Every X is Y’. M. Then what objec- tion do you make, looking at the way in which man thinks his thought and says his say, to the introduction of a sphere or universe, say U, on the same terms as X or Y: as material as they are, as unspecified with reference to ¢his or that as they are; allowing full right to consider, as one case, what I might perhaps denote by U==? What Logicus answered I could not even dream; so I awoke. be in thought, whether as possible or impossible, he invades the universe of a higher power, and will perhaps square the circle; a problem which a speculator of the last century re- duced to the following,—Construere mundum divine menti analogum. 2 When Aristotle practically dismissed the privative term under the name of aoris/, he had previously denied it to be the name of anything. My beliefis that he was inclined to deny that it isa ¢erm; he thought that not-man, for instance, takes in so much, and shuts out so little, that it is hardly distinctive If such were his idea, he would have refused, ἃ fortiori, the title of name to a word which designates the whole universe, both man and not-man; which shuts out nothing whatever. As to the aorist character, I should like to know, supposing a name to include just half the universe, which is the aorist, ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 455 The relations between terms, the only ones admissible because they wre terms and for no other reason, are those of applicable to some the same object, and not applicable to any the same olject. If X and Y be two terms, x and y their contraries, then, making full use of all our correlative alternatives,—namely X or x, Y or y, of joint application or not of joint application,—we shall obtain what must be all the forms of enunciation admissible into the system of relations between terms as terms. And from our purely onymatic enunciations we may decipher the common forms of identification, or of discrimination, in which the distine- tion of term and designated object of thought is afterwards lost to language! by the application of is and is not to the terms. The results are as follows: Onymatic relation between terms. Proposition. Relation. Symbol. X, Yhave joint application | Some Xs are Ys X partient of Y ΣΦΙ, δα X, Y have no joint application | No Χ is Y X external of Y ΧΟ ΠΟΥ X, y have joint application | Some Xs are not Ys X exient of Y X(-(Y X, y have no joint application | Every X is Y X species of Y ONG x,y have joint application | Some things neither Xs nor Ys Χ coinadequate of Y XCM x, y have no joint application | Everything either X or Y X complement of Y ΧΟΥΥ x, Y have joint application | Some Ys are not Xs X deficient of Y dC od) VE x, Y have no joint application | Every Y is X X genus of Y ΧΕ (ΟὟ The moment we begin to speak of part of a term, we are no longer using the term in the purest onymatic sense: we have made it stand for the collective group to each individual of which it applies as a designation. Before we introduce the word part, I observe that, as every relation has both its converse and its contrary, it is advisable in every case to examine both conversion and contradiction. One converse of ‘X, Y, have joint appli- cation’ is ‘there are objects to which both of the terms, X, Y, are applicable’, We have nothing to remark about this conversion except that it furnishes the most natural mode of reading the new propositions (-) and )(. The above table exhausts, I think demonstratively, all purely onymatic relation; that is, all in which the terms are names to be applied or not applied, not names used jor objects by conventional substitution. There is no notion of quantity in this system: the affirmatives—the assertions of joint application out of which the particulars spring—de- mand ‘one or more’ objects to which joint application is made. But this is only tanta- mount to ‘There exists that which...’ and its quantity is only the notion of one which precedes numeration in Omne quod est, eo quod est, singulare est. I have not space to that name, or its privative? This is the most nicely balanced | before an unpractised mind without warning, and reason may, question in logic, just as the following, which even Noles and Queries cannot answer, is the most nicely balanced question in geography. If all the northern hemisphere were land, and all the southern-hemisphere water, which should we have to say, that the northern hemisphere is an island, or the southern hemisphere a lake? 1 am Buridan’s ass in respect to both questions. 1 This distinction is usually obliterated in all cases in which the term has meaning. But let abstraction be placed Vou. X. Parr II. properly enough, refuse the identification of the terms by the substantive verb. A book on logic was presented to a young person of my acquaintance: after some time an account of pro- gress was asked for. “Oh!” was the answer, ‘‘I read as far as ‘ Every X is Y’, but I knew that wasn’t true, so I left off.”” Assuredly ‘no X is Y’: every child who learns the alphabet is plagued with 650 such negations. But it may chance that every [thing signified by] X is [also one of the things signified by] Y. 58 450 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND develope the objections which the pure onymatic system, as well as other views, furnish against the Hamiltonian doctrine that all enunciation is equation of quantity: but even those who would not admit their force will guess what they are. Let us now introduce the notion of multitude of objects which, considered as having a com- mon designation, give the idea of class, part of the universe separated from the rest. Each class —except when singular—has sub-classes which are its parts, and—except when penultimate— is a sub-class of classes which are its wholes. Any collection of objects which is itself only part of the universe may be called a class, as capable of receiving a common designation which is also distinctive. We shall find the eight onymatic forms starting up in the following simple appearance, without the reality, of system: this I say because, as shall be shewn, we have only a systematic selection from a complete system. The remainder, after the selection is made, will contain restrictive propositions, or their denials. And this will happen in all attempts to systematize which involve quantity, and which make a full use of all correlatives which are admitted at all. Observe that we do not admit the universe as distinctively a whole, because it is a whole of all terms, and not itself a term, Some class is part of both X and Y X partient of Y () No class is part of both X and Y X external of Y )-( Some class is whole of both X and Y X coinadequate of Y aC No class is whole of both X and Y X complement of Y (-) Some class is whole of X and part of Y | X species of Y ys) No class is whole of X and part of Y X exient of Y (-( Some class is part of X and whole of Y | X genus of Y aC No class is part of X and whole of Y X deficient of Y )-) Here we see terms without their contraries; ‘some’ with one terminal extreme, ‘none’, but without the other, ‘every’; conjunctions, as ‘both part of X and part of Y’, without the corresponding disjunctions, as in ‘either part of X or part of Y’; conjunctions of affirma- tions only, without the corresponding cases of one affirmation and one negation, or of two negations. If the whole system were formed, every case which does not reproduce one of the above, would either require terms coextensive with the universe, or penultimate, or singular ; er would deny propositions requiring such terms. But as this point will presently receive sufficient illustration, I shall proceed no further with it at present: I shall also presently have occasion to go some way into the extension. Both the preceding systems of enunciation have an exemplar character: in both the forms we see ‘there does exist an instance of...’ denied by ‘ there does not exist any instance of... I will now proceed to an exemplar system in which part or whole of one term is in affirmation identified with part or whole of the other; the unlimited selection any, and the possibly limited selection some, either or both, being used in all combinations. The restric- tive propositions will be denoted as follows: singular identity by (:=); penultimate identity by (=:); singular and penultimate identity by (72); singular and penultimate contrariety (3 ὁ). And that singular identity in which one term and the contrary of the other are singular and identical, may be denoted by (.—) or by (—-), as convenient. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 457 Part and Part. Whole and Whole. Any part of X is any partof Y :—= Any whole of X is any whole of Y =: Some part of X is not some part of Y :== denied | Some whole of X is not some whole of Y =: denied Any part of X is some part of Y_) ) Any whole of X is some whole of Y ( ( Some part of X is not any part of Y (-( Some whole of X is not any whole of Y )- Some part of X is any partof Y (( Some whole of X is any whole of Y ) ) Any part of X is not some part of Y_).) Any whole of X is not some whole of Y (-( Some part of X is some part of Y () Some whole of X is some whole of Y ) ( Any part of X is not any part of Y_ )-( Any whole of X is not any whole of Y (-) Only (-) ) ( excluded. Only ):( () excluded. Part and Whole. Whole and Part. Any part of X is any whole of Y ὃ: Any whole of X is any, ‘partiof Ye 8: Some part of Χ is not some whole of Y :: denied | Some whole of X is not some part of Y ¢: denied Any partof Xis some whole of Y := Any whole of X is some part of Y =: Some part of X isnot any whole of Y := denied | Some whole of X is not any part of Y =: denied Some part of X is any whole of Y =: Some whole of X is any part of Y := Any part of X is not some whole of Y —: denied | Any whole of X is not some part of Y :—= denied Some part of X is some whole of Y ( ( Some whole of X is some part of Y ) ) Any part of X is not any whole of Y )-) Any whole of X is not any part of Y (-( Only ((_ )-) included. Only )) (-( included. The symmetry and compensation of this table is an instance of what we shall always find whenever correlatives are fairly and equally used. By carrying the whole through Xy, xy, and xY, as well as XY, we produce the main system eight times, and complete the system of restrictives. We may call the system of part and part and of whole and whole by the name of balanced'; the others being unbalanced. The rules of distinction and identification of forms are as follows:—1. Balanced readings exclude from the general system nothing but any affirmed of any and some denied of some: unbalanced readings admit nothing but some affirmed of some and any denied of any. 2. When exclusion is not thereby made admis- sion, or vice versa, ‘any part’ and ‘some whole’ are convertible, as also ‘any whole’ and ‘some part’. Thus ‘Some part of X is some whole of Ὑ is the same proposition as * Any whole of X is some whole of Y’. There are two positions which have, alone or together, been expressed or implied in several distinct quarters. First, that the mere completed distribution of the quantifying words is the completion of a true logical system, dictated by the laws of thought. Secondly, that the eight forms first obtained by complete distribution of contrary terms through the old forms is an arbitrary system, which might have been something else if the framer had so pleased. I con- tend that these descriptions should be exchanged: that the arbitrary character, but not to so great an extent as asserted of the other, belongs to Hamilton’s system before the correc- tion which makes it simply the true extension of the real Aristotelian system; and that the 1 These useful terms, suggested by Hamilton, may be used in reference to any pair of correlatives. 58—2 458 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND extended cumular system is not in any sense arbitrary. Take what plan we please, carry the correlations fairly out, and we arrive at the eight onymatic forms, together with restric- tives and their denials. I shall take two more cases, observing that restrictives have appeared in every system except the original, in which nothing appears except terms as distinctive names, under the relation to objects of applicable or not applicable. First, I take the exemplar form in which some or other extent, or any extent—some class or any class—is identified, conjunctively or disjunctively, with both whole or part of X and whole or part of Y. This is that portion of extension which I previously announced that I should give. I write down only the apparent affirmatives, leaving the reader to con- struct the negatives: for brevity, I also write ‘some X’ for ‘some one part of X’, and ‘any X’ for ‘any one part of X°. And first of conjunctive comparisons. Any class is both any X and any Y Universe of one individual. No terms Some class is both any X and any Y X and Y singular and identical Any class is both any X and some Y Universe of one individual. No terms Some class is both any X and some Y X species of Y. ) ) Any class is both some X and some Y X and Y universal. No terms Some class is both some X and some Y X partient of Y. () Any class is both some X and any Y Universe of one individual. No terms Some class is both some X and any Y X genus of Y. ( ( Among these assertions and their denials we have the Aristotelian forms complete: and our assertions give the affirmatives, our denials the negatives. The disjunctive forms may now follow; either meaning either or both, the true’ contrary of neither. For ‘ both’ and ‘ and’ substitute ‘ either’ and ‘or’: none® but restrictives will be found. In going through all the varieties of application of part and whole, we come upon the complement, yet unseen, among the correlative affirmations of exclusion: as in ‘ Any class is either not any whole of X or not any whole of Y’. But the view opens as we proceed. Part and whole are but synonymes of species and genus: at our present point we may ask what would result if we were to examine all the cases of ‘ Any [or some] class is both — of X and — of Y’ when either blank may be filled up with any of the eight names of relation? I certainly should not have asked this question if the answer had required me to exhibit to the reader such a shaking of the pepperbox as would seem necessary. The truth is that I have all but answered the question in previous writings, as shall presently appear. I positively assert that the first of the preceding views® contains demonstration that the relations between terms, derived from their relations to objects, must be the eight forms, and no others. The postulates are that by a term we mean a distinguishing mark, the sign of some object or objects, not the sign of others; and that to any collection of objects which is not the whole universe, we have a right to assign a term. I contend, as in my last paper, 1 “Shall I bring both ?—No need, either will do.” Here | dividuals, one X and one Y: but “any individual is either the either is either or both. some X or some Y ” means X (-) Y, without restriction. 2 If individual were used instead of class, the restriction 8 The first idea of this mode of derivation is in my Formal would be removed from some propositions. Thus ‘‘any class | Logic (p. 105): but I did not then see either the import or the is either some X or some Y”’ enunciates a universe of two in- | importance of what was there given. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 459 for the right and duty of logic to treat of other relations between terms, derived from the relations of objects to one another: but my present concern is with onymatic relations only. I proceed to a more systematic connexion of the eight forms than I have yet given. Each universal is in two ways of a universal character, one of an active meaning, the other of a passive. Thus, X )) Y|X wholly included in and wholly incompletive of Y X (( Y |X wholly including and wholly uncompleted by Y X)-(Y |X wholly excluding and wholly excluded from Y X (-) Y |X wholly completive of and wholly completed by Y Each particular has also two characters: and by each character is inferentially attached Thus X()Y affirms that X is partially or wholly included in Y, and that X partially or wholly includes Y: and X (-(Y affirms that X is partially or wholly excluded to a universal, from Y, and is partially or wholly completive of Y. Again, four of the relations may be called greater, and four less. A greater relation is one which cannot be changed into its contrary without subtraction: a lesser relation is one which cannot be changed into its contrary without addition. The greater relations are ((, (-), (-(, (), being all of which the minor term is particular: the lesser relations are )), )-(, ).), )(, being all of which the minor term is universal. The Aristotelian collection includes the lesser universals and the greater particulars. Each universal has a contranominal, with which it may coexist; and two extreme’ contra- ries or extreme contradictories. Thus X))Y has the contranominal X((Y = x))y and the extreme contraries X )-( Y and X (-) Y. Hence we see the connexion of each universal with two inferred particulars. Each partial proposition asserts the existence of an indefinite share of the extreme extent by Thus ‘ wholly included in’ which is also ‘wholly incompletive of’, or )), necessarily contains ‘ partially which the universal is toto orbe divisum from one of its extreme contraries. included in’ and ‘ partially incompletive of’, () and )(, which are indefinite contraries (com- monly called contradictories) of )-( and (-), of each of which )) is an extreme contradiction. The connexion of the contranominals, through their extreme contraries and the particulars, is illustrated in the adjacent table (W., wholly; P., partially). backwards, the spicular symbols being still read forwards. The lines may also be read 1 T hold by the amalgamation of the words contradictory and contrary, in spite of the disapprobation of some who have approved various points of my system. And this I do first, because the common language makes synonymes of the two: he who contradicts maintains the contrary. And this even from the mouths of persons versed in technical logic. Dr Clarke said of Collier the idealist ‘‘he can neither prove his point himself, nor can the contrary be proved against him.” Secondly, the etymology does not support the distinction. Thirdly, the true opposition is that of any contradiction and the extreme or total contradiction. ‘All are’’; contradiction, some (perhaps all) are not; extreme contradiction, none are. Fourthly, the existing terms hide the distinction, and give a notion which makes a logician say, ‘‘so far from being the con- tradictory, it is not even the contrary.’” 400 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND )-¢ > W. excluded from = W. excluding (+) > W. completed by = W. completive of )i): deerinaks ieee (-( < W. included in P. excluded from (ClC macs ἘΣ.) - W. uncompleted by Ρ. completed by nis oes P. excluding W. including (GC sooscasepaos )) P. completive of | W. incompletive of )) > Oe a Gnas 2 ee 0 < (( W. included in P. included in W. excluded from = W. excluding P. including W. including (( > Wieck tapind orssonseeiae Hane ete eee aten SUC < 9) W. uncompleted by P.uncompleted by W. completed by = W. completive of Ne Gasca op a) < )) = W. excluded from P. included in W. included in = W. incompletive of P. incompletive of |W. incompletive of iC : SO) P. incompletive of W. completive of Θὺ------.- γ - (GC > Πρ Φπτ,ᾷ ῪΨ W. completed by P. uncompleted by W. uncompleted by = W. including P. including W. excluding W. excluded from Ρ. excluded from W. included in = W. incompletive of P. completiveof |W. completive of (+) > δι Ji eo ener eet sau Ey Ceres οὐ ξεν ὦ τ W. completed by P. completed by W. uncompleted by = W. including P. excluding W. excluding In this table contradiction is denoted by a dotted line; and ascent or descent by the algebraic signs for less and greater. Common language proceeds as if the part were more worthy than the whole, as a notion on which to base enunciation. Accordingly, we are familiar with inclusion, exclusion, and I have somewhere read of a speculator who maintained that every world has, in some other part of partience of both: but completion’ and coinadequacy are strange and heretical. space, a counterpart world of defects equal and opposite to its own. If his system be true— all questions about other stars or planets are quite open—there is somewhere a planet in which thought fixes upon whole in preference to part; in which the concept of penultimacy is more familiar than that of singularity ; in which the demonstrative pronoun is not ¢his, but some word of the force of all else; and in which, at this moment, some antimathematical logician—for the mathematical tendency is in excess in the logic of owr counterpart—endeavours to force attention to exclusion and partience upon a community which is too exclusively familiar with completion and coinadequacy. I have amused myself with constructing enunciations and syl- logisms as they are in the exemplar-counterpart forms of our counterpart planet; from which 1 Mr Spalding (vir1. 166) says that all the eight forms are set forth by Boethius. I cannot find them. Boethius does indeed apply the four to privatives, and so obtains equivalents of the eight onymatic forms: but I cannot detect him evolving relations between the given terms by help of their privatives. But he is rather prolix; and perhaps some reader may favour me with a definite reference to something which will support Mr Spalding’s assertion. If not, that assertion is one of a very numerous class, of the bad consequence of which no one can form an idea who is not familiar with the history of discovery. Two things are affirmed to be the same because the passage from one to the other is easy in the mind of the affirmant, after study of both: they are virtually the same, one amounts to the other, &c. This was Solomon’s practice, or he never would have said that there is nothing new under the sun. I once had a private discussion of several long letters which might have been spared if my correspondent had said at first what he said vat last, that certain two methods were the same 0 ail intents and purposes: he began by saying they were the same; which is quite a different thing. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 461 planet I myself may have come, if there be any truth in the doctrine of transmigration. If any of those who are too firmly rooted in our common notions will do the same, they will derive the same sort of benefit which young arithmeticians—who all think that 10 must be ten—derive from constructing systems on another radix. We want a better balance of logical correlatives. The original tendencies of language, partial, one-sided, stopping at just enough, have tied some of our mental muscles until they only act by special volition and a good deal of it. And we appeal to the defect in proof of the necessary character of the liga- tures; to the incapacity of the slave in proof of the inexpediency of emancipation. As to untying a ligature, that would be extralogical and material. Every universe of objects has its universe of relations, to which I now come. At the outset I am met by a difficulty which is shared by writers on perspective, and no way of escape is better than theirs. They cannot put solid objects into a book: so they draw a perspective figure to be the object, and then draw a tablet, a painter, and a collection of rays projecting the object on the tablet. Imitating this plan, let the symbols 1, 2, 3, 4, &e. be attached to the objects of the universe as, in the strictest sense, proper names: it being understood that these names imply no quality, and are assigned to the objects at hazard. Objects are thus distinguished by their ἀριθμοί, the word being used in the true Greek sense described in my last paper. What are commonly called proper names are frequently nothing but singular names, derived from notions of class; Horatius Flaccus shews both genus— or at least gens—and difference. We have a right to treat any collection of objects, from one inclusive upwards, as a class; to be distinguished from the contrary class, containing all other objects, by a mark. I am not afraid, at this time, of being met by the old dictum that the differentia of a species’ must be of the essence: but a little of the spirit of this demand may yet be left. disposed to think that selections exist—they will not say elasses—the individuals of which Some may be really have no common difference, nothing which distinguishes them, and them alone, from all other things. I challenge such a selection. While awaiting an answer I imagine an ac- 1 This is a question on which heretics have differed. Cicero affirmed Trojan and Theban to be species of man. Ludovicus Vives, heretic, and Johannes*Rivius, orthodox, declare Cicero wrong, on the ground that the species must have an essential difference. Marius Nizolius,a worse heretic, describes them as ‘“quorum uterque audet reprehendere Ciceronem’’, forgetting that Aristotle, on various points, is described by himself through four long books (De veris principiis et vera ratione philosophandi, contra Pseudo-philosophos, 1553) as Philoso- phaster and Pseudo-philosophus. I give his distinction of species, husk and all :—‘‘ Quis te docuit, O inepte grammati- cule, hominem, etiam si extra ordinem substantie non egredi- amur, non posse esse verum genus Thebani et Trojani...... Quare tu quoque disce vérum esse id quod dicit Cicero, Troja- num et Thebanum esse veras hominis species, si non essentiales at certe accidentales, et cognosce ea, que tu ex sterquilinio dialecticorum hauriens contra Ciceronem nugaris, nihil aliud esse nisi meras insanias,”’ Nizolius, great as the author of the Thesaurus Ciceronia- nus,—we have seen how sensitive he was on Cicero—is in logic a small handler of a large theme; and very scurrilous withal. G. L. (whom Tiraboschi and others assert to be Leibnitz, whose initials were G. G. L.) republished the De veris Prin- cipiis in 1674, with a preface. But G. L. according to the Bodleian catalogue, altered the title into ‘ Antibarbarus philo- sophicus; sive philosophia scholasticorum impugnata’: in other words, Leibnitz (?) saw that Nizolius was more useful against the schoolmen than in favour of truth. Tiraboschi leaves every one to decide for himself whether he will judge by the appro- bation of Leibnitz, or the disparagement of a modern writer, who expresses great surprise that Leibnitz should have pub- lished an edition. I judge by the book itself, which appears to me that of an emancipated slave, who made a new master of his liberty. Nizolius, arguing against what he supposes to be the scholastic doctrine, namely, that a genus contains only things present, strengthens the opposite opinion by the autho- rity (idem quogue confirmatur ab auctoritate) of Julius Pollux, who, in what he says περὶ γένουφ, includes both ancestors and posterity. = 402 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND ceptor; and I think I do nearly as well for him as he could do for himself, if I suppose him to select from the universe ‘ material object, past or present’, as a lot which he defies me to difference from all other things, the following miscellany ;—all men who have killed their brothers, the hundred largest ink-stands that ever were made, and Aristotle’s dinner on his twenty-first birthday. What is the class-mark of these objects? I answer that to . them alone belongs the epithet—‘ Selected by the fancy of (here insert name and date) in unsuccessful impeachment of the unlimited right of logical division’. I am willing to go further than Nizolius, and to divide species into essential, accidental, and perverse; affirming that the difference is extralogical. The more absurd such an instance as mine, the better does it make the claim asserted; Hamilton implied the like when he presented Newton and Leibnitz with their wigs awry. If the number of objects in the universe be 71. the number of possible collections which can be the selections denoted by terms is 2"-2, the number of pairs of collections is (g"~*—1)(2"—3) and the whole universe of relations, true and false, has 8(2"-*—1)(2"—3) instances, equally divided between true and false. Let the relations species, ewient, &c. be denoted by the symbols )), ((( &c: thus X)) and ((X both denote ‘species of X’, When a symbol of relation is placed between two others, let it be read in the singular exemplar method; and let the two extremes be read from the middle term, Thus (( (3 )-( or X(( (-) ) (ἃ means to assert that ‘ Any one class is either species of X or external of Y’: and X)) )) ))Y means ‘ Any one genus of X is some one species of Y*. Of such possible readings there are 8.8.8, or 512, of which half are restrictives, and half are not. I may be asked whether such methods of stating propositions are actually in use? I answer yes, sometimes in grave writing, and more often in rhetorical flourish, a kind of appeal to assent in which a little study of the characters of fallacy is not obviously needless. A certain sort of speaker wants to say that ald Englishmen are lovers of liberty: for your stump-orator deals in nothing but universals, be the name of his stump what it may; a proceeding forced upon him by the lovers of his style, who consider a man of rules with exceptions as an equivocator and a loophole-monger, He declaims as follows:—‘Show me any number of men, and I will say with confidence either that they will with one accord raise their voices for liberty, or that there are aliens among them.’ This figure of speech is X((Z expressed as X(((*) (‘(Z, where X is ‘lover of liberty ᾿ and Z is “Englishman’, Every proposition is a blank syllogism: that is, every true proposition is a conclusion which has middle eatents, whether the terms exist for them or not. Thus X))Y is X))0))Y, where for 0 may be written any genus of X which is also species of Y. It is also Χ)" (0(:)Y, where for 0 may be written the contrary of any such intermediate class. Even the useless ex- treme X))X may be written X))X))X. And the blank syllogism and the conclusion are con- vertible: thus X))Y is X))0))Y, and X))0))Y is X))¥. When the concrete middle term is inserted, this convertibility ceases: thus X))Y is deducible from X))A))Y, but not X))A))Y from X))Y. The essential of syllogism is the ewistence of the middle term, not its being this or that. The conclusion, as I have observed in a former paper, renounces all knowledge of the middle except its existence. ‘That ‘all man is mortal’ is established by every one who shall prove that a genus of man is a species of mortal: the physiologist may have ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 463 to think of the middle term ‘animal’, the theologian of the middle term ‘sinner’; but to both it is enough for the conclusion that a middle term exists, This explicit reduction of the middle term to mere existence is, I think, essential to the formal consideration of the syllogism, In such a proposition as )) )*( (-(, the spicule being, 12, 34, 56, let 12, 56, be primary relations, 34 the secondary relation, relation of the second order, or relation of relations, Let the spicule 3, 4, be means; 2, 5, adjacents; 1, 6, ewtremes. Take notice that the secondary relation is the common identification, or its denial: thus 12))56 is not ‘21X species of 56Y’, but ‘ Any 21X is one 56Y, some 56Y’, Of 512 secondary propositions, 256 are valid representations of unrestricted onymatic forms: the remaining 256 are either assertions or denials of restrictives. The unrestricted forms may! be obtained as follows: 32 of them are the forms of syllogism, with blank middle terms, and the secondary (); 32 more are the contradictions which deny that a middle term can be found, with the secondary )-(. Three other sets of 64 each are found by varying the readings of the first 64 in the same manner as X()Y and X)-(Y are varied by use of x and y for X and Y. Thus, the proposition X((Z being a necessary consequence of X(-) Y)-(Z is an equivalent of X(-) 0)-(Z, and of X(-) ())-(Z. That is, *X genus of Z’ is an equivalent of ‘Some complement of X is some external of Z’. The denial is ‘ Any complement of X is not any external of Z’, X(-) ).( )-(Z, which is denial of X((Z, or an equivalent of X).)Z, or ‘X deficient of Z’. The eight varieties, four of each proposition, are as follows, relaxing the exemplar form into ordinary reading. ΧΟ () ) (ὦ Some complement of X is external of Z ΧΟ ¢ ( ()Z Some complement of X is not partient of Z Χ) ( )( ()Z Some class is neither coinadequate of X nor partient of Z X)()*) ) (ὦ Some external of Z is not coinadequate of X. Here are four secondary ways of saying ‘ X is genus of Z’. Again, ΧΟ )γ() (ὦ No complement of X is external of Z X(‘) )) ()Z Every complement of X is partient of Z X)((*) ()Z Every class is either coinadequate of X or partient of Z X)( (( γὼ Every external of Z is coinadequate of X. Here are four secondary ways of saying ‘ X is deficient of Z’, We can now give meaning to the 32 compositions which fail to show valid conclusion: they are all denials of restrictives. For instance X (( Y )) Z gives no conclusion: and this is X (*( () ))4. There is a term, says the proposition, which is both deficient of X and species of Z. Of course there is, will be the first reply ; must every species of Z fill up X? Certainly not, unless every individual of Z be all X ; that is, unless Z and X be singular and identical, Consequently, X(( Y ))Z has a conclusion; it denies ‘Any X is any Z’; and we have one of Hamilton’s syllogisms, when the non-partitive ‘some’ is used. The secondary 1 J did not obtain them so easily, for I worked through the ; The reader may thus be made more sure of the completeness 512 cases separately and independently, before I saw what, | of my investigation. when seen, was also seen to be what ought to have been seen. | Vor. Xo) Pare Tr 59 494 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND form X(-( )-( ))Z affirms that no deficient of X is a species of Z, and affirms ‘any X is any Z°, or denies that ‘Some X is not some Z’. Again, X (.( ())-) Z expresses that some class is deficient both of X and Z. To deny it is to say that every class is genus either of X or of Z, which gives only two individuals to the universe, one X and one Z. The law which regulates these cases is very easily given, First, when the invalid form is either a universal and a particular, or two particulars of unbalanced middle terms, as }) 0. ):C):), or )-))-), CQ)G Let there be two singular and identical terms, of course with penultimate and identical contraries. When X is particular, let it be one of the singular terms; when X is universal, let it be one of the contraries: and the same for Z. The propo- sition which the (hitherto) invalid form denies is then constructed. ‘Thus, writing down the instances of the universe, with their designations, we have four cases, under which are written all the combinations which deny them. PRO: EX Ek οὖς BORE CORY he Kl ROE. ΧΕ Χο ν yas! ὅν, ΖΡ Hes Digna zee ZL Lie ALi. VA Shp SA phe (09) OO) 0 0) CC )-O4 COCOCO CO) OO = OTe. ἐπ 0)" oO) CQ) δ yO or τ Thus X )) () Ζ, or ‘a middle term is both genus of X and partient of Z’ denies that Z is singular and X its contrary: and the same of five others, )-()-) &c. Secondly, when the invalid form has two particulars with balanced middle terms, let terms and contraries be both singular; the cases in which X and Z have balanced quantities deny that X and Z are contraries, the cases in which X and Z have unbalanced quantities deny that X and Z are identical. Thus sors is denied by Ὁ Ὁ, OG COs CC τε ον ΤΟΣ is denied by (1-6 10 COG 0; To produce the forms which afirm the restrictives, we must have recourse to the secondary ):(. I return to the cases which are without restriction, There are three balances, which I I shall call primary, secondary, and tertiary. The primary balance is even when the primary relations are both universal or both particular; uneven in other cases. The secondary balance is even when the spicule of the secondary relation are both universal or both particular ; uneven in other cases. The tertiary balance is even when the primary relations are beth Aristotelian', or both otherwise; uneven in other cases. 1. When the primary and secondary balances are of the same name, both even or both uneven, the primaries agree with their adjacent means or dijer from them, according as the 1 These are species, exient, external, partient; )), (-(, )*(, (3 lesser universal or greater particular; the first spicula of the same name as the proposition. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 465 secondary is universal or particular. Thus (( )) () is admissible; both balances uneven, universal secondary, primaries of the same name as their mean adjacent spicule. If U and P denote wniversal and particular propositions, uw and p similar spicule, so that, for instance, uUp denotes a universal proposition of universal and particular spicule, as )), then the legitimate combinations in which the primary and secondary balances are of the same name are Wabi Usb; pUp es U;, up, PSP) puu, Ure, abn, P5°U; pPp, Us 2, bp, Us U, pPu, P. 2. When the primary and secondary balances are of different names, the tertiary balance must be even. 3. To determine the product, or resulting simple relation, take the extreme spicule, invert each one which has a universal mean nearest to it, and make the result negative when the data show one or three negatives. 4. Given a product, to determine all the cases of which it is the product. Choose a secondary ; treat the given spicule as in the last rule; distribute signs of negation so as to have none, two, or four, in all (product included); and supply adjacents in any manner which will satisfy the rules. For example () (-) () is valid; universal secondary, primary and secondary balances oth even, and particular primaries with particular mean spicule; and (.) results. That is »X complement of Y’ means that ‘Any term is either partient of X or of Y*. Again, (-) (()-) is valid: secondary relation particular, primary and secondary balances both un- even, universal and particular primaries with particular and universal mean spicule; the result is (-(. Also, (-) (-( (( is valid, for the primary and secondary balances are of different characters, and the tertiary balance is even, neither primary being Aristotelian; and the same of (( (-( )-( in which both primaries are Aristotelian. It must be remembered that the primaries are read from the secondary spicula. Thus the last is ‘some species of X is not any external of Y’, These rules are not complicated, considered as selecting 256 out of 512, and deciding on their results. But any one acquainted with the canons of onymatic syllogism will find it easier to change the secondary into )-( or (), according as it is universal or particular, and then to try the primaries by the rules of syllogism. For instance (( (.( (). If we contravert the right-hand mean spicula and primary, we have (( () )-(: and X ((Y)-(Z is a pair of premises with the valid conclusion X (-(Z. If space would permit, much might be said on the relations of the forms of syllogism in which the secondaries are () and )-(. The first must be used in practice, almost exclusively ; namely, the proof of the existence of a middle term by its actual production. The second is well known in thought, though its method of procedure, the denial of the existence of any middle term whatsvever, can but seldom be a direct means of establishing a conclusion. Thus X ):( Z, presented as X (()-()) Z, is a familiar type of thought: instead of ‘no X is Z’, we see that “Χ and Z have no species in common’. The assumption that inference must proceed upon a comparison of two terms with a third is shown to be only an incident of that bisection of system which begins in the refusal of privative terms. That there is no middle 59—2 400 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND term which will do might be made the regulator of our forms of inference. Thus Barbara, or )) () )), as commonly presented, is equally represented by (( )-( )-(, which affirms that no species of the minor term is an external of the major. And this last is but a strengthened form of () )-()-(, which affirms that no partient of the minor is an external of the major. When the middle term is only an intermediary, some term wholly indefinite, or any term, a notation which, for good reasons, I had dropped, may be amended and re-established. We may say )) ))=)), (( (-) G(=(, &e. Such notation as X))Y+Y))Z=X))Z is faulty in two! respects. First, the sign = is applied to an inconvertible relation: for, though X))¥+Y))Z gives X)) Z, X))Z does not give X )) Y +Y ))Z, but only X))?+?))Z, where both queries may be answered by the same term, when that term is known. Secondly, the premises are compownded, not aggregated. For both these errors I was indebted to the suggestion of the school of logicians, who not only aggregate concepts into concepts, but who sometimes go as far as animal + rational = man. I now assert that every onymatic syllogism can be announced in eight forms : and each 1 Mr Mansel (rv. 119) takes the following objection:— | set down as a ‘contradiction proving the psychological theory “ΑΞ little [as of Euler’s geometrical syllogism-figures] do we | of the conditioned’! The first of these was in a Lecture, with approve of the algebraical method adopted by Mr De Morgan, other things resembling it; the second and third were private in which the premises of a syllogism are connected by a plus, | notes. Mr Mansel replied (December 1 and 8, 1860), resting and their relation to the conclusion expressed by the sign of | mainly on the fact that Hamilton had taken the errors from equality, a method too redolent of the computation-theory noticed others. He also asks why Hamilton is to “ be tied down to an above, [either Hobbes or the arithmetically definite syllogism], | exactnessin the use of mathematical illustrations which professed and tending to confound the intuitive judgments of Arithmetic | mathematicians have not held themselves bound to observe.’” with the discursive inferences of logic. The algebraical equation His instance is as follows :—‘‘I find in Prof. De Morgan’s proper does not represent a syllogism, but a proposition which, | ‘ Formal Logic’ (p. 131) a syllogism in Barbara, expressed in like any other, may form part of a logical reasoning, but cannot the form Y)Z+X)¥Y=X)Z; an expression with which I with any propriety represent the whole.’’ To this I say first, shall not quarrel, as an algebraical metaphor, so to speak, that + and =are not signs peculiar either to arithmetic or to though 1 fancy that the author himself will hardly maintain algebra: +,—, and = are in genere aggregative, disaggrega- | that the relation between the premises and the conclusion of a tive, and equivalential. A person who has no counting, and | syllogism is, literally, identical with that between the two sides as yet no symbols, might be introduced at once to the symbo- | ofan equation.” To which I reply that, so soon as I quarrel lic aggregation of concrete lengths, which is seen in with the diferal application of a symbolic relation, I quarrel fh = 5 with the metaphor too. I rejected both in my third paper, for Secondly, a syllogism is a proposition ; for it affirms that a the reasons in the text. When I became master of the distinc- certain proposition is the necessary consequence of certain | tion between aggregation and composition, which the logicians others. An affirmation is not the less an affirmation because it | do not admit, I saw that thcre is generic agreement, with spe- affirms about other affirmations. Mr Mansel will not deny that | cific differences, between the connexion of two premises in a the following propositions are premises giving a valid conclu- | syllogism, and the operation symbolised in A x B (not A+B). sion. Minor ; ‘Every case in which all X is Y and all Y is Z | Accordingly, depriving x of the specific character, and retain- is a case in which all X is Z’. Major; Every case in which | ing only the generic, I now affirm the convertibility of (-) x )-) ‘all X is Z is a case in which all not-Z is not-X’; therefore | and(), and Isay (-) x )-) =(),or, using the usual abbreviation, &c. What is the first of these propositions but a syllogism 2 ΘΟ =. In the Atheneum Journal (Nov. 10 and 24, 1860) appeared With regard to the question why a person ignorant of ma- reviews of Hamilton’s Lectures, which Mr Mansel at once at- | thematics is to be tied down to a correctness of illustration tributed to me: in which he was correct, so far as any one can | which the proficient does not observe, the answer is easy: the be correct in giving to acontributor an article which, appearing | ignorant man is pretty sure to darken counsel, the proficient under editorial responsibility, passes through editorial hands | will probably illustrate the matter in hand, even though his before it is made public. Certain mathematical errors were | parallel be inaccurate with respect to what is not in hand, Let pointed out, which it appeared were mostly copied from others, | any one look at the manner in which Sir W. Rowan Hamilton though some of them were read to the class for twenty years | produced systematic truth out of the true side of my symbol, together. Such as that Euclid (1. 1) shows that his three lines | as shewn in my third paper; and then let him take the resem- constitute a triangle, and that the circles meet; such as men- | blance between an acute angle and a pyramid, and see what he tion of two lines which divaricate at an acute angle, ‘like a | can make of that: he will come out with some notion why Sir pyramid’; such as an inch equal to a foot, because both have William Hamilton is to be tied down much tighter than Sir an infinity of parts, and one infinity is not larger than another, | William Rowan Hamilton. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 467 conclusion has 24 ways in which it can be announced with a secondary of its own quantity ; and 8 ways with a secondary of the other quantity. I write down all the ways of announcing the conclusions )) and (). Conclusion ) ) τ συ οὐ τς ες τ OPP σα τς απ ως γ Gwe) τ Οὗ Ὁ Ὁ ΟΣ) ὦ» ))iCOC@GO):( CC ) 2) COPD). ECE 60)..-6 1) OD) OM) τοῦ HO). O@ Conclusion ( ) Se ECC ECE) eH Oa) NCE CC ee sO COC) eee) PC) Gr Οὐ CC). ( OC) CRO) OE COGN: "5" eee) OC) CCE) CCC HO) ey Bee INENGREC OC) (02) )C)) Bren) =) FOUR CC.) YRC Ὁ} χα eG GG GAD I 8 > 2) 2) The common syllogism has a conjunctive relation of premises: as in )) () )) which asserts terms—generally proved by assigning one—which are both genus of the minor and species of the major; or as in )-( () (-), ὁ. 6. there are terms which are both external of the minor and complement of the major. But there are disjunctive relations of premises: as ):(G) ©, any term is either external of the minor or partient of the major. When the secondary is thus disjunctive, the canon of validity is simply inverted as to universal and par- ticular, and the canon of inference as to affirmative and negative. Thus (-( (-) )-) gives infer- ence, because both premises are particular; and the conclusion is negative, (-): if every term be deficient, either of the major or the minor, these last are complements. Similar obser- vations may be made on the secondary )(. The eight methods offer a crowd of analogies which I shall not describe. Taking () for a standard secondary, the universal syllogisms comprise all the cases in which the primary and secondary balances are of the same character, and the tertiary balance uneven: when this last becomes even, we have the strengthened syllo- gisms. In the particular syllogisms, there is even tertiary balance. The fourth case, uneven tertiary balance accompanied by difference of character in the other two, does not give any unrestricted forms. I now make a selection from the 512 identifications, in illustration of the danger of assert- ing completeness without a very cautious examination. No one will deny, whatever he may think of the system, that it is @ system, and that no portion of it could be selected as com- plete in itself with reference to all correlatives employed. Should any one object, his objec- tion must affirm that the system is not yet complete, and that some higher power of 2 than 512 is the true number of cases, I take only one out of 32, as follows :— 408 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND X(()) ))¥=X))Y Any part of X is some part of Y X)) (( (CY =X)) Y Some whole of X is any whole of Y X(((-€)) ¥ = X(.CY Some part of X is not any part of Y X)))-) (CY =X(-(Y Any whole of X is not some whole of Y X((((.)) ¥ =X (CY Some part of X is any part of Y X)) )) (CY = X((Y Any whole of X is some whole of Y X(()-) )) Y = X)-) ¥ Any part of X is not some part of Y X)) ¢C (CY =X)-) Y Some whole of X is not any whole of Y X(():()) Y =X)-(Y Any part of X is not any part of Y ><) (QO S55 Every class is either some whole of X or of Y X((() )) Y=X()Y_ Some part of X is some part of Y X)))(CCY denies = Some class is neither whole of X nor of Y ας (ς ον» 5.-.- Ὁ Every class is either some part of X or of Y X)))-€((¥ =X(-) Y Any whole of X is not any whole of Y X(()()) Y denies s¢ Some class is neither part of X nor of Y X))Q (CY =X)(CY_ Some whole of X is some whole of Y This table selects all the cases in which the primaries are either genus and genus, or species and species; genus and species are called whole and part. The readings by part and part form a system analogous to that of Hamilton, and differing only from it in this, that the restrictive affirmed and denied is singular and penultimate contrariety, instead of singular identity. Both systems are only portions of larger wholes; and in both, other sections balance the irregularities of the sections here under review. And both give this lesson, that no system is complete until all its circumstances exhibit complete balance; any appearance of irregularity in one of the aliquot parts being thrown into symmetry by an inverted irregu- larity in another part. Taking the last table as a whole, and dismissing the restrictives to their proper sphere, we see that each unbalanced proposition has a reading of either kind; while each balanced pro- position has only one reading. And we see—again, having seen the same in the full exem- plar system,—that ) X or X ( is ‘ some part’ or ‘any whole’; while X_) or (Χ is ‘ any part’ or ‘ some whole’. In the preceding method the process of thought is absolute identification; or its denial: thus X (( )) ))Y affirms of any species of X that it is strictly identical, coextensive with, some species of Y. There is another set, of 512 cases, in which the comparison is made by assertion or denial of one class being the precise external complement, or contrary, of another: one instance would be ‘ Any species of X fs the contrary of some species of Y’. This system needs no more than a mention. I proceed to readings in which the copular notion no longer insists on complete identification or its extreme contrary, but assumes one of the eight terminally ambiguous forms: as seen in the presentation of ‘Every X is Y’ under the form ‘Any species of X is a species of any genus of Y’. I shall not lengthen this paper by a full discussion of the whole of this system: I shall confine myself to the cases in which the primary relations are genus and species. And these words may be dispensed with, since their correlatives are not to be employed: part and whole will be sufficient for the pur- Stank ὦ ὦ μὰ παν Ἀπ τὰν σας πο ον πὰ ἀν || ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 469 pose. Over and above the reinforcement of preceding notions, that purpose is the compari- son of extension and comprehension, or, as I prefer to say, of extent and intent. With the above limitations the following table, to be immediately explained, contains all that is necessary. i {I Il IV ))}) »). CEE Go) CEG CC) ) ED») I eo i ( IDmG@pw Dis wp 3: ( GS) spp) τὺ -Ξ ἡ ΠΡ -- πὴ Ρο }} ΟΡ Ξ ν- CC) pw τ wp De ) ©) (pp te ww ΒΞ )}ν pw Ρ τ): pp De ww )) ) wp : WwW Ξ: ( )C ( pp θὲς pwD-—|) (( (pp Ξ ὃν =|) () ) wp D_- ww Ὁ (-( ( wpD=: ww D=:|) (-) ) pp °3 pw -—|( )-) ) pp D:= pwD =|( )-( (wp --- ww OC pp =p =|) )}() pp Do; wpD—-|( (( ) pw = ww =:|( () ( pwD-— ww Ὁ: (() pp D:=wp D:=|( (-) (pp τὸ wp —-!) )-) (pw D=: wwD="/) )-( ) pw -— ww Take one of these readings, for instance I. 6 ) (-((, say X) (-( (Y, where X) may be either ‘ Any part of X’ or ‘some whole of X’ and the same of (Y. There are four read- ings; ww (whole and whole), wp, pw, and pp. Of ‘these pp and pw, about which no remark follows, are unrestrictive readings, and give ) (-( ( with the middle spicula erased, or X)-(Y. That is, ‘ Any part of X is exient of [not wholly contained in| any part of Y’ and ‘ Any part of X is exient of some whole of Y’ mean simply that ‘X is external of Y’, or that ‘no X is Y’; and the converse. But wp and ww are denials of a restrictive, and both simply deny (D) that X and Y are penultimately identical. That is, ‘some whole of X is exient of any part of Y’, and ‘some whole of X is exient of some whole of Y’, simply say, ‘It is not true that X and Y are coextensive and each taking up all the universe except one individual object.’ When X and Y are anything but coextensive, or, being coextensive, any- thing but penultimate, some whole of X, X or X and more, is not wholly within any part ΚΟ. Take the symbol IV. 4 as an example, X()-() Y. Omit the secondary spicule: we have X (-) Y, which is the proposition symbolized. Read the secondary )-( as ‘is out of’, ‘is entirely excluded from’, or ‘entirely excludes’, Four readings are possible, from part to part, pp, ἕο. These four readings are pps Some part of X is out of some part of Y; ww, Any whole of X is out of any whole of Y; pw, Some part of X is out of any whole of Y ; wp, Any whole of X is out of some part of Y. Of these the table tells us that only pw and wp are unrestrictive: that pp merely denies (D) that X and Y are singular and identical; and ww merely affirms that X and Y are singular and contrary. But pw, which affirms that Y and anything, up to penultimacy, leaves out some part of X, affirms, as we see, that all which is not Y is X, 470 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND The rules for detecting unrestricted readings are as follows :-— 1. (Lines 1 and 2.) When the primary and secondary spicule are alike, as in )))), ))-((, all the four readings are unrestricted. Thus X(-(Y follows from X((.( (Y in every case of some part 2 Ε an art of X is not in yP any whole some whole ( (-( ( 2. (Lines 3 and 4.) When the primary and secondary spicul are both balanced, there are two unrestricted readings, both unbalanced: when both unbalanced, two unrestricted readings, both balanced. Thus ) () (and ) (-) ( have the readings pw, wp, unrestricted: but ) (() and ) (-( ( have pp and ww unrestricted. OLN 3. (Lines 5, 6, 7, 8.) When the primary and secondary spicule are one balanced, and the other unbalanced, one of the extreme spiculz is of a different curvature from its neigh- bour: let this be the detached spicula. ‘Two readings are unrestricted; and the detached spicula has the same reading (part or whole) in both, That common reading is by part when the other extreme is particular in an affirmative, or universal in a negative; by whole, when the other extreme is universal in an affirmative, or particular in a negative. Take notice that the most Aristotelian combinations go together; part, particular affirmative, universal negative. Thus ())) has the first extreme spicula detached, the second particular in an affirmative: accordingly, ())) gives () in the readings pp, pw. But ()-)) gives (-) in the readings wp, ww. And ))-)(, in which the second spicula is detached has the readings pp, wp. Interchange of primary and secondary spicule produces no effect in any case on the modes of unrestricted reading: thus )(-(( and ():(( both give pp, pw, for unrestricted readings. Each relation is enunciated in ten ways: by secondary relation of its own name in four ways, and in two ways by each remaining relation of the same quality. Thus species enun- ciates species in four ways, and genus, partient, coinadequate, enunciate species in two ways each. The following are the ways of announcing’ that X is a species of Y. Any part of X is in some part of Y Any part of X does not complete any whole of Y Any whole of X does not complete any whole of Y | Any part of X is in any whole of Y | Some whole of X is in any whole of Y Any part of X takes in some part of Y | Some whole of X is in some part of Y Some whole of X takes in any whole of Y Any part of X is not out of some part of Y Any part of X is not out of any whole of Y Ὁ Tuse simple English verbs for the universal relations; és | great use of these simple verbs, and with a feeling of relief in, takes in, is out of, makes up or completes. ‘To eke out is | from the state trappings of technical terms; like the post-boy’s the purest English for to make up all the rest: but it has in our | horse in John Gilpin, I felt time too much the implication of pis aller and succedaneum. The particulars are merely the negations of the universals: 1 doubt if they ought ever to be anything else. I have made ΤΥ, right glad to miss The lumbering of the wheels. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 471 I finish this part of the subject by noticing that in any proposition one spicula’ may be read as a verb, subject to rejections on account of the production of restrictives asserted or denied. X), Nisin; X(, X takesin; X(-, Σ 15 not ins; X)., X does not take in. (-X, takes in X; )X, isin X; -)X, does not takein X; -(X, is not in X. The modes of reading are :— Examples are seen in X) )Y, All Xisin Y; X( -( Y, Some X is not in Y; Χ( ΟὟ, Any whole of X does not take in Y, and so on. Tt will not be worth while to reduce it to tables, because the complex syllogisms to which it leads are I shall now sketch out the whole of which the Hamiltonian attempt is a part. easily reduced to compositions of simple ones, and would really appear in this way, except only when they show the junctions of universals which are seen in the system of terminal preci- sion. The particulars of this system are of very infrequent occurrence compared with the universals. My object in giving this account is not to detail the system as for use, but to make it a lesson upon the necessity of giving full action and equal prominence to all sides of every correlation ; and further, to show that the defects of an incomplete system are magnified The Aristotelian table of enunciation, for instance, is a true bisection of system: it selects the lesser universals and when the part selected from the whole system is not an aliquot part. the greater particulars. But the system of syllogism is not a true bisection: nineteen syllo- gisms cannot be a real aliquot part of any system. I postulate—in my own mind I say I have demonstrated—that the eight onymatic forms are essential to any complete system of enunciation. We are to take in both ali and some-not-al/ as quantifiers: that is, ‘some affirmed to be At the outset then we are asked to select two out of three We know that Xs, if Xs enter into thought at all, enter as some; and this some is either affirmed as all, or neither affirmed nor denied as all, all’, and ‘some denied to be all’. alternatives, without allusion to the third. or denied as all. Any some must appear in enunciation under one, and only one, of these three relations to adl. The Aristotelian system makes a fair bisection of this set of alterna- tives. When there are three alternatives of which one is equally and symmetrically related to 1 The ‘‘ mysterious spicule’? make a powerful language. In using one symbol, ), as in X)Y, to denote both the total quantity of the subject and the particular quantity of the pre- dicate, I followed the plan by which a fraction is represented, in which one symbol distinguishes both numerator and deno- minator: and I ultimately marked the symbol twice. If a a - fraction had been denoted by 3 a@and ὁ would have been convenient symbols fora as a numerator and ὦ asa denominator; and might be made useful even as it is. Forgetting that I was not writing wholly for mathematicians, I used expressions on this subject which were misunderstood. In my magazine of animadversions (v1. 640*) there is a spirited criticism of my notation, the colouring of which is heightened by assuming to be one my two syllogistic notations, pictorial and arbitrary ; Vor. Parr ik the first only a study, the second a language for use. In it we find—‘* We need hardly, therefore, be surprised, that, in the end, Mr De Morgan should actually laud the farrago for ex- pressing diametrically opposite things (“the universality of the subject,” ‘“*the particularity of the predicate’’) by the self-same representation.’’ Had I held, with the logicians, the exclusive right of the onymatic relations to be logical forms, I should now have dropped the word spicu/ar, already borrowed from Hamilton, and have substituted farraginal, with the motto Quicquid agunt homines nostri est farrago libelli. If, as I suspect, I am on the way to a much wider use of the complex forms )), (-(, &c., the second adjective may yet find an introduction. 60 472 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND both of the others, that one may be repeated twice, once in relation to each extreme: and either extreme, with the common mean, is a symmetrical bisection of system. The Aris- totelian plan confines itself to ‘some affirmed to be all’ and ‘some not affirmed to be all’: one extreme, and the mean in relation to it. But if we take the two extremes, we must also take the mean in its relation to both. Again, as contrary terms must enter, whatever subdi- visions we make of ‘some X’ we must also make the same of ‘some x’. Accordingly, since a universal term gives a particular contrary, there is no proposition but must enter in four different ways. A term being universal, we must distinguish the case in which the particular contrary is to be some-or-all from that in which it is to be some-nof-all. Denoting all and some-or-all in my usual way, I shall denote by an accent that the particular term indicated is some-not-all, or else that the universal indicated has some-not-all for its contrary. There are then, besides the eight usual forms, 3 x 8 or 24 others, all formed, as we shall see, by con- junctions of two of the eight, or three. These last, by equivalences, are reduced to twelve ; which with twelve disjunctive denials, make a total of 32. Of these I shall, for brevity, consider only the common forms and the 12 conjunctives: syllogisms containing disjunctions can be dealt with by opponent reduction. First, a universal, such as )), is accompanied by ))’, )’), and )’)’. The three last are equivalent, and equivalent to the form )) joined with )-); or to )e). And the same of the others. So that the system of universals contains the simple universals )), ((, )-(, (-), and the double universals )9), (¢(, )o(, (9). Secondly, a particular, such as (), is accompanied by ()’,; (7) and (’)’.. These three have the following meaning. Each one consists of the proposition without the accent, joined to the proposition in which the unaccented term is contraverted. Thus ()’ is () and ).); () is () and (-(; (’)' is () and )-) and (-(. These may be denoted by ()-), (.(), and (-()-). Accordingly, we have ()’ and )-)’ mean ()-) () means (.()-) (Ὁ and ('-( mean (Ὁ (ὦ means )(0 δ 1 and (.(' mean )(-( )':)’) means ()-)( y( and )'-) mean )-)( , Ὁ means )-) (-( Remember that )-) (-( is the triple junction of )-), )(, (-(, ἅς. Contranominals do not appear in any double proposition: thus we have not equivalence, )) and ((, nor contrariety, )-( and (-), nor ()(, nor (-( and )-). The denials may be represented by the disjunctive comma: thus the denial of )’(’ is ‘either (( or (-) or ))’, represented by {((, (-), ))}. Of the whole number, Hamilton’s plan selects seven, when ‘Some X is not some Y” is properly treated, and adds the assertion of equivalence ‘ All X is all Y*. The seven are 25 CG, Gis eG (Ose) -):5 being two simple propositions, four double, and one triple. It thus includes all the cases in which the new forms, (-) and )(, are absent. To these it adds the equivalence, or junction of )) and ((, without adding the junction of (-( and )-), the denial of {)),(({. The ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 473 junction of contranominals is, in every one of its four cases, excluded from separate enunciation by the main principle of this system, If Hamilton had expressly started on the principle of allowing every Aristotelian proposition, every possible junction, and every disjunctive denial, with simple conversion, he would have come very near a system. The syllogisms of this system are, when it is fully taken,—1. The 32 forms with two single premises each. 2. Eight universal forms with double premises and double conclusion, being those I have called syllogisms of terminal precision; and 16 opponents, having one disjunctive premise and a disjunctive conclusion. 3. Various forms in which a single universal—or a double one, which gives only a strengthened form with the same conclusion,— with a double particular, give a double particular. or both of which are more than single give only a single conclusion. 4, Various forms in which premises one On these it is not neces- sary to dwell: any case is resolved at a glance by any one familiar with my notation. I cannot undertake, in the present paper, to give a full account, in relation to aggre- gation' and composition, of the distinction of ewtent and intent. I shall therefore confine myself to what I expect will be a termination of my controversy on this point, followed by a brief account of what I hold to be the true logical foundation of the distinction. No part of Hamilton’s system has received a more ready assent from his followers than his Ρ y mode of making the distinction—I say his substitute for the distinction—which Aristotle announced when he divided genus in species —class in class. that there zs a substitution. attribute in attribute—, from species in genus I shall not, after what is said in my third paper, offer any further proof It will be enough here to quote (VI. 642*) his governing? principle—*“ the predicate of the predicate is, with the predicate, affirmed or denied of the subject”—which ushers in the form of depth, or comprehension, ‘‘ All X is some Y”, as distinguished from that of breadth, or extension, ‘Some Y is all X”, 1 This distinction is not yet seen, nor, I fear, will it be seen until the logician is restricted to puddings which are only aggregates, and not compounds. It is remarkable that we have no word of pure English which designates the part as a com- ponent: element, constituent, ingredient, component, material, are all foreign. Shakspeare’s witches talk of ingredients; but they were scientific characters: the word can be distinctly shown to have been limited to medicines, charms, &c. What would a housewife of the time of Elizabeth have said, when she told her servant not to forget the for one ef those puddings which I would withhold from the logician. The things or the stuff will suggest themselves; but they only prove the absence of the truly distinctive word. For things may be aggregants as well as components: and stuff is the proper term when there is but one kind of material; though, for want of better, it occurs in such words as garden-stuff, kitchen-stuff. And the word applies equally to aggregants and components. 2 The word predicate is here loosely used. The first time it occurs it means ‘ predicate affirmed”: in the two other cases it means ‘predicate affirmed or denied’, Hamilton’s readers must be cautioned as to the very positive way in which he puts forward bran new principles as though they were gene- rally received, and convicts those who deny them of mistake, by appeal to the principles themselves. For example, (VI. 642*) :—“ This suffices to show how completely Mr De Mor- gan mistakes the great principle:—The predicate of the predi- When we say, says cate is, with the predicate, affirmed or denied of the subject.’’ Would not any one suppose it to be notorious that this great principle had been previously announced by others? Be this as it may—TI assert no negative, but I cannot find it—J had denied this principle, in effect, though I had never heard it, by proceeding on principles repugnant to it. When I say a horse has four legs, I ought to be taken as denying the great princi- ple of bipedality, not as counting each leg twice in an attempt to apply it. In the point before us the Port Royal logicians have repugned the ‘great principle’ as well as myself, and to all appearance with no more knowledge of its existence than myself; and Hamilton proves his knowledge of their opinion by citing them—very correctly, I believe,—as the restorers of the distinction of extension and intension. I may slightly mention another point. Hamilton thinks it sufficient to an- swer his opponent’s meaning by fixing another meaning on his opponent’s words. I designated a proposition of his as spurious, referring to a page of my own book for my own tech- nical use of that word. Hamilton (vr. 639") begins his answer thus, ‘* Spurious in law means a bad kind of bastard:” and on this definition he easily convicts me of absurdity. If, under a definition of my own, I had called his proposition goniome-~ trical instead of spurious, he might well have impeached my Greek; but it would have been of coequal absurdity if he had answered me by solemnly proving that he had not enunciated a theodelite. 60—2 47: Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND Hamilton, that Leibnitz, a mathematician, is not Newton, we deny ‘ mathematician” of Newton; not “any” mathematician, but the mathematician incorporated in Leibnitz; Newton is not the LmaEtheImaBNtiIciTanZ who is here spoken of. He means that each quality residing, inhering, in a subject is an object of thought, per se, as a quality, distinct from, though a component of, the subject of inhesion, and from the same quality in any other sub- ject. All this can be thought: what is its force as a distinct mode of enunciation? and what is its utility in logic? Show me the first, and I can undertake to find the second. Two objects are in a certain particular alike, so that if they were as much alike in all particulars they would be the same object. If Leibnitz, besides mathematician, had been English, Fellow of Trinity, Lucasian professor, ὅσο. &c. &c. he would have been Newton: and the proper name Leibnitz would have been but an alias of Newton. What! it will be asked, do you deny that in thought you can conceive two men, facsimiles in body and mind, thinking, speaking, and acting, exactly in the same way, &e. &c. &e., all through their lives ? If the querist mean that they are to differ in place or in time, I can conceive two different men, each the double of the other in all things except place or time. But if, among the other samenesses, they be to occupy the same place at the same time, I cannot call them two men. If I could, I should say there is no such thing as an individual; that each one man is a hundred, or a thousand, agreeing in all things, place included, at all times, and therefore without distinction. Suppose one individual to differ from another only in one quality, the first being AX YZ,,.and the second BXYZ,.., A and B being repugnant. Hamilton says they have two different X qualities, X, and X,: let it be so; the individuals are then AX,Y,Z,... and AX,Y.Z,... If A and B had chanced to be the same, these two individuals would have been wholly without distinction—would have been the same. Remember that we are supposed to have enumerated every concept under which either is viewed or which either receives or creates. If then, which I do not deny, X, and X, be really different examples of the same quality, all knowledge of this difference—the very difference itself, as to the esse quod habet in anima—is due to the difference of A and B. What then is the logical import of a method by which, because there are differences which distinguish, we read samenesses into different samenesses, and contend that agreements, as such, have differences of which only disagreements wholly independent of the agreements make us cognizant. Again, why do we give a class-mark, a term-name? to distinguish the objects of the class from all others, and (pro tanto) to confound them with one another. As against all other species, each is signatum by the class-mark; thus, though there be many men, I distinguish Newton and Leibnitz by the attribute humanity from each and every brute. But as against each other, this common class-mark is vagum: though I know each to be man, I do not know them to be different men till I have found another class-mark, the property of one, but not of the other. When the Irishman had caught the cluricaune, and made him show under which thistle out of many acres of them the treasure was buried, he tied his garter round the thistle—he added one? additional class-mark—and ran home for * The logician must not say that he merely distinguished | individual only, his own remaining leg, and many other legs, an individual, and did not include in a class: independently of | were fellov-members with the designated thistle. the truth, sometimes denied, that a class may consist of one ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 475 a spade. But the imp, while he was gone, tied a garter of exactly the same form and colour over every thistle in the field. When the poor man came back, he was made sadly sensible of the impossibility of distinguishing two individuals by the difference of their points of agreement. Hamilton would have described the situation to him as follows (VI. 643*) :—*‘ Let us consider what is meant by the proposition,—* This thistle has a garter.” “A garter” does not here imply all, every, or even any garter, but some garter,—qa certain garter; and this particulare,—he it vagum, be it signatum,—this some or certain garter which we affirm to be on this thistle, we do deny to be on that, in denying this to be that.” To which the Irishman might reply ;—*‘ True for you, your honour! but what will I be the better of that? Sure its the stgnatum I’m wanting, and the vagums are of no use at all at all.’ And this is the true answer. [ we only know that the garter on this thistle is not the garter on that, by (otherwise) knowing that this is not that, we have nothing that we can enunciate about garters as giving knowledge of this and that. Any one who can make a formal profit of the differenti of undistinguishable class-marks, may make a material profit of the cluricaune, if he can catch the creature. In the mean time, he may employ himself in studying how to advise the little boy who had two shillings, and was puzzled to find out which he ought to spend first, to make his money go farthest. I hold this distinction between ‘* Every man is in the class animal” and “ Every man is an object in which inheres one quality animal” to be of small meaning and no use. To make it the great distinction between the two sides of logic seems to me solemn trifling: to symbolize it by the inversion of phrase in ‘‘Some animal is all man” and “ All man is some animal ” is to bring distinction without difference in aid of difference without distinction. In my third paper I gave a generalization of the old distinction of extension and compre- hension (or intension) as the foundation of what I called the mathematical and metaphysical sides of logic. ‘Yo all there laid down I adhere; but I add that the logical skeleton of the metaphysical side is connected with whole in relation to part just in the same manner as that of the mathematical side is connected with part in relation to whole. Every attribute, or concept by which a class is distinguished, makes many portions of the universe to be so many wholes in relation to contained parts. If the class X be a part of the class Y, the class Y is a whole of the class X, the attribute Y is a component of the attribute X, whenever we mean by the attribute X the ¢otal attribute, the compound of all possible attributes, possessed by X. : The proposition is ‘X and every part of X’—not merely its distinct parts, but all possible parts—comes under Y and all its wholes. The correlation of part and whole has been so little examined that further detail may be necessary, There are classes, X and Y, containing 20 and 30 individuals: they aggregate into a class of not less than 30 nor more than 50 individuals; and I must know how many individuals belong to both classes before I can assign the aggregate number; that is, before I can ascertain the common whole of which X and Y are parts I must know the common part, if any, of which X and Y are wholes: this common part may be of any number of individuals not exceeding 20. This is the principle which Mr Boole has formulised in X+Y-—XY for the aggregate of X and Y3; and which determined my use of (X, Y) instead of X+Y in my Formal Logic. I call the common whole of two parts their aggregate; the common part of two wholes their compound, 476 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND Suppose a universe of six individuals, of which the proper names, the representatives This class It has the wholes, 123, 1234, 1235, 1236, 12345, 12356, of singular attributes, are 1, 2, 3, 4, 5, 6. Let us consider the class X, or 1, 2, 3. has the parts, 1, 2, 3, 12, 23, 31, 123, 12346. lowest parts, 1, 2, ἅς, and six highest wholes 23456, 13456, &c. Every selection is to have its But 123456, though a whole of 123, is not a term dividing the universe, which has six name, which may equally designate the class and the attribute by which the class is distin- guished ; being at once the instrument of cumulation and of distinction: of cumulation, when one individual of the class is coupled with another; of distinction, when one in the class is separated from one of the externals. Every point of correlation is seen, or will" be seen, to be perfect. The individuals being non-partient of each other, we may designate the class 1, 2, 3, by 1+2+3. We have no symbol in mathematics which may by analogy be employed to designate the attribute of this class; nothing which suggests ‘ the common mark of 1, 2, 3, and of them alone’: except so far as this, that Aj,.. is sometimes used in such a sense, inter alia; which may therefore denote Suppose we describe the class 1+2+43 as the aggregate of 1+2 and 3: what is the correlative mode of describing its attribute in terms of the compound of two? . the common attribute. The answer is that a class may be described by its contrary, and ‘that the alternative attribute of Agi, Ais? is the description of the attribute required. The relations of aggregation and composition are closely connected with those of direct and contrary: thus the propositions «Ὁ is aggregate of A and B’ and ‘ not-C is compound of not-A and not-B’ are convertible. The following is an example of the correlation of propositions. X):(¥. No X is Y; everything either x or y: X and Y have no common part: but, if not complements, have common wholes. Every individual is in some of the parts either of x or of y: and is either not in some whole of X, or not in some whole of Y. That is, no junction of a new attribute selects any part of one out of the other: everything wants some attribute of one or the other. X(:)¥Y. No x is y: everything either X or Y: X and Y have no common? whole: but, if not externals, have common parts. Every individual is in all the wholes either of X or of Y, and is either not any part of x, That is, no dismissal of an existing attribute makes any whole of or not any part of y. one a whole of the other: everything has all the attributes of one or the other. 1 A qualification rendered necessary by the smallness of the number who think of such distinctions. That esse is per- cipi is especially true of the esse in anima. The logical eye of the mathematician, and the mathematical eye of the logician, are yet to be opened. The cultivators of beth the sides of exact science seem to proceed upon the notion that distinct vision is not possible with both eyes together. Some contend for the right eye, some for the left: and the voice of mankind finds no utterance; for parmi les aveugles un borgne est roi, let him have which eye he may. 2 One word more on this stumblingblock. All terms have a common whole, the universe: but this is not a whole term. The logician does not see why the universe is to be excluded, nor can he see until his mathematical eye is open. But he excludes it in his own system, and very easily, by never in- venting a name for it. ‘Every thing that exists’, ‘the omne cogitabile’, are opposed in thought to their contraries, the non-existing, and the incogitabile. Where is the name that includes both the existing and the non-existing, the thinkable and the unthinkable? Let this name be shown, and shown in use, and then I shall be open to the charge of correcting the old logic: but I think I have only imitated it. In a full work on logic, the universal name might be discussed in the chapter which treats of res¢rictives and other extremes, not forgetting vacuous names. But in the logic of the term—distinclive name—the garter is as useless when all the thistles have it as it would be if none had it. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 477 I shall be asked whether, when the intensive proposition is thus reduced to its skeleton form, as a relation between wholes, I do not abandon the distinction of mathematical and metaphysical, as designative of the two sides of logic. Is not the idea of a whole including a smaller whole as mathematical as that of a part contained in a larger part? Certainly it is, but nevertheless I do not abandon the nomenclature, which loses none of its truth and The proposition of extent remains mathematical to the end; the proposition of intent becomes none of its utility: but the names must be held designative of a subsequent distinction. metaphysical in application. Even when man and brute are clothed with all their qualificative concepts, they make up animal just as the items of a tradesman’s bill make up the total of goods furnished. The individuals are plain counters in the formal enunciation, and painted counters in the material: but never anything except counters. But when, in the proposition of intent, the whole is recognized by its separating attribute, that attribute coalesces with others in each individual by a process of which we hide our ignorance when we call it ontological or metaphysical. The whole rational contains the whole man; the attribute rational goes to the composition of the attribute hwman: but, in spite of the logician, there is more than swimming up in this second process, Extensive quantity has partes extra partes, as they once said, and some will admit no other kind of quantity: de essentia quantitatis est habere partes extra partes, says Smiglecius (Disp. 1x, qu. 5). But extensive quantity has this quality objectively, permanently, and de essentia: intensive quantity has it only subjectively, pro re mata, as an accident of the thoughts. We can separate the rational in man from the animal in man, for the mind, by an act of the mind: we cannot but separate this man from that, save only when we think of the class as a unit, a process as subjective as that of separating the individual into concepts. First intentions give individuals which are compounds not yet decomposed, and ageregants not yet aggregated. Second intentions ) Ρ ἢ sereg 5 [- exhibit component attributes, and aggregate classes. The basis’ of these oppositions is seen in X))Y under the forms ‘ X and all its parts are parts of Y’—*Y and all its wholes are wholes of X.’ 1 Mr Mansel (rv. p. 117—119) has some remarks on exten- sion and intension, hinting opposition to Hamilton’s doctrine, and recognising the change of the quantities in passage from one to the other. With him ‘some A is all B”’ gives all the attributes of B as some of those of A; while ‘‘all A is some B” classes A under B. This mode of enunciation is very con- fusing: and from it follows that I owe Mr Mansel reparation for all but absolute misrepresentation in my article Logic (col. 344, note) in the English Cyclopedia; an inattentive reader would suppose I make him merely change the places of Hamil- ton’s forms, whereas he does more. Mr Mansel says (p. 119)— ‘©The problem which we wish to see satisfactorily solved by the advocates of Sir W. Hamilton’s doctrine may be stated as follows: To construct a synthetical proposition containing an equation or identification of subject and predicate in any other respect than that of the objects thought under the compared concepts.’’ My position is, either that this question is now solved, or that the given problem is net the one which should have been given. Mr Mansel criticises Bishop Thomson for not taking suf- ficient account of constitutive attributes as distinguished from simple characteristics : I hold that Dr Thomson—and others, including the author of my second paper—had taken too much account of this extralogical distinction; extralogical, so far as entrance into enunciation is concerned. For a term is held to be divided from its contrary before enunciation: while, in the proposition, an attribute is of the same import whether it be constitutive, or only characteristic. Hamilton, from whom I seldom differ in principle as to what is and is not logic—though in application we sometimes so widely disagree that, like a professor J have mentioned elsewhere, I do not grant him that the whole is greater than its part until I see what use he wants to make of it—replies as follows :—‘ ...In reference to Breadth and Depth, there is no difference whatever between ‘ constitu- tive’ and ‘attributive’, between necessary and contingent, be- tween peculiar and common. It is of no consequence, what has antecedently been known, what is newly discovered. These are merely material affections. We have only to consider what it is we formally think,” (vr. 643*). With reference to the quantities, Mr Mansel is answered by (fi44*, note)—‘S As others, besides Mr De Morgan, have misunderstood this mat- tereveres ” followed by a clear and dogmatical exposition of Hamilton's doctrine of breadth and depth, never till then given, and placing his error of quantity in broad daylight. 478 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND This basis of thought may now be introduced in my former papers as the mathematical substratum of the metaphysical notion: I need not enter into details. Both the systems of secondary relations may be adapted to it. In the second of the systems given above pp is the mathematical reading; ww the metaphysical; pw the physical; and wp the con- traphysical: according to the phraseology of my third paper. I need hardly say that, in like manner as any individuals may be selected, and con- stituted a class, so those same individuals may be distinguished by a term which denotes an attribute: we cannot put on a class-mark without acquiring a right to treat that mark When we pass from the arithmetical abacus to the use of terms of relation, mathematical or metaphysical, as species, dependent, &c. we shake our- as designative of a concept. selves free of many of the questions which I have discussed. Before we come to this point we feel a want as to (-) and )( of which we are inclined to complain until we see that only defective correlation prevented our feeling the corresponding want as to ).( and ( ) I have frequently heard it made an objection to X(-) ¥ that it appears as ‘ Everything is either X or Y’, of disjunctive character and apparently affirmative quality. in another wing of the subject. So long as we have a copula either of identification or inclusion, we cannot read either (-) or )( by part and part. For this objection, as an objection, I have never cared: those who acknowledge the existence, and admit the entrance, of a privative term must needs confess that X))Y and x (-) Y are equivalent. But I have always respected the complaint as merely directed against a blemish, and have awaited the time The reader will see that this time has now arrived: the forms ).( and () are subject to precisely the same difficulties when further consideration would provide further explanation. with reference to whole and whole. The following table of correlative readings will illus- trate this. xe) (Ye No part of X is any part of Y. Any part of X is not included in (and does not include) any part of Y. Some whole of X is external of some whole of Y. Every penultimate is whole either of X or of Y. Every penultimate includes either X or Δ ΩΣ Every individual is not included in some pen- ultimate either of X or of Y. XS (NG No whole of X is any whole of Y. Any whole of X does not include (and is not included in) any whole Y, Some part of X is complement of some part of Ye Every individual is part either of X or of Y. Every individual is included in either X or Y. Every penultimate does not include some in- dividual either of X or of Y. And so we might proceed, never failing to translate a reading of either proposition into a reading of the other, strictly correlative in every detail. I shall close this paper by attempting to procure for the quantification of the predicate an honourable acquittal from the charge of having disturbed the peace of the logical world. It has never been the subject of discussion, except by myself in the investigation of the numerical syllogism; an investigation of which the truth remains unquestioned, and in ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 479 which all quantification of the predicate is proved superfluous. Another notion, under the name of quantification, has stirred up controversy. I say nothing here about the mere question whether the quantity of the predicate should be expressed. The explicit demand for this expression was first made by Hamilton: it is a great step; and the logical world is pretty well agreed that its merit is quite distinct from the merit or demerit of the particular mode of quantifying which he adopted. What is quantification ? It means, or should mean, the giving or expressing of quantity ; and as quantity is essentially a more or a less, the giving of quantity cannot exist without the giving of a more or a less. But the quantity of the predicate, be it more or less, is always the quantity of the subject, or its complement, as well in negatives as in affirmatives; that is, so far as it is more or less. I postulate that a proposition is only a proposition, a pro- pounder, a challenger of assent or denial, in so far as, and with reference to, its assertion of For example, in ¢ Every X is Y’, it is a clear matter of affirmation, and so intended, that each X what might have been deniable, or its denial of what might have been assertable. is X, and each X not more than one Y: but these are not put forward as parts of the propo- sition, because they are not distinctive parts. In propositions, as in terms, all that belongs to the whole universe of propositions is to be tacitly rejected: I claim to make the rejection explicit, because it is sometimes tacitly refused. This being conceded, if I say ‘ Xs are Ys’ it is clear that the number of Ys spoken of is the same as the number of Xs: if that number If I say ‘ Xs are not Ys’, what I deny is that the ten (or fewer) Xs are any ten (or fewer) Ys: I do not mean to deny be ten at most, then ten (or fewer) Xs are ten (or fewer) Ys. that the ten are nine or eleven; for that I can deny by the form of thought, let X and Y be what they may. If I say ‘Everything is either X or Y’ neither X nor Y has quantity until quantity is assigned to the other: in a universe of 100 instances, if 40 Xs go to the veri- fication of this proposition, the number of Ys required for the same purpose is 60. In treating the numerical syllogism, it appears that ‘m Xs are found among n Ys’ is a Also that ‘m Xs are not found among n Ys’, Xs and Ys being ἃ and y in number, is spurious—that is, true inde- proposition importing no more and no less than ‘m Xs are Ys’. pendently of which are Xs and which are Ys—if m +m be less than y or less than w; and otherwise, of the same import as ‘m + Ἢ —y Xs are not Ys’ and m +n— « Ys are not Xs. Neither is the predicate of an affirmative more or less definite than the subject, as to quiddity', to revive an old term. If I say ‘All Xs are Ys’, I only fail to know whether 1 This word, which was but badly replaced by essentia, has been selected as a joke against the old logicians: but quantity and quality are in honourable use. The joke may be retorted upon a discerning public, which, while treating the word with ridicule, fell into the error of theory which it may be supposed to favour, to every extent short of the absolute maximum. All we know of quid is derived from quantum and quale: if man- kind had discarded guiddity on this ground, the race would have vindicated reason against philosophy with honour to itself, or at least would have shown an appearance of it. But, on the contrary, men in general assume a knowledge of things, res ips@, entities, essences, substances, natures, &c. ; and they Vou. X. Part II. claim to assert much about quiddity upon any the least know- ledge of quantity and quality. One exception, indeed, their modesty does reserve: it is admitted, enforced, and made pulpit doctrine, that the Almighty is known only by his attributes, in a manner which implies that his creatures can be otherwise known. ‘There was a time when educated persons, in numbers, had never heard of aitributes in any other way ; some may still be left. When a boy, I remember hearing mur- mured charges of irreverence against a person in company who spoke of the attributes of the vegetable world: my impression was that some of those present had a vague idea that the speaker might be a worshipper of leeks and onions. 61 480 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND this or that Y is spoken of by failure of definite knowledge of the Xs, If Isay ‘all man is animal’, and cannot say that I have spoken of that animal, it is only because I do not know whether that object was spoken of under ‘ man’. What notion, then, has been brought forward and discussed under the name of quantity ? Distinction between affirmation and non-affirmation of all or any: totality affirmed, totality not affirmed, without any reference to the quantity of the totum, the more or fewer indivi- duals existent in the class. To speak of X at all is to speak of the whole class, to speak of all X as a class, or not to speak of it as a class. ‘ Man is animal’: do you speak of man as a whole class? yes; do you speak of animal as a whole class? I say nothing about it; you are to take this proposition for purposes of inference without knowing whether I speak totally or partially; there is neither assertion nor denial, but reserve; it may be that I know the truth; it may be that you know the truth; but this proposition says nothing about it, and is intended to say nothing. This is all that is meant by ‘animal’ being particular. If this view had been taken from the beginning, the difficulties of the singular propo- sition and of the indefinite proposition would never have appeared, All the confusion which has arisen from want of care in stating the meaning of ‘some’ would have been avoided. The Hamiltonian quantification, if it had appeared at all, would have appeared in a sound form. It would have been remembered that affirmation and denial are not alternatives; and the three quantifiers of which I have shown the united effect would have been allowed full operation. To this I may add that Hamilton would never, even while denying its utility, have allowed ‘most’ (half plus some) to have been /egitimate. This importation from truly arithmetical quantification would have remained in its proper sphere, in company with other fractions. In my third paper I closed the controversy with my late opponent, as to every strictly personal matter: in this paper I hope to do the same with the purely logical questions, so far as his criticism on my own views is concerned. What remains of a polemical character— save only the question treated in the addition—concerns neither this logician nor that mathe- matician, but ¢he logician and the mathematician, I believe that the necessary laws of thought constitute as wide a study as the necessary matter of thought: and that Kant’s opinion on the finality of the Aristotelian system has as much truth and sense as any similar opinion—if any such were ever held—about the finality of the elements of Euclid. To the logician I say that the system which he owes to a mathematician, Aristotle by name, is a system of which none but mathematicians have ever shown a disposition to extend or vary the forms which has been followed by general respect: as Boethius, Ramus, Leibnitz, Lambert, Kant. There is but one logician of great note who, not having mathe- matical habit, has attempted to depart from routine in the construction of a system of infer- ence. It is not for me, appointed by himself his most prominent opponent, to pass sentence upon his system: but I suspect I have shown that system to be none the better for its author’s ignorance of the other branch of exact science. The growth of logic has been stunted by its separation from mathematics: I feel certain that my learned and acute antago- nist will be cited in time to come as the great champion of reunion, though appearing and intending to fight on the other side. ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 481 To the mathematician I assert that from the time when logical study was neglected by his class, the accuracy of mathematical reasoning declined. An inverse process seems likely to restore logic to its old place. The present school of mathematicians is far more tigorous in demonstration than that of the early part of the century: and it may be expected that this revival will be followed by a renewal of logical study, as the only sure preservative against a relapse. A. DE MORGAN, University Cottrer, Lonpon, April 14, 1862. *," This paper has been before the Council since May 19, 1862, though circumstances caused the deferment of the reading until May 4, 1863. ADDITION. Since the communication of the preceding paper I have obtained some notice of my criticisms from Mr 'T, Spencer Baynes, who I hoped might have been able to give evidence from his own personal recollections of Hamilton’s conversation and public teaching: this he does only on one of the points, referring the others to Hamilton’s printed works. Some account of his remarks is necessary: they do not induce me to alter anything I have written; but, as noticed, I omit the detailed proof of the falsehood and incompleteness of many of the syllogisms, because I find that no opposition will be made on this point. I Femain of opinion, and must so remain until further showing, of which I entertain no hope, that Hamilton did leave one set of syllogistic forms as recipients of both senses of ‘¢ some”, the old non-partitive sense, and his own doubly-partitive sense. That the neglect to make the necessary comparisons was a consequence of illness! I have no doubt. All the letters referred to appeared in the Atheneum journal: the dates are those of publication. As stated in the paper, I had brought forward Hamilton’s phrase “some at least (possibly therefore ald or none)”: failing all attempt at defence, I had (Dec. 28, 1861) given my own method of exeusing its occurrence. Mr Baynes defends Hamilton (Nov. 22, 1862): I abide by my explanation; and the matter is now left to opinion. The phrase carries its own condemna- tion with it; to those who cannot see this I have really nothing to say. But as my object in producing it was only to show the hurry of the article in which it appears; and as it belongs, not to Hamilton’s system, but to his account of the old one; and as I have omitted 1 Mr Baynes took no notice of my expressed conviction ; recent definitions of the quantifying words.” According to (Nov. 2, 1861) that “as to his [Hamilton’s] passing what I | Mr Baynes I have charged Hamilton with false reasoning : have called the Gorgon syllogism as valid inference, afier actual | the preceding quotations will show the sense in which the examination, there is no need to say that it was impossible he | charge was made. Be it remembered that these quotations are should have done it.” My whole position was that he had | no afterthoughts, but actual accompaniments of what is called allowed himself, without examination—and this probably | the “charge δὼ But 1 regret to say that the last proof of my owing to his illness—to take the whole application to syllogism | view of the subject, given near the end. of this addition as very for granted. I said (Nov. 2, 1861), “1 have no doubt that | recently discovered, shakes my confidence in Hamilton’s want when he returned to his studies after the seizure, he imagined | of examination, though I still hold that it is the more probable that he had tested the whole system of syllogism upon his most | hypothesis. 61—e 482 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND my own explanatory excuse from the preceding paper,—lI omit Mr Baynes’s defence from this addition. I may hereafter compare it with my own excuse, when something arises which the comparison will illustrate. I also asked (Nov. 2, 1861) for information as to whether Hamil- ton had given his own sense of ‘some’ from his chair. The silence of all his pupils on this point obliged me to think he might actually have taught this sense of ‘some’ as to be applied to the only forms of syllogism which I could—or can—ascertain that he had given. I therefore (Oct, 18, 1862) put the question in stronger terms, and by name to Mr Baynes, to Hamilton’s editors, and to his successor. In taking up this point, Mr Baynes of course felt it necessary to take up the others; but on this point his answer (Nov. 22, 1862) was explicit and satisfactory ; as follows. ‘ Within my experience of his class-teaching (up to the close of session 1853-4), Sir William did not, that I remember, depart from the ordi- nary meaning of ‘‘ some” in teaching the syllogism. But for years before this he was aecustomed to expound briefly from the chair his doctrine of immediate inference {in which one proposition ouly is concerned}, and of course as a part of it the different meanings of * some’.” This is to the purpose; and Mr Baynes is the best living witness on the matter. The remaining point is that of the application of the new meaning of ‘*some” to syllo- gism. On this Mr Baynes speaks—but without a single reference in proof of his statements —as follows: I put some words in Italics (Nov. 22, 1862). “The alleged invalidity of these syllogisms wholly depends on the use of the quantify- ing term ‘*some” in @ special sense. But Prof. De Morgan offers no proof whatever that it is so employed in the scheme he criticises, He states, indeed, what is perfectly true, that Sir William Hamilton signalised this particular meaning and contended for its partial use. {'This statement is not mine.] Sir William Hamilton, in applying his new doctrine to proposi- tional forms, disctisses the vague generality of ‘* some” in its ordinary use as a mark of quan- tity, points out that it may be taken in a narrower or more definite sense, and proposes the introduction of this new meaning ‘alongside of the other” in particular cases and for special objects. These objects, as Sir William defines them, ai/ relate to propositional forms. The partial use of the narrower ‘‘some” not only yields a complete and consistent scheme of opposition, but supplies certain valuable forms of immediate inference. For these reasons, Sir William introduces alongside the ordinary and vaguer ‘‘ some” (some at least) the more defimite ‘* some” (some at most) as a mark of quantity; but he carefully defines the con- dition of its use, and specifies the instances in which it is actually employed. From this partial and well-defined use of the more definite “some” in the treatment of propositional forms, Prof. De Morgan assumes that Sir William Hamilton not only carries it over into his scheme of syllogism, but applies it to every detail of that scheme.” In my reply (Nov. 29, 1862) I disposed of two of the Italic phrases by pointing out that I had given references to the successive papers on proposition and syllogism, which, writing to persons who had the book in their hands, and power to follow an implied argument in their heads, I took to be quite enough, I challenged Mr Baynes to support the remaining words in Italics from Hamilton’s writings, stating that this present paper was going to the printer, and desiring to couple with my statements the fullest account of his answer. Mr Baynes, acknowledging my references by substituting want of “definite proof” for statement that I ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 483 had made only “simple assertion”, refused (Dec. 6, 1862) to give any support whatever to what he admitted were his own unsupported assertions, until after the appearance of this memoir. He then pledged himself to reply if I should support my case by “ definite evi- dence”, by ‘anything like proof”, by “ anything indéed approaching to a plausible reason”. It would require, he said, “ ποῖ only a detailed statement, but a number of extracts”. I take Mr Baynes as admitting that no single extract, and no two extracts put together, would make a prima facie appearance of contradiction to my hypothesis. I fear I shall never see this defence: the mode of proceeding does not promise much. In Mr Baynes’s short opening letter (Nov. 1, 1862), he thinks his reply “« may be put into very moderate compass”; and that it “may be easily shown that Prof. De Morgan’s chief difficulties arise from a complete, though perhaps not very unnatural, misunderstanding of Sir W. Hamilton’s condensed form of expression”. Here the words “ easily shown” can hardly have meant that all the showing was to be assertion, without one single supporting reference to Hamilton’s writings. But when Mr Baynes finds that simple assertion will not be taken as showing anything, and that substantiating references are called for, and when he is told that I shall handle his reply in this paper, the moderate compass becomes detailed statement too long for the journal, and the tone becomes more sarcastic. In pointing this out I direct attention to all! that Mr Baynes will allow to be shown: high confidence with good humoured condescension changing, on demand for proof of statements, into what must be interpreted as confession of difficulty, with disparagements and ironies which seem intended to avenge the difficulty upon him who put it in the way. Iam quite content that Mr Baynes’s imitation of his great teacher’s tone of controversy shall continue, provided only that he will demand respectable references from every statement which applies for admission. Should he really attempt to redeem his conditional pledge, IT shall be much pleased: for I confidently expect that my views will be positively confirmed. But should I and the public hear nothing more from him, which from his recent retreat is too much to be feared, I must be content with the negative confirmation which his silence will inevitably be taken to afford. But I hope better things. I now proceed to point out how I came to arrive at so strange a conclusion as that Hamilton’s own new propositional forms, emerging out of his own new use of “some”, were intended to be used, as well as the old ones, in his own new system of syllo- gistic forms. In every book of logic the treatment of the proposition precedes that of the syllogism: and the forms of enunciation treated in the chapter on propositions are those used in the chapter on syllogism. This of course; for usually there is but one system of propositions, When, for the first time, we see two systems of propositional forms, of which 1 Perhaps not quite all, The assertions being dismissed | point: it was his habit to go on year after year without making which are to be established ‘‘some’’ day at latest (perhaps | any alterations. If he began to explain doubly partitive therefore never 3.) there remains the fact proved by Mr Baynes’s | enunciation to his class, with an intention of soon proceeding evidence, that Hamilton explained to his class the doubly par- | to investigate the syllogism belonging to it, there is good rea- titive ‘* some,” and (rx. ii. 268) the immediate inference thence | son to suppose that, though the execution of the intention were arising. There is also the fact proved by Mr Baynes’s silence, | delayed, he would still continue his imperfect statement. The that Hamilton did not therewith tell his hearers that the doubly | third Lecture on Logic has pages beginning with “1 would partitive enunciations would not validate the only syllogistic | interpolate some observations which 1 ought, in my last Lec- forms which he had given them. In my mind this adds pro- | ture, to have made before leaying...... ” These lectures were bability to the hypothesis that Hamilton had never tested the | read for twenty years without the alteration being made. 484 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND the new one is declared to be placed ‘ alongside” of the old’ one, we must needs infer— unless the contrary be expressly stated—that both sets of forms are to be used in syl- locism. If we be to have a thing so completely unheard of as a set of propositions which 5 have no syllogisms in which they combine, we feel that the writer will certainly give us warn- ing of what we are not to expect: especially when the new system is to stand alongside of the old one, side by side, in the same rank, Hamilton, writing on his own system, left a rough but very elaborate sketch of propositional forms, and another of syllogisms (Logie, 11. 277—284, 285—-289, appendixes (4) and (e)). without a word indicating that it had passed through their minds that, of the two sets of These are given consecutively* by his editors, propositional forms given, the new one was, or might be, unconnected with the one system of syllogism which appears to belong to both, for aught that Hamilton says to the contrary. These volumes are edited, as was said of them in a review, ‘in the best style of laborious and conscientious workmanship”: and they contain much more than a casual reader can appre- ciate of unpretending reference and comparison. The additional papers, on which this discus- sion arises, are put together in a manner which makes it clear that the trouble they cost must have left the editors in close possession of their details. Finding that the new sense of ** some” made syllogistic forms invalid, and having searched in vain for anything even congruent with the notion that this new sense was not to be used in syllogism, I publicly applied to Hamilton’s followers for information. a year. None was given for The editors* and Mr Baynes, who was Hamilton’s substitute during illness, remained 1 «Though it may not supersede’ the other: not ‘‘ must not,” nor ‘“‘ ought not to;’’ but only “may not.” The phrase is no more than permissive to the old system to remain, if others insist on it. That this was the leaning of Hamilton’s mind—nay more, that disapprobation accompanied the per- mission—is evidenced, I think, throughout his discussion. For instance, by his interpolation quoted in the body of the paper, to his reprint of the letter in the Atheneum journal: here he says that ‘as we shall see, two particulars in the affirmative and negative forms, ought to infer each other’. To this it must be added that (1x. ii. 254) he, in January 1850, demands it asa postulate of Logic that ‘‘ the some, if not otherwise qua- lified, means some on/y—this by presumption.” If we accept Mr Baynes’s statement, that some only (= some at most) was not intended to be introduced into syllogism, and if some, with- out qualification, be to mean some only, it follows that there is to be no formal syllogism in which the quantifying word “some” stands alone. That the old forms can be well spared, is clearly in Hamilton’s meaning: and if they go, what have we left? The new forms, without any syllogism? 2 It weighed much with me that one of the editors, Mr Mansel (iv. 113) came to his task with the conviction that Hamiltonhad a use of “some” different from that of Aristotle; and that this new sense of “some” was applied to some sort of syllogism. The quotation given in the body of the paper shows this. When the unpublished papers came into Mr Mansel’s hands, he, without any editorial remark, allowed the syllogisms in appendix (e) to follow the new and additional sense of ‘some’ propounded in appendix (d). I took it that he—who had shown his belief that Hamilton did apply some new ‘some’ to syllogism—had no reason to doubt that the syllogisms which he presented as editor were those which he had opposed to mine as reviewer. I divided the responsibility between Hamilton and the editors in the following words (Aug. 17, 1861)—* I do not say that Hamilton himself would have admitted this syllogism. But I do say that those who will accept his writings as they stand must admit it.” Mr Mansel did not impeach either my interpretation of Hamilton, or my implied interpretation of his own editorial proceeding. I consequently became fixed in the belief, which I still hold, that I had construed the editors rightly: and I believe that they were right as well as I. Though they had examined (e), which it was not their business to do, with reference to the validity of the connexion, they would not have been jus- tified in deviating from the course they have taken. They might have taken up my suspicion that Hamilton forgot, after his seizure, that he had not finished his investigation. They might have suggested what Mr Baynes asserts, but re- fuses to prove in time for this paper, that the new propositions were never intended to walk the world in pairs. But, whatever they might have thought, it would have been their duty to put the new syllogisms into that connexion with the new proposi- tions which the state of the papers seemed to require; leaving their caveat, if they had given one, to work its own effect on the reader’s judgment. 3 Mr Mansel, and Professor Fraser, Hamilton’s successor, have a right to the statement that they privately, in reply to applications from me, made after my letters were published, informed me that they had no more means of information than were open tomyselfin print. It was not for me to ask what opinion they had formed from these materials. The reader will understand that the second public application, especially ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 485 silent. Another appeal, relating to what Hamilton had orally taught, of a pointed and per- sonal character, brought out Mr Baynes on the whole question, with assertions which are —if he should see any plausibility’ in my reasoning—to be substantiated so soon as my most appropriate opportunity of discussing them shall have passed away. These things speak for themselves: I fully anticipate that any attempt to invalidate my conclusion will speak more plainly still. Something I have got; I have extracted the defence which is to be set up: namely, that the new sense of “some” is to be asyllogistic. Should any one point. out to me, publicly or privately, any passage in Hamilton’s writings on his sense of ‘some’ which expresses or implies that this was the case in his mind, or even agrees better with this supposition than with its contradiction, I will discuss that passage when I next take up the subject. In the mean time, I cannot too distinctly affirm that the most attentive consideration® has not enabled me to detect such a passage. I now come to a proof which I cannot claim as one of my original grounds, for I never noticed it until after this addition had been dated and signed. TI can easily understand how I came to miss it. I always read Hamilton’s paper (VI.) in the Discussions as his defence of himself; I gave it comparatively little of sharp scrutiny as his attack on me. I recommend to every one who has to read a mixed polemical argument to give separate readings, some treat- ing it solely as attack without reference to defence, some treating it solely as defence without reference to attack. The article (VI.) was written against my second paper. In that paper I had no notion® whatever that Hamilton had any other sense of ‘some’ than that of the logicians; this will be very apparent. I state that δὲν of his propositions agree with those of the old school; which is not true of any one; I add that the remaining two are “ peculiar propositions.” I set out the list of syllogisms symbolically ; I point out the differences between Hamilton’s system and my exemplar derivation from it ; especially the failure of the canon of made to four persons, was not on a question of opinion, but | pages, he proceeds to explain his diagrams, in which parallel on a question of fact; namely, as to what sense of ‘“‘some”’ | straight lines denote coextension so far as they run together, Hamilton taught from his chair; this question could bedecided | and coexclusion so far as they separate. Letters stand for only by testimony. terms, as usual: D and A for coextensives; Z and Q for total 1 Since the bulk of this addition was written, Mr Baynes | coexclusives; B and C for includent and included; C and Καὶ (Dec. 20) has given an unconditional assurance that he will | for partially co-including and co-excluding ; and something I attempt to substantiate his statements. It was drawn out by | am not sure I understand for ‘Some—is not some—’, This I a letter of mine (Dec. 13) in which I administered what I call | must explain to show that he is really symbolizing his own a tebuke, and he calls a personality, upon the tone of his pre- | peculiar forms, Then follows “the rationale of the letters is ceding letter. Here I need only say that I think my remark | manifest;...... **; it is so, and it is manifest that, so far as the was richly deserved, and that I know it was meant to be directly | different letters are distinctively symbolic, they typify cireum- personal. I have much reason to be pleased with the result, | stances peculiar to Hamilton’s own system. The sentence then namely, the withdrawal of the condition which left Mr Baynes | runs on thus: ‘and it is likewise manifest, that this principle at liberty to attempt proof of his statements, or to leave it | of notation may be carried out into syllogistic.’” Here is an alone, as should seem fit. 1 expect good from the discussion, | express reference to syllogism in connexion with the new sense which is really that of the question, argued upon an instance, | of ‘some.’ Any one who denies that the new propositions are whether one who is nof of a mathematical turn can safely at- | meant to be applied to syllogism must rebut, from elsewhere, tempt to meddle with the- forms of logic. My oppexents—all _ the presumption which this passage raises. at least who follow Hamilton—will hold that the word in 3 “But Sir William Hamilton is the first who published Italics ought to be omitted; and I readily accept this as the the idea of taking all phases of usual quantification, and issue, should it please them to take it. making them the basis of a systeni of syllogism” (§ 4 of my 2 The latest account which Hamilton gave of the proposi- | second paper, Vol. IX. p. 1). The word uswad implies anti- tions furnished by his own ‘some’ is in the Discussions | thesis, not to any other meaning of ‘some,’ but to the numerical (στ. 631*): to me itis also the clearest. After distinctly re- | quantification. legating the old system, indefinite definitude, to subsequent 480 Mr DE MORGAN, ON THE SYLLOGISM, No. V. AND inference. Hamilton sees all this (VI. 630*), speaks of my treatment of his syllogism, repre- hends me for my alleged mistake about his canon of inference, &e. But what of all this? Hamilton had the old system as well as the new. This is the point. He goes on to show that his head is so full of his own new plan that he cannot read an opponent in any other sense; that he cannot understand an opponent who knows nothing of his ‘some at most’, which he was then giving for the first time in print. He goes on to say (VI. 6815) “1 shall first consider the objections [1. 6. my objections] to the propositional forms, which I have pecu- liarly adopted. But it is proper to premise a general enumeration of these;...” He then proceeds to lay down what I have already called his clearest explanation of the forms involv- ing his new sense of ‘some’. Having done this, he proceeds with ‘*Of the four proposi- tional forms specially recognised by me (1, 3, 6, 8) Mr De Morgan questions only two... ;” Surely, because I took the other two, as I said, to be converted Aristotelians; but Hamilton clearly supposes that I had taken him in his own peculiar sense throughout, Thus when he comes (VI. 633*) to assail me for compounding “ All Xs are all Ys” out of * all Xs are some Ys” and “ Some Xs are all Ys” he charges me with compounding ‘‘ incompossible propositions” : that is, he supposes me to be taking his own propositions in his own sense. He proceeds thus—‘* But unless some be identified with ald [as it may be in the old system}, if either of the latter propositions is true the other must be false;—nay, in fact, if either be true, the very proposition which they are supposed to concur in generating is false likewise.” I now see what all this means: it says in effect—*‘ You pretend to argue about my propositions and their connexion, while you are advancing objections which are valid only on the supposi- tion that some of my forms are the old ones.’ It would have been absurd in Hamilton to have argued against me that my conjunction required some to be identified with all, unless he had supposed me to be employing a ‘ some’ which could not be so identified. It stands thus. I was representing Hamilton’s system to the best of my knowledge. Hamilton had not, so far as I knew, any but the common mean- ing of ‘some’. But he had another meaning, of which his own head was so full that he took it as of course that in my representation of him I adopted that meaning. He did not object to my collection of syllogistic forms—and they are identical with those on which this discussion has arisen. By failure of objection he accepts these forms, and quarrels with nothing but the form I had given of the canon of inference. If Mr Baynes be correct, Hamilton ought to have told me that his own new use of ‘some’ was partial; that it is for par- ticular cases and for special objects; that it is only for isolated propositions and immediate inference; that I was wrong in assuming it intended for syllogism at all, and still more wrong in carrying it into every detail. Instead of all this, he opens his fire by charging me with having taken the rule of mediate inference from Ploucquet, and then proceeds to a detailed exposition of his own new forms, of which he makes me receive six, and object to two. It is now for Mr Baynes to make Hamilton contradict me without making him contradict himself. There is one point which many persons may misconceive: and on which 1 therefore notice Mr Baynes again. Wishing to give an account of all the strength of his answer, I reminded him of the difficulty which would exist, a hundred years hence, in confronting the weekly journal with the scientific quarto; and I suggested that he should substantiate certain ON VARIOUS POINTS OF THE ONYMATIC SYSTEM. 487 assertions in time for me to present his whole case in this addition. In reply! he is jocose upon the idea of posterity knowing anything about the matter. He may think, as perhaps many do, that the whole question is about Hamilton and myself; I, from the beginning, in 1847, have never considered it in this light. I believe, and I am joined by many reflecting persons, among students both of logic and of mathematics, that as the increasing number of those who attend to both becomes larger and larger still, a serious discussion will arise upon the connexion of the two great brancies of exact science, the study of the necessary laws of thought, the study of the necessary matter of thought. The severance which has been widening ever since physical philosophy discovered how to make mathematics her own especial instrument will be examined, and the history of it will be written, A great contest of that future day will be seen to have had its origin in our day; the details of the controversy which began in 1847 will be sought for as matters of its early history ; the questions which have arisen between Hamilton and myself will be renewed between writers who will have a small public versed in both sciences to judge them. Let all else end how it may, it is clear that the great change to which Hamilton’s name must be attached, the expressed quantification of the predi- cate, must have its history. ΤῸ every one of our day his own opinion as to how the questions will be settled, or as to whether they will ever be settled at all: but I find that the reflecting of all sides are prescient of a discussion to come. Among them I doubt not I may place the administrators of our Society for the last twelve years: I cannot in any other way explain the publicity given by them to the controversial parts of this series of papers. While such anticipations exist among so large a number of thinking men, there is no reason to quail before those who joke the jokes which are stereotyped against all who avow that they take posterity into their calculations: there is as good a retort, not quite so commonplace. I have over them this undeniable advantage: if right, I shall be known to have been right; if wrong, I shall not be known to have been wrong. A. DE MORGAN. December 26, 1863. 1 Mr Baynes derives innocent amusement from the words | _dddition to page 18. From the list of those who lay down “scientific quarto.”” It may be worth while to inform those | nothing but exemplar readings Keckermann must be excluded. who do not know it that the scientific transactions are, almost | His universals are all laid down in the singular (except cuncti), without exception, printed in quarto form; while separate | and his particulars all in the plural (except non nemo). And works are almost always in octavo. Hence areference to quarto | these are employed, for the most part, in his instances of syllo- is—in the United Kingdom—rapidly coming to mean allusion | gism; universals in the singular, particulars in the plural. to publication in one of the sets of transactions. I have, a | But Ramus may be added to the exemplar list. I also find hundred times, heard such a phrase as ‘‘ That is not in his | that guidam is not so uniformly excluded as Mr Spalding sup- work; that is in the quarto memoir;” meaning that the author | posed: Stahl and Keckermann both give it. had not published in his separate writing something he had Addition to page 43. The restricted readings may be easily previously given in a memoir inserted in the transactions of | connected with the peculiar pairs in page 30, in which pp goes some scientific body. I fell into the phrase “ scientific quarto”? | with (-), )(; ww with )-(, (); pw with ((, )-); wp with )), (-(. as briefer than “transactions of a scientific body.’ It may | Take the secondary and concluding relation from any case in be useful to foreigners, who have more separate writings in | which restrictions exist; the letters to which they are attached quarto than ourselves, to notice this growing idiom of our | in the last sentence point out the restricted readings. ‘hus language. (-((, giving )( and (-(, has we and wp for restricted readings: Sper στ )(-)), giving )-), (-), has pw, pp, restricted. Vou, X. Parr II, 62 jen λέμε Bee: τὸ arate, = 0 pile wha πὲ 89} Ἷ εἰν oil’ 9 ἜΘΝΗ “πὰ ἢ piesa γῆν ‘wal 1 diode a sft iviyes ‘it au caus gai oH o J εἰν silt αὐ .]1 ; sesh. sabi alloy yeu τὸ ' ΤΡ δ i Ten eve! dt dda ltt ᾿ Π Grice i γι οἷν itty ον wil 4 μ ri. ers ginny “ad Ἐπ᾿ al) (eae may) 4 " ty ΠΝ pti ub ju 2” 4 yyul ἔπ εὐ ἢ} bo eet Toes ail 1 "“ εἶσ aoe i= nares lee el i Ὁ 7. eerie cog ee Halil Lull’, ὐμοῦι ἫΝ a ae MP P 2 ᾿ dhl ipa "πα wy τ᾿ re eden askin vl wal Ι, iti yale 2) 6] bbe We desde fe tip tiny al ΠῚ alle Ὃν egpiive wee we ott We eliiiesh aid) swab aie ἀξ Ἢ “oii, ἐφ ἢ olies 2st Ta # yee bil ΠΡ ed hu TT Lae ΠΝ 2 hij aidan tbe " Alive si dg ys [8 ! ιν ἢν Doel ἣ ; ᾿ 3 " γ : io Pek Ee | weve OF) inti Ay (PSB te vi) Here. Seb ih pa, Pa a vin eta Yowenwyeyen ἢ Jedota od Mae ὃ Po Δ. με 5 weet μὶ an αν} fp (yet Hides by aes ly ate ty . apt wth 1.0} hen Tom = let behagecetl ΥΥΝ ! rw Hy ἀν}... ἢ] 86] I jam pete 8 τον φύσις 0a ὩΣ ΤΟΝ ΕΣ ; - ; * ΙΗ Vin Ἢ, ἡ ἡ eR τ Δ AF yYiluw }: Jens >. ἢ εἶν alist ἢ ἀπ} Liv pest ee att) fen elie] ἵν Vay Cov lppiya af grail 1 ig ΠΥ δ θὲ Ἧ. pigs ads sib oeell ob amos 1 ὟΝ ly ae (= hy ils 4 TAAL IAL ob ᾿ a ναι, ται Δ, ὦν “Ξ ‘ t= Ἀ} ΜΝ ey ρος εἰν» 4} Ee!) isla ἐν μὲ hows ay νῷ pinyin ad] μα ΝΗ eure δ sepa ὁ πῇ ν φρο ‘areal ac Re ee ee ih NM) egaicew) Vig Al PH νυ PUD Bie! τ ate ὁ" wT αν ἂν ἡ δι ἢ} οὐ τΕῊᾺ πὴ νει γῇ} Mit lon olive Va γα dl ai α΄ να Pepin tent ἢ toll) ‘ie ὁ 4 :}} Hire υἱ rer ΗΠ oily a ΠΝ ΚΝ a ) δέν λυ ἡ ΠῚ nf Sail [4 Atld ah oe an, eke oh ioe ce ὉΠ. 4, awit i τ δ “Ἂν uta? ἢ on ψγ ννυ νυ OF ἣν του εν ΗΠ (lee Vubive A é : irom GA Add TARA iden . Slam "" mae Niger teers Oe Se εὐ πεθῦνΣ. hu εν ot a Bk ibd Lal τα Προς ae ite leeciee> - mn Hin lt: ἱ ἡ ΓΕ 1. | Ae ὧν ali A) dies τῷ ate ο γον Δ μϑεάνγι ἡ νὰ, oluseses ers gael iwtaia fone ciig ctdul σεν OnE ctytes epolisltves? Mild @ 5) {ὦ i ἣν ἥ ὡ μδὶ att ate > GL tate Sina) Coated aah SA] UP, τῶν elise wht τ μὰ, DD) μων 4 arte. Nd! = ee ey rece ὐ{}. i ᾿ oe rem ; . ν᾿ a . ja) ἡ; OAT AO’ a) μιῇ veil “ἢ δ ππλν γα Te, abt ait shone am at tnd eT ae a ea με ] ον Ἀτοιπὰ apo ΠῚ ΠΩ su parry ig “bed iat onan il a ι} μὲ εἰ nine. linha duping, 3 mi uel pied ol A ΗΒ ἢ ee ws one ἡ} Ye amin ἢ ΜΕΝ ι΄ ᾿ ΠΤ ΓῚ] ἍΜ} ΕΠ Ὁ] werta ot Tia, Pe τ RA 8 ead We ‘ μἰ. i ee ΠῚ vty @ Apes ' τ " ee yl ΜΠ AA Ne fH tty Medel 1h Ὧι Bry ody ΠΡ « iv tent εὐνῇ 2 ee own oi) ἢν ἡμεῖς δὶ nah bsaiina.s μὴν Lilyaoa, Ὁ Sine a ty wiathinnndt ei er) Be dash ak by il avail ‘ighilw ut 43 if wi angi ca Νὴ eo wp ΟΣ (vag ὅτι eae (ἢ Ἧς roe ἴα πὸ i i avi ARE ! ff OP ΕΠ ial dof al ell ‘wo ἮΝ} ὉΠ} ἢ iret bus “μι vad ἣν ἱ Re for fede { Re δὶ 4 Uhl te engl i vai 5 ciel γνῶ οι Γ᾿ in δ, a) kaatpsy heeies Bik aris αὐ μὲν, a cen.” - “ἐῴκει δὲ, ὃν Sal! vat τ ᾿ ine ἘΣ wae => rie os ϑ. pear oe INDEX TO \VOL@ME ΣΧ: ABELARD’S Dialectice, 152 Abnormities in voluntary Muscles, 240—247 Accelerations and velocities with respect to moving axes, 1—20 : —— radial, transversal, and azimuthal, expressions for, 6 —— tangential and in principal normal, expressions for, 7 Accent, Latin, Mr Munro on laws of, 377; Ritschl on, 382 Acrostichs, Latin, Mr Munro on, 376 JEschines, supposed statue of, 233 Aischylus, Agamemnon, explanation of passages in, 92, 93 Aggregation and composition, distinction of, 192 Aggregation, postulates and theorems relating to, 292 Arry, G. B., M.A. On the substitution of methods founded on Ordinary Geometry for Methods based on the General Doctrine of Proportions, in the treatment of some Geometrical Problems, 166; Euclid’s Doc- trine of Proportion the only one perfectly general, ἐδ... but in special cases often cumbrous from its generality, 7b.; can be avoided when geometrical lines alone are the subject of investigation, ἐδ. ἢ by a new treatment of a theorem equivalent to Huclid’s simple ex a@quali, ib.; and of doctrine of similar triangles, 7.; series of propositions suflicient for this purpose proved, 167—170; their use illustrated by application to a well-known theorem, 171; Adden- dum. New Proof by Prof. de Morgan of Euclid’s Theorem of ex wquali in ordine perturbata, 172 — Suggestion of a Proof of the Theorem that every Algebraic Equation has a Root, 283 — Supplement to a Proof of the Theorem that every Algebraic Equation has a Root, 327; objection to Proof (p. 283 et seq.) that it is obtained by the use of imaginary symbols, 327, § 19; the problem divested of the idea of imaginary roots is to shew “that every algebraic expression can be divided without remain- der by 2*—2p cos 6a + p*,” § 20; the actual division by this expression effected and the remainder obtained in the case of an expression of the Sth degree, § 21, 22; the condition of evanescence of the remainder leads to two equations of condition, the possibility of satisfying which has been demonstrated in the former Memoir, § 23, 24; Cotes’ Theorem demonstrated by a method indicated, § 26; the method of the Memoir perfectly general, § 27 Airy’s Integral | cos 5 (08 — mw) dw, remarks upon, 0 105; the differential equation to which it leads dis- cussed, and the values of the arbitrary constants de- termined, 115; its complete integral geometrically illustrated, 116 Algebra, a branch of thought in which the process is visible, 179 Alimentary Canal, manifestation of current force during secretions in, 250—252; theoretical remarks on, 251, 252 Ambiguity, terminal, mathematical view, 199; meta- physical view, 200 Ameinocles, inventor of Trireme, 84 Ampere’s Laws of Electromagnetism, 55 Analogies, physical, 28 Animal Electricity, remarks on, 248 Antisthenes, criticised in Thezetetus, 158, 165; his hatred of Plato, 159; criticised in Philebus, 160 Argument, on origin and proper use of word, by Dr Donaldson, 317; etymology of, ib.; classical and tech- nical uses of the word, 319; used in logic to denote the middle term, 321; three meanings of, 324; ought not to be used for a process of reasoning, 326 Aristides, supposed statue of, 231 Aristotle, his allusions to Plato, 147; De Anima, Pas- sage in, ἐδ. ; reference to Politicus, 148; Politics, pas- sage in, 7.; method of dichotomy, ἐδ. ; other allu- sions to Plato, 149 ; Posterior Analytics, 162 Aristotle's system of logic, exemplar, 443 ; misconceived by recent writers, 7b. Aristophanes, Equites, explanation of passage, 92 Arithmetical whole, in logic, 190, 194, 209, 212 ‘Arnold on Homer, Munro on, 403 62—2 490 Arrows, velocity of, 370; range of, 373 Athenzeus, comic fragment preserved by, 161 Atom, meaning of, as applied to a proposed division of octave, 130 ἄτρακτος, explanation of, 308 Attraction, formule of, analogy between, and those of heat, 28 Attribution, unity of, discarded in favour of plurality of qualification, 185 Average, of &® power of sum of values, theorem relating to, 412; not merely the mean value but also the mean supposition as to the mode of obtaining value, 416 Axes, moving, theory of, 1—20 Baxter, H. F., M.R.C.S.L. On Organic Polarity, 248; reasons for choice of this title, 7.; history of previous researches, 248, 249; subject of the paper, Zhe Mani- Jestation of Current Force during the processes of Secretion: (i) during the formation of the secretions in the mucous membrane of stomach and intestines, 250—252; Wollaston’s conjectures and experiments on this subject, 253; (ii) during biliary secretion, 254 ; (iii) during urinary secretion, 255, 256; (iv) du- ring mammary secretion, 257; (vy) during respiration, 257, 258; concluding remarks (containing reference to Graham’s researches on Osmose), 258—260 Baynes, T. Spencer, Essay on the new Analytic, 448 ; and Hamilton’s system, 481 Beams, motion of, under passing load, 360 Beats of imperfect consonances, on the, 129—145; his- tory of theory of, 129—135; Dr Smith’s theory and » his formulze deduced, 135—141 —— of two kinds, Tartini’s and Smith’s, 131, 132; Tar- tini’s used by Sauveur, 131; the two kinds confounded by Young, Chladni, and probably by Robison, 133 —135 Biliary secretion, manifestation of current force during, 254, 255 Blakesley, on his copy of an inscription, by Mr Munro, 374 Bow, motion of vibrating, 359; power of steel, 373 Breadth and Depth, use of, by Sir W. Hamilton, 225 note, 229 Butler, Professor Archer, on History of Ancient Phi- losophy, 94; statement of Platonic philosophy, 95 Cauchy’s theorem on the limits of imaginary roots de | monstrated, with an extension thereof, 265, 266 Charges of propositions and syllogisms in ordinary use, 181 Charts, Mercator’s and Gnomonic Projection, 272; for great circle sailing, construction of, 278; use of same, 279; examples, 280 Chest, deformity in, will not arise from the defect or absence of the pectoral and serrati muscles if the skeleton be sound, 246 Chladni, his confusion of Tartini’s beats with Smith’s, INDEX. 134, 135; disputed the claim of Tartini to the disco- very of the grave harmonic, 132 n. Circles, theorem concerning tangents to each pair of three unequal, proved, 171 Cirta, metrical inscription at, 374 Coil-machine, spherical electromagnetic, theory of, 79, SO Commodian, his poems, Mr Munro on, 376 Concert-pitch, cause of ascent of, considered, 131 Conduction of current electricity, theory of, 46 equations of, 39; reference to memoirs by Stokes and Thomson, 40 Conservation of momentum, principle of, 9, § 16 Continuity, ordinal, illustrated, 24 of value and permanence of form, distinetion be- tween, 22 Copula, of cause and effect, 179: other than “is” need- ed, 193 Core of electromagnet, effect of, 77, 78 Correlatives, right of all in any set to equal fulness of treatment, 452 Cotes’ theorem, a demonstration of, indicated, 329, § 26 Current force, manifestation of, during secretion, 248. See Baxter, H. F. on Organie Polarity Currents, electric, action of, at a distance, 48; pro- duced by induction, 50 Curves, course and latitude, construction of, 282 De Moivre’s Doctrine of Chances, appearance of e-2* in, 418 De Moreay, A. on the Beats of Imperfect Consonances, 129—141; History of the theory of Beats, 129— 135; Theory of Beats of Imperfect Consonances, 135 —141: obseryations on tuning and on temperament (Postscript), 141—145; Dr Robert Smith’s work on Harmonics, 129; two kinds of beats, Tartini’s beats and Smith’s beats, 131, 132 J umfavourable opinion of Young on Smith’s theory shewn to result from con- fusing Tartini’s beacs with Smith’s, 133, 134; the same mistake made by Chladni, and, probably, by Robison, 134, 135; Tartini’s beats used by Sauveur, 131; and Smith’s accusation that Sauveur confounded the beats of an imperfect consonance with the flutter- ings of a perfect one shewn to be not true, 132, 133; on the division of the octave and on musical intervals, 129—131; new division proposed, and a new term atom defined, 130; various problems solved, as on the number of mean semitones in a number of atoms, ὅτο. ib.; Tartini’s grace harmonic, 131; Tartini’s beats considered in connexion with Smith’s, 135—137; for- mul obtained, 138; Smith’s formulze deduced there- from, 139 —— On the general principles of which the composi- tion or aggregation of forces is a consequence, 290; tendency defined, 291; four postulates stated as the INDEX. 491 grounds of every method of aggregation known in mechanics and five theorems following from them, 292; whence the laws of aggregation of forces either meeting in a point, 293, or parallel, are deduced, 294; applications to translations, 296; and rotations, 297; velocity of ditto, 298; statical pressure, 299; dynamical pressure, 302 De Morgan, A. On the question, what is the solution of a Differential Equation? A supplement to the third section of a paper, on some points of the Integral Calculus, printed in Vol. 1x. Part IL, 21—26. The object of the paper is to shew that common theorems about the singularity of the constant of integration must be transferred from differential egwations to differential re/ations, 22; correction which the com- mon theory requires, 25; Differential Equation, what is meant by solution of, 21—26 —— A proof of the Existence of a Root in every Alge- braic Equation: with an examination and extension of Cauchy's Theorem on Imaginary Roots and Re- marks on the Proofs of the existence of Roots given by Argund and by Mourey, 261; prefix to Sturm’s demonstration of Cauchy’s theorem on the limits of imaginary roots which establishes the existence of roots, 261—263; algebraical substitute for a geome- trical step in Sturm’s proof, 264, 265; Cauchy’s theorem with an extension thereof, 265, 266; Ar- gand’s Proof of the existence of Roots, 267; Mourey’s proof, 268, 269; postseript on divergent series and spherical triangles, 269, 270 —— New Proof of Euclid’s proposition ex wquali in ordine perturbato, 166, 172 On the Syllogism No. III, and on Logie in general, 173; modern definition of logic relates to a distinction more familiar to mathematicians than logicians, 175 ; logic both science and art, 181; charges of propositions and syllogisms, ἐν. ἢ objections to use of mathematical symbols in, discussed, 183; the only science which has grown no symbols, 1§4; hitherto confined to logico-mathematical field but to be ex- tended to the metaphysical field, 184; extension and intension claimed for both the mathematical and metaphysical sides of, and symbolised, 184; dis- tinction of extension and comprehension m:sconceived by recent logicians, 187; onymatic relations, 190; fourfold mode of thought, denoted and symbolised, ἐδ... arithmetical whole in logic, 190—194; form, quality and quantity of propositions, 194; use of force to express quantity in metaphysical reading, 197; schetical system, 199; spicular notation, 198; mathe- matical and metaphysical views of terminal ambi- guity and precision, 200; first elements of a system of logic, 206 — On the syllogism No. IV., and the logic of Rela- tions, 331; influence of the schoolmen on language, 332; difficulties in applying laws of thought to un- familiar matter, 334; logie not to be confined to the onymatic form, 335; no purely formal proposition except this, ‘there is the probability a that X is in the relation Z to Y . 339; necessity of taking ac- count of combinations involving a sign of inherent quantity, 341; table of forms of syllogism, 350; ex- tension to quantified propositions, 352; cases of con- vertible and transitive relations, 353; technical exhi- bition of the syllogism not necessary in reasoning, 356; syllogisms of transposed quantity, *355 —— On Syllogism No, V., and on various points of the Onymatie system, 428; criticism of Hamilton’s sys- tem, 7b.; explanation of Aristotle's system, 442; which is affirmed to be exemplar, 443; and misconceived by recent writers, 7b. ; right of both correlatives in any pair, and of all in any set to equal fulness of treat- ment, 452; application to the distinction of affir- mation, and non-affirmation, ἐδ. ; syllogism of inde- cision, 453; eight onymatie forms deduced from purely onymatic meaning, 455; alleged demonstration of the necessity and completeness of these forms, ib. ; restrictive propositions, their affirmation and denial introduced in every view except the purely onymatic whenever complete treatment of all correlatives is allowed, 456; completion of exemplar system, 457; extended comparison of the onymatic relations, 459 ; system of primary aud secondary relations by copula of identification, 463; the same when the copula is any one of the simple onymatic relations, 469; the full system at which the Hamiltonian quantification aims, 471; logical basis of extension and compre- hension, 475; addition on a recent phase of the con- troversy, Dec. 1562, 481 On the Theory of Errors of Observation, 409 : probability and facility contrasted, ib.; difficulty of passing in argument from the finite to the infinite or infinite to finite, 410; theorem that “all values being positive and the number of letters taking value being infinite, the average of the &* powers of the sum of values is the Δ power of the sum of average values,” 412; laws of facility of value discussed, 413; deficiency in Laplace’s investigation, 415; first prin- ciples, number of observations finite, 416; average defined, ἐδ. ; law of error deduced from observations, 418; mode of deducing probable results, 421; diffi- culties in probability corresponding to indiyisibles in geometry, 424; process of solving them, 7.; origin of fluxional notion, 426 Demosthenes, Statue of Solon mentioned by, 231, 235 Dialectic, Platonic, 152; Aristotelian, ib.; of School- men, 7b. Diamagnetic Induction, Theory of, 44 Diamagnetic or Paramagnetic Sphere, effect of, in a uniform magnetic field, 70,71; case when the field is the terrestrial magnetic field, 71 Dielectrics, Theory of, 43 492 Gu idu Differential Equation dt wu =O discussed, x da 121; the complete integral according to ascending series given, 122; the same integral also expressed by descending series, 7b.; the relation between the constants in these two integrals determined, 123; the discontinuity of the arbitrary constants deter- mined, 122 Divisive Method, invention of, attributed to Plato, 162 Donaupson, Dr J. W. On the origin and proper use of the word Argument, 317; etymology of Argument, 2b. ; borne out by the classical usage and technical appli- cation of the word, 319; the proper use of the word in logic to denote the middle term, 321; Aristotelian enthymeme may be rendered approximately by argu- | 299. ment, 322; how argument and topic came to denote the subject of a discourse, 324; three meanings of argument in English writers, (1) Proof, (2) process of reasoning, (3) subject of discourse proved by exam- ples, 325; the second use ought to be excluded from scientific language, 326 —— On Plato’s Cosmical system, 305; translation of passage in Republic, ἐδ. ; examination of points in Greek text of same, 807; explanation of meaning of same, 310 —— On the Statue of Solon mentioned by schines and Demosthenes, 231; a statue in WZuseo Borbonico supposed to represent Aristides the Just, 7b.; or Elius Aristides, 232; or schines, 233; these theo- ries refuted, ἐδ: shewn to be that of Solon, 235 On the structure of the Athenian Trireme, 84 Dynamics of material system, with application to motion of body of invariable form, 7—20 Dynamics and Statics, their relation to each other, liz. Fandi, Vassali, Letter from M. Delaméthrie (Journal de Physique, 1799), 256; his experiments in support of existence of animal electricity, 7b. Elastic rod, motion of, fixed at one end, 365; bent within breaking limit may be broken by rebound on being set free, 369 Eleatic, doctrine, 97; dialogue, the Parmenides an, 100, &c.; Stranger, 154 ; school, 7b.; logic, confusion at root of, 155; ontology, nothingness of, 156 Electric Currents, on Quantity and Intensity as Proper- ties of, 52—54 Electrical science, present state of the theory and requirements of, 27; the true method of investiga- tion, 7b. Electrical and Magnetic problems with reference to spheres, 68—S83; I. Theory of Electrical Images, 68—70; u. On the effect of a paramagnetic or dia- magnetic sphere in a uniform field of magnetic force, INDEX. 70, 71; ut. Magnetic field of variable intensity, 71, 72; 1v. Two spheres in uniform field, 73; v. Two spheres between the poles of a Magnet, 73, 74; VI. On the Magnetie Phenomena of a Sphere cut from a substance whose coefficient of resistance is different in different directions, 74—76; vu. Permanent mag- netism in a spherical shell, 76; vi. Electromagnetic spherical shell, 77; 1x. Effect of the core of the electromagnet, 77, 78; x. Electrotonie functions in spherical electromagnet, 78, 79; x1. Spherical elec- tromagnetic Coil-Machine, 79, 80; xu. Spherical shell revolving in magnetic field, $1 Electricity, current, conduction of, 46 Electrodynamics, Weber’s physical theory of, stated, 66, 67 Electromagnet, effect of core of, 77, 78 Electromagnetism, 55—57; Ampére’s laws of, 55 Electromagnetic, spherical, coil-machine, theory of, 79, 80 : Hlectromotive forces, 46 Hlectrotonic functions, or components of the Electro- tonic intensity, 63 Electrotonic intensity round a curve, 65 Electrotonie state, meaning of, explained, 51, 52; of Faraday, discussion of, 51—67 Electrotonie state, conditions of conduction of currents within the medium during changes in, 64; summary of theory of, 65—67 Emerson, account of his connexion with the theory of beats and the formula which he obtained, 135 and note Enthymeme, instances of use of term, 321 Equation, Differential, meaning of solution of, 21—26 — dy?—adx*=0, Euler’s solution of, viz. (y—aa+b) (y+ax+c)=0, with opinions of Lacroix and Cauchy thereon, 21; the assertion that the generality of * Euler’s solution is not restricted by the supposition b =c examined, 24 Equation and Relation, proper meaning of terms, 21 Equation of conduction, 39 “Equation, proof of theorem, that every algebraic equa- tion has a root, 283, 327 Er the Pamphylian, 305; same as Zoroaster, 313 Errors of observation, theory of, 409; deficiency of La- place’s investigation, 415; first principles of, 416; law of, deduced, 418 Euclid, Siath Book, Props. 1—xm1, can be proved in a different manner, 171; which is of general application to a number of Theorems involving proportions of straight lines (not areas, &e.), 7b. Euler’s equations of rotation obtained, 14 Euripides, Hercul., explanation of passage in, 93 Ex equali, new treatment, equivalent to Euclid’s theo- rem of, 167, 168 —— in ordine perturbatd, new proof of theorem of, by Prof. De Morgan, 172 INDEX. Exemplar, and cumular reading, 212; Aristotle’s sys- tem affirmed to be, 443; system completed, 457 Extension and comprehension, distinction made by Aristotle, 187; misconceived by recent lo- gicians, 188; logical basis of, 475 Extension and intension claimed for both the mathe- matical and metaphysical sides of logic, and symbo- lised, 184; former predominates in the mathematical whole, 191; latter in metaphysical, 192 Facility, contrasted with probability, 409 ; modulus of, 413, 418; of value, laws of, discussed, 414 Faraday’s lines of Force, 27 —— Experimental Researches, the mathematical foun- dation of the modes of thought in, expressed in six laws, 65, 66 —— Electrotonie state, 51 —— researches on the origin of the power in voltaic circle suggest certain researches on animal secretions, 249 Field, Magnetic, of uniform or variable Intensity, on effect of sphere placed in, 70—76; revolving spherical shell in, 8I—83 Winite to infinite, difficulty of arguing from, 410 Fluid, Incompressible, Theory of Motion of, 30—33 Fluxion, notion of, included in intension and remission of schoolmen, 426 Force, Intensity of, Method of representing by velocity of an imaginary incompressible fluid in fine tubes of variable section, 30 —— lines of, defined, 29 Forces, laws of aggregation of, 993; proofs of parallelo- gram of, not mere mathematical playthings, 301 Force, to express quantity in metaphysical reading, 197 Form and matter, distinction of, 174; misconceived by recent writers, 177 Foucault’s experiment on heat produced in spherical shell revolving in magnetic field, referred to, 83 Fraser's Magazine, criticism of an Article in, by Mr Munro, 403 8 Fresnel’s Integrals i cos G ots, &e.; also integral 0 t 2 J a = et dé are connected with the integral | e” da, 0 0 112 Functions, Electrotonic, defined, 63; in spherical elec- tro-magnet, 78 ynyeveis (of Sophista), on, 165 Goprray, H., M.A. On a chart and diagram for facili- tating great circle sailing, 271; proposal for chart on Central or Gnomonic projection, 272; on windward sailing, 273 ; on composite sailing, 277; on construction of the chart, 278; description of course and distance diagram, 279, on use of same, 7b., examples of use, of, how | 493 280; on construction of course and latitude curves, 282 Graham, researches of, on Osmose considered, 259 Grave Harmonic of Tartini, 131 Gyroscope, Theory of, 18—20 Hamilton’s, Sir W., criticism on De Morgan’s logic answered, 223, 229 ; his system criticised, 429 ; forms expressed in Aristotelian forms, 434; quantification, full system at which it aims, 471 Harmonics, Dr Smith on, 129 Haywarp, R. B., M.A. On a direct Method of esti- mating velocities, accelerations and all similar quan- tities with respect to axes moveable in any manner in space, with applications. The Method with some Kinematical applications, Section 1. pages 2—7; Dy- namical applications, Sect. τι. 7—20 Heat, formule of, analogy between, and those of attrac- tion, 28 Helmholtz, Method of, in Memoir on Conservation of Force, applied to conditions of conduction of currents within the medium during changes in the electro- tonic state, 64 Hexameter, Latin, Mr Munro on, 387; English, the same on, 403 Holder, Dr W. His claim to be considered the imme- diate predecessor of Sauveur in the theory of beats, 131 n. Iambic, Latin, Mr Munro on, 383 Ideas, Platonic Theory of, 94 Images, Electrical, Theory of Sir W. Thomson’s, 68—70 Imaginary symbols, use of, not strictly logical, 327 Indivisibles, 424 Induction, Paramagnetic and Diamagnetic, Theory of, 44 —— Magnecrystallic, Theory of, 45 Inductive capacity, 54 Inertia, measure of, 12; relatively to translation is mass, and to rotatioa a guaternion, in Sir W. Hamilton’s sense, 7b. Inscription at Cirta copied by Mr Blakesley, Mr Munro on an, 374; another copy of same, 403 Integral, Airy’s, 105, 115; mode of discontinuity of constants in, determined and discussed, 117; Geome- trical illustrations, 118; the complete integral ex- pressed, 119; two different forms of the integral, 116; the two linear relations which connect the two con- stants in these different forms determined, 119—121 Integrals, Fresnel’s, 112 ‘00 Integral \ e—™ sin θαυ dx considered, 106; determina- 0 tion of the arbitrary constant of this integral and its mode of discontinuity ascertained, 107; the constant determined by a second method confirmatory of pre- vious results, 109 494 Intervals, Musical, Theory of, 130, 131; various pro- blems on, 130; application of logarithms to, 130 and note; Woolhouse’s essay on, referred to, 135 Invariable system, motion of, investigated, 11—1S Kinematical problems, 5—7; a. Relative velocities of a point in motion with respect to revolving axes, 5; 6. Accelerations, radial, transversal in the vertical plane, and perpendicular to that plane, 6; c. Accele- rations parallel to the tangent, principal normal, and normal to osculating plane of any curve, 7 Language, influence of schoolmen on, 332 Lines (straight), properties of, investigated by means of a series of propositions, 167—170; which involve a new treatment of Euclid’s Theorem of the simple er equali, 166, and are proyed by Elementary Geome- try, ib. Lines of Force defined, 29 Logic. See De Morgan. Milton’s, 181 2.; Ramus’, ἐδ. ; Sanderson’s, 227; | Port Royal, 1S8—228; Watts’, 228; Hospinian’s, 229 general considerations, 173; Aristotle’s logic only a beginning, 174 — not to be confined to the onymatic form, 335; of Relation, 331 Logicians, the first were also mathematicians, 175 Logico-mathematical, logico-physical, logico-metaphysi- eal, and logico-contraphysical modes of thought sym- bolised, 190, 210 Magnecrystallie Induction, Theory of, 45 Magnetic Field, Uniform or Variable, considered, 70— 76 —— Quantity and Intensity, 54 Magnets, theory of action of, on magnetic or diamag- netic spheres considered and explained, 70—76 —— Permanent, Theory of, 44 Mammary Secretion, Manifestation of Current Force during, 257 Material, some objections arising from its introduction into logic considered, 177 Material System, on Motion of, 7—10 Maxwe t, J. CLerk. On Faraday’s Lines of Force, Preli- | minary explanations, 27—30; I. Theory of the Motion of an Incompressible Fluid, 30—33, § 1—9; II. Theory of the uniform motion of an imponderable incompressible fluid through a resisting medium, 33— 42, § 10—33; Application of the Idea of Lines of Force, 42, 43; Theory of Dielectrics, 43, 44; Theory | of Permanent Magnets, 44; Theory of Paramagnetic and Diamagnetic Induction, 45; Theory of Magne- erystallic Induction, 45, 46; Theory of the conduction of Current Electricity, 46; on Electromotive Forces, 46—48; on the Action of Closed Currents at a Dis- | tance, 48—50; on Electric Currents produced by In- | INDEX. duction, 50, 51; Part II. on Faraday’s “ Electrotonic State,” 51; on Quantity and Intensity as Properties of Electric Currents, 52—54; Magnetic Quantity and Intensity, 54; Electromagnetism, 55; Analytical Theo- rems, 57—62; bearing of the foregoing theorems on the theory of magnetism, 62; Summary of the Theory of the Electrotonic State, 65—67; Electrical and Magnetic Problems with reference to spheres, 68—83 Mode of thought, four-fold, how denoted and symbo- lised, 190 Momentum, 7 ὃ 10; Conservation of, 9 § 16 Momenta, linear and angular, defined, 8 ὃ 11 Motion of beams and elastic rods, 359 Moving Axes, theory of, 1—20 Munro, H. A. J., M.A. Ona metrical Latin Inscription at Cirta, 374; instances of Latin acrostichs, 376; Com- modian’s poems, ἐδ; on rules of Latin accent, 377; on the Saturnian, hexameter and iambic verses, 383 ; discussion of the inscription, 397; Appendix, 403 ; criticism on an article in Fraser's Magazine, tb.; on the English Hexameter, 7b. Muscles, voluntary, remarkable abnormities in, 240— 247 ; varieties in, most frequent in parts the office of which is different in different animals, 245; cases ex- emplifying primary defects in, 240, 247 Musical Intervals, 129, 130; measured by ratio, not by difference, 129 1 3 Names, uses of, 185; various to one relation, 201; four applications of, 206; formed from other names by in- _ tension, extension, and combination, 207 Nautical Magazine, on windward sailing, 274 Notation, spicular, in logic, extension of, 198 Observation, theory of errors of, 409; Deficiency in La- place’s investigation of, 415; first principles of, 416 Octave, on the division of, 129, 130; new division of, proposed, 130 Onymatic relations, 190, 209 logic not to be confined to, 335; extended com- parison of, 459; system, 428; forms, deduced from purely onymatic meaning, 455, and demonstrated to be necessary and complete, 7b. Organic Force, a Polar Force, 248, 260 | —— Polarity, 248; History of certain researches on, 248, 249 Osculating plane, acceleration normal to, shewn to yanish, 7. | Osmose, phenomena of secretion compared with, 259 ; Graham’s researches considered, 2. Pacer, G. E., M.D. Instances of Remarkable Abnor- mities in the voluntary Muscles, 240—247 ; description of two cases of, in brothers, 240—243; the abnormities in these cases of two opposite kinds, characterised by defect and excess of muscular development, 243; INDEX. the former congenital, the latter probably had their origin after birth and were a consequence of the for- mer, 243, 244; cases, which present some of the same features, mentioned by Messrs Quain, Poland, and James Paget, 244, 245; all these cases illus- trate the fact that varieties in muscles are most fre- quent in parts the office of which is different in different animals, 245; they probably exemplify also primary defects in muscles, ἐδ. another case com- municated by Mr James Paget, 246; no deformity of the chest existed in any of the cases; hence chest deformity, as in chicken-breast, will not ensue ἐγ the | skeleton be sound, though the Pectoral and Serrati muscles be wholly wanting, 246; Pathology of the cases shortly considered, 246, 247 Paramagnetic or Diamagnetic sphere, effect of, in a uni- form magnetic field, 70, 71; case when the uniform field is that due to terrestrial magnetism, 71 Induction, theory of, 44 Parmenides (Plato), account of, 97, probably not a Pla- tonie dialogue, 100; not referred to by Cicero or Aristotle, 104; referred to by Athenzeus, 104; School and theories of, 154; confusion in root of Eleatic logic, 155 Penteconter, 85 Periodic Series, Theory of, referred to, and certain re- sults on the discontinuity of arbitrary constants com- pared therewith, 112 Permanence of Form and continuity of value, distinction between, 22 Phedrus, method of Division in, 163 Philebus, treatment of Cynical ethics in, 160 Pitch (concert), cause of ascent of, considered, 131 Plato, Cosmical System of, by Dr Donaldson, 305; his ’ theory of sacred numbers, 310; source of his mate- rials, 313 —— his theory of Ideas, 94; Sophista and Politicus, on the genuineness of, 146; alluded to by Aristotle, 147, &c.; Thezetetus, connected with Sophista, 151; Gi- gantomachy, 157; Inventor of Divisive Method, 162; Pheedrus, 163 Poinsot’s “couple d’impulsion,” same in meaning as angular momentum, 14 n.; his Théorie de la Rota- tion referred to, 18 Polar Force, reasons for considering Organie Force as such, 248, 260 Polarity, organic, 248; history of certain researches on, 248, 249 Politicus (Plato), genuineness of, 146; reference to, by Aristotle, 148; quotation from, 160 Prevecilius, an inscription by, Mr Munro on, 375 Precision, terminal, 200; relation of the two, 213 Pressure, Statical, laws of, 299; Dynamical ditto, 302 Principal Axes, the only permanent axes of rotation of a body acted on by no forces, 13, § 24 Vor, S¢, Parr ΠῚ 495 Principles, general, of which the composition of forces is a consequence, 290 Probable results, mode of deducing from observations, 421 Probability, contrasted with facility, 409; case of in an infinite number of events, 410; negative, 420, 421 n.; indivisible of, 424 Proof, considerations on, 214 Proportion, Euclid’s Doctrine of, referred to, 166 ; necessarily general in certain special cases, ἐδ. as in the investigation of properties of geometrical lines, ib. Proposition, form, quality and quantity of, 194, 208; affirmative and negative, 214; universal, 215; restric- tive, introduced whenever complete treatment of cor- relatives is allowed, 456 Propositions, series of Geometrical, 167—170 ; and appli- cation to a well-known Theorem, 171 Prout, views of, on Secretion, 248 un- Quantity of no fundamental account when inclusion and exclusion are opposed to one another and combined with assertion and denial, 195; of extension and in- tension, how expressed, 197; attachment to mathe- matical view of, 204 Rameau, account of, 132 ”.; relation of his views on Harmonics to those of Tartini, ἐν. Relation, in logic, defined, 208; onymatic, 190, 209; logic of, 331; convertible and transitive, 353; system of primary and secondary, by copula of identification, 463; and when the copula is any one of the simple onymatic relations, 469 Relation and Equation, proper meaning of terms, 21 Republic (Pilato), translation of passage in, 305; discus- sion of points in same passage, 507 Resisting Medium, theory of uniform motion of an im- ponderable incompressible fluid through, 33—42 Respiration, Manifestation of current Force during, 257, 258 Rhumb, sailing on a, advantages and disadvantages of, 271 Rhythm, Greek and Latin, Mr Munro on, 383 Rigid body, meaning of term, 11 7. Ritschl on Latin Accent, 382 Rours, J. H., M.A. On the Motion of Beams and thin Elastic Rods, 359 ; the original object of the investi- gation to determine the motion of a vibrating bow, %.; this problem closely connected with that of the vibra- tion of a Railway Girder under the action of a passing load, investigated by Professor Stokes, ἐδ. ; the latter problem easily solved by the aid of Fourier’s Fune- tions when the deflection of the girder is very slight and the pressure of “the load constant, 360—363; method of solution indicated when pressure varies, 63 496 364; and applied to determine deflection of a girder suddenly loaded in the middle, 364, 365; to deter- mine the motion of an elastic rod fixed at one end and free at the other, 365—370; Fourier’s func- tions do not apply to this case, 365; a bent rod with- in the breaking limit at the centre may be broken by the rebound after it is set free, 369; to determine the velocity of an arrow discharged from a bow, 370—372; the amount of displacement of the centre of the cord assumed to be four times that of the ends and to be constant during the motion, 370; any two bows of same material and of similar figure will impart the same maximum velocity to an arrow, provided the arrows and cords are also similar and proportional, 372; experiments on the range of the “bolts” and the general power of the steel cross-bow, 373 Root, proof of theorem that every Algebraic equation has a, 283, 327 Roots of equations, Argand’s proof of existence of, 267 ; Mourey’s proof, 268, 269 Rotation of a body about its centre of gravity, solution of problem of, 14—17; case when no forces act, 17 Rotations, laws of, 297 Sailing, great circle, 271; difficulties attending, ἐδ. ; windward sailing, 273; composite sailing, 277 Sauveur, the beats commonly called after Tartini, in reality used by. Smith’s accusation against, unfound- ed, 132, 133 Schetical words needed, 193; system of, 199 Secretion, Manifestation of Current Force during. Baxter, H. F., on Organic Polarity, 248 Secretion, phenomena of, compared with Osmose, 259 σελίς, 91, 92 Series divergent, property of, 269 Shakespeare, his use of the word argument, 324; versi- fication of, 387 Ships, different forms of, Greek, 85 Simplicius, on Plato’s definition of φαντασία, 147 Smith, Dr Robert. His work on Harmonics, 129; beats of, 132; his formulz deduced in an independent manner, 139; his accusation against Sauveur un- founded, 132, 133 Socrates, in the Parmenides, 97, &c.; his exposition of tenets of Ephesian followers of Heraclitus, 158 Solon, statue of, 231; an elegiac poet, 237 Solution of equation, what requirements as to continuity are conveyed in the term, 23 Some and all, use of, in Hamilton’s system, 433 Sop/ista (Plato), on the genuineness of, 146 ; a critique of the doctrines of three other schools, 147; arguments in favour of genuineness, ἐδ. ; continuation of Theze- tetus, 151; connexion with, 152; ‘earthborn’ of, 165 See INDEX. Spheres, electrical and magnetic, problems with refer- ence to, 68—S3 } Spherical shell, permanent magnetism in, 76; effect of filling up the shell with magnetic or diamagnetic matter, 7b.; electromagnetic, 77; revolution in a uniform magnetic field, $1 Spherical Triangles, classification of species of, as to character of sides and angles with respect to a right angle, 269, 270 Spicular notation in logic, 198 Statics and Dynamics, their relation to each other, 11 7. Stoxss, G. G., M.A., D.C.L. On the discontinuity of Arbitrary Constants which appear in divergent de- velopments, 104; principles of the investigation dis- cussed, § 1—13 Sturm’s proof of Cauchy’s theorem on imaginary roots, algebraical substitute for a geometrical step in, 264, 265; proposed prefix to, 261—263 Σύμμιξις, ἃ Platonic word, 147 r Συμπλοκή, an Aristotelian not Platonic word, 147 Supercontrary distinguished from contrary, 201 Syllogisms, complicated by charges, 181; defined, 217; strengthened, ib. ; opponent, 218; relations of forms of, ἐδ. Syllogism, No. III., 173 Syllogism, No. IV. 331; table of forms of, 350; exten- sion of, to quantified propositions, 352 ; technical ex- hibition of, not necessary in reasoning, 356; of trans- posed quantity, *355 No. V., 429; of indecision, 453 Symbols, mathematical, use of, in logic, 183; logic the only science which has grown none, 154 System, cosmical of Plato, by Dr Donaldson, 305; source of materials for, 313 Tangents, theorem concerning pairs of, drawn to three unequal circles proved, 171 Tartini, account of, 132.; D’Alembert’s opinion of his treatise on harmony, ἐδ. ; beats of (so-called), 132; in reality due to Sauveur, 131; his grave harmonic, 131; his beats considered in connexion with Smith’s, 135—137 Ν Temperament and tuning, observations on, 141—145 Tendency, defined, 291; postulates relating to, 292 Thalamitze, 88 Theeetetus (Plato), 151 ; connexion with Sophista, 152; materialists described in, 158 Theorems, Analytical, connected with magnetism, 57 —62 Tompson, Prof. On the genuineness of the Sophista of Plato, 146; arguments in favour of genuineness, 147; defence of Politicus against Dr Whewell, 150 ; explanation of connexion between Sophista and Thez- tetus, 152; on Eleatic Ontology, 155; on the two sects in the Gigantomachy, 157; on remarks of Dr Whewell’s, 163; on the γηγενεῖς of Sophista, 165 INDEX. Thomson’s (Sir W.), electrical images, theory of, 68—70 — Papers by, on electricity and magnetism, referred to, 58, 60, 67, 68, 70, 73, 76 Thranitee, 89 Translations of a point, laws of, 296 ; velocities of, 298 Triangles, similar, new treatment of doctrine of, 169 Trireme (Athenian), Dr Donaldson on, 84; rowers of, 85, 87; their arrangement, 85, 86; explanation of difficulties in Tragedians, 92, 93. Tuning and temperament, observations on, 141—145 Universe of a proposition, limitation of, contended for, 190 Ὑποζώματα, explanation of word, 307 Urinary secretion, manifestation of current force during, 255, 256 Velocities of a point with respect to revolving axes, 5, 6 Velocities and accelerations with respect to moving axes, 1—20 Weber's physical theory of electrodynamics, assump- tions of, stated and discussed, 66, 67 Wuewe tt, W., D.D. On the Platonic Theory of Ideas, 94; two primary parts of theory, 97; the Parmenides, 497 an Eleatic dialogue, 100; on Sophista and Politicus of Plato, 146, 150; on method of division in Phze- drus, 163 Whole, arithmetical, in logic, 190, 194, 209, 212; proba- ble, not composed entirely of probable parts, 411 Wholes, three logical, 209 Wollaston, views of, on animal secretions, 248, 253; his paper (Phil. Mag. Vol. xxxiu. p. 488) quoted, 253; his conclusion that secretion is the efzet of a power similar to that which exists in a voltaic circle, tb. Woolhouse, his Lssay on Musical Intervals, referred to, 135 Xenophanes, founder of Hleatic school, 154 Young, Dr Thomas, unfayourable opinion of, on Smith’s work on Harmonies, 133, 134; his opinion shewn to be unfounded, owing to his having confused Vartini’s beats with Smith’s, 133, 134 Zeno in the Parmenides, 97, sqq. Zoroaster, same as Er the Pamphylian, 313 Zygitee, 87. CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. 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