THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES
 
 MATHEMATICAL PSYCHICS 
 
 AN ESSAY ON THE APPLICATION OF MATHEMATICS TO 
 THE MORAL SCIENCES 
 
 by F. Y. EDGEWOBTH, M.A. 
 BABRISTEB-AT-LAW
 
 MATHEMATICAL PSYCHICS 
 
 AN ESSAY ON THE 
 
 APPLICATION OF MATHEMATICS TO 
 
 THE MORAL SCIENCES 
 
 BT 
 
 F. Y. EDGEWORTH, M.A. 
 
 BARRISTtH- AT- LAW 
 
 LONDON 
 C. KEGAN PAUL & CO., 1 PATERNOSTER SQUARE 
 
 1881
 
 
 ^yyi ' 
 
 INTEODUCTOBY 
 
 ntacmmov of 
 
 CONTENTS. 
 
 Mathematical Psychics may be divided into two parts — 
 Theoretical and Applied. 
 
 In the First Part (1) it is attempted to illustrate the 
 possibility of Mathematical reasoning without numerical 
 data (pp. 1-7) ; without more precise data than are 
 afforded by estimates of quantity of pleasure (pp. 7-9). 
 (2) An analogy is suggested between the Principles of 
 Greatest Happiness, Utilitarian or Egoistic, which con- 
 stitute the first principles of Ethics and Economics, and 
 those Principles of Maximum Energy which are among 
 the highest generalisations of Physics, and in virtue of 
 which mathematical reasoning is applicable to physical 
 phenomena quite as complex as human life (pp. 9-15). 
 
 The Calculus of Pleasure (Part 11.) may be divided 
 into two species — the Economical and the Utilitarian ; 
 the principle of division suggesting an addition to Mr. 
 Sidgwick's 'ethical methods' (p. 16). 
 
 The first species of Calculus (if so ambitious a title 
 may for brevity be applied to short studies in Mathe- 
 matical Economics) is developed from certain Definitions
 
 VI INTEODUCTORY DESCRIPTION OF CONTENTS. 
 
 of leading conceptions, in particular of those connected 
 with Competition (pp. 17—19). Then (a) a mathematical 
 theory of Contract unqualified by Competition is given 
 (pp. 20-30). (13) A mathematical theory of Contract de- 
 ter mined by Competition in a perfect Market is given, or at 
 least promised (pp. 30-33, and pp. 38-42). Reference 
 is made to other mathematical theories of Market, and 
 to Mr. Sidgwick's recent article on the * Wages-Fund ' 
 (pp. 32, 33, and Appendix V.) (y) attention is concen- 
 trated on the question — What is a perfect Market? It 
 is argued that Market is imperfect. Contract is indeter- 
 minate in the following cases : — 
 
 (i.) Wlien the number of competitors is limited 
 (pp. 37, 39). 
 
 (ii.) In a certain similar case hkely to occur in con- 
 tracts for personal service (pp. 42, 46). 
 
 (i. and II.) When the articles of contract are not 
 perfectly divisible (p. 42, 46). 
 
 (ill.) In case of Combination^ Unionism ; in which 
 case it is submitted that (in general and abstractly 
 speaking) unionists stand to gain in senses contradicted 
 or ignored by distinguished economists (pp. 44, 47, 48). 
 (iv.) In a certain case similar to the last, and likely 
 to occur in Co-operative Association (pp. 45, 49). 
 
 Tlie indeterminateness likely from these causes to 
 aflect Commercial Contracts, and certainly afiecting aU 
 sorts of Political Contracts, appears to postulate a prin^ 
 ciple of arbitration (pp. 50—52). 
 
 It is argued from mathematical considerations that 
 the basis of arbitration between contractors is the greatest 
 possible iitility of all concerned; the Utihtarian first 
 principle, which can of course afford only a general
 
 INTRODUCTORY DESCRIPTION OF CONTENTS. VU 
 
 direction — yet, as employed by Bentham's school, has 
 afforded some direction in practical affairs (pp. 53-56). 
 
 The Economical thus leads up to the Utilitarian 
 species of Hedonics ; some studies in which already 
 published ^ (under the title of ' Hedonical Calculus ' — 
 the species being designated by the generic title) are 
 reprinted here by the kind permission of the Editor of 
 * Mind.' 
 
 Of the Utilitarian Calculus (pp. 56-82) the central 
 conception is Greatest Happiness^ the greatest possible 
 sum-total of pleasure summed through all time and 
 over all sentience. Mathematical reasonings are em- 
 ployed partly to confirm Mr. Sidgwick's proof that 
 Greatest Happiness is the end of right action ; partly to 
 deduce middle axioms, means conducive to that end. 
 This deduction is of a very abstract, perhaps only nega- 
 tive, character ; negativing the assumption that Equality 
 is necessarily imphed in Utilitarianism. For, if sentients 
 differ in Capacity for happiness — under similar circum- 
 stances some classes of sentients experiencing on an 
 average more pleasure {e.g. of imagination and sym- 
 pathy) and less pain (e.g. of fatigue) than others — there 
 is no presumption that equality of circumstances is the 
 most felicific arrangement ; especially when account is 
 taken of the interests of posterity. 
 
 Such are the principal topics handled in this essay 
 or tentative study. Many of the topics, tersely treated 
 in the main body of the work, are more fully illustrated 
 in the course of seven supplementary chapters, or 
 APPENDICES, entitled : 
 
 • Mind, July 1879.
 
 viii INTRODUCTOEY DESCRIPTION OP CONTENTS. 
 
 PAQB 
 
 I. On TJNiojMfajiCAL Mathematics .... 83-93 
 
 II. On the Importance op Hedonical Calculus . 93-98 
 in. On Hedonimetrt 98-102 
 
 IV. On Mtxkd Modes of Utilitaklanism . . . 102-104 
 
 V. On Pkofessob Jevons's FoBMULiE of Exchange . 104-116 
 
 VI. On the EreOBS of the ayewfitrpjirol . . . 116-125 
 
 VII. On the Pbesent Crisis in Ireland . . . 126-148 
 
 Discussions too much broken up by this arrangement 
 are re-united by references to the principal headings, in 
 the Index ; which also refers to the definitions of terms 
 used in a technical sense. The Index also contains the 
 names of many eminent men whose theories, bearing 
 upon the subject, have been noticed in the course of 
 these pages. Dissent has often been expressed. In so 
 terse a composition it has not been possible always to 
 express, what has always been felt, the deference due to 
 the men and the diffidence proper to the subject.
 
 MATHEMATICAL PSYCHICS. 
 
 ON THE APPLICATION OF MATHEMATICS TO 
 THE MORAL SCIENCES. 
 
 The application of mathematics to Beliefs the calcuhis 
 of Probabihties, has been treated by many distinguished 
 writers ; the calculus of Feeling, of Pleasure and Pain, 
 is the less famihar, but not in reality ^ more paradoxical 
 subject of this essay. 
 
 The subject divides itself into two parts ; concerned 
 respectively with principle and practice, root and fruit, 
 the apphcability and the apphcation of Mathematics to 
 Sociology. 
 
 PAET I. 
 
 In the first part it is attempted to prove an affinity 
 between the moral and the admittedly mathematical 
 sciences from their resemblance as to (1) a certain 
 general complexion, (2) a particidar salient feature. 
 
 (1) The science of quantity is not ahen to tlie study 
 of man, it will be generally admitted, in so far as actions 
 and effective desires can be numerically measured by 
 way of statistics — that is, very far, as Professor Jevons ^ 
 anticipates. But in so far as our data may consist of 
 
 ^ Cf. JevoDS, Theory, p. 9. 
 
 ' Introduction to Theory of Political Economy.
 
 2 MATHEMATICAL PSYCHICS. 
 
 estimates other than numerical^ observations that some 
 conditions are accompanied with greater or less pleasure 
 than others, it is necessary to reahse that mathematical 
 reasoning is not, as commonly ^ supposed, limited to 
 subjects where numerical data are attainable. Where 
 there are data which^ though not numerical are quan- 
 titative — for example, that a quantity is greater or less 
 than another, increases or decreases, is positive or nega- 
 tive, a maximum or minimum ^ there mathematical 
 reasoning is possible and may be indispensable. To 
 take a trivial instance : a is greater than h, and h is 
 greater than c, therefore a is greater than c. Here is 
 mathematical reasoning apphcable to quantities which 
 may not be susceptible of numerical evaluation. The 
 following instance is less trivial, analogous indeed to an 
 important social problem. It is required to distribute 
 a given quantity ? of fuel, so as to obtain the greatest 
 possible quantity of available energy, among a given 
 set of engines, which differ in efficiency — efficiency being 
 thus defined : one engine is more efficient than another 
 if, whenever the total quantity of fuel consumed by the 
 former is equal to that consumed by the latter, the total 
 quantity of energy yielded by the former is greater than 
 •that yielded by the latter. 
 
 In the distribution, shall a larger portion of fuel be 
 given to the more efficient engines ? always, or only in 
 some cases ? and, if so, in what sort of cases ? Here is 
 a very simple problem involving no numerical data, yet 
 
 ' The popular view pervades much of what. Mill (in his Loyic), after 
 Comte, says about Mathematics applied to Sociology. There is a good 
 expression of this view in the Saturday Revieio (on Professor Jevons's 
 Theory, November 11, 1871.) The view adopted in these pages is expressed 
 by Coumot, Recherchfs.) 
 
 2 Or, a given quantity pei- unit of time, with corresponding modification 
 of definition and problem.
 
 UNNUJfERICAL MATHEMATICS. 3 
 
 requiring, it may be safely said, mathematics for its 
 complete investigation. 
 
 The latter statement may be disputed in so far as 
 such questions may be solved by reasoning, which, 
 though not symboHcal, is strictly mathematical ; 
 answered more informally, yet correctly, by undis- 
 ciphned common sense. But, firstly, the advocate of 
 mathematical reasoning in social science is not con- 
 cerned to deny that mathematical reasoning in social, 
 as well as in physical, science may be divested of symbol. 
 Only it must be remembered that the question how far 
 mathematics can with safety or propriety be divested of 
 her pecuHar costume is a very deUcate question, only 
 to be decided by the authority and in the presence of 
 Mathematics herself. And, secondly, as to the suf- 
 ficiency of common sense, the worst of such unsynibolic, 
 at least unmethodic, calculations as we meet in popular 
 economics is that they are apt to miss the character- 
 istic advantages of deductive reasoning. He that will 
 not verify his conclusions as far as possible by mathe- 
 matics, as it were bringing the ingots of common sense 
 to be assayed and coined at the mint of the sovereign 
 science, will hardly realize the full value of what he 
 holds, will want a measure of what it will be worth in 
 however shghtly altered circumstances, a means of 
 conveying and making it current. When the given 
 conditions are not sufficient to determinate the problem 
 — a case of great importance in Pohtical Economy — 
 the dy€w/x€T/)i7Tos is less hkely to suspect this deficiency, 
 less competent to correct it by indicating what con- 
 ditions are necessary and sufficient. All this is evident 
 at a glance through the instrument of mathematics, but 
 to the naked eye of -'common sense partially and ob-
 
 4 MATHEMATICAL PSYCHICS- 
 
 sciirely, and, as Plato says of unscientific knowledge, 
 in a state between genuine Being and Not-Being. 
 
 The preceding prol^lem, to distribute a given quan- 
 tity of material in order to a maximum of energy, with 
 its starting point loose quantitative relations rather than 
 numerical data — its slippery though short path almost 
 necessitating the support of mathematics — illustrates 
 fairly well the problem of utilitarian distribution.^ To 
 illustrate the economical problem of exchange, the maze 
 of many dealers contracting and competing with each 
 other, it is possible to imagine ^ a mechanism of many parts 
 where the law of motion, which particular part moves 
 off with which, is not precisely given — with symbols, 
 arbitrary functions, representing not merely not nu- 
 merical knowledge but ^ ignorance — where, though the 
 mode of motion towards equilibrium is indeterminate, 
 the position of equihbrium is mathematically deter- 
 mined. 
 
 Examples not made to order, taken from the common 
 stock of raatliematical physics, ^vill of course not fit so 
 exactly. But they may be found in abundance, it is 
 submitted, illustrating- the property under consideration 
 — mathematical reasoning without numerical data. 
 In Hydrodynamics, for instance, we have a Thomson or 
 Tait * reasoning ' princii)les ' for ' determining P and Q 
 icill be given later. In the meantime it is obvious that 
 each decreases as X increases. Hence the equations of 
 motion sliow ' — and he goes on to draw a., conclusion of 
 
 ' See p. G4. ^ See p. 34. 
 
 ' lynontfiim vf Co-ordinate* (Thoinsou and Tait, Nnturtd mimophy, 
 2iid edition), is appropriate in many soeial problems where we only know in 
 part. 
 
 '' Thomson and Tait, Treatise on Nntxiral l*hilo$ophy, p. 320, 2nd edition. 
 The italics, which are ours, call attention to the unnumerical, louse quantita- 
 tire, rdatitm whieh rojiatitntes tlie datum oi'tlie mathematieal reasoninp.
 
 UNNUMERICAL JtfATHEMATICS. 5 
 
 momentous interest that balls (properly) projected in 
 an infinite incompressible fluid will move as if they 
 were attracted to each other. And generally in the 
 higher Hydrodynamics, in that boundless *ocean of 
 perfect fluid, swum through by vortices, where the 
 deep first principles of Physics are to be sought, is not 
 a similar unnumeirical^ or hi/per arithmetical method there 
 pursued ? If a portion of perfect fluid so moves at any 
 time that each particle has no motion of rotation, then 
 that portion of the fluid Avill retain that property for 
 all time ^ ; here is no application of the numerical 
 measuring-rod. 
 
 No doubt it may be objected that these hydro- 
 dynamical problems employ some precise data ; the very 
 definition of Force, the conditions of fluidity and con- 
 tinuity. But so also have our social problems some 
 precise data : for example, the property of uniformity 
 of price in a market ; or rather the (approximately 
 realised) conditions of which that property is the de- 
 ducible effect, and which bears a striking resemblance to 
 the data of hydrodynamics : ^ (1) the fulness of the market: 
 that there continues to be up to the conclusion of the deal- 
 ing an indefinite number of dealers ; (2) the fluidity of the 
 market, or infinite dividedness of the dealers' interests.. 
 Given this property of uniform price, Mr. Marshall and! 
 M. Walras deduce mathematically, though not arith-* 
 metically, an interesting theorem, which Mill and Thorn- 
 ton failed with unaided reason to discern, though they 
 were quite close to it — the theorem that the equation 
 of supply to demand, though a necessary, is not a suffi- 
 cient condition of market price. 
 
 To attempt to select representative instances from each 
 
 1 Stokfls, Mathttnafkal Paptn, p. 112. 
 » See r- IS-
 
 6 MATHEMATICAL PSYCHICS. 
 
 recognised branch of matlieniatical inquiry would exceed 
 tlie limits of tins paper and the requirements of the argu- 
 inent. It must suffice, in conclusion, to direct atten- 
 tion to one species of Mathematics which seems largely 
 affected with the property under consideration, th# 
 Calculus of Maxima and Minima, or (in a wide sense) of 
 Variations, The criterion of a maximum ^ turns, not 
 upon the amount, but upon the sigyi of a certain quan- 
 tity.''^ We are continually concerned^ with the ascer- 
 tainment of a certain loose quantitative relation, the 
 decrease-of-rate-of -increase of a quantity. Now, this is 
 the very quantitative relation which it is proposed to 
 enij)loy in mathematical sociology ; given in such data 
 as the law of diminii^hing returns to capital and labour, 
 the law of diminishing utility, the law of increasing 
 fatigue; the very same irregular, unsquared material 
 which constitutes the basis of the Economical and the 
 Utilitarian Calculus. 
 
 Now, it is remarkable that the principal inquiries in 
 Social Science may be viewed as mcuvimum-prohlems. 
 For Economics investijrates the arrangements between 
 agents each tending to his own maximum utility; and 
 Politics and (Utilitarian) Ethics investigate the arrange- 
 ments which conduce to the maximum sum total of 
 
 ' Ma.x-imu7n in ihls paper is employed axjcording to the context for (1) 
 3frtx?M/wwj in the proper mat hemntical sense; (2) G r eat st possible; (3) tta- 
 tionanj ; (4) where mmimum (or leoM jwssible) might have been expected ; 
 upon the principle that even' minimum is the correlative of a maximum. 
 Thus Thomson's Minimum tlieorem is correlated with Bertrand's Maximtmi 
 thforem. (Watson and Burhury.) This liberty is taken, not only for 
 brevity, but also for the sake of a certain suggestiveness. * Stationary^ for 
 instance, fails to suggest the siipcrlativcness which it connotes. 
 
 •* The second terra of Variation. It may be objected that the otJier con- 
 dition of a maximum equation of the first terhi to zero is of a more precise 
 characler. St^e, however, Appendix I., p. 92. 
 
 ^ E.g., Todhunter's Mcsearches on Caladnt of Variations, pp. 21-30, 80, 
 117, 286, &c.
 
 HEDONIMETRY. 7 
 
 Utility. Since, then, Social Science, as compared with 
 the Calculus of Variations, starts from similar data — 
 loose quantitative relations — and travels to a similar con- 
 clusion — determination of maximum — why should it not 
 pursue the same metjiod, Mathematics ? 
 
 There remains the objection that in Physical Calculus 
 there is always (as in the example quoted above from 
 Thomson and Tait) a potentiality, an expectation, of 
 measurement ; while Psychics want the first condition 
 of calculation, a unit. The following ^ brief answer is 
 diffidently offered. 
 
 Utility, as Professor Jevons'^ says, has two dimen- 
 sions, intensity and time. The unit in each dimension is 
 the jusf perceivaBIe^ increment. The implied equation 
 to each other of each m inimum ,gffl-<??A?7^i«; -a first prij»rip1f> 
 inc apable of pro of. It resembles the equation to each 
 other of undistinguishable events or cases,^ which con- 
 stitutes the first principle of the mathematical calculus 
 of belief. It is doubtless a principle acquired in the 
 course of evolution. The implied equatabihty of time- 
 intensity units, irrespective of distance in time and 
 kind of pleasure, is still imperfectly evolved. Such is 
 the unit 'of econojnical calculus. 
 
 For moral calculus a further dimension is required ; 
 to compare the happiness of one person with the happi- 
 ness of another, and generally the happiness of groups Ij 
 of different members and different average happiness. 
 
 Such comparison can no longer be shirked, if there 
 
 ' For a fuller discussion, see Appendix III. 
 * In reference to Economics, Theory, p. 51 . 
 
 •^^Cf. Wundt, Physiological PsycJtology ; below, p. 60. Our ' ebenmerk- 
 
 if ' minim is to be regarded not as an infinitesimal differential, but as a 
 
 ■1 , -mall difference ; a conception which is consistent with a (duly cau- 
 
 ployment of infinitesimal notation. 
 a sim^ lace, Essair—Probahilitics, p. 7. 
 
 insta
 
 8 MATHEMATICAL PSYCHICS. 
 
 is to be any systematic morality at all. It is postulated 
 by distributive justice. It is postulated by the population 
 question ; that horizon in which every moral prospect 
 terminates ; which is presented to the far-seeing at every 
 turn, on the most sacred and the most trivial occasions. 
 You cannot spend sixpence utilitarianly, without having 
 considered whether your action tends to increase the 
 comfort of a limited number, or numbers with limited 
 comfort ; without having compared such alternative 
 utilities. 
 
 T^ virtue oLjwhfrf-TITut is such comparison possible ? 
 It is here submitted : ^"Yir^^^^i?a^ PTrppriPTipinnr a 
 unit of plea«tire=mtensity during a unit of time ts to 
 * counL-ibr_QneJJ — Etihty, then^has-4Ar«9<g-dimeai8iQns ; 
 a mas§_jQf_-utility, ' lot __of pleasu re/ is greater than 
 another when it has more {ntensity=titMzimniber units. 
 Tlie third dimension is doubtless an evolutional acqui- 
 sition ; and is still far from perfectly evolved. 
 
 Looking back at our triple scale, we find no peculiar 
 difficulty about the third dimension. It is an affair of 
 census. The second dimension is an affair of clock- 
 work ; assuming that the distinction here touched, be- 
 tween subjective and objective measure of time, is of 
 minor importance. But the first dimension, where we 
 leave the safe ground of the objective, equating to unity 
 
 -l i^ js. Atoms of pleasure are not easy to distinguish and 
 disceiiu _niore~c ontinuflii«~th"Tl san5I^Iwfvrp^T?TgpT^^ 
 llcfted- L as it were nuclei of^ the just-pefceivable, em- 
 bedded in circumambient senGd-conscioustrgss. """"^-^ 
 
 We cannot count the golden sands of life ; we canr\ *. 
 nitmber tlie * innumerable smile ' of seas of love ; but vc 
 
 ^ In the Pure, for a,fraciioH, in the Impure, imperfectly evolved, Ut* 
 rianism. See p. 16.
 
 M.\XIMUM ENERGY. 9 
 
 seem to be capable of observing that there is here a 
 greater^ there a less^ multitude of pleasure-units, mass of 
 happiness ; and that is enough. 
 
 (2) The apphcation of mathematics to the world of 
 soul is countenanced by the hypothesis (agreeable to the 
 general hypothesis that every psychical phenomenon is 
 the concomitant, and in some sense the other side of a 
 physical phenomenon), the particular hypothesis adopted 
 in these pages, that Pleasure is the concomitant of 
 Energy, Energy may be regarded as the central idea 
 of Mathematical Physics ; maximum energy the object of 
 the principal investigations in that science. By aid of 
 this conception we reduce into scientific order physical 
 phenomena, the complexity of which may be compared 
 with the complexity which appears so formidable in 
 Social Science. 
 
 Imagine a material Cosmos, a mechanism as com-?- 
 posite as possible, and perplexed with all manner of 
 wheels, pistons, parts, connections, and whose mazy 
 complexity might far transcend in its entanglement the 
 webs of thought and wiles of passion ; nevertheless, if 
 any given impulses be imparted to any definite points 
 in the mechanism at rest, it is mathematically deducible 
 that each part of the great whole will move off" with a 
 velocity such that the energy of the whole may be the 
 greatest possible ^ — the greatest possible consistent with 
 the given impulses and existing construction. If we 
 know something about the construction of the mechan- 
 ism, if it is ' a mighty maze, but not without a plan ; ' 
 if we have some quantitative though not numerical 
 datum about the construction, we may be able to deduce 
 a similarly indefinite conclusion about the motion. For 
 instance, any number of cases may be imagined in 
 
 B 
 
 ' Beitrand's Theorem.
 
 10 MATHEMATICAL PSYCHICS. 
 
 which, if a datum about the construction is that certain 
 parts are less stiff tlian others, a conclusion about the 
 motion would be that those parts ^ take on more energy 
 than their stiflTer fellows. This rough, indefinite, yet 
 matliematical reasoning is analogous to the reasoning 
 on a subsequent page,'^ that in order to the greatest 
 possible sum total of happiness, the more capable of 
 pleasure sliall take more means, more happiness. 
 
 In the preceding illustration the motion of a 
 mechanism was supposed instantaneously generated by 
 the apphcation of given impulses at definite points (or 
 over definite surfaces) ; but similar general views are 
 attainable in the not so dissimilar case in which we 
 suppose motion generated in time by finite forces acting 
 upon, and interacting between, the particles of which 
 the mechanism is composed. This supposition includes 
 the celebrated proI)lem of Many Bodies (attracting each 
 other according to any function of the distance) ; in 
 reference to which one often hears it asked what can 
 be expected from Mathematics in social science, when 
 she is unable to solve the problem of Three Bodies in 
 her own department. But Mathematics can solve the 
 problem of many bodies — ^not indeed numerically and 
 explicitly, but practically and philosojjhically, affording 
 approximate measurements, and satisfying the soul of 
 the philosopher with the grandest of generalisations. 
 By a principle discovered or improved by Lagrange, each 
 particle of the however complex whole is continually so 
 moving that the accumulation of energy, which is consti- 
 tuted by adding to each other the energies of the mechan- 
 ism existing at each instant of time (technically termed 
 Action — the time-integral of Energy) should be a^ maxi- 
 
 ' Cf. "Walson and Bnrbury, Geiiej-aliscd Co-ordinates, A.rt, 30, and pre- 
 ceding. ^ P. 64. ^ See note, p. 6.
 
 MAXIMUM PLEASURE. 11 
 
 mum. By the discovery of Sir William Rowan Hamilton ^ 
 the subordination of the parts to the whole is more 
 usefully expressed, the velocity of each part is regarded 
 as derivable from the action of the whele ; the action is 
 connected by a single^ although not an explicit or in gene- 
 ral easily interpretable, relation with the given law of 
 force. The many unknown are reduced to one un- 
 known, the one unknown is connected with the known. 
 
 Now this accumulation (or time-integral) of energy 
 which thus becomes the principal object of the physical 
 investigation is analogous to that accumulation of 
 pleasure which is constituted by bringing together in 
 prospect the pleasure existing at each instant of time, 
 the end of rational action, whether self-interested or 
 benevolent. The central conception of Dynamics and 
 (in virtue of pervading analogies it may be said) in 
 general of Mathematical Physics is oiher-sidedly identical 
 with the central conception of Ethics ; and a solution 
 practical and pliilosophical, although not numerical 
 and precise, as it exists for the problem of the inter- 
 action of bodies, so is possible for the problem of the 
 interaction of souls. ; 
 
 This general solution, it may be thought, at most is 
 appHcable to the utilitarian problem of which the object 
 is the greatest possible sum total of universal happiness. 
 But it deserves consideration that an object of Eco- 
 nomics also, the arrangement to which contracting agents 
 actuated only by self-interest tend is capable of being 
 regarded upon the psychophysical hypothesis here 
 entertained as the reahsation of the maximum sum- 
 total of happiness, the relative maximum^ or that which 
 is consistent with certain conditions. There is dimly 
 discerned the Divine idea of a power tending to 
 
 ' PJulosophical Trananctiont, 1834-5. - See pp. 34, 142.
 
 12 MATHEMATICAL PSYCHICS. 
 
 the greatest possible quantity of happiness ^ under con- 
 ditions-, whether the condition of that perfect disinte- 
 gration and unsympathetic isolation abstractedly as- 
 sumed in Economics, or those intermediate ^ conditions 
 of what Herbert Spencer might term integration on to that 
 perfected utihtarian sympathy in which the pleasures of 
 another are accounted equal with one's own. There are 
 diversities of conditions, but one maximum-principle ; 
 many stages of evolution, but ' one increasing purpose.*" 
 
 * Mecanlque Sociale ' may one day take her place 
 along with ' Mecanique Celeste,' throned each upon 
 the double-sided height of one maximum principle,^ the 
 supreme pinnacle of moral as of physical science. As 
 the movements of each particle, constrained or loose, in 
 a material cosmos are continually subordinated to one 
 maximum sum- total of accumulated energy, so the 
 movements of each soul, whether selfishly isolated or 
 linked sympathetically, may continually be reahsing 
 the maximum energy of pleasure, the Divine love of 
 the universe. 
 
 'Mecanique Sociale,' in comparison with her elder 
 sister, is less attractive to the vulgar worshipper in that 
 she is discernible by the eye of faith alone. The 
 statuesque beauty of the one is manifest ; but the fairy- 
 hke features of the other and her fluent form are 
 
 ' Cf. Mill, Essays on Nature and Religion. ^ See p. 16. 
 
 ^ The mathematical reader does not require to 1)6 reminded that upon the 
 principles of Lagrange the whole of (conservative) Dynamics may be pre- 
 sented as a Maximum-Problem ; if without gain, at any rate without loss. 
 And the great principle of Thomson (Thomson & Tait, arts. Cf. Theory of 
 Voj-tices, by Thomson, Royal Society, Edinburgh, 1865), with allied maxi- 
 mutn-prina'ples, dominating the theory of fluid motion, dominates Mathema- 
 tical Physics with a more than nominal supremacy, and most indispensably 
 efficacious power. Similarly, it may be conjectured, the ordinary moral rules 
 are equivalently e.xpressed by the Intuitivist in the (grammatically-speakinp), 
 positive degree, by the Utilitarian in the superlative. But for the higher 
 moral problems the conception of maxiinum is indispensable.
 
 MfiCANIQUE SOCIALE. 13 
 
 veiled. But Matliematics has long walked by the 
 evidence of things not seen in the world of atoms (tlie 
 methods whereof, it may incidentally be remarked, 
 statistical and rough, may illustrate the possibilit}^ of 
 social mathematics). The invisible energy of electricity 
 is grasped by the marvellous methods of Lagrange ; ^ 
 the invisible energy of pleasure may admit of a similar 
 handling. 
 
 As in a system of conductors carrying electrical 
 currents the energy due to electro- magnetic force is to 
 be distinguished from the energy due to ordinary dyna- 
 mical forces, e.^.,gravitation acting upon the conductors, 
 so the energy of pleasure is to be distinguished not only 
 from the gross energy of the limbs, but also from such 
 nervous energy as either is not all represented in con- 
 sciousness {pace G. H. Lewes), or is represented by 
 intensity of consciousness not intensity of pleasure. As 
 electro-magnetic force tends to a maximum energy, so 
 also pleasure force tends to a maximum energy. The 
 energy generated by pleasure force is the physical con- 
 comitant and measure of the conscious feelincf of delight. 
 
 Imagine an electrical circuit consisting of two rails 
 isolated from the earth connected at one extremity by a 
 galvanic battery and bridged over at the other extremity 
 by a steam-locomotive.^ When a current of electricity 
 is sent through the circuit, there is an electro-magnetic 
 force tending to move the circuit or any moveable part 
 of it in such a direction that the number of hnes of force 
 (due to the magnetism of the earth) passing through the 
 circuit in a positive direction may be a maximum. 
 The electro-magnetic force therefore tends to move the 
 
 * See Clerk Maxwell, Electricity and Magnetism, on the use of Lagrange'e 
 Generalised Co-ordinates, Part iv., chaps. 5 and 6. 
 '^ Clerk Maxwell has a similar construction.
 
 14 MATHEMATICAL PSYCHICS. 
 
 locomotive alon<T the rails in that direction. Now tliis 
 dehcate force may well be unable to move the ponderous 
 locomotive, but it may be adequate to press a spring 
 and turn a handle and let on steam and cause the loco- 
 motive to be moved by the steam-engine in the directum 
 of the electro-magnetic force, either backwards or forwards 
 according to the direction in which the electrical cur- 
 rent Hows. The delicate electro-magnetic force is placed 
 in such a commanding position that she sways the 
 movements of the steam-engine so as to satisfy her own 
 yearning towards maximum. 
 
 Add now another degree of freedom ; and let the 
 steam-car governed move upon a plane ^ in a direction 
 tending towards the position of Minimum Potential 
 Electro-Magnetic Energy. Complicate this conception ; 
 modify it by substituting for the principle of Minimum 
 Force-Potential the principle Qi Minimum^ Momentum- 
 Potential ; imagine a comparatively gross mechanism of 
 innumerable degrees of freedom governed, in, the sense 
 adumbrated, by a more delicate system — itself, however 
 inconceivably diversified its degrees of freedom, obedient 
 still to the great Maximum Principles of Physics, and 
 amenable to mathematical demonstration, though at first 
 sight as hopelessly incalculable as whatever is in life 
 capricious and irregular — as tlie smiles of beauty and the 
 waves of passion. 
 
 Similarly pleasure in the course of evolution has 
 become throned among grosser subject energies — as it 
 were explosive engines, ready ^ to go off at the pressure 
 
 • See p. 24. 
 
 - Moinentuin-rote-nfial n^pon the analogy of Fe/oci^/y-Po^e/iifiW (Thomson 
 on Vortex Motion, § 31) ; nnd Minimum, as I venture to think, in virtue of 
 certain analogies between theories about Enenjy and about Action. 
 
 •* See the account of the Mechanism of Life, in Balfour Stewart's Con- 
 iernntion of Enerou.
 
 MAN A PLEASURE-MACHINE. 15 
 
 of a hair-spring. Swayed by the first principle, she 
 sways the subject energies so as to satisfy her own yearn- 
 ing towards maximum ; * her every air Of gesture and 
 least motion ' a law of Force to governed systems — a 
 fluent form, a Fairy Queen guiding a most comphcated 
 chariot, wheel within wheel, the" ' speculative and active 
 instruments,' the motor nerves, the limbs and the envi- 
 ronment on which they act. 
 
 A system of such charioteers and chariots is what 
 constitutes the object of Social Science. The attractions 
 between the charioteer forces, the collisions and com- 
 pacts between the chariots, present an appearance of 
 quantitative regularity in the midst of bewildering com- 
 plexity resembhng in its general characters the field of 
 electricity and magnetism. To construct a scientific 
 hypothesis seems rather to surpass the powers of the 
 writer than of Mathematics. 'Sin has ne possim 
 naturae accedere partes Frigidus obstiterit circum prse- 
 cordia sanguis ; ' at least the conception of Man as a 
 pleasure machine may justify and facilitate the employ- 
 ment of mechanical terms and Mathematical reasoning 
 in social science. 
 
 PAET n. 
 
 Such are some of the prehminary considerations by 
 which emboldened we approach the two fields into 
 which the Calculus of Pleasure may be subdivided, 
 namely Economics and Utilitarian Ethics. The Econo^"^ 
 mical Calculus investigates the equihbrium of a system , 
 of hedonic forces each tending to maximum individual 
 utility ; the Utihtarian Calculus, the equihbrium of a 
 system in which each and all tend to maximum uni-
 
 16 MATHEMATICAL PSYCHICS. 
 
 versal utility. The motives of the two species of agents 
 correspond with Mr.^Sidgwick's Egoistic and Universal- 
 istic Hedonism. Biit the correspondence is not perfect. 
 For, firstly, upon the principle of ' self limitation ' of a 
 method, so clearly stated by Mr. Sidgwick, so persistently 
 misunderstood by critics, the Pure Utihtarian might 
 think it most beneficent to sink his benevolence towards 
 competitors ; and the I^eductive Egoist might have need 
 of a Utihtarian Calculus. But further, it is possible that 
 the moral constitution of the concrete agent would be 
 neither Pure Utihtarian nor Pure Egoistic, but /tt/cr^ 
 Tt5. For it is submitted that Mr. Sidgwick's division of 
 Hedonism — the class of * Method' whose principle of 
 action may be generically defined maximising happiness 
 — is not exhaustive. For between the two extremes 
 Pure Egoistic and Pure Universahstic there may fee an 
 indefinite number of impure methods ; wherein the 
 happiness of others as compared by the agent (in a 
 calm moment) with his own, neither counts for nothing, 
 not yet 'counts for one,' but counts for a fraction. 
 
 Deferring controversy,^ let us glance at the elements 
 of the Economic Calculus ; observing that the connota- 
 tion (and some of the reasoning) extends beyond the 
 usual denotation ; to the political struggle for power, as 
 well as to the commercial struggle for wealth. 
 
 ECONOMICAL CALCULUS. 
 
 ' Definitions. — The first principle of Economics ^ is 
 that every agent is actuated only by self-interest. The 
 workings of this principle may be viewed under two 
 aspects, according as the agent acts without, or with, the 
 
 ' See Appendix IV. 
 
 ^ Descriptiona rather, but sufficient for the purpose of tbeee teutatiye 
 studies.
 
 ECONOMICAL DEFINITIONS. 17 
 
 consent of others affected by his actions. In "vvide 
 senses, the first species of action may be called war ; the 
 second, contract. Examples : (1) A general, or fencer, 
 making moves, a dealer lowering price, without consent 
 of rival. (2) A set of co-operatives (labourers, capital- 
 ists, manager) agreed nem. con. to distribute the joint- 
 produce by assigning to each a certain function of his 
 sacrifice. The articles of contract are in this case the 
 amount of sacrifice to be made by each, and the_ principle 
 of distribution. 
 
 * Is it peace or war ? ' asks the lover of * Maud,' of 
 economic competition, and answers hastily : It is both, 
 pax or pact between contractors during contract, war., 
 when some of the contractors without the consent of others 
 recontract. Thus an auctioneer having been in contact 
 with the last bidder (to sell at such a price if no higher 
 bid) recontracts with a higher bidder. So a landlord on 
 expiry of lease recontracts, it may be, -with a new 
 tenant. 
 
 The field of competition with reference to a contract, 
 or contracts, under consideration consists of all the 
 individuals who are willing and able to recontract about 
 the articles under consideration. Thus in an auction 
 the field consists of the auctioneer and all who are 
 effectively wilhng to give a higher price than the last 
 bid. In this case, as the transaction reaches determi- 
 nation, the field continually diminishes and ultimately 
 vanishes. But this is not the case in general. Suppose 
 a great number of auctions going on at the same point ; 
 or, what comes to the same thing, a market consisting 
 of an indefinite number of dealers, say Xs, in commodity 
 .r, and an indefinite number of dealers, say Ys, in com- 
 modity y. In this case, up to the determination of 
 equihbrium, the field continues indefinitely large. To
 
 18 MATHEMATICAL PSYCHICS. 
 
 be sure it may be said to vanish at the position of 
 equilibrium. But that circumstance does not stultify 
 the definition. Thus, if one chose to define the field of 
 force as the centres of force sensibly acting on a certain 
 system of bodies, then in a continuous medium of 
 attracting matter, the field might be continually of 
 indefinite extent, might change as the system moved, 
 might be said to vanish when the system reached 
 equilibrium. 
 
 There is free communication throughout a normal 
 competitive field. You might suppose the constituent 
 individuals collected at a point, or connected by tele- 
 phones—an ideal supposition, but sufllciently approxi- 
 mate to existence or tendency for the purposes *of 
 abstract science. 
 
 A perfect field of competition professes in addition 
 certain properties pecuharly favourable to mathematical 
 calculation ; namely, a certain indefinite multiplicity and 
 dividedness, analogous to that infinity and infinitesimality 
 which facilitate so large a portion of Mathematical 
 Physics (consider the theory of Atoms, and all appUca- 
 tions of the Differential Calculus). The conditions of 
 a perfect field are four ; the first pair referrible ^ to the 
 heading multiplicity or continuity, the second to divicled- 
 ness or fluidity. 
 
 I. Any individual is free to recontract with any out 
 of an indefinite number, e.g., in the last example there 
 are an indefinite number of Xs and similarly of Ys. 
 
 n. Any individual is free to contract (at the same 
 time) with an indefinite number ; e.g.., any X (and simi- 
 larly Y) may deal with any number of Ys. This con- 
 dition combined with tlie first appears to involve 
 
 » See p. 5.
 
 ECONOMICAL DEFINITIONS. 10 
 
 tlie indefinite divisibility of ^ each article of contract 
 (if any X deal with an indefinite number of Ys he must 
 give each an indefinitely small portion of x) ; which 
 might be erected into a separate condition. 
 
 III. Any individual is free to reconiract with another 
 independently of, without the consent being required of, 
 any third party, e.g., there is among the Ys (and simi- 
 larly among the Xs) no combination or precontract be- 
 tween two or more contractors that none of them will 
 recontract without the consent of all. Any Y then may 
 accept the offer of any X irrespectively of other Ys. 
 
 IV. Any indiv^idual is free to contract with another 
 independently of a third party ; e.g.., in simple exchange 
 each contract is between two only, but secus in the 
 entangled contract described in the example (p. 17)> 
 where it may be a condition of production that there 
 should be three at least to each bargain. 
 
 There will be observed a certain similarity between 
 the relation of the first to the second condition, and 
 that of the third to the fourth. The failure of the first 
 involves the failure of the second, but not vice versa ; 
 and the third and fourth are similarly related. 
 
 A settlement is a contract which cannot be varied 
 with the consent of all the parties to it. 
 
 A final settlement is a settlement which cannot be 
 varied by recontract within the field of competition. 
 
 Contract is indeterminate when there are an indefinite 
 number oi final settlements. 
 
 ' This species of imperfection will not be explicitly treated here ; partly 
 because it is perhaps of secondary practical importance : and partly because 
 it has been ^sufficiently treated by Prof. Jevons {Theory, pp. 135-1^7). It 
 is .important, as suggested in Appendix V., to distinguish the effects of this 
 imperfection according as the competition is, or is not, supposed perfect 
 in othei- respects.
 
 20 MATHEMATICAL PSyCHICS. 
 
 The PEOBLEM to which attention is specially directed 
 in this introductory summary is : Hov:) far contract is 
 indeterminate — an inquiry of more than theoretical im- 
 portance, if it show not only that indeterminateness 
 tends to prevent widely, but also in what direction an 
 escape from its evils is to be sought. 
 
 Demonstrations.^ — The general answer is — (a) Con- 
 tract without competition is indeterminate, (/3) Contract 
 with perfect competition is perfectly determinate, (7) 
 Contract with more or less perfect competition is less or 
 more indeterminate. 
 
 (a) Let us commence with almost the simplest case 
 of contract, — two individuals, X and Y, whose interest 
 depends on two variable quantities, which they are 
 agreed not to vary without mutual consent. Exchange 
 of two commodities is a particular case of this kind 
 of contract. Let x and y be the portions interchanged, as 
 in Professor Jevons's example.^ Then the utility of one 
 party, say X, may be written $1 (a — x) + Fj (y) ; and 
 the utihty of the other party, say Y, $2 {^) + ^2 (^ ~" y) '■> 
 where ^ and W are the integrals of Professor Jevons's 
 symbols <^ and i/j. It is agreed that x and y shall be 
 varied only by consent (not e.g. by violence). 
 
 More generally. Let P, the utility of X, one party, 
 = F(a?2/), and IT, the utility of Y, the other party, 
 = $ [x y). If now it is inquired at what point they 
 will reach equilibrium, one or botli refusing to move 
 further, to what settlement they will consent ; the answer 
 is in general that contract by itself does not supply 
 sufRcient conditions to determinate the solution ; sup- 
 plementary conditions as Avill appear being supplied by 
 
 * Conclusions rather, the matheoiatical demoustration of which is not 
 fully exhibited. 
 
 ■^ Theory of Political Economy, 2nd ed., p. 107.
 
 PURE coNi'iL'^crr. 21 
 
 competition or ethical motives, Contract will supply 
 only one condition (for the two variables), namely 
 
 d^dn_d'Pdn pi I. 
 
 da; dy ~ dy dx 
 
 (corresponding to Professor Jevons's equation 
 
 <^i {a- x) ^ 4,^{x) 
 
 V*! {y) " V'2 (^ ~ y) 
 
 Theory p. 103), which it is proposed here to inves- 
 tigate. 
 
 Consider P — F (^ 2/) = as a surface, P denoting 
 the length of the ordinate drawn from any point on the 
 plane of x y (say the plane of the paper) to the surface. 
 Consider 11 — ^ [x y) similarly. It is required to find a 
 point {xy) such that, in whatever direction we take an 
 infinitely small step, P and U do not increase together, 
 but that, while one increases, the other decreases. It 
 may be shown from a variety of points of view that the 
 locus of the required point is 
 
 dV dn_ dF dn^Q. 
 dx dy dy dx 
 
 which locus it is here proposed to call the contract- 
 curve. 
 
 (1) Consider first in what directions X can take an 
 indefinitely small step, say of length p, from any point 
 {x y). Since the addition to P is 
 
 p cos 6 being — d x^ and p sin 6 =^ dy^ it is evident that 
 X will step only on one side of a certain line, the line 
 of indifference^ as it might be caUed ; its equation being
 
 22 MATHEMATICAL PSYCHICS. 
 
 And it is to be observed, in passing, that the direction in 
 which X will prefer to move, the line of force or line of 
 preference^ as it may be termed, is perpendicular to the 
 line of indifference. Similar remarks apply to IT. If 
 tlien we enquire in what directions X and Y will con- 
 sent to move together^ the answer is, in any direction 
 between their respective Unes of indifference, in a direc- 
 tion positive as it may be called for both. At what 
 ])oint then will they refuse to move at all ? When their 
 lines of indifference are coincident (and lines of preference 
 not only coincident, but in opposite directions) ; whereof 
 the necessary (but not sufficient) condition is 
 
 \dxJ ^dy-^ \dyJ \dx J 
 
 (2) The same consideration might be thus put. Let 
 tlie complete variation of P be DP=/o I ( -r- ) cos 6 + 
 
 (-J-) sin 6 and similarly for IT. Then in general 6 can 
 
 DP 
 
 be taken, so that ^ — should be positive, say = g^^ and 
 
 so P and 17 both increase together. 
 
 dV _ i dn 
 
 n dx d X 
 tan. & = — -—=r ; — 
 
 dF _ 2 ^ 
 Ty ^ dy 
 
 But this solution fails when I -r- ) ( ^— ) 
 
 ^dx^ _^dx^ 
 
 (dY\ " (dn\ 
 ^dy^ \dy^ 
 
 In fact, in this case =-— is the same for all directions.
 
 PURE CONTRACT. lio 
 
 DP 
 
 If, then, that common value of =— - is negative, motion 
 
 is impossible in any direction. 
 
 (3) Or, again, we may consider that motion is pos- 
 sible so long as, one party not losing, the other gains. 
 The point of equilibrium, therefore, may be described 
 as a relative maximum, the point at which e.g. II being 
 constant, P is a maximum. Put P = P — c (IT — IT'), 
 where c is a constant and 21' is the supposed given value 
 of II. Then P is a maximum only when 
 
 whence we have as before the contract-curve. 
 
 The same result would follow if we supposed Y in- 
 duced to consent to the variation, not merely by the 
 guarantee that he should not lose, or gain infinitesimally,- 
 but by the understanding that he should gain sensiWy 
 with the gains of P. For instance, let IT = FP where k 
 is a constant, certainly not a very practicable condition. 
 Or, more generally, let P move subject to the condition 
 that DP = ^ X DIT, where ^ is a function of the co- 
 ordinates. Then DP, subject to this condition, vanishes 
 only when 
 
 where c is a constant ; 
 
 whence©(l.c)-o^(g) =
 
 24 MATHEMATICAL PSYCHICS. 
 
 whence as before { -r- ) ( i- ^ ~ ( -r-) \ -r- )— ^^ 
 ^djy \d y J ^dyJ ^dx ^ ■ 
 
 No doubt the one theory which has been thus dif- 
 ferently expressed could be presented by a professed 
 mathematician more elegantly and scientifically. What 
 appears to the writer the most philosophical presenta- 
 tion may be thus indicated. 
 
 (4) Upon the hypothesis above shadowed forth, ^ 
 human action generally, and in particular the step 
 taken by a contractor modifying articles of contract, 
 may be regarded as the working of a gross force governed^ 
 let on, and directed by a more delicate pleasure-force. 
 From wliich it seems to follow upon general dynamical 
 principles applied to this special case that equilibrium 
 is attained when the total pleasure-energy of the contractors 
 IS a maximum relative,"^ or subject, to conditions ; the 
 conditions being here (i) that the pleasure-energy of X 
 and Y considered each as a function of (certain values 
 of) the variables x and y should be functions of the 
 same values : in the metaphorical language above em- 
 ployed that the charioteer-pleasures should drive their 
 teams together over the plane of xy ; (ii) that the joint- 
 team should never be urged in a direction con- 
 trary to the preference^ of either individual; that the 
 resultant line of force (and the momentum) of the 
 gross, the chariot, system should be continually in- 
 termediate between the (positive directions of the) 
 lines of the respective pleasure-forces. [We may 
 without disadvantage make abstraction of sensible mo- 
 mentum, and suppose the by the condition joint- 
 system to move towards equilibrium along a line of 
 resultant gross force. Let it start from the origin. And 
 
 ' See pp. 13-15. -' See note, p. 11. ' See p. 22.
 
 PURE (JONTIL\CT. 25 
 
 let ua employ an arbitrary function to denote the un- 
 known principle of compromise between the parties ; sup- ' 
 pose the ratio of the sines of angles made by the 
 resultant line with the respective lines of pleasure- 
 force.] Then, by reasoning different from the pre- 
 ceding only in the point of view, it appears that the 
 total utility of the system is a relative maximum at any 
 point on the pure contract-curve. 
 
 It appears from (1) and (2) there is a portion of the 
 
 1- © © 7 (S © = «' -- SI ^^ 
 
 + , not therefore indicating immobility, au contraire, the 
 impure (part of the) contract-curve, as it might be 
 called. This might be illustrated by two spheres, each 
 having the plane of the paper as a diametral plane. 
 Tlie contract curve is easily seen to be the line joining 
 tlie centres. Supposing that the distance between the 
 centres is less than the less of the radii, part of the 
 contract-curve is impure. If the index, as Mr. Marshall 
 might call it, be placed anywhere in this portion it will 
 run up to a centre. But between the centres the con- 
 tract-curve is pure; the index placed anywhere in tliis 
 portion is immovable ; and if account be taken of the 
 portions of the spheres underneath the plane of the 
 paper, tlie downward ordinates representing negatix'e 
 pleasures^ similar statements hold, mutatis mutandis. 
 It appears that the pure and impure parts of the 
 
 contract-curve are # demarcated by the points where 
 
 DP . . . DX^ 
 
 =-— changes sign, that is (in general) where either y- 
 
 or ■. — {d(T being an increment of the length of the 
 a <T 
 
 contract-curve) either vanishes or becomes infinite. 
 
 Accordingly the maxima and minima of P and U present
 
 26 MATHEMATICAL PSYCHICS. 
 
 demarcating points ; for example, the centre ol each 
 sphere, which corresponds to a maximum in reference 
 to the upper hemisphere, a minimum in reference to 
 the lower hemisphere. The impure contract curve is 
 relevant to cases where the commodity of one party is 
 a discommodity to the other. 
 
 But even in the pure contract-curve all points do 
 not in the same sense indicate immobihty. For, accord- 
 ing to the consideration' (3) [above, p. 23], the contract- 
 curve may h& treated as the locus where, U being 
 constant, P is stationary^ either a maximum or 7ninim,um. 
 Thus any point in our case of two intersecting spheres 
 affords a maximum in relation to the upper hemisphere ; 
 but the same point (it is only an accident that it should 
 be the same point — ^it would not be the same point if 
 you suppose slightly distorted spheres) affords a mini- 
 mum in relation to the lower hemisphere. This pure^ 
 but unstable (part of the) contract-curve is exemplified 
 in certain cases of that ^ unstable equilibrium of trade, 
 which has been pointed out by Principal Marshall and 
 Professor Walras. 
 
 The preceding theory may easily be extended to 
 several persons and several variables. Let Pj = Fj 
 {x y z) denote the utihty of one of three parties, utihty 
 depending on three variables, xy z \ and similarly 
 ^i — ^^t 1*3=^3. Then the contract-settlement^ the 
 arrangement for the alteration of which the consent of 
 all three parties cannot be obtained, will be (subject to 
 reservations analogous to those analysed in the pre- 
 ceding paragraphs) the Eliminant. 
 
 d^ d^i d^ 
 
 dx dy d z 
 
 ' Mr. MarahallB figure 9 but not his figure 8 ; for the delicate relation 
 between the conceptions — ^instability of Trade (where per feci royr, petition is 
 presupposed) and instability of conf-ract in ijen-prii — m jn-i ujio oi identity.
 
 PURE 
 
 CONTR 
 
 ACT. 
 
 ^P^ 
 
 dV, 
 
 dV, 
 
 dx 
 
 dy 
 
 dz 
 
 dV, 
 
 rfPs 
 
 dT, 
 
 dx 
 
 dy 
 
 dz 
 
 27 
 
 In general let there be m contractors and n subjects 
 of contract, n variables. Then by the principle (3) 
 [above, p. 23] the state of equihbrium may be con- 
 sidered as such that the utility of any one contractor 
 must be a maximum relative to the utihties of the other 
 contractors being constant, or not decreasing ; which 
 may be thus mathematically expressed : 
 
 D (Ij Pi + Is Pj -I- &c. -f Z„ P „)=0, where D repre- 
 sents complete increment and Zj l^ &c., are indeterminate 
 multiphers ; whence, if there be n variables x^x^ . . . x^, 
 we have n equations of the form 
 
 from which, if n be not less than ?», we can ehminate 
 the (m— 1 independent) constants / and obtain the con- 
 tract-system consisting of n—{m—l) equations. 
 
 The case of n being less than m may be sufficiently 
 illustrated by a particular example. Let the abscissa x 
 represent the single variable on which the utihties P 
 and JI of two persons contracting depend. Then if /? 
 and 77 are the maximum points for the respective plea- 
 sure-curves (compare the reasoning, p. 22) it is evident 
 that the tract of abscissa between n and p is of the 
 nature of pure contract-curve ; that the index being 
 placed anywhere in that tract will be immovable ; 
 secus on either side beyond n and p. Similarly it may 
 be shown that, if three individuals are in contract 
 about two variables x y, the contract locus or region is 
 (the space within) a curvilinear triangle in the plane xy
 
 2S AIATHExMATICAL rSYCHICS. 
 
 bounded by the three contract-curves presented by suc- 
 cessively supposing each pair of individuals to be in 
 contract with respect to x and y. And similarly for 
 larger numbers in hyperspace. 
 
 It is not necessary for the purpose of the present 
 study to carry the analysis further. To gather up and 
 fix our thoughts, let us imagine a simple case — Robinson 
 Crusoe contracting with Friday. The articles of contract : 
 wages to be given by the white, labour to be given by the 
 black. Let Robinson Crusoe =X. Represent y, the labour 
 given by Friday, by a horizontal hne measured north- 
 icard from an assumed point, and measure x, the remu- 
 neration given by Crusoe, from the same point along 
 an ea.stLcard line (See accompanying figure 1.). Then 
 
 Fig. 1. 
 
 
 any point between these lines represents a contract. It 
 will very generally be the interest of both parties to 
 vary the articles of any contract taken at random. But 
 there is a class of contracts to the variation of which 
 the consent of both parties cannot be obtained", of settle-
 
 PURE COXTRACT. 29 
 
 ments. These settlements are represented by an inde- 
 finite number of points, a locus, the contract-curi>e CC, or 
 rather, a certain portion of it which may be supposed 
 to be wholly in the space between our perpendicular 
 lines in a direction trending from south-east to north- 
 west. This available portion of the contract-curve lies 
 between two points, say t^o ^o north-west, and y^ ^^ south- 
 east ; which are respectively the intersections witli the 
 contract-curve of the curves of indifference^ for each 
 party drawn through the origin. Thus the utility 
 of the contract represented by vjo-^ois for Friday zero, 
 or rather, the same as if there was no contract. At 
 that point he would as soon be off with the bargain — 
 work by himself perhaps. 
 
 This simple case brings clearly into view the charac- 
 teristic evil of indeterminate contract, deadlock, m\- 
 decidable opposition of interests, a.KpLTo^'^ cpts koI 
 rapaxn- It is the interest of both parties that tliere 
 should be some settlement, one of the contracts repre- 
 sented by the contract-curve between the limits. But 
 which of these contracts is arbitrary in the absence of 
 arbitration, the interests of the two adversd jmpiantia 
 fronte all along the contract-curve, Y desiring to get as 
 far as possible south-east towards yolo? X north-west 
 toward yjoJ-q. And it further appears from the preceding 
 analysis that in tlie case of any number of articles (for 
 instance, Eobinson Crusoe to give Friday in the way of 
 Industrial Partnership a fraction of the produce as well 
 as wages, or again, arrangements about the mode of 
 work), the contract-locus may still be represented as a 
 sort of line, along which thfe pleasure-forces of the con- 
 tractors are mutually antagonistic. 
 
 An accessory evil of indeterminate contract is the 
 
 ^ Sk-e p. 22. ■■' DemnFibenes, Be Corona.
 
 30 MATIIKMATICAL PSYCHICS. 
 
 tendency, greater than in a full market, towards dissimu- 
 lation and objectionable arts of higgling. As Professor 
 Jevons ^ says with reference to a similar case, ' Such a 
 tran.saction must be settled upon other than strictly 
 economical frrounds. . . . The art of bargaining consists 
 in the buyer ascertaining the lowest price at which the 
 seller is willing to part with his object, without dis- 
 clo.sinii, if possible, the highest price which he, the 
 buyer, is willing to give.' Compare Courcelle-Seneuil's^ 
 account of the contract between a hunter and a wood- 
 man in an isohited region. 
 
 With tliis clogged and underground procedure is 
 contrasted (/8) the smooth machinery ofthe open market. 
 As Courcelle-Seneuil says, ' k mesure que le nombre 
 des concurrents augraente, les conditions d'echange de- 
 viennent plus necessaires, plus impersonelles en quelque 
 sorte.' You might suppose each dealer to write down ^ 
 his donand, how much of an article he would take at 
 e;u*h price, without attenii)tnig to conceal his require- 
 ments ; and these data having been furnished to a sort of 
 market-macJiine, the price to be passionlessly evaluated. 
 
 That contract in a state of perfect competition is 
 determined by demand and supply isgenerally accepted, 
 but is hardly to be fully understood without mathe- 
 matics. The mathematics of a perfect market have been 
 worked out by several eminent writers, in particular 
 Messrs. Jevons, Marshall, Walras ; to whose varied cul- 
 tivation of the mathematical science, Catallactics^ the 
 reader is referred who wishes to dig down to the root of 
 first principles, to trace out all the branches of a com- 
 ])lete system, to gather ftuits rare and only to be 
 reached by a mathematical substructure. 
 
 • Theonj, p. \M. * Traits, book ii. 
 
 ' Cf. Walras, Elements, Art. 50.
 
 PERFECT COMPETITION. 31 
 
 There emerges amidst the variety of construction and 
 terminology ttoWcov ovoy^drcav ju,op<^77 ja/a, an essentially 
 identical graphical form or analytical formula express- 
 ing the equation of supply to demand ; whereof the 
 simplest type, the catallactic molecule, as it might be 
 called, is presented in the case above described in tlie 
 definition of perfect competition.^ The famihar pair of 
 equations is deduced ''^ by the present writer from the 
 first principle : EquHibrium is attained when the ex- 
 isting contracts can neither be varied without recontract 
 with the consent of the existing parties, nor by recon- 
 tract within the field of competition. The advantac^e 
 of this general method is that it is appHcable to the par- 
 ticular cases of imperfect competition ; where the con- 
 ceptions of demand and siq^ply at a price are no lonn^er 
 ap])ropriate. 
 
 The catallactic molecule is compounded, when we 
 suppose the Xs and Ys dealing in respect each of several 
 articles Avitli several sets of Zs, As, Bs, &c. ; a case re- 
 solved by M. Walras. 
 
 Thus the actual commercial field might be represented 
 by sets of entrepreneurs Xs, Ys, Zs, each X buyin"^ labour 
 from among sets of labourers, As, Bs, Cs, use of capital 
 from among sets of capitalists, Js, Ks, Ls, use of land from 
 among sets of lando\vners, Ps, Qs, Es, and selling pro- 
 ducts among a set of consumers consisting of the sum of 
 the three aforesaid classes and the entrepreneurs of a 
 species different from X, the Ys and Zs. As the demand 
 of the labourer is deducible from considering his utility 
 
 ' See p. 17. It muBt be carefully remembered that Prof. Jevons's 
 Formulae of Exchange apply not to bare individuals, an isolated couple, but 
 (as he himself sufficiently indicates, p. 98), to individuals clothed with the 
 properties of a market, a typical couple (see Appendix V.). The isolated 
 couple, the catallactic atmn, would obey our (a) law. 
 
 ' See p. 38.
 
 32 MATHEMATICAL ISVCHICS. 
 
 as a function of wages received and work done, so tlie 
 demand of the entrepreneur is deducible from consider- 
 ing his utility as a function of (1) his expenditures on 
 the agents of production; (2) his expenditures in the way 
 of consumption ; (3) his receipts from sale of produce ; 
 (4) his labour of superintendence. The last-named 
 variable is not an article of contract ; but there beingf 
 supposed a definiLc relation connecting the produce with 
 agents of production and entrepreneur's labour, the 
 ru,talla("tic. fwrmuhc become applicable. This is a very 
 ai)stract representation (abstracting e.g. risk, foreign 
 trade, the migration from one employment to another, 
 e.q. Xs becoming Ys," (fcc), yet more concrete than that 
 of M. Walras, who aj)parently makes the more abstract 
 supposition of a sort of y'/vWio/z/t^^^i- entrepreneur, ' faisant^ 
 )ii perte ni benefice!' 
 
 From the point of view just readied may with ad- 
 vantage be contem[)lated one of the domains most 
 recently added to Economic Science — Mr. Sidgwick's 
 contribution to the 'Fortnightly Eeview,' September, 
 1879. The indirectness of the relation betweeii wages and 
 intereM which Mr. Sidgwick has so clearly demonstrated 
 in words is self-evident in symbols. The predetej-minate- 
 ?v.s,>,' of the iragefund, which has received its covp de 
 grace from Mr. Sidgwick, must always, one would think, 
 have appeared untenable from the humblest mathemati- 
 cal point of view, the consideration of the simplest 
 type ^ of perfect competition ; from which ^also it must 
 be added tiiat Mr. Sidgwick's — perhaps inadvertent, 
 perhaps here misinterpreted — statement, * that contract 
 
 ' Th\9 permenbility between employments (such as explained in Econo)}ncs 
 of Industry with reference to the supply of unskilled and skilled labour and 
 of business power) tends to a level of utility. 
 
 2 Elements, Arts, "i;^!, 242, kc. ^ See pp. 17, 31. 
 
 '• Fortmyhtly Rtvieir, It<70, pp. 410 (end) 411 (beginuiug).
 
 PERFECTT COMPETITION. 33 
 
 between employer and operative even in the case of 
 what is here called ^ jc»^r/<?ci competition, is indeterminate, 
 does not, it is submitted, appear tenable. It is further 
 submitted that Mr. Sidgwick's strictures ^ on Prof. 
 Jevons are hasty ; for that by a (compound) employ- 
 ment of the Jevonian (or an equivalent catallactic) for- 
 mula, the complex relations between entrepreneur, capi- 
 tahst, and labourer are best made clear. And so ' there 
 is a priori ground for supposing that industrial compe- 
 tition tends to equahze the rate of profit (as well as 
 interest) on capitals of different amount.'^ That ' the 
 labour of managing capital does not increase in propor- 
 tion to tlie amount managed ' is so far from creating any 
 peculiar difficulty, that it is rather of the essence of the 
 theory of exchange ; quite congruent with the familiar 
 circumstance that the disutility of (common) labour 
 (labour subjectively estimated) does not increase in pro- 
 portion to work done (labour objectively estimated). 
 That the labour of managing capital increases not only 
 not at the same but at a less rate-of-increase than the 
 amount managed, as IVIr. Sidgwick seems to imply, is in- 
 deed a peculiar circumstance ; but it is of a sort with 
 which the Jevonian formula, the mathematical theory of 
 catallactics, is quite competent to deal, with wiiicli in 
 fact ]\Ir. Marshall has dealt in his second class of 
 Demand-Curves. 
 
 ' See Defin, p. 18. 
 
 ■^ Fortnightly Reoietc, pp. 411, 412. 
 
 ' As the g:ain per unit of produce is the same for one X as l'or*another 
 X, and the gain per unit of capital lent ib the same for one J as for another 
 J ; so, if there is in the field in addition to the classes prescinded, a class of 
 capitalist-enlrepreneurB, e.g. (J K)s, the gain per unit of pixxiuce is the same 
 for one (J K) as for another (JK). But no equ.ation is made between the 
 gain of a (J K) and the sum of the gains of a J and a K : even if to pimplif)' 
 the comparison -we abstract rent. ( Gain of course in this statement mea- 
 sured objectively, say in money, not subjectively iu utility).
 
 d4 MATHEMATICAL PSYCHICS. 
 
 But it is not the purport of the present study to 
 attempt a detailed, much less a polemical, discussion of 
 pure Catallactics, but rather (y) to inquire how far con- 
 tract is determinate in cases of imperfect competition. 
 It is not -necessary for this purpose to attack the general 
 problem of Contract qualified by Competition, which is 
 much more difficult than the general problem of un- 
 qualified contract already treated. It is not necessary 
 to resolve analytically the composite mechanism of a 
 competitive field. It will suffice to proceed synthetically, 
 observing in a simple typical case the effect of con- 
 tinually introducing into the field additional com- 
 petitors. 
 
 I. Let us start, then, from the abstract typical case 
 above put (p. 28), an X and Y deahng respectively in 
 X and 1/. Here .r represents the sacrifice objectively 
 measured of X ; it may be manual work done, or com- 
 modity maiuifactured, or capital abstained from during 
 a certain time. And y is the objectively measured 
 remuneration of X. Hence it may be assumed, accord- 
 ing to the two first axioms ^ of the UtiHtarian Calculus, 
 tlie law of increasing labour, and the law of decreas- 
 
 ing utility, that P being the utility of X, (1)^ — is 
 
 continually negative, - positive; (2) _,, -^„ ^^-^, 
 
 continually negative. (Attention is sohcited to the inter- 
 pretation of the third condition.) No doubt these latter 
 conditions are subject to many exceptions, especially in 
 regard to abstinence from capital, and in case of pur- 
 
 ' See these laws stated in the companion calculus. The proofs TPere 
 offered in .Mind, without acknowledgment, because without knowledge, of 
 the cum\ilati\o proofs already adduced by Prof. Jevons. 
 
 '^ Cf. Appendix V.
 
 IMPERFECT COMPETITION. 35 
 
 chase not for consumption, but with a view to re-sale ; 
 and in the sort of cases comprised in Mr. Marshall's 
 Class n. curves. Still, these exceptions, though they 
 destroy the watertightness of many of the reasonings in 
 this and the companion calculus, are yet perhaps of 
 secondary importance to one taking a general abstract 
 view. 
 
 This being premised, let us now introduce a second 
 X and a second Y ; so that the field of competition con- 
 sists of two Xs and two Ys. And for the sake of illus- 
 tration (not of the argument) let us suppose that the 
 new X has the same requii'ements, the same nature as 
 the old X ; and similarly that the new Y is equal- 
 natured with the old. 
 
 Then it is evident tliat there cannot be equilibrium 
 unless (1) all the field is collected at one point ; (2) that 
 point is on tJie contract-curve. For (1) if possible let 
 one couple be at one point, and another couple at 
 another point. It \^dll generally be the interest of the 
 X of one couple and the Y of the other to rush together, 
 leaving their partners in the lurch. And (2) if the com- 
 mon point is not on the contract-curve, it will be the 
 interest of all parties to descend to the contract-curve. 
 
 The points of the contract-curve in the immediate 
 neighbourhood of the hmits yo^o and t^qX^ cannot be Jinal 
 settlements. For if the system be placed at such a point, 
 say slightly north-v^est of ^/ofo^ it will in general be pos- 
 sible for one of the Ys (without the consent of the 
 other) to recontract with the two Xs, so that for all 
 those three parties the recontract is more advantageous 
 than the previously existing contract. For the right 
 fline joining the origin to (the neighbourhood of) y^io 
 will in general He altogether within the indifference- 
 cvi've drawn from the origin to y„^9. For the indif-
 
 36 MATHEMATICAL PSYCHICS. 
 
 ference-curve is in general convex to the abscissa. For 
 its differential equation is 
 
 "".7/,. V 
 
 V dy 
 
 fdY(xy 
 
 ^) 
 
 
 whence 
 
 - r (^^\ + {M.\^^i {ii.\ + (iz'\ r TAI^ + ^ ^1 
 
 L V rfxV Vdlrrfy; dxJ\di/J \dx J L \dxdy) dy- dx } 
 
 w 
 
 which is perfectly positive. Therefore the indifference- 
 curve (so far as we are concerned with it) is convex to 
 the abscissa. 
 
 Now, at the contract-curve the two indifference-curves 
 for X and Y touch. Thus the figure 1, page 28, is proved 
 to be a correct representation, indicating that a point af'i/ 
 can be found both more advantageous for Y than the 
 point on the contract-curve 2/1^1 (on an intenor indif- 
 ference-curve, as it may be said), and also such that its 
 co-ordinates are the sums (respectively) of the co-ordi- 
 nates of two other points, both more advantageous for 
 an X. These latter points to be occupied by Xj and Xj 
 may be properly regarded (owing to the symmetry 
 
 . a/ v' 
 
 and competition) as coincident-^ with co-ordinates ~ jr- 
 
 Ji A' 
 
 Further, it appears from previous reasonings that there 
 will be a contract-relation between {^'y') and f ^ ;|-); 
 
 
 namely ' ;' . •',{ = '-' "' ; where F'^^ is put for the
 
 IMPERFECT (COMPETITION. 37 
 
 first partially derived function f — -^ — ^'j 
 
 When this relation is satisfied the system of three 
 
 might remain in *he position reached ; but for Yj who 
 
 has been left out in the cold. He will now strike in, 
 
 with the result that the system will be worked down to 
 
 the contract-curve again ; to a point at least as favour- 
 
 a/ v' 
 able for the Xs as -^ —. Thus the Ys will have lost some 
 
 of their original advantage by competition. And a cer- 
 tain process of which this is an abstract tyj^ical repre- 
 sentation will go on as long as it is possible to find a 
 point a! y' with the requisite properties. Attention to the 
 problem will show that the process will come to a stop 
 at a point on the contract-curve y^ ^21 such that if a line 
 joining it to the origin intersect the curve, the supple- 
 mentary contract-curve as it might be called, 
 
 F,/v 
 
 (I-2) 
 
 in the point x y' then 4> (^2 ^2) = ^ (.2/ y'), provided that 
 ^ ^ falls within the indifference-curve for Y drawn 
 
 V2 2) 
 
 through (I2 yz)- If otherwise, a slightly different system 
 of equations must be employed. 
 
 K now a third X and third Y (still equal-natured) be 
 introduced into the field, the system can be worked 
 down to a point ^33/3 ;^ whose conditions are obtained 
 
 from those just written by substituting for -^~- -A. . 
 
 For this represents the last point at which 2 Ys can re- 
 contract with 3 Xs with advantage to all five. Analyti- 
 
 - ' Compare the analysis in Appendix VII.
 
 38 MATHEMATICAL PSYCHICS. 
 
 cal geometry will show that this point is lower down (in 
 respect of the advantage of Y) than ^^ y^. In the hmit, 
 when the Xs and Ys are indefinitely (equally) multiplied, 
 we shall have {a/ 1/) coincident with (^^ y^), or as we may 
 say for convenience (^ 77), satisfying one or other of the 
 alternatives corresponding to those just mentioned. 
 In case of the first alternative we have 
 
 ^4>;(^,7) + ^*',(^,7) = 
 
 For 4» (1 77) = <I> (^' y) = * ( (1 + A) f (1 + h) rj). 
 In the limiting case h is infinitesimal. Whence by dif- 
 ferentiating the above equation is obtained. And the 
 
 1 ^ not falling within the indif- 
 
 ference-curve of Y) is not to be distinguished from the 
 first in the limiting case. 
 
 If this reasoning does not seem satisfactory, it would 
 be possible to give a more formal proof; bringing out 
 the important result that the common tangent to both 
 indillerence-curves at the point £ 13 is the vector from 
 the origin. 
 
 By a parity of reasoning it may be shown that, if 
 the system had been started at the north-west extremity 
 of (the available portion of) the contract-curve, it would 
 have been worked down by competition between the Xs 
 to tlie same point ; determined by the intersection with 
 tiie contract-curve of ^F' .r + lyF'y = ; for the same 
 j)oint is determined by the intersection of either curve 
 with the contract-curve. For the three curves evidently 
 intersect in the same point. 
 
 Taking account of the two processes which have 
 been described, the competing Ys being worked down for 
 a certain distance towards the north-west, and similarly 
 the competing Xs towards the south-east : we see that
 
 PERFECT COMPETITION. 39 
 
 iu general for any number sliort of the practically 
 infinite (if such a term be allowed) there is a finite length 
 of contract-curve, from ^^ y^ to x^ t)^^ at any point of 
 which if the system is placed, it cannot by contract or 
 recontract be displaced ; that there are an indefinite 
 number of final settlements, a quantity continually dimi- 
 nishing as we approach a perfect market. We are 
 brought back again to case (yS), on which some further 
 remarks have been conveniently postponed to this place. 
 (For additional illustrations see Appendix V.) 
 
 The two conditions, ^^'x + i^^'^, = and ^F^ + yjYy 
 = 0, just obtained correspond to Professor Jevons's two 
 equations of exchange. His formulae are to be regarded 
 as representing tlie transactions of two individuah i?i, or 
 subject to, the laic of, a market. Our assumed u?iity of 
 nature in the midst of plurahty of persons naturally 
 brings out the same result. The represented two 
 curves may be called demand curves, as each expresses 
 the amount of dealing which will afibrd to one of the 
 dealers the maximum of advantage at a certain rate of 
 
 exchan.ge a value of ^ . This might be elegantly ex- 
 pressed in polar co-ordinates, tan 6 wiH then be the 
 rate of exchange, and, if P be the utihty of X, 
 
 C-1-) = is the demand-curve. By a well known 
 
 property of analysis ( i— j = represents not only 
 
 maximum points, but minimum points ; tlic lowest 
 depths of valley, as well as the highest elevatiojis, wliicli 
 one moving continually in a fixed right line from the 
 origin over the utility-surface would reacli. This mini- 
 mum portion of the demand-curve corresponds to Mr. 
 Marshall's Class IE. We see that the dealer at any given
 
 40 ^[AT1IEMATICAL PSYCHICS. 
 
 rate of exchange, far from resting and having his end at 
 a })ornt on this part of the curve, will tend to move away 
 from it. It has not the properties of a genuine demand- 
 curve. 
 
 The dealing of an individual in an open market, in 
 wliich there prevails what may be called the law of 
 price, the relation between the individual's require- 
 ments and that quantity c<?//^c/2U6Zy-demanded-at-a-price, 
 usually designated by the term Demand^ between Httle 
 d and big D in M. Walras's terminology, is elegantly 
 exliibited by that author. Compare also Cournot on 
 ' Concurrence.' 
 
 Here it is attempted to proceed without postulating 
 the phenomenon of uniformity of price ^ by the longer 
 route of contract-curve. When we suppose plurahty of 
 natures as well as persons, we have to suppose a plurality 
 of contract-curves (which may be appropriately con- 
 ceived as grouped, according to the well-known loga- 
 rithmic law, about an average). Then, by considerations 
 analogous to those already employed, it may appear 
 that the quantity of :final settlements is diminished as 
 the number 9f competitors is increased. To facilitate 
 conception, let us suppose that the field consists of two 
 Xs, not equally, but nearly equally, natured ; and of two 
 Ys similarly related. And (as in the fifth Appendix) 
 let the indifference curves consist of families of concen- 
 tric circles. Then, instead of a single contract-curve, 
 we have a contract-region, or bundle of contract-curves ; 
 tianiely the four lines joining the centres of the circle- 
 systems, tlie lines CjCi, CjC's, CjC'i, CjCj ; wherein Ci, Cj 
 are the centres of X, and X2, supposed close together ; 
 and similarly Q,\ and C'2 for the Ys. 
 
 ' The term will sometimes ie uaed here for rate of exchange in tjentral, 
 afi by M. Walras.
 
 PERFECT COMPETITION. 41 
 
 What corresponds here to that settUment of the whoU 
 field at a single point in the contract- curve, which we 
 had under consideration in reasoning about equal- 
 natured Xs, may thus be indicated. Take a point ^'iVi 
 on one of the contract-lines, say CjCi ; and let Xj and Y, 
 be placed there. Let X2Y2 be placed at a neighbouring 
 point, ^'\r)\, on the line CjCs ; such that (] ) ^'i^'i ^s 
 outside the two indifference curves drawn for X, and Yj 
 respectively tlirough ^iTj'i ; (2) ^jt^'i is outside the two 
 indifference-curves drawn for Xj and Y2 respectively 
 through ^'^r/'^. 
 
 Fig. 2. 
 
 Then the settlement cannot be disturbed by an X and 
 a Y simply changing partners, rushing into each other's 
 arms, and leaving tlieir deserted consorts to look out for 
 new alliances. Ee-con tract can now proceed only by 
 one Y moving off Avith the two Xs, as in the previous 
 case ; by which process the system may be worked 
 down to a neighbourhood describable as ^2^2- ^ the 
 limit, when the number of Xs and Ys are increased 
 indefinitely, but not necessarily equally (suppose mX, 
 and nY, where m and n are indefinitely large) ; if 
 x^y^ represent the dealings of any X, viz. X^ and simi- 
 larly ^ and -q be employed for the deahngsof the Ys, we 
 should find for the 2 w -h 2 tz variables the following 
 2m + 2n equations: 
 
 (1) m + n equations indicating that each Xand each
 
 42 iLA^THEMATICAL PSYCHICS. 
 
 Y is on his individual demand-curve (compare the con- 
 dition stated below, p. 48), e.g. 
 
 '' d x^ ^'' dy^ 
 
 (the differentiation being of course partial). 
 
 (2) m + n — \ equations indicating uniformity of 
 
 price ^ = -f^ = &c. = I' = J2 = &c. 
 
 (3) A last condition, which might perhaps be called 
 par excellence tlie equation of Demand to Supply, 
 namely, either S.i = ^ ^, or Sy = Xri. Thus the deal- 
 ings of each and all are completely determinate and de- 
 termined. 
 
 If we transform to polar co-ordinates, we might 
 write any individual demand-curve, as p—f^ (6) ; and 
 tlience obtain two collective demand-curves p = S/(^) and 
 p = S (f>{0) ', substantially identical with those collec- 
 tive demand curves so scientifically developed by M. 
 Walras, and so fruitfully applied by Mr. Marshall. 
 
 Thus, proceeding by degrees from the case of two 
 isolated bargainers to the limiting case of a perfect 
 market, we see how contract is more or less indeterminate 
 according as the field is less or more affected with the first 
 imperfection^ hmitation of numbers. 
 
 II. Let there be equal numbers of equal-natured 
 Xs and equal-natured Ys, subject to the condition that 
 eacli Y can deal at the same time with. only wXs, and 
 similarly each X with only n'Ys. First let w = n'. 
 Then, in the light of the conceptions lately won, it ap- 
 pears that contract is as indeterminate as if the field- 
 consisted of only nXs and wYs; that is to say, there 
 are as many and the same final settlements as in that case, 
 represented by the same portion of the contract-curve
 
 COMBINATIONS. 43 
 
 between (say) ^y and x-q. Let n' increase. Conti-act 
 becomes less indeterminate : f moving north-west, and 
 the quantity oi final settlements being thereby diminished. 
 The subtracted final settlements are most favourable to 
 the Ys. Let n' diminish. Contract becomes more inde- 
 terminate ; ^ moving south-east, and the quantity of final 
 settlements being thereby increased. Tlie added final 
 settlements are more favourable to the Ys than those 
 pre\iously existing. 
 
 The theorem admits of being extended to the 
 general case of unequal numbers and natures. 
 
 III. Let there be an equal number N of equal- 
 
 natured Xs and equal-natured Ys, and let each set be 
 
 formed into equal coiihinations, there being wXsin each 
 
 X combination, and n'Xs> in each Y combination. First, 
 
 let n = n'. Then contract is as indeterminate as if the 
 
 N N" 
 
 field consisted of — Xs and — Ys ; in the same sense as 
 n n 
 
 that explained in the last paragraph. Let tz' diminish. 
 Contract becomes less indeterminate, in the same sense as 
 in the last paragraph. Let n' increase. Contract be- 
 comes more indeterminate ; the added final settlements 
 being more favourable to the Ys than those previously 
 existing. 
 
 The theorem is typical of the general case in which 
 numbers, natures, and combinations are unequal. 
 Combination tends to introduce or increase indeter- 
 minateness ; and the final settlements thereby added are 
 more favourable to the combiners than the (determinate 
 or indeterminate) final settlements previously existing. 
 Combiners stand to gain in this sense. 
 
 The worth of this abstract reasoning ought to be 
 t€sted by comparison with the unmathematical treat- 
 ment of the same subject. As far as the writer is aware.
 
 44 MATHEMATICAL PSYCHICS. 
 
 a straightforward answer has never been offered to the 
 abstract question, What is the effect of combinations on 
 contract in an otherwise perfect state of competition, as 
 here supposed ? Writers either ^ ignore the abstract 
 question altogether, confining themselves to other as- 
 })ects of Trade Unionism ; its tendency to promote 
 communication, mobility, &c. ; in our terms, to render 
 the competition more normal^ and more perfect in respect 
 of extent (diminishing our first imperfection, for such is 
 tlie effect of increased mobility, ahke of goods and men). 
 Or, while they seem to admit that unionism would have 
 the effect of raising the rate of wages ^ they yet deny that 
 tlic total remuneration of tlie operatives, the wage fund 
 (in the intelligible sense of that term), can be increased. 
 But if our reasonings be correct, the one thing from an 
 abstract point of view visible amidst the jumble of 
 catallactic - molecules, the jostle of competitive crowds, 
 is tliut those who form themselves into compact bodies 
 by combination do not tend to lose, but stand to gain in 
 the sense described, to gain in point of utihty, which is 
 a function not only of the (objective) remuneration, but 
 also of the labour, and which, therefore, may increase, 
 although the remuneration decrease ; as ^Ir. Fawcett 
 well sees (in respect to the question of unproductive 
 
 ' Mr. Sidgwick iadeed (if the passage already referred to, Fortnightly 
 Revieic, p. 4ll, ante, p. 33, might be thus construed ?) — at any rate some 
 other.s have observed the momentous dead-lock resulting from the complete 
 solidijicit.icm of the whole operative-interest and the whole employer-interest; 
 our (a) case, contract unqualified by competition. But this hardly affords 
 any indication of what would happen, or what the writers suppose would 
 happen, when contract is qualified, however slightly, by competition ; as if, 
 for instance, there were two or three combinations on one side and two or 
 three on the other; which in view of foreign competition is likely, one might 
 think, to be long the concrete case. 
 
 "^ Cf. Cadmes on Trades Uniont (first sections) ; Courcelle-Seneuil on 
 Coali'ions,
 
 COMBINATIONS. 45 
 
 consumption. — ' Manual,' cli. iv.), though he gives so 
 
 uncertain a sound about Trades Unionism. And if, as 
 
 seems to be imphed in much that has been written on 
 
 this subject, it is attempted to enforce the argument 
 
 against Trades Unionism by the consideration that it 
 
 tends to diminish the total national produce^ the obvious 
 
 reply is that unionists, as ' economic men,' are not 
 
 concerned with the total produce. Because the total 
 
 produce is diminished, it does not ^ follow that the 
 
 labourer's share is diminished (the loss may fall on the 
 
 capitahst and the entrepreneur, whose compressibility has 
 
 been well shown by I^Ir. Sidgwick in the article already 
 
 referred to) ; much less does it follow (as aforesaid) that 
 
 there should be diminished that quantity which alone 
 
 the rational unionist is concerned to increase — tlte 
 
 labourers utility. If this view be correct, it would seem 
 
 as if, in the matter of unionism, as well as in that of the 
 
 predeterminate wage-fund, the ' untutored mind ' of the 
 
 workman had gone more straight to the point than 
 
 economic intelligence misled by a bad method, reasoninf^ 
 
 without mathematics upon mathematical subjects. 
 
 IV. Let there be an equal number N of equal- 
 
 natured Xs and Ys; subject to the condition that to 
 
 every contract made by a Y at least n Xs must be 
 
 parties, and similarly for an X n' Ys. First, let n^n'. 
 
 Contract is as indeterminate as if the field consisted of 
 
 N N 
 
 — Xs and — Ys. Let n' increase. Contract becomes 
 
 n n 
 
 more indeterminate, and the Ys Hand to gain. And con- 
 versely. 
 
 To appreciate the quantity of indeterminateness 
 hkely to result in fact from these imperfections (opera- 
 ting separately and together) would require a knowledge 
 ' See the remarks in Appendix Vn.
 
 46 MATHEMATICAL PSYCHICS. 
 
 of concrete phenomena to which the writer can make 
 no claim. 
 
 The first imperfection appHes to Monopolies. It is 
 perliaps chiefly important, as supplying a clue for the 
 solution of the other cases. 
 
 The second imperfection may be operative in many 
 cases of contract for personal service. Suppose a 
 market, consisting of an equal number of masters and 
 servants, ofTerinir respectively wages and service ; sub- 
 ject to the condition that no maTj can serve two masters, 
 no master employ more than one man ; or su])pose 
 equilibrium already established between such parties to 
 be disturbed by any sudden influx of wealth into the 
 hands of the masters. Then there is no determmate^ and 
 very generally ^ unique, arrangement towards which the 
 system tends under the operation of, may we say, a 
 law of Nature, and which would be predictable if we 
 knew beforehand the real requirements of each, or of 
 tlie average, dealer; but there are an indefinite number 
 of arrangements a priori possible, towards one of wliich 
 the system is urged not by the concurrence of innume- 
 rable (as it were) neuter atoms ehminating chance, but 
 (abstraction being made of custom) by what has been 
 (tailed ti;e Art of Bargaining — ^^higghng dodges and 
 designing obstinacy, and other incalculable and often 
 disreputable accidents. 
 
 Now, if managerial work does not admit of being 
 distributed over several establishments, of .being sold in 
 bits, it v/ould seem that this species of indeterminateness 
 affects the contract of an entrepreneur with foreman, 
 of a cooperative association of workmen (or a com- 
 bination) witli a manager. This view must be modified 
 
 ' Exceptions are the multiple intersections of Demand-Cur vps ehown fcy 
 Mr. Mamhall and M. Walrap.
 
 COMBTNATIOXS. 47 
 
 in so far as managerial wages are determined by the 
 cost of production (of a manager !), or more exactly 
 by the equation ^ between managerial wages and the 
 remuneration in other occupations, w^here the remu- 
 neration is determined by a process of the nature of 
 perfect competition ; and by other practical considera- 
 tions. 
 
 The third imperfection may have any degree of 
 importance up to the point where a wliole interest 
 (labourers or entrepreneurs) is solidified into a single 
 competitive unit. This varying result may be tolerably 
 well illustrated by the case of a market in wjiicli an 
 indefinite number of consumers are supplied by varying 
 numbers of monopolists (a case properly belonging to 
 our first i/nperfectiou : namely, hmited nui/tber of dealers). 
 Starting with complete monopoly, we shall find the 
 price continually diminish as the number of monopolists 
 increases, until the point of complete fluidity is reached. 
 This gradual ' extinction ' of the influence of monopoly 
 is well traced by Cournot in a discussion masterly, but 
 limited by a particular condition, which may be called 
 uniformity of price, not [it is submitted) abstractedly neces- 
 sary in cases of imperfect competition} Going beyond 
 Cournot, not without trembling, the present inquiry 
 finds that, where the field of competition is sensil)ly 
 imperfect, an indefinite number of final settlements are 
 possible ; that in such a case different final settle- 
 ments would be reached if the system should run 
 down from different initial positions or conti-acts. The 
 
 ' In virtue of pertneabilifi/ hetvfeen occupations; postulating (1) free- 
 dom of choice between different occupations, (2) knowledfre of circuni- 
 fllances determining choice. With the latter sort of knowledge (so warmly 
 impugned by Mr. Cliff Leslie) our free comnumication about uriidrf vf wn- 
 irttct (in vonnal market) is not to be confounded. See p. IH. 
 
 •^ Cf. Walras's Elements, s. .352.
 
 48 MATHEMATICAL PSYCHICS. 
 
 sort of diflerence which exists between^ Dutch and 
 English auction, theoretically unimportant in perfect 
 competition, does correspond to different results, different 
 final settle jueiiti in imperfect competition. And in general, 
 and in the absence of imposed conditions, the said final 
 settlements are not on the demand-curve, hut on the con- 
 tract-curve. That is to say, there does not necessarily 
 exist in the case of imperfect as there does in the case 
 of perfect competition a certain property (which some 
 even mathematical writers may appear to take for 
 granted), namely, that — in the case all along supposed 
 of Xs and Ys dealing respectively in x and y — if any 
 X X give X in exchange for y„ he gets no less and no 
 more y than he is willing to take at the rate of ex- 
 change — . 
 
 If, however, this condition, though not spontaneously 
 generated by imperfect as by perfect competition, should 
 be introduced ab extra, imposed by custom and con- 
 venience, as no doubt would be very generally the 
 case, nevertheless the property of indeterniinateness, 
 plurality of final settlements, will abide. Only the final 
 settlements will now be by way of demand-curve, not 
 contract-curve. If, for instance, powerful trades unions 
 did not seek to fix the quid pro quo, the amounts of 
 labour exchanged for wealth (which they would be 
 quite competent to seek), but only the rate of exchange, 
 it being left to each capitalist to purchase as much 
 labour as he might demand at t}iat rate, there would 
 still be that sort of indeterminateness favourable to 
 unionists above described. The geometry of this case 
 may be understood from an attentive consideration of 
 
 • As Thornton suggests. Now we believe, but not because that un- 
 matheuiatical writer has told us.
 
 COOPERATIVE ASSOCIATIONS. 49 
 
 the typical illustration at the end of Appendix Y., 
 fig- 4. 
 
 The fourth imperfection would seem likely to operate 
 in the case of cooperative associations up to the time 
 when the competitive field shall contain a practically 
 infinite number of such bodies ; that is, perhaps 
 for a long time. To fix the ideas, suppose associa- 
 tions of capitalist- workmen, consisting each of 100 
 members, 50 contributing chiefly capital, and 50 cliiefly 
 labour. Let the field of competition consist of 1,000 
 indi\dduals. The point here indicated is that, not- 
 withstanding the numerical size of the field, contract 
 will not be more determinate (oAving to the fact that all 
 the members of the association are in contract with each 
 other — not, as now usual, each for himself contracting 
 with employer) than if the field consisted of 10 indi- 
 viduals. And a similar result would hold if, with more 
 generality, we suppose members contributing labour 
 and capital in varying amounts, and remunerated for 
 their sacrifices according to a principle of disij'ibuiion ; 
 in the most, or, at any rate, a sufficiently general case, 
 a function of the sacrifices, the form of the function 
 being a contract- variable, or what comes to much tlie 
 same thing, there being assumed a function of given 
 form containing any number of constants, which are 
 articles of contract, subject, of course, to the condition 
 that the sum of the portions assigned is equal to the 
 distribuend. And, similarl}^ if we introduce different 
 kinds of labour and other concrete complications. 
 
 The Determinateness will depend not so much upon 
 tlie number of individuals as upon the number of 
 associations in the field. As cooperative association 
 becomes more prevalent, no doubt, cceteris paribus, the 
 indeterminateness liere indicated would decrease.
 
 50 MATHEMATICAL PSYCHICS. 
 
 Nevertheless, in consequence of the great variety of 
 cooperative experiments, the sundry kinds of contract 
 and divers species of articles, the field of competition 
 being thus broken up, it is submitted that the rise of 
 cooperative association is likely to be accompanied 
 with the prevalence of ^ indeterminateness, whatever 
 opinion we may form about the possible regularity in a 
 distant future. 
 
 Altogether, if of two great coming institutions, trades- 
 unionism is affected with the third imperfection, and 
 cooperative association with the fourth, and both with 
 the .second, it does not seem very rash to infer, if not for 
 the present, at least in the proximate future, a consider- 
 able extent of indeterminateness. 
 
 Of this inference what would be the consequence. 
 To impair, it may be conjectured, the reverence paid to 
 competition ; in whose results — as if worked out by a play 
 of physical forces, impersonal, impartial — economists 
 have complacently acquiesced. Of justice and huma- 
 nity there was no pretence ; but there seemed to com- 
 mand respect the majestic neutrality of Nature. But if 
 it should appear that the field of competition is deficient 
 in that continuity of fuid^ that multiety of atoms which 
 constitute ^ the foundations of the uniformities of Physics; 
 if competition is found wanting, not only the regularity 
 of law, but even the impartiahty of chance — the throw 
 of a die loaded with villainy — economics would be 
 indeed a ' dismal science,' and the reverence for com- 
 petition would be no more. 
 
 ' There has been, I believe, observed in cooperative associations, with 
 regard to the comparative remuneratioas of capitah and labour, that dispute 
 without any principle of decision which is the characteristic of contract. 
 
 ' Above, pp. 5, 18, 
 
 ' Theory of Vortices and Theory of Atoms.
 
 NEED OF AHBITRATION. 51 
 
 There would arise a general demand for a fninciple 
 of arbitration. 
 
 And this aspiration of the commercial world would 
 be but one breath in the universal sigh for articles of 
 peace. For almost every species of social and political 
 contract is affected vn{\\ an indeterminateness like that 
 which has been described ; an evil which is likely to be 
 much more felt when, with the growth of intelligence 
 and hberty, the principle of contract shall have replaced 
 both the appeal to force and the acquiescence in custom. 
 Throughout the whole region of in a wide sense contract., 
 in the general absence of a mechanism like perfect com- 
 petition, the same essential indeterminateness prevails ; 
 in international, in domestic politics ; between nations, 
 classes, sexes. 
 
 The whole creation groans and yearns, desiderating 
 a principle of arbitration, an end of strifes. 
 
 Corollary. — Where, then, would a world weary of 
 strife seek a principle of arbitration ? lujnMice, replies 
 the moralist ; and a long hne of philosophers, from Plato 
 to Herbert Spencer, are ready to expound the principle. 
 But their expositions, however elevating in moral tone, 
 and of great hortative value for those who already 
 know their duty, are not here of much avail, where the 
 thing sought is a definite, even quantitative, criterion 
 of what is to be done. Equity and ' fairness of division ' 
 are charming in the pages ^ of Herbert Spencer, and 
 delighted Dugald Stewart with the appearance^ of mathe- 
 matical certainty ; but how would they be applicable to 
 the distribution of a joint product between coopera- 
 tors ? Nor is the equity so often invoked by a high 
 authority on cooperation much more available ; for 
 why is the particular principle of distribution recom- 
 
 , ' Data of Ethics, p. 1G4. "^ Imoi/s, Book 11.
 
 52 MATHEMATICAL PSYCHICS. 
 
 mended by Mr. Holyoake (operatives to take net pro- 
 duct, paying therefrom a salary to manager, roughly 
 speaking, and to say nothing of capital) more equitable 
 than an indefinite number of Other principles of distri- 
 bution (e.g. operatives to take any fraction which might 
 have been agreed upon, manager the remainder ; eithei^ 
 party, or neither, paying wages to the other). 
 
 J?uitice requires to be infoimed by some more defi- 
 nite principle, as Mill ^ and Mr. Sidgwick reason well. 
 The star of justice affords no certain guidance — for 
 those who have loosed from the moorings of custom — 
 
 c 
 
 unless it reflect the rays of a superior luminary — utili- 
 tarianism. 
 
 But, even admitting a disposition in the purer wills 
 and clearer intellects to accept the just as f.nis litium, 
 and the useful as the definition of the just; admitting 
 that there exists in the higher parts of human nature a 
 tendency towards and feeling after utihtarian institu- 
 tions ; could we seriously suppose that these moral 
 Considerations were relevant to war and trade ; could 
 eradicate the ' controlless core ' of human selfishness, or 
 exercise an appreciable force in comparison with the 
 impulse of self-interest. It would have to be first shown 
 that the interest of all is the interest of each, an illusion 
 to which the ambiguous language of Mill, and perhaps 
 Bentliam, may have lent some countenance, but which 
 is for ever dispelled by the masterly analysis of Mr. 
 Sidgwick. Mr. Sidgwio-k acknowledges two supreme 
 principles — Egoism and Utilitarianism ; of independent 
 authority, conflicting dictates ; irreconcilable, unless 
 indeed by religion. 
 
 It is far from the spirit of the philosophy of pleasure 
 to depreciate the importance of religion ; but in the 
 
 ' See review of Thornton on Labour (as well as Utilitarianism).
 
 PRINCIPLE OF ARBITRATION. 53 
 
 present inquiry, and dealing with the lower elements of 
 human nature, we should have to seek a more obvious 
 transition, a more earthy passage, from the principle 
 of self-interest to the principle, or at least the practice, 
 of utilitarianism. 
 
 Now, it is a circumstance of momentous interest — 
 visible to common sense when pointed out by mathe- 
 matics — that one of the in general indefinitely nume- 
 rous settlements'^ between contractors is the utilitarian 
 arrangement of the articles of contract, the contract 
 tending to the greatest possible total utihty of the con- 
 tractors. In this direction, it may be conjectured, is to 
 be sought the required principle. For the required 
 basis of arbitration between economical contractors is 
 evidently some settlement ; and the utilitarian settle- 
 ment may be selected, in the absence of any other 
 principle of selection, in virtue of its moral peculiarities : 
 
 ■ Where the cantract-curve is (^.) (^°) - (^) (^) = 0, 
 the utilitarian point has co-ordinates determined by the equations 
 
 the roots of which evidently satisfy the contracts-equation. The theorem is 
 quite general. 
 
 Rere may be the place to observe that if we suppose our contractors t<J 
 be in a sensible degree not 'ecouomic' agents, but actuated in en'ectite 
 moment* by a sympathy with each other's interests (as even now in domestic, 
 and one day perhaps in political, contracts), we might suppose that the 
 object which X (whose own utility is P), tends — in a calm, etiective moment 
 — to maximise, is not P, but P + X n ; where X is a coefficieni. of effect ii^e 
 tympathy. And similarly Y — not of course while rushing to self-grfltiti ca- 
 tion, but in those regnant moments which characterise an elh'cal ' method ' 
 — may propose to himself as end n + /i P. What, then, will be the con- 
 tract-curve of these modified contractors ? The old contract curve between 
 narrower limits. In 6g. 1, j/o |o ^H have been ^displaced in a noi th-wcsterly 
 and »jo ^o in * south-easterly direction. A? the coefficients of sympathy in- 
 crease, utilitarianism becomes more;jurc, (of. pp. 12, 17), the contract-curve 
 narrows dawn to the utilitarian point.
 
 54 MATHEMATICAL PSYCHICS. 
 
 its satisfying the sympathy ^ (such as it is) of each with 
 all, the sense of justice and utilitarian equity.^ 
 
 These considerations might be put clearest in a 
 particular, though still very abstract, case. Let us sup- 
 \)O^Q that in consequence of combinations competition 
 fails to determine the contract between entrepreneur 
 and operatives. The case becomes that described under 
 (a) — deadlock between two contracting parties. One 
 of the parties is indeed here collective; but it is allow- 
 able for the sake of illustration to make abstraction of 
 this circumstance, to abstract also the correlated bar- 
 gains with capitalists, landowners, &c., and to suppose a 
 single entrepreneur in dealing with a single operative. 
 
 And, first, let it be attempted to arbitrate upon 
 some principle oi doctrinaire jiustice — some metaphysical 
 dogma, for instance, of equality : that the entrepreneur 
 shall have an ' equal ' share of the produce. Now, 
 there is no presumption that this ' fair division ' is 
 utilitarianian ; in view of the different character of 
 the entrepreneur's sacrifice^ in view also (if one may be 
 allowed to say so) of a possible difference in the entre- 
 preneur's capacity :^ suppose, for instance, that a more 
 highly nervous organisation required on the average a 
 higher ininiiiiuni of means to get up to the zero of 
 utility. As there is no presumption that the proposed 
 arrangement is utilitarian, so there is no presumption 
 that it is on the contract-curve. Therefore, the self- 
 interests of the two parties will concur to bulge away 
 from the assumed position ; and, bursting the cobwebs 
 
 ' Assuminff as economists assume (sew Mill, book II. chap. xiv. 6. 7, 
 Walker on Wages, &c.), an however slight clinamen from the rectilinearity 
 of the ' economic man.' 
 
 ^ Whereof the unconsciously implicit first principle is : Time-intensity 
 units of pleasure are to be equated irrespective of persons. 
 ^ See p. 58.
 
 PRINCIPLE OF ARBITRATION. 55 
 
 of doctrinaire justice, to descend with irresistible force 
 to some point upon the contract-curve. Suppose that 
 by repeated experiences of this sort the contract-curve 
 has been roughly ascertained — a considerable number of 
 final settlements statistically tabulated. Now these 
 positions he in a reverse order of desirability for each 
 party ; and it may seem to each that as he cannot have 
 his own way, in the absence of any definite principle of 
 selection, he has about as good a chance of one of the 
 arrangements as another. But, rather than resort to 
 some process which may virtually amount to tossing 
 up, both parties may agree to commute their chance of 
 any of the arrangements for the certainty of one of 
 them, which has certain distinguishing features and 
 peculiar attractions as above described — the utilitarian 
 arrangement. 
 
 Or perhaps, considering the whole hne of possible 
 arrangements, they might agree ' to ' spHt the difference,' 
 and meet each other in the neighbourhood of the cen- 
 tral point — the ' quantitative mean,' as it might be 
 called. Well, first, this quantitative mean would Hkely 
 to be nearer ihan the extremes to the utihtarian point ; 
 and, further, this very notion of mean appears to be the 
 outcome of a rudimentary 'implicit' justice, apt in a 
 dialectical atmosphere to bloom into the ' quahtative ^ 
 mean ' of utilitarian equity. 
 
 ' See p. 136. 
 
 ' Aristotle's metaphysical theory that \irtiie is a mean between two 
 vices is analogous to the mathematical theory that a maximum of pleasure 
 is a mean between two minima. 
 
 So also Aristotle's notion of two species of excellence (aprrr)), and more 
 generally all cases in which there seem to be two (or more) best ways 
 of acting (using the superlative in a sense analogous to the proper mathe- 
 matical sense of ' maximum '), may be cases of muhiple solutions of a problem 
 in the Calculus of Vat-iatums, the problem of majimum utility. 
 
 Jt if difficult to allude to Mr. Todhunter's beautiful and delicate problems
 
 56 MATHEMATICAL PSYCHICS. 
 
 Or less specifically may we say that in the neigh- 
 bourhood of the contract-curve the forces of self-interest 
 being neutralised^ the tender power of sympathy and 
 right would become appreciable ; as the gentler forces 
 of the magnetic field are made manifest when terrestrial 
 magnetism, by being opposed to itself, is eliminated. 
 
 Upon the whole — omitting what it is obvious to 
 understand about the spirit in which very abstract 
 reasonings are to be regarded : a star afibrding a 
 general direction, not a finger-post to specify a by- 
 path — there may appear, at however great a distance, a 
 general indication that competition requires to be sup- 
 plemented by arbitration^ and the basis of arbitration 
 between f<elf-interested contractors is the greatest possible 
 sum-total utility. 
 
 Tlius the economical leads up to the utilitarian cal- 
 culus ; the faint outlines of which, sketched in a pre- 
 viously published paper, may be accepted as the second 
 subdivision of our Second Part. 
 
 UTILITARIAN CALCULUS. 
 
 Problem. — To find (a) the distribution of means and 
 {P) of labour, the {y) quahty and (S) number of popu- 
 lation, so that there may be the greatest possible hap- 
 piness. 
 
 Definitions. — (1) Pleasure is used for * preferable 
 feeling ' in general (in deference to high authority, 
 though the general term does not appear to call up with 
 equal facility all the particulars which are meant to be 
 
 without once more in%'itiQg attention to the versatile features and almost 
 human complexion of that species of Calculus which seems most directly 
 applicable to the affairs of men ; so differant from the brutal rigour abcribed 
 to Mathematics by men who are acquainted only with its elements.
 
 UTILlTARLi> CALCULUS. 57 
 
 included under it, but rather the grosser ' feelings than 
 for instance the 'joy and fehcity ' of devotion). The 
 term includes absence of pain. Greatest possible hap- 
 piness is the greatest possible integral of the differential 
 ' Number of enjoyers x duration of enjoyment x degree 
 thereof (cf. axiom below).^ 
 
 (2) Means are the distributable proximate means of 
 pleasure, chiefly wealth as destined for consumption and 
 (what is conceivable if not usual in civilisation) the un- 
 purchased command of unproductive labour. 
 
 (3) An individual has greater capacity for happiness 
 than another, when for the same amount whatsoever of 
 means he obtains a greater amount of pleasure, and also 
 for the same increment (to the same amount) whatsoever 
 of means a greater increment of pleasure. 
 
 This -'- definition- of-a-t-hing ' is doubtless (like Euclid's) 
 ii» pci'fectly leali& ed. One imperfection is that some 
 individuals may enjoy the advantages not for any amount 
 of means, but only for values above a certain amount. 
 This may be the case with the liiglier orders of evolu- 
 tion. Again, one individual may have the advantages 
 in respect of one kind of means, another of another. 
 But, if one individual has tlie advantages in respect of 
 most and the greatest pleasures, he ma y be treated as 
 ha ying more capacity for pleasure in gen eral. Thirdly, 
 the two advantages may not go together. If ' the higlier 
 pleasures, such as those of affection and virtue, can 
 
 ' C!ompare the "base associatione of ' Utilitarianism.'' Surely, as Mr. 
 Arnold sayp, a pedant invented the term. 
 
 ' The greatest possible value ^^ I I I ^P ''^ '^^ (where dp corresponds 
 
 to a juat perceivable increment of pleesure, dn to a sentient individual, dl to 
 an mstant of time). The limits of the time-integi-ation are and od, the 
 present and the indefinite future. The other limits are variable, to be deter- 
 mined by the Calculus of Yariations. 
 
 E
 
 58 MATHEMATICAL PSYCHICS. 
 
 liardly be said to come from pleasure- stuff at all' (as 
 Mr. Barratt says in his able Note in ' Mind X.,' often 
 cited below), it is possible (though not probable?) that 
 the enjoy ers of the higher pleasures should derive 
 fro m the zero^^ ^BrI Lt ^lhei '"X^ert airi_jmi^^ 
 (and a fortiori for all superior values) ..^i^-ameunt- of 
 pleasure {i^rea ter than an other class of enjoyers, say the 
 sensual, can obtain for any amount whatsoever oTmeans ; 
 whilti at the same time tlie sensual obtain greater incre- 
 ments of pleasure for the same increments of means 
 (above the minimum). In such a case the problem 
 would be complicated, but the solution not compromised. 
 Eougldy speaking, the first advantage would dominate 
 the. theory of population ; the second the distribution 
 of means. A fourth imperfection in the statement of 
 the definition is that the units whose capacities are com- 
 pared are often (/roups of individuals, as families. With 
 these reservations the reality of the definition jqiay be 
 allowed. 
 
 But it may be objected that differences of capacity, 
 though real, are first not precisely ascertainable, and 
 secondly artificial, being due to education. But, first, 
 even at present we can roughly discriminate capacity 
 for happiness. If the higher pleasures are on the whole 
 most pleasurable — a fact of which the most scientific 
 statement appears to have been given by Mr. Sully ^ 
 — then those who are most apt to enjoy those pleasures 
 tend to be most capable of happiness. And, as Mr. 
 Barratt says, it 'seems (speaking generally) to be the 
 fact that, the higher a being in the scale of evolution, 
 the higher its capacity for pleasure ; ' while greater pre- 
 cision might be attainable by improved examinations and 
 hedonimetry. Further it will be seen that some of the 
 
 ' Pessimism, note to chap. xi.
 
 DEFINITIONS. 59 
 
 applications of the problem turn upon s2ij)poseJ, ratlier 
 than ascertained, differences of capacity. Tlie second 
 objection, William Thompson's, would liardly now be 
 maintained in face of what is known about heredity. 
 But it is worth observing that his conclusion, equahty of 
 distribution, follows from his premiss only in so far as a 
 proposition like our first postulate (below) is true of 
 wealth and labour applied to education^ in so far as it is 
 true that improvement is not proportionately increased 
 by the increase of the means of education. 
 
 (4) An individual has more capacity for work than 
 another,^ when for the same amount whatsoever of work 
 done he incurs a less amount of fatigue, and also for the 
 same increment (to the same amount) whatsoever of work 
 done a less increment of fatigue. 
 
 This fourth definition may present the same imper- 
 fections as the third. Indeed the fourth definition is but 
 a case of the third ; both stating relation between means 
 and pleasure. The third definition becomes the fourth, 
 if you change the sigjis of me'dus B.nd pleasure, put means 
 produced for means consumed and the pains of produc- 
 tion for the pleasures of consumption. Or not even the 
 latter change, in so far as labour is sweet (which is very 
 far according, to Fourier). It is submitted that this 
 identification confirms the reahty of the third definition, 
 since the reahty of the fourth is undisputed. Of course, 
 if we identify the definitions, we must bear in mind that 
 they are hable to be separated in virtue of the second 
 imperfection above noticed. 
 
 Axiom. — Pleasure is measurable, and aU pleasures 
 are commensurable ; so much of one sort of pleasure 
 
 ' Or this : When the same amount of fatigue corresponds to a greater 
 amount of work done, and the same increment (to the same amount) of 
 fatigue to a greater increment of work.
 
 60 jVIATHEMATlCAL PSYCHICS. 
 
 felt by one sentient being equateable to so much of 
 other sorts of pleasure felt by other sentients. 
 
 Professor Bain has shown ^ how one may correct 
 one's estimate of one's own pleasures upon much the 
 same principle as the observations made with one's 
 senses ; how one may correctly estimate the pleasures 
 of others upon the principle ' Accept identical objective 
 marks as sliowing identical subjective states,' notwith- 
 standinj*^ jxirsonal difl'erences, as of activity or demon- 
 s^trativeness. This ' moi'al arithmetic ' is perhaps to be 
 siipplementcid by a moral differential calculus, the Fech- 
 nerian method apphed to pleasures in general. For 
 Wundt has shown that sensuous pleasures may thereby 
 be measured, and, as utihtarians hold, all pleasures are 
 commensurable. The first principle of this method 
 miglit be : Just-perceivable increments of pleasure, of 
 all pleasures for all persons, are equateable.^ Imph- 
 cated Avith this principle and Bain's is the following : 
 Equimultiples of equal pleasures are equateable ; where 
 the multiple of a pleasure signifies exactly similar plea- 
 sure (integral or differential) enjoyed by a multiple 
 number of persons, or through a multiple time, or 
 (time and persons being constant) a pleasure whose 
 degree is a multiple of the degree of the given pleasure. 
 Tlie last expression is open to question (though see 
 Delboeuf * Etude psychophysique,' vii. and elsewhere), 
 and is not here insisted upon. It suffices to postulate 
 tlie practical proposition that when (agreeably to Fech- 
 nerian conceptions) it requires n times more just-per- 
 ceivable- increments to get up to one pleasure from zero 
 than to get up to another, then the former pleasure 
 enjoyed by a given number of persons during a given 
 
 ' Emotions and Will, 3rd edition. 
 
 ' Cf. Wundt, Phys. Psych., p. 295; above, p. 8, Appendix III.
 
 THE UNIT OF PLEASURE. 61 
 
 time is to be sought as much as the latter pleasure en- 
 joyed by n times the given number of persons during 
 the given time, or by the given number duiing the 
 multiple time. Just so one cannot reject the practical 
 conclusions of Probabilities, though one may object 
 with Mr. Venn to speaking of belief being numerically 
 measured. Indeed these principles of jxeTprjTLKT) are 
 put forward not as proof against metaphysical subtle- 
 ties, but as practical ; self-evident a priori^ or by what- 
 ever iTrayoiy) or c^tcr/xos is the method of practical 
 axioms. 
 
 Let us now approach tlie Problem, attacking its 
 inquiries, separately and combined, with the aid of 
 appropriate postulates. 
 
 (a)^ The first postulate appropriate to the first in- 
 quiry is : Tlie rate r f increase of pleasure decreases as 
 its means increase. The postulate asserts that the 
 second differential of pleasure w4th regard to means is 
 continually negative. It does not assert that the first 
 differential is continually positive. It is siipposablc 
 (though not probable) that means increased beyond a 
 certain point increase only pain. It is also supposable 
 that 'the higher pleasures ' do not ' come from pleasure- 
 stuff at all,' and do not increase with it. Of course 
 there are portions of the utilitarian whole unaffected by 
 our adjustments ; at any rate the happiness of the 
 stellar populations. But this does not invalidate the 
 postulate, does not prevent our managing our ' small 
 peculiar' for the best, or asserting that in respect 
 thereof there tends to be the greatest possible liappiness. 
 The proposition thus stated is evidenced by every-day 
 experience ; experience well focused by Buffon in his 
 
 ' See the cumulative proofs of this postulate adduced bj- Professor 
 JevODB in Theory of Ptditical Economy.
 
 C2 MATHEMATICAL TSYCniCS. 
 
 ' Moral Arithmetic,' Laplare in Ins ' Essay on Proba- 
 bilities,' William I'hompson in his 'Inquiry into the 
 Distribution of Wealth,' and JMr. Sidgwick in the 'Me- 
 thods of Ethics.' 
 
 This empirical generalisation may be confirmed by 
 ' ratiocination ' from simpler inductions, partly common 
 t(^ tlie followers of Fechner, and partly peculiar to 
 Professor Delboeuf. All the forraulce suggested for the 
 relation between quantity of stimulus and intensity of 
 sensation agree in possessing the property under con- 
 sideration ; which is true then of what Professor Bain 
 would describe, as pleasures of mere intensity ; coarse 
 pleasures indeed but the objects of much expenditure. 
 Thus pleasure is not proportionately increased by in- 
 creased glitter of furniture, nor generally by increased 
 scale of establishment ; whether in the general case by 
 analoofy from the Fechnerian experiments on the senses^ 
 or by a more a priori ' law of relation ' in the sense of 
 Wundt. 
 
 But not only is the function connecting means and 
 pleasure such that tlie increase of means does not pro- 
 duce a })ro]>ortionate increase of pleasure ; but this 
 eflect is heightened by the function itself so varying (on 
 repetition of the conditions of pleasure) that the same 
 means produce less pleasure. The very parameter in 
 virtue c>f which such functional variation occurs is 
 exhibited by Professor Delboeuf in the case of eye-sen- 
 sations ; - that a similar variation holds good of pleasures 
 in <Teneral is Bain's Law of Accommodation. Increase 
 of means then, affording proportionately increased re- 
 petition of the conditions of pleasure, does not afford 
 proportionately increased pleasure. Doubtless there 
 
 ' Cf. Fechner, Psychophysik, vol. ix. p. 6. 
 ■^ Etvde psychophysique, &c.
 
 LAW OP DIMINISHING UTILITT. 63 
 
 are compensations for this loss ; echoes of past pleasures, 
 active habits growing up in the decay of passive im- 
 pressions. Indeed the diJBerence of individuals in res- 
 pect to these compensations constitutes a large part of 
 the difference of capacity for pleasure. 
 
 It may now be objected : increased means do not 
 operate solely by repeating old pleasures, but also by 
 introducing to new (e.g. travel) ; also the ' compensa- 
 tions ' may more than counterbalance the accommodations. 
 It is generally replied : In so far as a part only of hap- 
 piness increases only proportionately to its means, the 
 second differential of happiness with regard to means 
 does not cease to be negative. That second differential 
 cannot be condnuaUy negative. Its being negative for 
 a space may not affect the reasoning. K it does affect 
 the reasoning, one conclusion, the inequahty of distri- 
 bution, would probably (if the pleasure-curve is not 
 very comphcated) become a fortiori. Not only would 
 the less capable receive then stiU less means, but even 
 the equally capable might then not all receive equal 
 means. 
 
 This being postulated, let us mark off the degrees 
 of capacity for happiness on an abscissa (supposing that 
 capacity is indicated by the values of a single variable ; 
 if by the values of a function of several variables, the 
 proof differs only in complexity). At each degree erect 
 an ordinate representing the number of individuals of 
 that degree of capacity. On the rectangle correspond- 
 ing to each individual it is requii-ed to construct a 
 paraUelopiped representing his means. Let us proceed 
 \o impart the distribuend means — in the first inquiry a 
 given distribuend to given distributees doing each a given 
 amount of labour — by way of small increments. Let 
 us start with the assumption that each individual has
 
 04 MATHEMATICAL PSYCHICS. 
 
 and shall retain that minimum of means just sufficient 
 to bring him up to the zero-point of happiness (a con- 
 ception facilitated by, though not quite identical with, 
 the economical ' natural minimum of wages '). There- 
 after who shall have the first increment of means ? By- 
 definition an individual of the highest capacity (at least 
 supposing the minimum to be the same in all capa- 
 cities). Who shall have the next increment of means ? 
 Another individual of the highest capacity, in preference 
 to the same individual by the postulate. Thus a first 
 dividend will be assigned to the first section (all the 
 individuals of the highest capacity) exclusively. But 
 they will not continue sole assignees. Their means 
 only, being continually increased, must by the postulate 
 reach a point such that an increment of means can be 
 more felicifically assigned to an individual of the second 
 section (the next highest capacity) than to one of the 
 first. The second section will then be taken into distri- 
 bution.^ Thus the distribution of means as between the 
 equally capable of pleasure is equality ; and generally is 
 such that the more capable of pleasure shall have more 
 7neans and more pleasure. 
 
 The law of unequal distribution is given by a plane 
 curve, in tlie plane of tlie capacities and means, say 
 a megistfiedone. To different distribuends correspond 
 megisthedones differing only by a constant. For it is 
 educible from the postulate that there is only one 
 family of raegistliedones. We may have any number 
 of ma.cima by tacking between different mehibers of the 
 family; but tlie greatest possible value is afibrded by the 
 continuous solution. 
 
 If we now remove the condition that each individual 
 shall retain his minimum, what happens ? Simply that 
 
 ' Compare the renaoning in the ordinary Theory of Rent..
 
 "DISTRIBUTION OF l^IEAXS. 65 
 
 the megi..:ueclones may now dip below the minimum 
 line. But it is improbable that tlie}' should dip very 
 low under the minimum at the lower end while they 
 rise very high above the minimum at the higher end ; 
 since excessive physical privations cannot be counter- 
 balanced, by any superfluity of refined pleasures. In 
 fact, if ^ve assume that the zero of means corresponds 
 to injiitite pain of privation [cf. Wundt's curve of 
 pleasure and pain), then by investigating the radius of 
 curvature it is shown that, as the distribuend dimin- 
 ishes, the megisthedone tends to become a horizontal 
 line. In famine the distribution even between unequals 
 is equahty — abstracted ulterior considerations, as of 
 posterity. 
 
 These conclusions may be affected by the imperfec- 
 tions of the third definition. By the first imperfection, 
 if the ' minimum ' line were not horizontal. Secondly, 
 suppose that the individuals who have less capacity for 
 pleasures in general have a special capacity for parti- 
 cular pleasures. The bulk of means will be distributed 
 as before, but there will be a residue distributed 
 according to a second megisthedone. The second megis- 
 thedone superimposed upon the first \\dll more or less 
 deform it. Lastly, the unit distributee is often a group 
 {e.g., a married couple, in respect of their common 
 menage). The conclusions may be afiected, in so far as 
 the most capable groups are made up of individuals 
 not most capable as individuals. 
 
 [P) The distribution of labour (to which attention 
 has been called by Mr. Barratt) is deduced by a parity 
 of reason from the parallel second axiom : that the rate 
 of increase of fatigue increases as the work done in- 
 creases, which is proved hj common experience and 
 (for muscular work) by the experiments of Professor
 
 66 MATHEMATICAL PSYCHICS. 
 
 Delboeuf (* Etude Psychophysique '), As appears in- 
 deed from Professor Delbceuf 's formulie, the first and 
 second postulates are to a certain extent implicated 
 (whereby the first postulate gains strength). Let us 
 now arrange our individuals according to their capacity 
 for work^ and proceed as before. Who shall do the 
 first increment of work ? Of course one of the most 
 capable of work. And so on. The distribution of 
 labour as between thefCqually capable of work is equality^ 
 and generally is such that the most capable of work shall 
 do more work — so much more work,^ as to sufier more 
 fatigue. 
 
 The inquiry presents the same declensions as the first. 
 In particular, cooperatives are to be compared not inter 
 se, but with the similar operatives in similar cooperative 
 associations : except, indeed, so far as the work done 
 is a symmetrical function of the effort of fellow-workers: 
 It is deducible that the rowers of a 10709 etcriy? shall have 
 equal fatigue ; but the fatigue of the pilot is not to be 
 equated to that of the oarsman. All the while it is to 
 be recollected that the fatigue or pain of work under 
 consideration may be negative, 
 
 {afi) To combine the first and second inquiries, 
 determine by the Differential Calculus the constants of 
 a megisthedone and a brachistopone such that the means 
 distributed by the former may be equal to the work 
 distributed by the latter and that the (algebraical) sum 
 of the pleasures of consumption and the pains of pro- 
 duction may be the greatest possible. Or,' ab initio, by 
 the Calculus of Variations, we may determine the means 
 and fatigtte as independent variable functions satisfying 
 those two conditions. 
 
 ' This inference requires the second form of the fourth definiticui, given 
 in the Note. 
 
 I
 
 DISTRIBUTION OF WORK. 67 
 
 Let V ^f'n [n^.y)-p-e{y -f{xp)]y^^ 
 
 where .t is degree of either capacity, or more elegantly, 
 if possible, a third variable in terms of which both ca- 
 pacities may be expressed ; x^^ and Xq are the given limits 
 of integration (the number and quality of the distributees 
 being not in the present inquiry variable) ; n is the number 
 of each section ; F(;ry) is a unit's pleasure of consump- 
 tion, being a function of x his quality (capacity for 
 pleasure) and the independent variable y his means ; p 
 is the unit's pain of work, another independent variable 
 function ; c is the constant incidental to problems of 
 relative maximum ; f(xp) is the work done by the unit, 
 being a function of his quahty (capacity for work) and 
 fatigue (effort). 
 
 Greatest possible happiness = greatest possible value 
 
 n [F(a? y) — p']dx =^ 
 
 Xq 
 
 greatest possible value of V, c being taken so that 
 / '^n[y-f{xp)]dx=^. 
 
 */ Xq 
 
 Tlie second term of the variation of V, 
 
 is continually negative by the postulates. Therefore 
 the greatest possible value of V is when its first term 
 of variation vanishes. The first term of variation, 
 
 «%[d-f)-]-«^[<^^)-i].
 
 68 MATHEMATICAL PSYCHICS. 
 
 vanishes only when both 
 
 If these equations hold, the two rules (a and )S) hold. 
 Q.E.D. The combined solution takes for granted that 
 the means of pleasure and the pain of work are inde- 
 pendent variables. And to a certain extent this may- 
 fail to be the case. An individual may want strength or 
 time to both enjoy the means and do the work which 
 the double rule assigns to him. In that case there will 
 be a compromise between the two rules. 
 
 (y) The thir-d pofitulate simplifying the third inquiry 
 is fhat capacity for pleasure and capacity for work 
 generally speaking go together ; that they both rise 
 with evolution.^ The quality of population should be the 
 highest possible evolution — provided ^ that the first im- 
 perfection of the third definition does not give us pause. 
 To advance the whole population by any the same 
 degree of evolution is then desirable ; but it is probably 
 not the most desirable application, given quantity of a 
 of means of education. For it is probable that the 
 highest in the order of evolution are most capable of 
 education and improvement. In the general advance 
 the most advanced should advance most. 
 
 (8) The fourth postulate essential to the fourth in- 
 quiry is tliat, as population increases, means (the dis- 
 tribuend) increase at a decreasing rate. This is given 
 by the Malthusian theory with regard to the products 
 of extractive labour. And this is ""' sufficient. For the 
 second differential of the whole means with regard to 
 
 ' See New and old Methods of Ethics (by the present writer), p. 72, 
 » Ibid. p. 77. 
 
 ' This is not quite accurate. For a part of the distribuend may increase 
 more than proportionately in virtue of econoraiea effected by increased pro-
 
 UTILITARIAN SELECTION. 69 
 
 population is still negative, even though a part of means 
 increase proportionately to the number of population ; 
 for instance, im productive labour requiring little or no 
 materials {e.g.^ ballet-dancers), or those manufactured 
 articles of which the cost is not appreciably affected by 
 the cost of the raw material. From this Malthusian 
 premiss it is deduced that population should he limited ; 
 but the hedonical conclusion is not necessarily of the 
 same extent as the Malthusian (cf. below ayS). A simple 
 inquiry under this head is the following. Assuming that 
 all the sections (degrees of capacity or orders of evolu- 
 tion) multiply equally, and that each section reproduces 
 exactly his kind, to find the (utilitarian) rate of increase ? 
 
 {yh) A more important inquiry is : not assuming that 
 all sections multi})ly equally^ to find .the average issue 
 for ench section, so that the happiness of the next gcne- 
 rati(m may be the greatest possible. 
 
 First let us introduce a conception more appropriate 
 than was possible under the preceding head ; namely, 
 that each section does not reproduce exactly its kind, 
 but that the issue of each (supposed endogamous) sec- 
 tion ranges on either side of the parental capacity, as 
 thus — /^ \2 
 
 V — pe X ^^ ; where f is the capacity of the 
 
 parental section, n its number ( = something like Ae — y 
 
 a , 
 
 duction. In thesanle manner, and for the same reason as a demand-curve 
 may have a plurality of intersections with a vector from the origin {Cf. Mr. 
 Marshall's theorem) corresponding alternately to ma^dmum and minimum 
 utility, so there may be a plurality of values for the sought number of 
 population, corresponding alternately to utilitarian and pessimistic arrange- 
 ment. The highest value which satisfies the equation to zero of the first 
 I'jrm of variation must correspond to a maximum. 
 
 The imperfection of this postulate does not affect the reasoning based 
 upoii ihe other poetulaTes.
 
 70 MATHEMATICAL PSYCHICS. 
 
 since the parental generation is to be conceived as 
 ranging under a curve of possibility ; cf. Galton, Que- 
 telet, &c.), V is the number of issue of capacity x. Per- 
 haps h is constant for all the curves of issue ; the 
 variation of y8 alone determines the natural maximum, 
 or artificial limit, of the average issue. But neither 
 the symmetry of the curves of possibihty, nor the par- 
 ticulars of this conception, are postulated. 
 
 T\iQ fifth postulate appropriate to tliis case is that to 
 substitute in one generation for any number of parents 
 an equal number each superior in capacity (evolution) 
 is beneficial for the next generation. This being granted, 
 either analytically with tlie aid of Mi*. Todhunter's 
 'Eesearches,'^ or by unaided reason, it is deduced that 
 the average issue shall be as large as possible for all 
 sections above a determinate degree of capacity, but 
 zero for all sections below that degree. 
 
 But can we be certain that this method of total selec- 
 tion as it might be termed holds good when we provide 
 not only for the next generation, but for the indefinite 
 future ? In the continuous series of generations, wave 
 propagating wave onward through all timCj it is required 
 to determine wliat wavelet each section of each wave 
 shall contribute to the proximate propagated wave, so 
 that the whole sum of hght of joy which glows in the 
 long line of waves shall be the greatest possible. If in 
 the distant future, agreeably to the views of Herbert 
 Spencer, population tends inartificially to become nearly 
 stationary ; if to the contemplator of all time genera- 
 tions fade into differentials ; we may conceive formed a 
 differential equation connecting the population of one 
 generation with the population of its successor and in- 
 
 * See Appendix I. p. 9^.
 
 UTILITARIAN SELECTION. 71 
 
 volving an independent variable function^ the average issue 
 for each section. By the Calculus of Variations (if the 
 calculator is not at sea) it is educed that the average 
 issue shall be as large as possible for all sections above 
 a (for each time) determinate degree of capacitj^ but 
 zero for all sections below that degree. But a further 
 postulate is required for so long as the movement of 
 population is not amenable to infinitesimal calculus ; 
 while the present initial irregular disturbances are far 
 from the tranquil waves of the ' stationary ' state. Tliis 
 sixth postulate might be : To substitute in one generation 
 for any number of parents an equal number each supe- 
 rior in capacity (evolution) is beneficial for all time. 
 This ]iostulate being granted, if jiossible let the most 
 beneficial selection be not total. Then a total selection 
 can be arranged more beneficial ! 
 
 If only we have swum through the waves to a terra 
 Jirma, our position need not appear outlandish. For, 
 first, these rules are veiy general, founded on very 
 abstract tendencies, and requiring to be modified in 
 practice. Thus our principle of selection might be 
 modified, in so far as endogamy should not be the rule, if 
 the higher orders of evolution have a greater tendency to 
 reversion (in violation of the fifth and sixth postulates), 
 and so forth. Again, since to exclude some sections 
 from a share of domestic pleasures interferes with the 
 principle of (a), it could not be expedient to sacrifice 
 the present to the future, without the highest scientific 
 certainty and pohtical security. Again to indicate an 
 ideal, though it can only be approached a.v6ponTivo)<;^ 
 may be useful. What approach is useful in such cases 
 is to be determined by Mr. Todhunter's principle.^ 
 Again, mitigations might be provided for the classes not 
 
 ' Researchet ; below p. 93.
 
 72 MATHEMATICAL PSYCHICS. 
 
 selected.^ In particular, they might have the benefit of 
 rule (/3) now almost cut away by the struggle of com- 
 petition. Again, emigration might supplement total 
 selection ; emigration from Utopia to some unprogres- 
 sive country where the prospect of happiness might be 
 comparatively zero. 
 
 (ayS) In tlie preceding analysis (yS) the distribution 
 of means (and labour) was supposed given. But the 
 reasoning is unaffected, if the distribution of means is 
 supposed variable, provided that the later postulates are 
 not affected by tliat distribution. And this they might 
 be on Mr. Doublcday's hypothesis. But in Herbert 
 Sj^encer's more probable view of the relation of affluence 
 to po[)ulousness, tlic first rule (a) will become a fortiori. 
 
 Under this head may be considered the question : 
 What is the fortune of the least favoured class in the Utili- 
 t(iri<in coiiununitij? Let us consider first the case of 
 enii(/r(itio7i for the benefit of the present generation. Let 
 us start with the supposition, however inappropriate, 
 that the distribuend does not vary with population ; as 
 in an isolated island where the bounty of nature could 
 not be alTected by liuman exertion. 
 
 The happiness of the present generation may be 
 symboiised 
 
 n [Y{x y) — cyldj: + c D 
 
 r 
 
 where T) is the given distribuend and the rest of the 
 notation is as above (a^). By the third postulate Xy is 
 given as the highest existing degree of capacity. What 
 remains variable is .ro, the abscissa of emigration. At 
 
 ' Cy. Galton, ' The weak could find a welcome and a refuge in celibate 
 juonawferies,' &c. ; also Sully, Peisimisin, p. 392.
 
 THE LEAST FAVOURED CLASS. 73 
 
 the limit F{xg y^ ^ cyQ — 0. Now c is positive, for it 
 
 equals /^-r— ), the first differential of pleasure with 
 \dyJ 
 
 regard to means, which (presupposed a utilitarian intel- 
 ligence) is probably never negative (above Postulate I.). 
 
 But this is not postulated. Only, if (-^ — j is negative, 
 
 we are deahng with the external case of the inquiry, 
 determining what sections shall immigrate from our 
 * unprogressive country.' For if the Utopians have such 
 a plethora of means that their happiness would be 
 increased by a diminution of their means, then immi- 
 gration will set in until the point of satiety be at least 
 repassed. Then c is positive, and y is essentially posi- 
 tive. Therefore F(a?oyo) is positive. It cannot be zero, 
 the zero-point of pleasure corresponding to a positive 
 minimum of means. 
 
 In this case the condition of the least favoured class is 
 positive happinesfi. This conception assists us to con- 
 ceive that a similar answer would be obtained if the 
 increase of the distribuend with increasing population 
 were small. 
 
 Small in relation to the megisthedonic share 
 of the least favoured class. Write the distribuend 
 
 nf{xp^)da: ; where p is the effort of each unit 
 
 ^0 
 
 worker, so far supposed given as a function of ;r ; N is 
 
 nd,c. Differentiate the 
 
 distribuend with regard to Xq. Substitute x for Xq and 
 call the curve so presented the Malthisian. Then the 
 condition of the least favoured class is positive^ zero^ or
 
 74 MATHEMATICAL PSYCHICS. 
 
 negative happiness^ according as at the limit the ordi- 
 nate of the Malthusian is less than, equal to, or greater 
 than that of the megi3thedone. 
 
 Our uncertainty as to the condition of the lowest 
 class increases when we consider the case of selection for 
 the benefit of the next generation. 
 
 Let n=<f>(x) be the curve of possibility for the pre- 
 
 sent j^eneration. Let v = Be/—^ — j-,~^ x ^ be the curve 
 
 6 2 
 
 of issue fo¥_eapacity ^ ; where B is the natural maximum 
 
 of issue. Then n^, the line of possibility for the next 
 
 -^2 
 
 i 
 
 12 
 
 generation, is^ i ^ 4. ^ ff)(x-\-z)dz, where by the 
 
 fifth postulate x^ is given as the highest existing degree 
 of capacity ; what is variable is x^^ the abscissa of total 
 selection. The happiness of the next generation 
 
 . QC 
 
 B}= / [n}(F{xt/)—cy)]dx + c'D, where oc is a con- 
 
 »/ — QC 
 
 venient designation for the utmost extent of variation—-' 
 variation in the Darwinian sense. Xq is given by the 
 
 equation =0 ; from which it is by no means clear 
 
 that the condition of the least favoured in the second 
 generation is above zero. 
 
 In fact, the happiness of some of the lower classes 
 may be sacrificed to that of the higher classes. And, 
 again, the happiness of part of the second generation 
 may be sacrificed to that of the succeeding generations. 
 Moreover (it is convenient, though out of order, here to 
 add) our uncertainty increases when we suppose the 
 laboriousness also of population variable. Nothing 
 indeed appeal's to be certain from a quite abstract point of
 
 THE LEAST FAVOURED CLASS. 75 
 
 view, except that the required limit is above tlie starving- 
 point ; both because in tlie neighbourhood of that point 
 there would be no Avork done, and — before that con- 
 sideration should come into force and above it — because 
 the pleasures of the most favoured could not weigh 
 much against the privations of the least favoured. {Cf. 
 Wundt's pleasure-curve.) 
 
 It may be admitted, however, that a limit below t1ie 
 zero of happiness, even if abstractedly desirable, would 
 not be humanly attainable ; whether because discomfort 
 in the lower classes produces political instabihty (Aris- 
 totle, &:c.), or because only through the comfort of the 
 lower classes can population be checked from sinking to 
 the starving-point (Mill, <fcc.). Let politics and political 
 economy fix some such limit above zero. If now Hedo- 
 nics indicate a limit still superior (in point of comfort) 
 — M'ell. But if abstract Hedonics point to a limit beloic 
 that hard and fast line which the consideration of human 
 infirmity imposes, what occurs ? Simply that popula- 
 tion shall press up against that line without pressing it 
 back. 
 
 (^yS) Under this head should be considered whether 
 rule (^) does not interfere with rule (yS). And this 
 upon Mr. Herbert Spencer's theory of population it 
 would do.^ The present then may have to be sacrificed 
 to the future ; though in general how much of the 
 present it is expedient to sacrifice to the future must be 
 as nice a question in pohtical, as in personal prud- 
 ence. 
 
 {a/Sy^) Contemplating the combined movements we 
 seera to see the vast composite flexible organism, the 
 play and the work of whose members are continually 
 readjusted, by degrees advancing up the line of evolu- 
 
 ' Contrast, ho-wever, Champagny, Lts Antonint, iiL p. 277,
 
 76 MATHEMATICAL PSYCHICS. 
 
 tioii ; the parts about the front advancing most, the 
 members of the other extremity more slowly moving on 
 and largely dying off. The final shape of the great 
 organism, whether its bounding line of possibihty shall 
 be ultimately perpendicular, whether the graduation of 
 (ui a Greek sense) aristocracy^ or the level of modern 
 revolution, is the ideal of the future, is still perhaps a 
 subject more for prejudice than judgment. Utilitarian- 
 ism, indifferent about the means, with eye undistorted by 
 ])repossessions, looks only to the supreme end. 
 
 Corollaries. Tlie application of these inquiries is 
 (I.) to first principles (11.) to subordinate rules of con- 
 duct. 
 
 I. The end of conduct is argued to be Utilitarianism, 
 as exactly deflnpd m the ' Mctliods of Ethics,' by deduc- 
 ing from tliat general principle maxims of common 
 sense ; perliaps as tlie constitution of matter is proved 
 by deducing from the theory experimental laws. What 
 inferior accuracy in the moral universe indeed ! But 
 before that inferiority should prejudice, let it be settled 
 wliat degree of accuracy was here to be expected. No 
 one would listen to Professor Clerk Maxw^ell inOavoko- 
 yovvTof; about the atoms without a mathematical corres- 
 pondence of his theory and the facts. But we have a 
 large experience of the progress of Physics ; it is well 
 seen liow slie goes ; but is the movement of Morals so 
 familiar that the true science should be manifest by her 
 method ! Whatever the method- — for Universal Eud^e- 
 monism prescribes no dogma about the origin of her 
 supremacy ; affiliated as readily to practical reason as 
 pure passion, the ' Faith ' of a Green or ' Ideals ' of a 
 Grote — whatever our faith, when we descend from 
 faith to works, requiring a criterion for alternative 
 actions, it may be divined that we shall not far err in
 
 PROOF OF UTILITARIANISM. t i 
 
 following, however distantly, the procedure of the 
 * Methods of Ethics.'* 
 
 Consider first then Equality, the right of equals to 
 equal advantages and burdens, that lai'ge section of 
 distributive justice, that deep principle which continually 
 upheaves the crust of convention. 
 
 ij3' m Koi Xvcrfi' rov ydp Kparos eari p.cyurTov. 
 
 All this mighty moral force is deducible from the prac- 
 tical principle of exact Utilitarianism combined with 
 the simple laws of sentience (a and ^). 
 
 But Equality is not the whole of distributive justice. 
 There may be needed an a^ia for unequal distribution. 
 Now inequalities of fortune — abstracted the cases of 
 governor and general and every species of trustee for 
 the advantage of others — are generally explained by 
 utilitarians as the consequence of conventions clear and 
 fixed and preventing confusion and encouraging pro- 
 duction, but not otherwise desirable, or rather of which 
 the necessity is regretted. Yet in the minds of many 
 good men among the moderns and the wisest of the 
 ancients, there appears a deeper sentiment in favour of 
 aristocratical privilege — the privilege of man above 
 brute, of civilised above savage, of birth, of talent, and 
 of the male sex. This sentiment of ricfht has a orpound 
 of utilitarianism in supposed differences of capacity. 
 Capacity for pleasure is a property of evolution, an 
 essential attribute of civilisation (a). The grace of life, 
 the charm of courtesy and courage, which once at least 
 distinguished rank, rank not unreasonably received the 
 
 • Pp. 90, 34G, 392, &c., 2nd edition. Cf. Buffon, Moral ArMniPiK : 
 ' Le sentiment n est en general qu'un raisonnement implicite moins clair 
 niais souvent plus fin et toujours plus sur que le produit direct de la raison.' 
 (He is proving our first postulate.)
 
 78 MATHEMATICAL PSYCHICS. 
 
 means to enjoy and to transmit (a). To lower classes 
 was assigned the work of which they seemed most 
 capable ; the work of the higher classes being different 
 in kind was not to be equated in severity.^ K we sup- 
 pose that capacity for pleasure is an attribute of skill 
 and talent (a) ; if we consider that production is an un- 
 sijmuietrical function of manual and scientific labour ()8) ; 
 we may see a reason deeper than Economics may afford 
 for tlie larger pay, though often more agreeable work, 
 of the aristocracy of skill and talent. The aristocracy 
 of sex is similarly grounded upon the supposed superior 
 capacity of the man for happiness, for the ivepy^lai of 
 action and contemplation ; upon the sentiment — 
 
 Woman is the lesser man, and her passions unto mine 
 Are as moonlight unto sunlight and as water onto wine. 
 
 Her supposed generally inferior capacity is supposed 
 to be compensated by a special capacity for particular 
 emotions, certain kinds of beauty and refinement. Agree- 
 ably to such finer sense of beauty the modem lady 
 has received a larger share of certain means, certain 
 luxuries and attentions (Def 2 ; a sub finem). But gal- 
 lantry, that ' mixed sentiment ^ which took its rise in the 
 ancient chivahy,' has many other elements. It is ex- 
 ])lained by the polite Hume as attention to the weak,^ 
 and by the passionate Eousseau <f>vcriKciiTep(o<;^ Now 
 attention to the weaker sex, and woman's right not only 
 to certain attentions in polite society but to some exemp- 
 tion from the harder work of life, are agreeable to the 
 utiUtarian theory : that the stronger should not only do 
 more work, but do so much more work as to suffer^ 
 more fatigue where fatigue must be suffered (y8). It 
 
 « Cf. Livy, ii., p. 32, ^. " Burke. » JSssat/, 14. 
 
 * Emile, iv. * See note, p. 66.
 
 PEOOP OF UTILITARIANISM. 79 
 
 may be objected : consideration should equally be due 
 from the stronger to the weaker members of the same 
 sex. But in the latter case there is wanting a natural 
 instinct predisposing to the duties of benevolence ; there 
 has been wanting also a fixed criterion of strength to 
 fix the associations of duty ; and, lastly, competition 
 has interfered, while competition between man and 
 woman has been much less open (and much less ob- 
 viously useful to the race). Altogether, account being 
 taken of existing, whether true or false, opinions about 
 the nature of woman, there appears a nice consilience 
 between the deductions from the utihtarian principle 
 and the disabihties and privileges which hedge round 
 modem womanhood. 
 
 Utilitarian also is the custom of family life, among 
 other reasons, in so far as (contrasted with communistic 
 education) it secures for the better-bom better educa- 
 tional influences (y) ; in particular a larger share of 
 good society in early life. The universal principle of 
 the struggle for life, as Mr. Barratt may suggest, conduces 
 to Utihtarian selection. This being borne in mind, there 
 appears a general correspondence between the popula- 
 tion-theory above deduced (yS) and the current ethics 
 of marriage, which impose ^ only a precedent condition, 
 success, hereditary or personal, in the struggle for hfe. 
 Concerning the classification of future society, common 
 sense anticipates no Utopia of equahty. Physical pri- 
 vations are pitied ; the existence of a subordinate -and 
 less fortunate class does not seem to accuse the bounty 
 of Providence.^ With the silence of common sense accords 
 the uncertain sound of exact Utihtarianism (ayS). 
 
 But, if egoist or intuitivist are not to be altogether 
 
 ^ In nepect to population. 
 
 " C£. Borke on the * labouring poor,' in Jtefficide P«icc, 3.
 
 80 MATHEMATICAL PSYCHICS. 
 
 converted by the deductive process of Mr. Sidgwick, at 
 least the deaUng with his exact definition may tend to 
 mark out and reclaim from the indefinite one large 
 common field of conduct, one of the virtues of the in- 
 tuitivist, one of the gratifications of the egoist — rational 
 benevolence. For can there be a rational wish to please 
 without a wiUingness to estimate the duration of the 
 pleasure, the susceptibility, as well as the number, of 
 the pleased ? 
 
 Exact Utilitarianism may also, as Mr. Barratt thinks 
 plausible, present the end of Politics ; of Pohtics as 
 based upon self-interest.^ A political ' contract ' for the 
 adjustment of conflicting interests should have two 
 qualities. It sliould be clear and fixed, universally 
 interpretable in the same sense. It should be such that 
 4,he naturally more powerful class, those who, though 
 fewer, outweigh the more numerous by strength, ability, 
 and capacity to co-operate, should not have reason to 
 think that they would fare better under some other 
 contract. Two contracts present these quaHties; the 
 rough and ready wocratical, the exact possibly aristo- 
 cratical, Utilitarianism. . The first contract excels in the 
 first quality ; the second in the second. 
 
 n. That the same reasonings should lead up to a 
 general 'principle and down again to its appUcations — 
 that the theory should be tolerably certain, the practice 
 indefinitely remote — is not more paradoxical than that 
 the demonstrator of the atom-theory should foresee the 
 remote possibility of its apphcation, no less a possibility 
 than to triumph over the second law of Thermodynamics.^ 
 The triumphs of Hedonics, if equally conceivable, are 
 equally remote ; but they do not so certainly become 
 
 ■ Compare the Corollary of the Eeonontic Calculus. 
 =* Clerk-Maxwell, Theory of Heat, p. 308.
 
 APPLICATIONS OF UTILITARIANISM 81 
 
 more conceivable when considered more remote ; for 
 what if in the course of evolution the subtlety of science 
 should never overtake the subtlety of feeling ! Faint 
 and vague and abstracting many things which ought 
 not to be abstracted, the Hedonical Calculus supplies 
 less a definite direction than a general bias, here brief!}' 
 and diffidently indicated. 
 
 The end of action being defined as above, the Jacobin 
 ideal ' All equal and rude,' J. S. Mill's ideal ' All equal 
 and cultivated,' are not necessarily desirable, not para- 
 mount ends to be sought by revolution or the more 
 tedious metliod of depopulation. Pending a scientific 
 hedonimetry, the principle ' Every man, and every 
 woman, to count for one,' should be very cautiously ap- 
 phed. In communistic association (if such should be) 
 the distribution of produce should be rather upon the 
 principle of Fourier than of Owen. Universal equal 
 sufirage is less likely to be approved than plural votes 
 conferred not only (as Mill thought) upon sagacity, 
 but also upon capacity for happiness. 
 
 The play of the struggle for life is to be encouraged, 
 in the present state of society, within limits, without 
 prejudice to the supremacy of the supreme principle. 
 Mr. Barratt indeed from the same premisses, the utility 
 of competition, infers a different conclusion : that Utih- 
 tarianism should resign in favour of Egoism. But surely 
 the inference is, not that the Utihtarian should change 
 his destination from Universal to Egoistic Hedonism 
 (points toto coelo apart, as the chart of Sidg%vick shows) ; 
 but that, while constant to his hfe's star, he should tach 
 (in the present state of storm at least) more considerablj^ 
 than the inexperienced voyager might advise. No one 
 can misunderstand this ' self-hmitation ' of Utilitarianism 
 — for it has been explained by Mr. Sidgwick ; least of all
 
 S2 MATHEMATICAL PSYCHICS. 
 
 the Egoist — ^for a similar delegation, without abdication, 
 of the supreme command is much more necessary in the 
 case of the supremacy of self-love (Butler, &c.). 
 
 Lastly, while we calculate the utiHty of pre-utihtarian 
 institutions, we are impressed with a view of Nature, 
 not, as in the picture left by Mill, all bad, but a first 
 approximation to the best. We are biassed to a more 
 conservative caution in reform. And we may have here 
 not only a direction, but a motive, to our end. For, as 
 Nature is judged more good, so more potent than the 
 great utihtarian has allowed ^ are the motives to mo- 
 rahty which religion finds in the attributes of God. 
 
 ' Mill, Esaayit on Nature and IMitfion.
 
 APPENDICES. 
 
 ON UNNUMERICAL MATHEMATICS. 
 
 It seemed undesirable to load our opening pages with a multi- 
 plicity of illustrations which, if the writer's views are correct, 
 would be superfluous to the mathematician, and, in any case, 
 might be uninteresting to the arfswixsrpTjTos. Indeed, the 
 nature of the subject is such that a single instance — by a 
 sort of ' mathematical induction,' as it has been called — a 
 single ' representative-particular ' authenticated instance of 
 mathematical reasoning without nmnerical data is sufficient to 
 establish the general principle. However, it may be well to 
 add a few words of exposition after first precising the point at 
 issue by citing on our side the father of Mathematical Econo- 
 mics, as the representative of the contrasted view the very able 
 author of a review (on Prof. Jevons' * Theory ') already re- 
 ferred to. 
 
 Coumot says : ' — ' L'une des fonctions les plus importantes 
 de I'analyse consiste precisement a assigner des relations deter- 
 mineesentre des quantites dont les valeurs numeriques, et meme 
 les formes algebriques, sont absolument inassignables. 
 
 ' D'une part, des fonctions inconnues peuvent cependant jouir 
 de proprietes ou de caracteres generaux qui sont connus, par 
 exemple, d'etre indefiniment croissantes ou decroissantes, ou 
 d'etre periodiques, ou de n'etre reelles qu'entre de certaines 
 limites. De semblables donnees quelque imparfaites qu'elles 
 paraissent, peuvent toutefois, en raison de leur generalite meme, 
 et a I'aide des signes propres a I'analyse, conduire a des relations 
 egalement generales, qu'on aurait difficilement decouvertes pans 
 
 ' Thforie des Richetats, p. 51. See also Preface, p. xiii.
 
 84 APPENDICES. 
 
 ce secours. C'est ainsi que, sans comiaitTe la loi de decroisse- 
 ment des forces capillaires, et en partant du seul principe que 
 ces forces sont insen§ibles a des distances seusibles, les geo- 
 metres out demontre le^ lois generales des phenom^nes de la 
 capillarite, lois confirmees par I'observation.' 
 
 The * Saturday Eeview ' (Nov, 11, 1871):— . . . * We can 
 tell that one pleasure is greater than another ; but that does not 
 help us. To apply the mathematical methods, pleasure must 
 be in some way capable of numerical expression ; we must be 
 able to say, for example, that the pleasure of eating a beef- 
 steak is to the pleasure of drinking a glass of beer as five to four. 
 The words convey no particular meaning to us ; and Mr. Jevons, 
 instead of helping us, seems to shirk the question. We must 
 n'mind him that, in order to fit a subject for mathematical in- 
 quiry, it is not sufficient to represent some of the quantities 
 concerned by letters. If we say that G represents the confi- 
 dence of Liberals in Mr. Gladstone, and D the confidence of 
 Conservatives in Mr. Disraeli, and y the number of those 
 parties ; and infer that Mr. Gladstone's tenure of office depends 
 
 ,. - 1. d G jcZD t. 1 
 
 upon some e(|uation involvmg - and -^ -, we have merely 
 
 wrapped up a plain statement in a mysterious collection of 
 letters.' The reader is referred to th« whole article as typical 
 of the literary method of treating our subject. Thus, again, 
 'the equations . . . ., assuming them to be legitimate, seem 
 to us to be simply useless so long as the functions are obviously 
 indeterminable. They are merely a roundabout way of express- 
 ing what may be better said in words.' And, again, ' he wraps 
 up liis mysterious conclusions in symbols which are mere ver- 
 biiige, as they contain functions which neither are nor can be 
 determined.' 
 
 Compare Mill : — * Such principles (mathematical) are mani- 
 festly inapplicable where the causes on which any class of phe- 
 nomena depend are so imperfectly accessible to our observation, 
 that we cannot ascertain by a proper induction their numerical 
 laws.' ' 
 
 Compare also the spirit of his remarks * upon algebra and 
 its exclusive ' adaptation to the subjects for which it is cpm- 
 
 / Logic, book iii. chap. xxiv. p. 9. '* Book iv. chap. vi. p. 6.
 
 U^'^TMER1CAL MATHEMATICS. 85 
 
 monly employed, namely, those of which the investigations 
 have been already reduced to the ascertainment of a relation 
 between numbers.' Compare also the ^^ews of Comte to which 
 he refers. 
 
 A single instance — that already cited in the text — seems 
 sufficient to oppose to this popular impression about the limits 
 of mathematics. Thomson and Tait, in their ' Treatise on 
 Natural Philosophy,' p. 320, discuss the problem of a ball set in 
 motion through a mass of incompressible fluid extending infi- 
 nitely in all directions on one side of an infinit-e plane, and 
 originally at rest. After constructing the Lagrangian equations 
 from (what may be called in reference to ntmierical measure- 
 ments) a priori considerations, they go on : ' principles suffi- 
 cient for a practical solution of the problem of determining P 
 and Q will be given later. In the meantime, it is obvious that 
 each decreases as x increases. Hence the equations of motion 
 show ' several deductions which are truly ' most remarkable and 
 very suggestive,' e.g. (in an analogous problem), that two balls 
 properly projected in a perfect incompressible liquid will seem 
 to attract one another. It is suggested, I think, that a certain 
 hypothesis as to the ultimate constitution of matter corresponds 
 with the observed phenomena of attraction. 
 
 Now here is the type of mathematical psychics. The ' prac- 
 tical solution of the problem of determining P and Q,' func- 
 tions denoting quantities of pleasure in terms of external ob- 
 jects (means, &c.), is not yet given. But certain properties of 
 such functions are given. Thus, if P be a person's pleasure 
 considered as a function of x his means, it is obvious (compare 
 the premises of Thomson and Tait's reasoning) that P increases 
 
 d P 
 
 as X increases, but at a decreasing rate ; whence con- 
 
 dx 
 
 ■ . d T 
 
 tinually positive, ?■ continually negative. And from such 
 
 data, mathematical reasonings show several interesting results. 
 It has ' been suggested that a certain hypothesis as to the ulti- 
 mate principle and supreme standard of morals corresponds (to 
 an extent not usually noticed) with the observed phenomena of 
 human action. 
 
 ' Above, p. 4.
 
 86 APPENDICES. 
 
 One can imagine how facetious the ' Saturday Reviewer ' 
 might be in criticising the method employed by Thomson and 
 Tait in the above example, namely, mathematical deduction 
 without numerical measurement. As we are not able to say 
 that P is to Q as 5 to 4, the argument 'conveys no particular 
 
 meaning to us.' In employing -^ -^, ' we have merely 
 
 wrapped up a plain statement in a mysterious collection of let- 
 ters.' Doubtless, I reply, what we know of P and Q might have 
 been stated unmathematically in a roundabout literary fashion ; 
 but that statement, as compared with Thomson's, would not be 
 a plain statement^ nor appropriate nor serviceable. For this 
 same symbol-speech, so harsh and crabbed as compared with 
 literary elegance, is gifted with a magical charm to win coy 
 truth ; the brief and broken language which the love of abstract 
 truth inspires, no doubt foolishness to those who have no 
 sympathy with that passion.' 
 
 What need to multiply illustrations of what is self-evident 
 that mathematics, of which the very genius is generalisation, 
 without liipping into particulars, soors from generality to gene- 
 rality ! I shall attempt, however, to illustrate a little more 
 fully the method of mathematical physics, hoping that the 
 professed mathematician would pardon in an amateur particular 
 errors, ' JSi modo plura mihi bona sunt,' if only the general 
 view is correct. 
 
 (,)n the theory of sound we .obtain an expression for an 
 atmospheric wave involving two (almost) indejpe ndcnt arhitrat^ 
 functious,^ j> (n 6 at — x) + yp- {ndat + x). Without sajjpo- 
 sing the forms of <f) and yjr to be kno'Wii, we may deduce sub- 
 
 ' Here may be the place to notice the Saturday Revinrs criticism upon 
 I'rnfestior Jovous's formulie for tlie ' law of indifference : ' that his symbols 
 needlessly complicate the plain and simple facts of the market. But the 
 most potent instruments of research are open to similar criticism. The so- 
 called 'equation of continuity' may no doubt appear to literary common 
 sense a very artiticial and complicated statement of some such simple fact, 
 as that matter carmot enter or leave a given space without crossing its 
 boundary. But how fruitful of deductions is this formula in connection 
 with other symbolic statements, needs not to tell to any one, even moderately 
 acquainted with the kinetics of fluids. . 
 
 * Airy on Sound, pp. 23, 28.
 
 UNNUMERICAL MATHEMATICS. 87 
 
 stantial conclusions ; as that, when a tube is stopped at both 
 ends, the forward and backward waves are of identical form.' I 
 would not, however, insist too much on this particular instance, 
 and the very large class of siuular physical problems, as in 
 all respects typical of psychical reasoning. For no doubt it 
 may be said that the data from which the expression for wave- 
 distvirbance was deduced, the differential equation express- 
 ing the motion of a particle of air ' -^— = k -A_j that this 
 
 cut CLQu^ 
 
 premiss is of the nature of numerical precision ; k is made up of 
 factors supposed at least approximatively measurable ; whereas 
 (some of) the data of psychics consist of loose general relations, 
 the fact of increase or decrease, positive or negative, possessing 
 not even that degree of grossly approxiTnative accuracy,^ beyond 
 which even Professor Jevons in his illustrations of mathematical 
 reasoning does not appear to extend his view. At the same 
 time, if we consider as premiss the integral equation for the 
 disturbance, then the method of psychics is faiily well exem- 
 plified by the employment in the theory of sound and elsewhere 
 of arbitrary functions; a conception, one might suppose, 
 which had never been entertained by those who object to 
 mathematics' inability to deal with the complexities of social 
 science ; as if any degree of complexity might not be attributed 
 to an arbitrary function. 
 
 But it would exceed the ability and requirements of the 
 present writer to justify the method above postulated (deduc- 
 tion from loose and numerically indefinite relations) by a 
 general review of the uses of arbitrary functions ; it will suffice 
 to show the validity of the method in two provinces of mathe- 
 matics least distant from the sphere of psychics — I,, the theory 
 of natural forces and energy ; and II., the calculus of varia- 
 tions. 
 
 I. The hypothesis of natural forces assumes, directly or by 
 implication, as a first or proximate prindple, that the attraction 
 or repulsion between two particles is some fuTiction of the 
 distance between them. From this loose indefinite relation, 
 without knowledge of the form of the function, the most im- 
 
 ' Airy, p. 78. = Id. p. 21. ' Prinei^ of ScUncr.
 
 88 APPE^'DICES. 
 
 portant coacluaions may be deduced. As a very simple example 
 take the motion of a particle round a centre of force. Without 
 knowing the form of the force-function, we deduce that equal 
 areas are swept out by the particle in equal times, that the 
 motion is one plane, that the velocity is inversely proportional 
 to perpendicular from centre upon tangent, and so forth. 
 
 No doubt it may be objected that while there is something 
 indefinite and loose in the premisses, the hypothesis of naturzd 
 forces, there is also something definite and precise, for instance, 
 the very conception of uniform acceleration. But firstly, the 
 hypothesis in question would generally be admitted to hold of 
 the systems of matter immediately concomitant with mental 
 phenomena, so that the deductions therefrom may well be of 
 great psychophysical interest (especially in view of the analogies 
 to be suggested between energy and pleasure). And again, it 
 is not to be supposed that the data of social science have 
 nothing 'precise. While there is something in them indefinite 
 and loose, there is also something definite and precise ; for in- 
 stance,' the * law of indifference,' that there is only one price in 
 a market, a proposition which possesses that degree of at least 
 approximative precision, which is generally, and supposed to be 
 universally, characteristic of applied mathematics. And statis- 
 tical data, as Professor Jevons has pointed out, admit of tht 
 same sort of precision. In fine, the objection applies at most 
 to our dynamical illustrations, not to those which will be pre- 
 sented by pure analysis, by the calculus of variations. 
 
 The great theories relating to energy present abundantly 
 mathematical reasoning about loose indefinite relations. Con- 
 servation of energy is implicated with such a relation, the 
 mutual attraction of particles according to some function of 
 the distance between them. The principle of conservation of 
 energy affords instances of what is vulgarly supposed a contra- 
 diction in terms, of reasoning at once mathematically and 
 TrayvXfofy obtaining by mathematical deduction a general id/f 
 a state of motion. Suppose a swarm of particles so moving under 
 natural forces that they are now aU clustered near each other 
 now all fly asunder to a distance, then from the principle of till, 
 conservation of energy we obtain the genei'al idea that thfe 
 
 ' As aforesud, p. 5.
 
 UNNUMERICAL MATHEMATICS. 89 
 
 movements of the particles are on an average more rapid, or 
 more correctl)' their kinetic energy is greater, when they swarm 
 together than when they are widely dispersed. 
 
 Peculiarly typical' of psychics are the great principles of 
 maximum and minimum energy. That a system tends to its 
 least potential energy, this principle affords us m innumerable 
 instances a general idea of the system's position of rest ; as in 
 the very simple case of equilibrium being stable when the 
 centre of gravity is as low as possible. Thus, without knowing 
 the precise shape of a body, we may obtain a general idea of 
 its position of equilibrium. 
 
 From the principle of least action we infer that a particle 
 under any (natural) forces constrained to move on an equi- 
 potential surface will so move that its path from point to point 
 is of maximum or minimum length. Without knowing the 
 precise law of the forces, the precise shape of the potential 
 surface, we may thus obtain a general idea of the motion. 
 
 The great Bert rand-Thompson maximum-minimum prin- 
 
 ' The coiBpanson between pleasure and enerjn" may be viewed under 
 two aspects; first (than which not more b a&serted here), as not known to 
 be more than a metaphor, vet elegant and convenient, like the hypothesis of 
 tldds in electricity, or the ' now abandonsd but still interesting ' (Thomson 
 it Tait) corpuscular theory of light ; secondly, as in the text (pp. 9-15J a deep 
 and real analog}-, the maxvnum of pleasure in peychics being the eliect or 
 concomitant of a maaim-um physical energy. 
 
 The comparison assists us to conceive what appears to some inconceivable, 
 that pq^oility is not a necessary condition of greatest happiness. Energv- is 
 the product of mass and the square of velocity. Therefore the importance 
 of any part of a system, with respect to the total energy, depends not only 
 oil its mass, but on its velocity. In the system, consisting of discharged 
 rifle and shot bullets, there lives more energj- in the little whiffling bullet 
 than the heavy recalcitrant rifle. And, indeed, the smaller the bullet, the 
 greater ceteris panl>uA its energy. So, in the social system, we must accus- 
 tom ourselves to believe that the importance in respect to the utilitarian 
 greatest possible quantity of each class is not necessarily iti proportion to 
 iw numbei-a. More energy of pleasure, more tvepye'icu in the Oracular lan- 
 guage of ArL«totle, may exist in one poet than many boors ; in Athens than 
 the rest of Hellas, in Hellas than Barbaria; in a century of the age of 
 Phidias, than a thousand years of the declining Roman Empire. 
 
 No doubt this property is implicit in the definition of integral pleasure 
 HH defined, for instance, in tie third Appendix. But the conception of an tn- 
 tegral b not, perhaps, so familiar to the unmathematical as not to desiderate 
 illustration. 
 c
 
 90 APPENDICES. 
 
 ciples and their statical analogues present abundant instances 
 of raatheToatical reasoning about loose, indefinite relations. 
 We know, in each case, that the energy of a system to which 
 impulses (or finite forces) have been applied is the maximum 
 or minimum consistent with certain data. Without knowing 
 the data precisely, we may obtain certain general ideas of the 
 arrangement of energy in the system under consideration. 
 Thus, if the masses of any part or parts of a material system 
 are diminished, the connections and configuration being un- 
 altered, the resulting kinetic energy under given (however 
 complex and nndefined) impulses from rest must be increased.* 
 If the stiffness in any part or parts of the system be diminished, 
 the connection remaining unchanged, the potential energy of 
 deformation due to given force applied firom without will be 
 increased.* Diminution in the premisses, increase in the 
 conclusion, loose, indefinite relations ! So again, I think, if 
 certain velocities be imparted by impulses to the bounding 
 surface of an incompressible liquid, we may obtain, without 
 having more than a general idea of the distribution of these 
 given velocities, a general idea of the resulting motion, by 
 reasoning, from the Thomsonian principle, that the motion of 
 the liquid is un-rotatory, that the motion of each particle is 
 perpendicular to a certain velocity-potential surface passing 
 through it, one of the series of such surfaces being the 
 bounding surface, &c. Compare with the last two paragraphs 
 the reasonings in moral science. By first principles the 
 arrangement (of social institutions, &c.) productive of maxi- 
 mum pl«3asure holds. Without deducing precisely what this 
 best arrangement is, we may obtain mathematically a general 
 idea of it as that one arrangement is better than another. 
 
 Upon analogous principles in statical electricity, we know 
 that, if there be a given distribution of electricity over the 
 conductors in a field, the strains throughout the dielectric are 
 such that the potential energy of the whole system is a mini- 
 mum.' We may not know the precise form of the functions 
 which express the distribution of electricity over the conduc- 
 tors; much less, if we had these data, would we be able to 
 
 ' WRtaon & Burbury, Gcnernliiied Co-ordinates. 
 
 • Ibid. 3 Clerk-MaxweU, Electricity, Arts. 08, 00.
 
 UNNUMERICAL MATHEMATICS. 91 
 
 calculate the potential, the function whose respective dififer- 
 entials shall give the strain in each direction at any point.' 
 Yet it is something both tangible and promising to know 
 mathematically that the potential energy is a minimum. That 
 something is the type of what mathematical psychics have to 
 teach. Analogous remarks are applicable to the somewhat 
 analogous theorem of ^ minimum energy of electric currents ; 
 in a higher dimension, as I think it may be said, and of the 
 nature of what may be called moriientuTn-jpotential rather than 
 force-potential. 
 
 II. It is the first principle of the calculus of variations that 
 a varying quantity attains a maximum when the first term of 
 variation vanishes, while the second term is negative (mutatis 
 mutandis, for a minimum). The latter condition is one of 
 those loose, iTidefinite relations which we have been aU along 
 describing. In the simple cases which in the infancy of Mathe- 
 matical Psychics are alone presented in these pages,^ we 
 know by observation not what the second term is, but that 
 it is continually negative. In more complicated cases the re- 
 sources of mathematics are exhausted in calculating, not a 
 definite nnmerical, but a loose, indefinite relation, the. sign 
 of the second term. The reader should consider Jacobi's 
 method of discrimination, as stated, for instance, by Mr. 
 Todhunter ; * and Mr. Todhunter's application of the same 
 to a particular problem,* and realise how a mathematical 
 reasoning may turn upon the loose, indefiinite relations of 
 positive or negative, convex or concave. Consider also the many 
 of Mr. Todhunter's ' Miscellaneous Observations ' directed 
 to the same relation. All through the calculus of varia^- 
 tions the relation is of paramount importance, constituting, 
 indeed, all the difference between a maximum and mini- 
 mum. You find continually, in the statement of a problem, 
 
 ' Compare Mill's or rather Comte's double objection against Mathematics 
 in Social Science : that the premisses are unattainable, and the reasoning- 
 impossible. — Logic, book iv. ch. 24, p. 9. 
 
 2 Clerk- Marwell, Art. 283. 
 
 ' See above, pp. 61-65. 
 
 * Researches in the Calculus of Variations, pp. 21-26. 
 
 * Ibid, pp. 2fr-30.
 
 92 APPENDICES. 
 
 the condition that a required curve shall be, or shall not be, 
 convex ; ' so rough and unshaped are the materials with which 
 mathematics is able to build. Now this very relation of con- 
 cavity, not a whit, more indefinite in psychics than in physics, 
 constitutes a main pillar of utilitarian calculus ; quarried 
 from such data as the law of decreasing utility, of increasing 
 fatigue, of diminished returns to capital and labour ; for the 
 exact statement and proof of which the reader is referred to 
 the economical writings of Professor Jevons and Principal 
 ^Marshall. 
 
 It may be said that the former condition of a maximum 
 mentioned lately, the equation of the first term of varia- 
 tion to zero, is of a definite precise rather than a loose indefinite 
 character. But, again, it is to be repeated that all the data of 
 mathematical psychics are not indefinite, but only (as in the case 
 of physics) some. Accordingly, from this equation to zero, 
 combined with an irulefinite. datum^ the increase of one quan- 
 tity with another, of capacity for happiness with evolution, we 
 niay deduce another indefinite quantitative relation, namely, in- 
 crease,^ or diminution of share oimeaTia in utilitarian distribution. 
 
 There are two other leading principles of the calculus of 
 variations which seem calculated to illustrate the method of 
 psychics. First, a consideration of first principles C prior, it may 
 be observed, to any particular measurements or determination 
 of the forms of functions), shows that if the ' Haupt Gleichung,' 
 as Stranch calls it, the leading — in general differential — eqiia- 
 tion, which must be satisfied in order that the first term of 
 variation should vanish, breaks up into factors, there are, or 
 rather may be,^ several solutions, several different functions, 
 each corresponding to a maximum or minimum. (In the 
 simple cases alone presented in these pages, or rather in the 
 companion paper, in which the expression whose maximum is 
 sought does not involve any differential co-efficients, say 
 
 TT = I F (y x) d X between limits, where y is an independent 
 
 variable function ; then, if -=— breaks up into factors, there 
 
 dy 
 
 ' Researches in the Calculus of Vanatums, pp. 80, 117, 286. 
 ' Above, p. 68. ^ Todhunter's Researches, p. 262.
 
 UNNUMERICAL MATHEMATICS. 93 
 
 will in general, I think, be multiple solutions.") A curve 
 between two given points required to fulfil some maximum 
 condition may be discontinuous, may be made up of the different 
 solutions, one step according to one law, and the next step 
 according to another law.' But the different laws or function, 
 though they may thus be employed successively, are not to be 
 mixed and compounded. Any one portion of the required curve 
 mast (in general and subject to the exceptions of the following 
 paragraph), obey soTne one of the laws supplied by the solution 
 of the Haujpt Gleichung. It is submitted that this property 
 has its counterpart in human affairs ; the fact that there are 
 sometimes two best ways of attaining an end — if the superlative 
 best may be employed in a technical sense analogous to the 
 superlative maximuvi. To realise the best, one or other 
 course must be adopted, not a confusion of the two. 
 
 The subject of discontinuity leads up to another general 
 remark. It is not universally necessary that the first term of 
 variation should vanish. It suffices for a maximum that the 
 first term of variation should be known to be negative (and 
 obversely for a minimum). Such knowledge is generally the 
 result of imposed conditions ; as in Mr. Todbunter's problems 
 that a curve must not pass outside a given boundary, must not 
 exclude a given point, must be convex. It is submitted that 
 such complicating imposed conditions have some analogy with 
 the conditions imposed by necessity upon practical politics and 
 applied utilitarianism. For ^p6v7)<7ts has often to be con- 
 tent not with the best course, but the best subject to existing 
 conditions. Compare the subtle spirit of Mr. Todbunter's 
 calculus of variations with the subtle, and as the * plain man ' 
 might almost suppose, sophistical spirit of Mr, Sidgwick's 
 method of utilitarianism, when it comes to be applied to the 
 actual world in -which we live. The abstract maximum, in 
 psychics as well as in physics, is comparatively simple ; but 
 the concrete is complicated by imposed conditions; and the 
 complexion of a wise benevolence, in view of each established 
 constitution, custom, church, is affected with a congenital re- 
 semblance to the wily charms of the calculus of variations. 
 
 ' Todhunter, ^ffMtMi,
 
 94 APPENDICES. 
 
 n. 
 
 ON THE IMPORTANCE OF HEDONICAL CALCULUS. 
 
 It may be objected that mathematical psychics, though 
 possible, are not valuable ; I say valuable rather than, what 
 might be understood in a too restricted sense, useful. For 
 no philosophical objector would maintain that the love of the 
 soul for the universal is then only legitimate, when it has been 
 blessed with the production of the useful. 
 
 The love of the soul for the universal is undoubtedly capable 
 of extravagance, as in the devotion of Plato to the idea. ' Amor 
 ipse ordinate amandus est.' But the limits are to be traced by 
 a loving hand, and not to be narrowed by a too severe construc- 
 tion of utility. The great generalisations of mathematics have 
 perhaps been pursued and won less for the sake of utility to be 
 produced, than for their own charm. Certainly the superior 
 genius who reduced the general dynamical problem to the 
 discovery of a single action-function was as much affected by 
 the ideal beauty of 'one central idea,' ' as by the practical con- 
 sequences of his discovery. In the example first cited from 
 Thomson and Tait, it might have happened that the generalised 
 (.-o-ordinates employed did not yield that 'first vindemiation ' of 
 truth above described (p. 85). Yet the Lagrangian conception 
 <»f considering the energy of the whole system as a function of 
 the position and velocities of the immersed bodies would still" 
 have been legitimate, and great, and promising. The Gossenian, 
 the Jevonian thought of referring economics to pleasure as the 
 central ide^ might be equally splendid, though unfruitful. 
 And so Mr. G. H. Darwin, in his review of Professor Jevons's 
 ' Political Economy,'^ appears, not without reason, to prefer the 
 mathematical method on theoretical, abstracted from practical, 
 grounds. 
 
 Professor Cairnes * himself admits that the mathematical 
 method might be useful, though not indispensable. If so, the 
 
 • Sir William R. Hamilton, Philosonhiccd Trimsactions, 1834, 1836. 
 
 * Fortnightly Review, 1876. 
 
 ^ Preface to Logical Method.
 
 IMPORTANCE OF HEDONICS. 95 
 
 position of the mathematical method in economics might be 
 compared, perhaps, to that of quaternions, which calculus, even 
 if it conduct to no theorem not otherwise deducible, yet, in the 
 opinion of some ' competent judges, deduces theorems already 
 known more elegantly and, as it may be said, naturally and 
 philosophically, than the blind and elephantine formulae usually 
 employed for the purpose. At any rate, is it for one who is 
 not conversant with both methods to oflfer an opinion on their 
 relative value ; to declare forbidden, without having himself 
 trodden, the sublimer path ? 
 
 But is the method unfruitful in social science ? The black 
 list in our appendix may show the possibility that mathema- 
 tical ' reason is here no guide, but still a guard.' But I go 
 further, aud challenge the a7fa)/i.fT^77Toy to answer the following 
 exajnination paper. 
 
 Social Problems to be solved without Mathematics. 
 
 1. A communistic sr>ciety owns land of varj-ing degrees of 
 fertility, which land it cultivates so as to obtain with a given 
 quantity of labour the maximum of produce. Suppose the 
 quantity of labour at the disposal of the community to be 
 suddenly increased, how will the new labour be distributed ? 
 Will more or less additional labour be employed on any acre 
 according as it is more or less fertile, or otherwise ? 
 
 2. WTien Fanny Kemble \-isited her husband's slave planta- 
 tions, she found that the same (equal) tasks were imposed on 
 the men and women, the women accordingly, in consequence of 
 their weakness, suffering much more fatigue. Supposing the 
 husband to insist on a certain quantity of work being done, and 
 to leave the distribution of the burden to the philanthropist, 
 what would be the most beneficent arrangement — that the men 
 should have the ssune fatigue^^ or not only ynore task, but viore 
 fatigue ? 
 
 3. Commodities being divided into two species, those whose 
 expenses of production (do not diminish or) increase as the 
 
 * Cf. Tait, Edinburgh Philosophical Transactions, 1825. 
 
 * Cf. Mill's Theory of equal sacrifice in taxation.
 
 96 APPENDICES. 
 
 amount increases and those whose cost of production diminishes 
 with the amount produced ; show that it is abstractedly expe- 
 dient to tax one of these species rather than the other, and 
 even to tax one so as to bounty the other (Marshall's 
 theorem). 
 
 4. Commodities being divided into two species^ according as 
 a slight decrease of price is, or is not, attended with a consider- 
 able increase of demand, which species is it abstractedly pre- 
 ferable to tax ? ' 
 
 5. The labour market, from an indefinite number of masters 
 and men competing on each side, is transformed by trades- 
 unions and combinations of masters into a small number of 
 competing (corporate) units on each side. Can this transform- 
 ation be advantageous to both sides ? 
 
 6. It has been said that the diMnbution of net produce 
 between cooperators (labourers and capitalists associated) is 
 arbitrary and indeterraiyiate. Discuss this question, 
 
 7. Mr. Sidgwick in the * Methods of Ethics ' (iv. chap, i.), 
 having defined the utilitarian end as the greatest possible sum 
 of pleasures, proceeds to observe that with a view to this end 
 pgiial distribution of happiness, though not necessarily of the 
 means of happiness, is desirable. Assuming what the author's 
 not« seems to imply (cf. ' Methods of Ethics,' p. 256, 2nd 
 edition), that individuals have their happiness differently re- 
 lated io means, derive different amounts of happiness from the 
 f^ame means ; show that to attain the end defined happiness 
 and its means must be either both equally or both unequally 
 distributed. 
 
 There are those no doubt who see nothing in all this, tum- 
 'n\<r away contemptuously from such questions, as the dog when 
 you try to put him on a scent which nature or •discipline has 
 made to him insignificant. The professed mathematician, it 
 must be owned with regret, is not unlikely to be in this num- 
 ber. But the professed mathematician, however infallible a 
 guide upon the purely mathematical side (and sure to find many 
 errors in these pages should they be so fortunate as to come 
 
 ' See Notes on Exchange Value, by II. Cunynghaiue, p. fl.
 
 IMPORTANCE OF HEDONICS. 97 
 
 under his notice) is not necessarily an infallible guide over the 
 untrodden pass here supposed to exist between the heights of 
 physics and psychics, supposing that his attention has not been 
 directed to psychological problems. Nevertheless, great is the 
 authority of the masters of the supreme science. 
 
 The authority of the mere metaphysician need give us 
 much less pause. The noble Hegelian, from the transcendental 
 heights whence he looks down upon Newton, might smile at 
 the attempt to estimate qxiantitatively pleasure. A notable 
 authority forsooth, this demolisher of Newton, upon the science 
 of quantity and its limits ; and notable authorities and judges 
 of authority are those his followers, whose chosen philosopher 
 and guide is not only blind to truth in her clearest manifesta- 
 tion, but also, what is even more un philosophical, is ignorant 
 of his ignorance and vain of his inanity. Nonragionam di lor. 
 As the 01}'mpian Zeus, defied by Here and Athena, addresses 
 his. rebuke not to the inveterately obstinate one, but only to the 
 rebellious goddess of wisdom — 
 
 al(\ yap ot ttiidtv fVLKXav om vorjarj' 
 
 so a serious argument is addressed not to the incorrigible 
 mystic. 
 
 Common sense is addressed and may be persuaded, it is 
 hoped, to forego its prejudices against this sort of calculus. 
 Tht-re is the old prejudice still reviving, however often slain, 
 against the reign of law in psychology, as incompatible with the 
 higher feelings. But it is too late. The reign of law is estab- 
 lished, and will not become more oppressive to feeling by be- 
 coming mathematical. And again, common sense, catching 
 sight of such terms as hedonism, is apt to dismiss the 
 whole affair as rnetaphysical. But, it is to be insisted, the 
 materials with which exact social science is concerned are no 
 metaphysical shadows, but the very substance of modem civili- 
 sation, destined, doubtless, ere long to become embodied in 
 practical politics and morals. Quantity of labour, quantity of 
 pleasure, equality of sacrifice and enjoyment, greatest average 
 happiness, these are no dreams of German metaphysics, but the 
 leading thoughts of leading Englishmen and corner-stone con-
 
 98 APPENDICES. 
 
 ceptions, upon which rest whole systems of Adam Smith, of 
 Jeremy Bentham, of John Mill, and of Henry Sidgwick. 
 
 Are they not all quantitative conceptions, best treated by 
 means of the science of quantity ? 
 
 m. 
 
 ON HEDONIMETRY. 
 
 It has been shown that some of the data of physics are as in- 
 definite as some of the data of psychics. And yet it may 
 be admitted that there is a potentiality of precision about 
 even the looser physical demonstrations which gives them a 
 certain prestige. In physics, when we deal with an indefinite 
 P and Q (to revert to an earlier example), there is some 
 \mderstanding that ' principles sufficient for a practical solution 
 of the problem of determining P and Q will be given later.' 
 Whereas in psychics we are so far from expecting, that it seems 
 doubtful whether we can even conceive precise measurement. 
 Yet the conceivability at least may be thought necessary to 
 mathematical reasoning. We must then carefully consider this 
 possibility, or, what is much the same thing, the existence and 
 nature of a unit of -pleasure. 
 
 There is, no doubt, much difficulty here, and the risen 
 science is still obscured by clouds ; and hedonism may still be 
 in the state of heat or electricity before they became exact 
 sciences, as described by Professor Jevons.' Let us, however, 
 following in his footsteps, endeavour to gain as clear a view as 
 may be. At least it is hoped that we may sight an argumen- 
 tuni ad /io)ni?iC>?i, an argument to the man who (with Professor 
 Jevons), admitting mathematical reasoning about.self-regarding 
 pleasures, denies the possibility of mathematically comparing 
 ditferent persons' pleasures. Let us accordingly, with reference 
 to this question of fiETptjrtKrj and pleasure-unit consider sepa- 
 rately the quantitative estimate which a man can form (I.) of 
 his own pleasure, (U.) of other people's. 
 
 * Theory of Political Economy, p. 9.
 
 HEDOXIMETRY. 99 
 
 I. *Utilitj/ says Professor Jevons (writing exclusively of 
 the first sort of measurement), * may be treated as a quantity 
 of two dirroensiona.' ^ Now, when it is Eisked, * I n virt iip of- 
 wha t unit is one intensity said to be grea ter than another?'^ 
 the answer must b e, I think, * Just perceivable increments o f 
 p leasure are eqiiatabie^ ' which may be shown, perhaps, by that 
 sort of internal experience and handling of ideas which seems 
 to be the method of attaining m^athematical axioms.^ Vnr if 
 poss ible let on e jnit ]i i i i i i i ll r in r TTi"Tit b" yr-nfay^a,^ i^r. 
 another . Then it must be preferred in virtue of some differ- 
 ence of pleasurability (non-hedonistic action not existing, or 
 not being pertinent to the present inquiry). But., if one of the 
 i ncrements exceeds the other ^i pIPQcnraViilif-y , fV>pn th^f one 
 i fi not a. ju.9t p f ^rr.p.i rn. hlp ^r\p.rpf n f^ J ]\ .^ bnt r,nnf;isit,fi of at leastA wo 
 oii^K ^•p).|-pTr.oT^fo Of course such a way of turning the subject 
 has no pretence to cZt-duction. The stream of thought ' mean- 
 ders level with its fount.' Turn the matter as we please, there 
 must, I think, be postulated some such equation as the above, 
 which may be compared, perhaps, to the first principle of 
 probabihties,' according to which cases about which we are 
 equally undecided, between which we perceive no material dif- 
 ference, count as equal ; a principle on which we are agreed to 
 act, but for which it might be hard to give a reason. 
 
 It must be confessed that we are here leaving the terra 
 fimia of physical analogy. It may plausibly be objected, the 
 just perceivable increment, the minimum sensibile, is not treated 
 as a unit in the cases with which physics deal. Let us suppose 
 that for the same objective increase of temperature or weight 
 (as estimated by the approved methods of physics) I have at 
 different times, or with different organs of my body, different 
 subjective estimates. In one sense, certainly more usual, the 
 quantities are the same. In another sense, the minima sensi- 
 bilia being equated, wftat is felt is. And this latter sense, it 
 is contended, not without hesitation, is appropriat-e to our sub- 
 ject. 
 
 ~Thn iprrnmrntn in r[unrti^n nrp/T thinlr^ tin lir rinvrd as 
 
 * Tlieory of Political Economy, p. 51. 
 
 ^ Cf. Plain on Axioms. 
 
 ' Laplace, Euai P/u'loftophique mr let Probahilitis, 5th edit., p. 7.
 
 A 
 
 100 
 
 APPENDICES. 
 
 1 
 
 finite difFerences, ra ther thap_ a§,gmmiae- diffcrcntial s (a concep- 
 ti^g ^hich iia«j; ;goF militAt^ witli the enipkj maent of ihp. Tint.a - 
 tion of the diff erential calculus).^ The conception might be 
 illustrated by that oflTTorce just sufficient to turn a balance 
 overcoming friction. Why, however, each inclination of the 
 will is treated as equal by the rational intelligence, of this, as 
 already intimated, no proof is to be expected. 
 
 Indeed, the equation, or equatability, in question exists not 
 s o jnuch in fact as in the limit of perfect evj ilution. .Jifi-ioa- — 
 perfect intellig ence does not trea t_a _nnit nf pl r mTivr in the 
 fn?5x elas pqtKtl~To one in the prese nt. 
 
 bstracting from the 
 imnpi-f-.^inty of thp. fiitnr e. the mere circim3stan^ IjQi_iaturity 
 aff eote the e^timato of a - plea sure ; which depreciation the 
 Jevonian factor q ^ denotes, as I understand. Now it is only 
 in theideaJLli mit that - ^ Hfecomes equal to unityr ' 
 
 50 far about the dimension of intensity. As to the dimen- 
 sion of time a similar line of remark is open. The same ob- 
 jective (say horological) t ime may correspond to different ra tes 
 o f thought and J egling nt different p^rif7f|°j n,° Locke intimates.^ 
 
 It is COnceiva ^^^ th^lt ^^" rf-^f^o ^ pff gpnt inpr tn r^n-nanin ^Ka- 
 
 sh ould differ in this ratp n^ A^-^ And perhaps some states, 
 intellectual exercise in particular, which philosophers have dis- 
 tinguished as more good, t hough not i Tr"^'''^ ^leasurablea than 
 others, may so differ. I n dreams, the rate seems high, t jhj^ 
 in tensity _Jow. A nd r q j^ plpagiTrr Trnul^ hfiY" yi^t '^j]]j jh yu 
 d imension s, as Professor Jevons says, but three dimensions, 
 namely, objective time, subjp.ctivp timp, and intensitj r. 
 
 And yet the correction may not seem very important, for 
 probably it is more competent to consciousness to combine into 
 a single mark the two considerations of rate and intensity. 
 Suppose one state presents about three pleasufe-increments, ' 
 another about two, above zero, that the rate of the former is 
 double that of the latter, their objective duration being the 
 
 ' See the remarks of Clerk- Maxwell, * Essay on Atoms/ Encyclopcsdia 
 Britnnnica, p. 38. 
 
 - Theory of Political Economy, p. 78. 
 
 ' Compare As You Like it. Act iii. sc. 2, and elsewhere. Cf. Mr. Sullj^s 
 remarks on Illusions of Perspective.
 
 HEDOJJIMETRY. 101 
 
 same, is it better to give two marks to each state, say three and 
 two to the former, two and one to the latter, and then to mul- 
 tiply the marks of each ; or by a sort of imconscious multiplica- 
 tion to mark at once six and two — about ; for tiif^ (^.^mpnricnn 
 of ni l i i iii ii r 1 n''i f" ■| ii n ii n'tj if] hrrf Tiflmittrd t^ W vngn^ ; not^ 
 vagher perhaps tha n the comparisons made bv an examiner -a s 
 to excellence, where nnmenp^l mArks are usefnlly prnployt^H. 
 
 To precise the ideas, let there be granted to the sc ience of 
 pic ture what is grant^^ ^^ ^bf^ gfienrp ff PTifrgy •; ' to imagine 
 an ideally perfect instrument, a psychophysical machine, con- 
 tinually registering the height of pleasure experienced by an 
 individual, exactly according to the verdict of consciousness, or 
 rather diverging therefrom according to a law of errors. From 
 moment to moment the hedonimeter varies; the deUcate 
 index now flickering with the flutter of the passions, now 
 steadied by intellectual activity, low sunk whole hours in the 
 neighbourhood of zero, or momentarily spjinging up towards 
 infinity. The continually indicated height is registered by 
 photographic or other frictionless apparatus upon a uniformly 
 moving vertical plane. Then the quantity of happiness between 
 two epochs is represented by the area contained between the 
 zero-Hne, perpendiculars thereto -at the points corresponding to 
 the epochs, and the curve traced by the index ; or, if the cor- 
 rection suggested in the last paragraph be admitted, another 
 dimension will be required for the representation. The in- 
 tegration must be extended from the present to the infiaitely 
 future time to constitute the end of pure egoism. 
 
 II. Now it is here contended that there are as many, and 
 the same sort of difficulties, in this estimate of pleasures by 
 the sentient himself (which is yet admitted by Professor 
 Jevons, and substantially by common sense), as in the estimate 
 of other people's pleasures. We have only to modify our axiom 
 thus : Apy lust perceivable pleasure-increment e jperienoed by 
 a-ny sentjent at any tfrnt hrvn thn nmrrr^-nlnr The same primal 
 mystery of an ultimate axiom hangs, no doubt, over this utili- 
 tarian, as over the egoistic, first principle. 
 
 The equation is only true in the limit of perfect evolution. 
 
 The variation of s ubjective time fo r flifypr<^nf iTtH^ yi'Hiialg^ 
 
 ^ Se« Clerk-Maxwell, TTwory of Heat, p. 139.
 
 102 APPENDICES. 
 
 p resents no gprftat er difficulty than the variation for one in- 
 dividual. ■ 
 
 T^e integration may be equally well illustrated by ideal 
 mechanism. We have only to add another dimension express- 
 ing the number of sentients, and to integrate through all 
 time and over all sentience, to constitute the end of pure 
 utilitarianism. 
 
 It raa;jbe obje cted that the just perceivable in^ rfrn'^n^''? 
 given by consciousness in the case of one's own pleasures, only 
 i nfun ' cd in the uiac - ^ -others.' It may be replied, greater 
 uncertainty of hedonimetry in the case of others' pleasures n;ay 
 be compensated by the greater number of measurements, a 
 wider average ; just as, according to the theory of probabilities, 
 g Tipatf.r accuracy may be attained by rnore numerous -o bserv^a- 
 tiaBa_ jyith a lesg^ ^ie iibot inst r u jBaent. Th e proposition. * the 
 
 fjnr plf^snrp/ is proved hy taking a wide a^^^igg^gjjb^^ tb^^ ^J 
 ihf^ self-obsefvatJTiST'nSowever accurate, of—t t oingl ^y perhaps 
 -exceptionalfindividual. 
 
 IV. 
 
 OiV MIXED MODES OF UTILITARIANISM. 
 
 The distinction between egoism and utiUtarianism has been 
 drawn with matchless skill by Mr. ■ Sidgwick. But it has not 
 been observed that between these two extremes, between the 
 frozen pole of egoism and the tropical expanse of utilitarianism, 
 there has been granted to imperfectly-evolved mortals an inter- 
 mediate temperate region ; the position of one for whom in a 
 calm moment his neighbour's happiness as compared with his 
 own neither counts for nothing, nor yet * counts for one,' but 
 counts for a fraction. We must modify the utilitarian integral 
 as defined above (Appendix III.) by multiplying each pleasure, 
 except the pleasures of the agent himself, by a fraction — a 
 
 ' This is a distinction insisted on hy Mr. Herbert Spencer, in his remarks 
 on utilitarianism. — Data of Ethics, p. 57.
 
 MIXED UTILITAEIANISM. 103 
 
 factor doubtless diminishing with what may be called the social 
 distance between the individual agent and those of whose 
 pleasures he takes account. 
 
 There is not much more diflSculty about this intermediate 
 conception than about the extremes. The chief diflRcultj is 
 one which is common to the extremes, presented by the phe- 
 nomena which ]\Ir. Sidgwick describes as the self-limitation of 
 a method. For example, in a hfe ordered according to the 
 method of pure utilitarianism there may be tracts of egoistic 
 action, times when the agent gives fall swing to self-int-erest, 
 leaving out of sight his utilitarian creed. The test whether 
 such an agent is really a pure utilitarian would be, I suppose, 
 whether on having his attention directed to the alternative 
 between methods, having collected himself, in a cool moment, 
 he would or would not calmly and deliberately sacrifice his ovm 
 greatest happiness to that of others. It seems superfluous to 
 labour a point which has been explained by JNIr. Sidgwick. 
 
 Yet that there is some difficulty about this rhythm between 
 sovereign and subordinate method may be inferred from the 
 expressions of able thinkers. Thus, Mr. Spencer appears to 
 employ ^ as an argument against utilitarianism the utilities of 
 self-indulgence. ' For his wife he has smiles, and jocose 
 speeches,' and so forth — the self-indulgent non-utihtarian. 
 But, if self-indulgence and the not taking account of the 
 general good has such an agreeable effect, the intelligent 
 utilitarian will cultivate a temporary relaxation and forgetful- 
 ness of his supreme principle. It never was meant that he 
 should wrap himself up in his utilitarian virtue so as to become 
 a wet ^ blanket to his friends. It never was meant, as Austin 
 says, that the sound utilitarian should have an eye to the 
 general good while kissing his wife. In order that one's life 
 should be subordinated to the general good, it is not necessary 
 that the general good should be always present to conscious- 
 ness. If I have an hour to prove a theorem at an examination, 
 I shall do well not to keep the quod est demonstrandum 
 continually before the mind, but to let the mind range among 
 theorems which may serve as premisses. If a man has a day to 
 •write an article, though the whole time may be consecrated to 
 
 > Data of Ethic*, chap. xi. » See Mr. Spencer's gloomy picture.
 
 104 APPENDICES. 
 
 the purpose, it may be expedient to banish the purpose during 
 refreshment or exercise. You cannot disprove the authority of 
 utilitarianism by proving the utility of egoistical, or any other, 
 practice. 
 
 To argue, then, that the utilities described by IVIr. Spencer 
 could not be grafted upon pure utilitarianism would imply a 
 different conception of a * method of ethics ' from that vrhich 
 may be derived from Mr. Sidgwick's great work. That as a 
 matter of fact the utilities of egoistic action do not now spring 
 from a root of pure utilitarianism would be freely here admitted ; 
 agreeing with the view suggested that the concrete nineteenth 
 century man is for the most part an impure egoist, a mixed 
 utilitarian. 
 
 And the reconciliation between egoisTn and altruism^ 
 gradual process and ideal limit beautifully described by Mr. 
 Spencer, would be upon the view suggested here, the transfor- 
 mation of mixed into pvu"e utilitarianism, the psychical side of 
 a physical change in what may be dimly discerned as a sort of 
 hedonico '-magnetic field. 
 
 V. 
 
 ox PROFESSOR JEVONS'S FORMULA OF EXCHANGE. 
 
 Professor Jevons's formula, nli — ^^^^ = -,- is almost id en- 
 
 , F' (x v^ V 
 
 tical with our ^') '•< -( = ^, Almost ; for the notation here 
 
 employed is slightly more general. The utility is regarded as 
 a fimction of the two variables, not the sum of two functions of 
 each. The inquiry suggested at p. 34, near -foot, could not 
 have been suggested by Professor Jevons's formula. Our for- 
 mula also is adapted to take account of the labour of produc- 
 tioUy the 'compliciited double adjustment' glanced at by Pro- 
 fessor Jevons.' 
 
 Let X manufacture the article which he exchanges for y. 
 
 ' Above, p. 14. * Theory, p. 108. » Theonj, p. 203.
 
 FORMULAE OF EXCHAJfGE. 105 
 
 Then (by a violent but not dangerous abstraction) his utility 
 may be written 
 
 V^F{f{e)-x,y)^J {e) 
 
 where e is the objective measure of labour {e.g. time of work) ; 
 
 / (e) is the subjective measure of work, the toilsomeness of 
 
 fatigue ; / (c) is the produce, corresponding to e. Now, as e is 
 
 not an article of contract, it appears that (-,- ) the partial 
 
 differential with regard to e must always be equated to zero. 
 Hence, by eliminating e we come on oiu- old form F (x y)^ or 
 F ( — x, y\ as it is convenient here to write. 
 
 This ' complicated double adjustment ' maybe illustrated by 
 a brief reference to that interesting phenomenon pointed out 
 by both Mr. Marshall and ^Mr. Walras, unMable eqiuUhriura of 
 trade. From the point of \-iew here adopted the utility of a 
 dealer in x may be written P = F ( — rr, y). Transformed to 
 polar co-ordinates Ps=F(--p cos 0, p sin 6) ; when tan 6 expresses 
 
 the rate of exchange. The demand-curve is [-7-]—^' ^"or 
 
 this locus erprepses the utmost amount of dealing to which the 
 dealer wiJJ consent at any given rate of exchange, the amount 
 for which bis utility is a maximum at that rate. But the locus 
 also expresses positions for which the utility is a minimum at 
 any given rate. And this part of the locus is not in a genuine 
 demand-curve. Each point represents a position not which the 
 dealer will not consent to change, but which he would by all 
 means wish to change. 
 
 By a general property of analysis the maximum and minimum 
 points are arranged alternately along any vector. This property 
 is closely connected with the property of alternately stable and 
 unstable equilibriuTn of trade. There are, however, I think, 
 
 unstable positions where [ — — ) = does not correspond to a 
 
 miTiimum, e.g. Mr. Marshall's figure 8. 
 
 But the most important sort of instability is perhaps that 
 which may be presented in the case of (]Mr. Marshall's) Class 
 II ; of which, as I take it, the definition connects two properties
 
 106 
 
 APPENDICES. 
 
 (1) diminution of value in exchange upon increase of exports, 
 with (2) diminution in the expense of production upon increase 
 of wares produced for exportation. It is interesting to see from 
 olur individualistic point of view how these two properties are 
 connected. The analytic condition of the first property is 
 
 ( Ll_ ] = + . For this condition must hold from the point P., 
 
 where the property in question sets in (see figure) to the point 
 
 Fig. 3. 
 
 Pj, w^here the property ceases. At each of these points -^ = 0. 
 
 The analytic condition of the second property of Mr. Marshall's 
 
 definition (the first in the order of his statement) is - %\ - -= + J 
 
 where (as before) e is the objective measure ^ of labour, /(e) is 
 the amount of product corresponding to e. 
 
 It may be shown, then, that [ -^ j can only be positive 
 when -j^^ is positive. For, agreeably to previous notation, put 
 
 Or 6 
 
 ' Other than that which the produce itself presents ; e.g., length of time 
 during which a uniform muscular energy is put forth by a workman.
 
 FORMULA OF EXCHANGE. 107 
 
 P=r i^ (J(^e)—p COS 6, p sin 6)— J (e). Then we have always the 
 
 condition ( -= — ) = 0, and we have to find -^ subject to this 
 \deJ Idp'^J '' 
 
 condition. Now, as 6 is throughout treated as constant, 
 
 whereas e is considered as a variable, dependent on p, it will be 
 
 d P 
 
 convenient to denote the object of ovir inquiry as -/— without 
 
 d p^ 
 
 brackets, denoting by brackets differentiation, which is partial 
 
 with respect to p, does not take account of e's variation. With 
 
 tHi. noeauon, since (|Z) = 0,g = (g) . . (A^ J g 
 
 where -— is to be found from the equation to zero of 
 dp 
 
 \d eJ \deJ~ T ~ dp" \dpdeJ 
 
 de 
 
 fd,F^ dj 
 
 Therefore iZ= f^Jl) - 2 f ^Y 
 
 dp^ \dpy -- \dpdeJ 
 
 ■P 
 
 m-'-d 
 
 de" 
 
 Now we may be certain this expression can only be positive 
 when -J-'-^ is positive, ij we are certain of the laws of sentience 
 
 which were postulated on a previous ' page. For, writing a for 
 /(?) (the a employed in Professor Jevons's equation of exchange), 
 and y for p sin 6^ we have 
 
 fd^F\ d.F ^.f,^^ d^F . a n , d,F . ^ 
 j% = j^, eos^ + 2 3-^— sm ^ cos ^ + -^ sm 0. 
 \dp^J da^ dady dy^ 
 
 (dZ\^d^Tdjn\dF dJ 
 \de^J da^ IdeJ da de^ 
 
 where it does not seem necessary to bracket the differentials on 
 the right-hand side. Substituting these values in the expression 
 
 ' Page 34.
 
 108 APPEKDICES. 
 
 d P 
 
 for -J— we see that that expression is certainly negative upon 
 
 these conditions : 
 
 d P d W 
 (1) -7*-, -^2 (both) continually not positive. 
 
 (2) 
 
 d^F 
 dady 
 dF 
 
 (3) — continually not negative. 
 
 (4) 
 
 de^ 
 
 d,f 
 
 (5) -J.J not positive. 
 
 ie 
 
 The first condition is secured by Professor Jevons's law of 
 diminishing utility, our first postulate (see p. 61). 
 
 The second condition is an interesting variety of the same ; 
 that the rate of increase of utility derived from one sort of 
 wealth diminishes with the increase of other sorts of wealth. 
 
 The third condition imports that utility at least does not 
 decrease with increase of wealth ; which in a civilised country 
 may be allowed. 
 
 The fourth condition is Professor Jevons's law of increasing 
 toilsomeness of labour,* our second axiom (see p. 65). 
 
 d P 
 
 If then these laws of sentience hold, — "^ can onl}' be posi- 
 
 d p'^ 
 
 d f 
 tive when -^-^ is positive. It is submitted that this subordina- 
 d e. 
 
 tion — in however abstract and typical a form — of the more 
 complicated phenomena of the market to the simple laws of 
 sentience is not without interest. 
 
 But to return to Professor Jevons : the formulae here em- 
 ployed, along with a general, and perhaps it ought to be added 
 a filial, resemblance to his, present two points of contrast 
 which deserve especial attention : (1) Graphical illustration 
 has been more largely employed here. Now in some sense pure 
 Analysis may appear to be the mother-tongue of Hedonics ; 
 which soaring above space and number deals with quantities of 
 
 ' Theonj, p. 185.
 
 POR^nJL.E OF EXCHANGE. 109 
 
 pleasure, employing the Calculus of Variations, the most sub- 
 lime branch of analysis,' as Comte, Caiaphas-like, called the 
 branch most applicable to Sociology. But on the other hand 
 the differential equations which occur in the theory of exchange 
 are of such a peculiar character that it is rather difiBcult, as 
 may presently appear, to handle them without geometrical 
 apparatus. In this respect at least Mr. Marshall's preference 
 for geometrical reasoning would seem to be justified.'^ 
 
 (2) It has been prominently put forward in these pages 
 that the Jevonian ' Law of Indifference ' has place only where 
 there is competition, and, indeed, perfect competition. Why, 
 indeed, should an isolated couple exchange every portion of 
 their respective commodities at the same rate of ezchanfre? 
 Or what meaning can be attached to sUch a law in their case ? 
 The dealing of an isolated couple would be regulated not by 
 the theory of exchange (stated p. 3l), but by the theory of 
 simple contract (stated p. 29). 
 
 This consideration has not been brought so prominently 
 forward in Professor Jevons"s theory of exchange, but it does 
 not seem to be lost sight of. His couple of dealers are, I take 
 it, a sort of typical couple, clothed with the property of ' In^ 
 difference,' whose origin in an 'open market' is so lucidly 
 described ; ^ not naked abstractions like the isolated couples 
 imagined by ;. De Quincey or Courcelle-Seneuil in some solitary 
 region. Each is in Berkleian phrase a * representative parti- 
 cular;' an individual dealer only is presented, but there is 
 presupposed a class of competitors in the background. This 
 might safely be left to the intelligence of the reader in the 
 general case of exchange. But in deahng with exceptional 
 cases (pp. 132, 134), a reference to first principles and the pre^ 
 supposition of competition would have introduced greater pre- 
 cision, and suggested the distinction submitted in these pages 
 "(pp. 19, &c.), namely, that exchange is indeterminate, if eitJter 
 (l)one of the trading bodies {qua individual or qua union) 
 or (2) the commodity supplied by one of the dealers, be iiidi- 
 visible or not 'perfectly divisible. 
 
 The whole subject of the mathematical theory of exchange 
 
 ' riiUosophie Positive, I^efon 8. "^ Foreign Trade, p. 19. 
 
 ' Theory, pp. 08, 90.
 
 110 APPENDICES. 
 
 would be put in a clearer light by considering the objections 
 which have been broxoght against Professor Jevons's theory by an 
 able critic in the 'Saturday Review' (Nov. 11, 1871). The 
 Reviewer says : ' When Mr. Jevons proceeds to apply this equa- 
 tion to the solution of his problem, he appears to us to fall into 
 a palpable blunder. Translated into plain English, the equation 
 
 ^ = .-y^ means, as we see, simply that, however much com 
 X . dx 
 
 A gives to B, he will receive a proportionate quantity of beef 
 
 in exchange. If he doubles the amount of com, that is, he 
 
 will receive twice as much beef. But the other quantities are 
 
 obtained on the contrary supposition, namely, that the rate of 
 
 exchange will vary according to some complex law, determinable, 
 
 if we could tell precisely what effect will be produced on the 
 
 mind of the parties to the bargain, by the possession of varying 
 
 quantities of beef and com. In fact x is now a function of y, 
 
 as might easily be foreseen from Mr. Jevons's statement of the 
 
 case, in quite a different sense from what it was before. The 
 
 substitution, therefore, of - for — ^ is a mistake.' 
 
 X dx 
 
 I submit (1) the following is a significant problem. Given 
 
 two differential equations F, { xy —^ ) = 0, Fj ( a;^ -j^ ) = 0, find 
 
 V d xJ \ d xJ 
 
 X and y two quantities such, that if each differential equation 
 
 be solved, and thereby 2/ "for each be found as a function of a;, 
 
 and thence for each— y-^ be derived as a function of x ; then, if 
 dx 
 
 X be substituted in both (functional) values of y^ and both 
 (functional) values of -,^, (a) the two (quantitative) values of 
 y are equal to each other equal to y, and (6) the two (quantita- 
 tive) values of -^ ^ are equal to other. 
 
 (2) The following is a solution of this problem. Eliminate 
 
 ^ V- between the equations Fi (x y^^ = 0, F, (x y -^) ^ ; 
 dx \ dxJ \ dx/ 
 
 the resulting equation in x and y is the locus of the required 
 
 point. 
 
 (3) The problem and solution correspond to Professor 
 Jeyons's problem and solution.
 
 FORifULiE OP EXCHANGE. Ill 
 
 Let US take these propositions in order. 
 
 (1) This proposition by its extreme bumblediness illustrates 
 what was above said about the advantages of graphical illustra- 
 tion. For the geometrical equivalent is simply : Eequired a 
 point at which two curves each given by a differential equation 
 (of the first order) meet and touch. Or even more briefly : 
 Find the locus of contact between members of two families. 
 
 The conception thus introduced is not only legitimate, but 
 femiliarly employed in the Calculus of Variations, in those pro- 
 blems where we have multiple solution subject to the condition 
 i\at there shall be no abrupt change of direction. The reader 
 will find any number in Mr. Todhunter's ' Kesearches.' 
 
 I am not concerned to show that Mr. Todhunter's problems 
 are exactly parallel to ours. They could not well be so involv- 
 ing second, where they involve first, differentials. But it is easy 
 to construct an exactly parallel problem with curves presented 
 by maximum analysis, the source of our economical ciu-\'es. 
 Take the straight line and the cycloid, the shortest line and line 
 
 Fig. i. 
 
 of quickest descent. A cycloid is generated by a circle of given 
 diameter rolling on a given horizontal line, the starting-point 
 of the circle — that is where the generating point M is on the 
 horizontal line — being arbitrary. Find (the locus of) a point P 
 on the cycloid such that if a particle starting from rest slide
 
 112 APPENDICES. 
 
 down the cycloid from the horizontal line as far as P, and there 
 fly oflf at a tangent, it will pass through a given point 0. 
 
 (2) The solution above offered is easily verified. Having 
 
 eliminated -,- between Fj and Fj, take any point x y on the 
 
 eliminant, and draw through it a curve of each family. Then 
 
 Fj (x ypi) = 0; where p^ is the value of -^ for the first curve 
 
 when X is substituted for x. Since the point is on the elimi- 
 nant FjC^ yiP\) = ^' ^^^^ ^2 (f y Vi)—^' Therefore pi=_p2- 
 Q.E.D. ' ' 
 
 In the particular case just put let the differential equation 
 
 of the cycloid' be _i= a /- " , and the differential equa- 
 
 tion of the line _^ = ^(.~^ where p and g* are the co-ordinates of 
 dx x — q 
 
 the given point. Then the required locus is 
 
 V 
 
 y ^-q' 
 
 a curve of the third degree passing through the given point, 
 as it evidently ought, if it can ; for the given point may be too 
 far from the horizontal line to be reached by generating circle 
 or generated cycloid. In this last case the point is still the 
 scene of contact between a cycloid and line, only the cycloid is 
 imaginary. The mathematician is prepared for such freaks of 
 analysis } the economist should be prepared for somewhat simi- 
 lar freaks ^ on the part of his similarly obtained ' demand- 
 curve.' 
 
 To avoid misconstruction it may be as well to add that this 
 
 solution by elimination of ^ would Tioi have been admissible if 
 •^ dx 
 
 ' See Todhunter's Diferential Calculus, p. 342. 
 
 * Thus the origin, though nn intersection of the demand-cui-ves, is not in 
 any sense a position of equilibrium ; not e\en being on the contract-curve. 
 An-ain the altei-nate intersections of the demand-curves are (as Messrs. Mar- 
 shall & Walras have shown) positions of trade-equilibrium only in name. 
 And we have seen that similar caution is required in handling the analytical 
 expresaioo of the contract-curve (p. 26).
 
 FORMULAE OF EXCHANGE. 113 
 
 there had been otJiefi' differentials besides those of the first 
 order. Elimination would in this case have resulted in that 
 sort of mongrel differential equation, ' Mixtumque genus pro- 
 lemque biformem,' which the Reviewer may be supposed to 
 have had dimly in view. 
 
 (3) An attentive consideration of Prof. Jevons's problem will 
 show that it is a case of the problem here proposed, whether in 
 the language of pure analysis or of geometry. I take the latter 
 for brevity and to illustrate its convenience. Taking for origin 
 the point at which the deahng begins where x and y are zero, 
 Ave see (a) by the law of indifference * that each dealer must 
 move along a straight line given by the differential equation 
 
 ■^ = - (the Reviewer sees this much). Again under the head- 
 ing * Theory of Exchange ' ^ we may learn (6) that the 
 -r^ which expresses the dealer's change of position is at the 
 
 CJLjC 
 
 point of equilibHum= ^^ ^ ~ ^ ' But by (a) the ^ which ex- 
 
 ti (2/ ^^ ^ dx 
 
 presses the dealer's change of position is continually^ - 
 Therefore by the principles just now laid down the locus of the re- 
 quired point is found by eliminating -=^ between (a) and (6) ; 
 
 whence ?i-l_^ ^== " which is none other than our old friend the 
 
 -^1 iy) ^ 
 
 * demand-curve.' 
 
 We may recognise another old friend in the equation 
 
 -■,-^=?' ^ , , ^ considered as an ordinary differential equation. 
 dx ^,(2/) •" ^ 
 
 It is the differential equation of our * curves of indifference.'' 
 
 The problem under consideration may be expressed : Find the 
 
 locus of the point where lines from the origin touch curves of 
 
 indifference. If (as before supposed) the ciirves of indifference 
 
 consist of a series of circles round d. point C, then the locus of 
 
 the point of contact to any curve of a tangent from is the 
 
 locus of vertices of right-angled triangles described on OC ; that 
 
 is, a semicircle described in OC, a result which of course might 
 
 ' Theory, p. 98, et neq. ^ J>. 103, fqo.
 
 114 
 
 APPENDICES. 
 
 be obtained analytically according to the method here described. 
 Transforming to the point of bisection of OC, and putting c= 
 I OC, the equation of any indiflference-curve is (3/— c)'+a;*=r*. 
 
 "Whence the differential equation of the family - -^ = — 
 
 dx y — c. 
 
 And the differential equation of a straight line from is 
 
 Eliminating -j^ upon the principle here de- 
 fended, we have x^ + y"^ = c^ the equation of a circle whose 
 
 dx 
 
 X 
 
 Fig. 5. 
 
 "^./ 
 
 
 --^^TV" ^' 
 
 -^. 
 
 
 
 / / / \ 
 
 / / ^'f 
 
 / '' i 
 
 
 \ 
 
 \ 
 
 diameter is OC. Q E D. The determination of a point by the 
 intersection of the locus thus obtained, with another locus 
 similarly obtained, presents no diflBculty. The conjoint deter- 
 minate problem may, as we have already seen, be thus ex- 
 pressed. Draw from the origin a straight line, whiph at the same 
 point touches two curves of indifference. As we have seen, the 
 problem of determinate exchange may be turned in a great 
 variety of other ways. Turn it as you will, the essential cor- 
 rectness of the formula under consideration emerges clearer! 
 
 Meraea profundo ; pulchrior evenit. 
 Luctere ; multa proruet integrum 
 Cum laude Tictorem.
 
 FORMULAE OF EXCUANGE. 115 
 
 The remaining objections of the Saturday Reviewer 
 against this formula are based upon the interpretation already 
 shown to be erroneous that the formula is applied to solitary 
 couples, such as those which political economists delight to 
 place in lonely islands. It happens, indeed, that the Reviewer 
 is not enabled by his literary method to deduce correct conclu- 
 sions from these premisses of his own assumption.' But we are 
 here concerned not with his fallacious reasoning from assumed 
 premisses, but with his undue assumption of premisses or igno- 
 rantia elenchi. "We are only concerned to show that his ob- 
 jection does not apply to a typical couple in a market. 
 
 He puts the case of A and B, dealing respectively in corn 
 
 and beef, and supposes that at a certain rate 5 of com to 1 of 
 
 beef A would exchange 20 of com against 4 of beef and no 
 
 more. Kow, in so far as this objection might apply to the 
 
 typical formula which we have been building — I do not say 
 
 that the Reviewer aimed at this structm-e, but I am concerned 
 
 to show that he does not hit it — it might import that a typical 
 
 dealer would refuse to deal if the price of his article were to be 
 
 raised, would not consent to such a rise of price, which surely 
 
 requires no refutation. In symbols, P being the utility of 
 
 dP 
 dealer in x, and tan 9 the rate of exchange, -=-^ is continually 
 
 + ; it being understood, of course, that movement is along the 
 demand-curve of P ; for, as we are here concerned with typical 
 indi^aduals in a market, there is no talk of movement other 
 than along demand-curves, and the case put shows that the 
 position of the index is on P's demand-curve, say at the point 
 q (on the last figure). 
 
 Well, then, subject to "this condition, namely i—^ — j= 0, 
 
 P increases continually with 6. For 
 
 dV fdV\^fdV\dp fdV\ dF.^dF . 
 
 d-e = ^-dl) ^ K-djm^ {dw)= d^''''^''d^'''^ 
 
 [P being here supposed = F (a — p cos 0, p sin 0) ], which is 
 
 ' An attentive consideration of his Lypothesis will show that he sup- 
 poses that there can he a settlement not <m the confrart-cnrve ; which is 
 imtenahle.
 
 116 APPENDICES- 
 
 continually + , unless it can be supposed that wealth can so 
 increase as to become a disutility. Q. E. D. 
 
 But, it may be said, and not without plausibility, of course 
 A would be willing enough to maie the change you describe, 
 but B, though by hypothesis he is willing to make changes in 
 some direction, is not willing to make a change in that direc- 
 tion. And, true enough, a mere B, unclothed with the proper- 
 ties of a market, might well be unwilling to make that change. 
 Referring to the same figure, let us suppose that B's curves of 
 indifference are circles with C at centre. Then we see that 
 for all points above Q where a curve of indifference of B touches 
 the demnand-curve of A, it will not be for the interest of the 
 individual B to move up the demand-curve of A. But the 
 tjipical competitive Vrpresentative B cannot help himself. The 
 force which moves him is not his maximum utility barely, but 
 subject to competition ; the best that he can get in short. And 
 this play of the market, as fully explained here and by other 
 writers, leads to the formulae ' which have been so often retiurned 
 to our inquiry. 
 
 VI. 
 
 Oy THE ERRORS OF THE aytu,fUTpjiroi. 
 
 * EcQUiD tu magnum reprendes Homerum,' ' Egregio in corpore 
 njpvos,' and whatever adage is applicable to carping smallness, 
 might occur at sight of the undermentioned names, if the 
 critic did not hasten to disclaim any disrespect for these great 
 mimes, and to explain that the argument of this work, to 
 
 ' If, however, the competition between the Bs is not pet feci, it may hap- 
 pen that they cannot force each other up to T, the intersection of the demand 
 cuTTCd ; but that the system will reach a final settlement at some interme- 
 diate point q (as intimated at p. 48), supposing that the s\-8temis coTisti-ain-ed 
 to move along the demand-curve of A (our old X) ; for in the absence h 
 of tbis imposed condition it would run down to a final settlement on the con- 
 tract-curve, not necessarily nor even probably T, the point where the demand 
 curve intersects the contract-curve (in this case a straight line), CO'.
 
 CRITICISM OF BENTHAM. 117 
 
 vindicate the mathematical method in Social Science, could 
 only, or would best, be completed by showing that the pro- 
 foundest thinkers would have thought more clearly upon 
 Social Science if they had availed themselves of the aid of 
 Mathematics. 
 
 And, if after all it appears to the reader that the list of the 
 accused and that the accusation are not of very formidable 
 length, he will please to consider — with reference, at least, to 
 the two first and the two last of the reviewed authors — both 
 who they are who are here suspected to have erred, and t<;^ai 
 the subject of their error. If these have erred from want of 
 mathematical aid, what shall we expect from the unaided reason 
 of others ? And, if there is obscurity about the conception of 
 the ends of action, must there not be error and confusion about 
 the means — about all the middle axioms of morality ? * If the 
 light that is in thee be darkness, how great is that darkness.' 
 
 Bentham. 
 
 That the great' Beutham should have adopted as the creed 
 of his life and watchword of his party an expression which is 
 meant to be quantitatively precise, and yet when scientitically 
 analysed may appear almost unmeaning, is significant of the 
 importance to be attached to the science of quantity. ' Greatest 
 happiness of the greatest number ' — is this more intelligible 
 th?ji ' greatest illumination with the greatest number of lamps ' ? 
 Suppose a greater illumination attainable with a smaller 
 number of lamps (supplied with more material), does the 
 
 ' I am aware that Bentham is said by Bowring {Deontology, p. 328) to 
 Lave corrected this phrase in later life. It was not, however, corrected in 
 his latest ■works {Constitutional Code, chs. ii. vii.). And at any rate, as our 
 contention is not for victory but for the sake of instruction, ov ntpl rplnoBot 
 'AXXa ircpl ^vx^js, it may be useful to note the errors of genius, even if they 
 were at length self-corrected. 
 
 If after the preceding, and in view of a subsequent (p. 130), admission, 
 the criticism in the text appears hypercritical, let it be applied only to such 
 of Bentham's followers as may have been led by Bentham's incautious use 
 of the phrase {e.p. Fallacies of Confusion, ch. iii. f. 2) into exaggerating the 
 democratic or isocratic tendencies impUcit in Utilitarianism ; to Bentham's 
 predecessors also, Priestley, and Beccaria, with his ' La massjma felicita 
 divisa uel maggior numero.'
 
 118 APPENDICES. 
 
 criterion in this case give a certain sound? Nor can it be 
 contended that variation of number could not have been con- 
 templated in Benthatn's day. For, supposing the number of 
 distributees fixed, and as before a fixed distribuend, might not 
 the sum-total of happiness be greatest when the greatest part of 
 tlie sum-total, or at any rate larger portions, were held by a 
 few ? \Miich perhaps the. aristocratic party, if they would ex- 
 press themselves precisely, might contend. 
 
 The principle of greatest happiness may have gained its 
 popularity, but it lost its meaning, by the addition * of the 
 greatest numk^r-^ ' 
 
 J. S. Mill. 
 
 Nor is Mill any clearer about the definition of the Utili- 
 tarian End ; indeed, darkens the subject (as many critics seem 
 to have felt), by imposing the condition of equality of distribu- 
 tion. Suppose that * equality of sacrifice,' which he lays down 
 as the principle of taxation, should not correspond to ' least 
 possible sum-total of sacrifice,' what then ? 
 
 In the Political Economy of Mill occur some fallacies of the 
 species under notice, on which it is unnecessary to dwell, since 
 they have been more than abundantly exposed by Professors 
 Jevons and Walras.^ It might be possible, indeed, to maintain 
 that these critics have been unnecessarily severe, and that the 
 ione of Mr. Marshall improving upon Mill by the aid of ]\Iathe- 
 matics is more proper.' Thus Mill's definition of Value appears 
 to be the same, though not always, perhaps, so well expressed, 
 as that of Professor Jevons. And again, it might be possible 
 for jVIill to have a saving knowledge of the mysteries of Supply 
 and Demand, even though he may have acknowledged, not two 
 equations, but one equation."* For it is possible mathematically 
 to subsume several equations in one condition. Thus the 
 e(iuation to zero of Virtual Velocities includes in the general 
 
 ' See this point examined in New and Old Methods of Etkicg, by the 
 present writer. 
 
 ' Thewy of Political Economy, 2nd edition ; Element,s dEconomie Poli- 
 tique. 
 
 * Theory of Pure Trade, ch. i, pp. 4, 12. 
 
 * Theory of Political Evonomy.
 
 CRITICISM OF MILL. 119 
 
 case of a free rigid body six, and may include any number of 
 equations. And thus we have seen reason to suppose that all 
 the equations of Political Economy, however numerous, may be 
 subsumed under one.' And, to come nearer the mark, we have 
 seen above that the conditions of trade-equilibrium are not 
 necessarily stated in a bilateral and symmetrical form, but may 
 be subsimied in a single solitary condition, the equation of 
 Dernand to Supply ; presupposed and understood — what, in 
 fact, economists only too readily'* presuppose and take for 
 granted— -two sets of conditions, which might be described as 
 (1) the /act, (2) the uniformity of price.' 
 
 But it is none of our part, Agamemnon-like, ' through the 
 camp to go and rob an ally,' rather than * despoil a foe.' * 
 
 If an author will use unmathematical language about mathe- 
 matical subjects, he must expect a doubtful interpretation and 
 fame. 
 
 Pkofessor CaIKvES. 
 
 Professor Catrnes's substantial contributions to the matter of 
 Political Economy might surely have been enhanced • by being 
 framed in a more miathematical form. 
 
 It will be found very difficult to seize the connotation of the 
 phrase * increase in the aggregate amount of values.' ' The 
 denotation, the two instances immediately preceding, does not 
 appear to afford any significant common attribute to constitute 
 a definition. 
 
 The amazing^ blindness of this author in view of the 
 mathematical theory of exchange, his inability to contemplate 
 scientifically the psychical mechanism underlying the pheno- 
 menon ' of exchange, must vitiate, one should think, what he 
 
 ' Mr. WaJras has discerned the all-comprehensive character of the piin- 
 ciple of Maximum {Elhnents, Le9on 15) ; though he has not ventured, as 
 far as I am aware, to identify Hedonical with Physical Maximum. 
 
 ^ If our reasouings are right. See Index mb voce Price. 
 
 ^ Above, p. 42. 
 
 •• Pope, Iliad, i. 
 
 * Leading Principles, p. 5. 
 
 « See the only too lenient criticism of Mr. Geo. Darwin in FortniyhtJy 
 JRevietc, 1875. 
 
 ' Ibid. p. 15.
 
 120 APPENDICES. 
 
 has to tell us of * demand ' in ponderous phrase, or of ' supply, 
 as the desire for general purchasing power.' . . . This is a 
 subject as to which he who despises the science of quantity is 
 not likely, as Plato would say, to be himself evapud^os. 
 
 No doubt he occasionally detects a vulnerable point in IVIill 
 (p. 116) which had already been more clearly exhibited by Pro- 
 fessor Jevons. Still I venture to think that the contentions of 
 Professor Caimes about the definition of Supply and Demand 
 are much more a dispute about words than could be evident to 
 one who had no grasp of the forces determining a market. Let 
 the facts, with sufficient accuracy for the present purpose, be 
 summed up in Professor Jevons's symbolic statement, ^ 
 
 4>,(a - x) _ y _ cf)^{x) 
 
 where </> yjr are the first differentials of 4> '^, and e.g., "^j (y) 
 represents the utility to dealer No. 1 of the quantitv y of com- 
 modity No. 2 ; in the simplest abstract case the pleasure to be 
 at once obtained by the consumption of y, but in the general 
 case the pleasure to be obtained both in the immediate and 
 more distant future, reduced to the common measure so to 
 speak of present pleasure (by way of the Jevonian factors for 
 risk and remoteness)^ the pleasxxre I say to be thus obtained 
 from having nx)w the quantify of y (whether to be consumed 
 gradually or perhaps exchanged for other commodities). 
 
 When the fact expressed by the symbolic statement has 
 been grasped, it is only a dispute about words, whether we 
 define 
 
 (1) Supply of commodity No. I. = a. 
 Supply of commodity No. II. = 6.' 
 
 (2) Demand of commodity No. I. at rate of exchange 
 
 (•'\ = a: (the usual definition, I think). • 
 
 Demand of commodity No. II. at rate of exchange 
 
 © = 
 
 
 ' Theory, p. 108. ' Thfory, pp. 36, 38. 
 
 •■' Cl'. Caimes, p. 117.
 
 CRITICISM OP CAERNES. 121 
 
 (3) Demand for commodity No. I. is measured by the 
 quantity y exchanged for ic.' (?) 
 
 (4) Demand is the desire for commodities, &c? Such 
 language is justified, though it is not pretended that Caimes 
 uses it with any definite meaning, by the first intention of the 
 term * demand.' ^ In this case the demand for y might perhaps 
 be represented by i/r (?/). 
 
 But I know what angry susceptibilities are awakened by 
 the dogmatic terms Supply and Demand, and decline a contest 
 in a region which has been darkened by such clouds of dust. 
 
 Professor Caimes's whole contention that ' cost means sacri- 
 fice,' &c. (p. 60), may seem an unconscious tribute to the im- 
 portance of the quantification and measiirement of the sense of 
 sacrifice, subjective labour. If it is admitted that on the 
 whole he uses his ' sacrifice ' and ' cost of production ' ^ as an 
 objective not a subjective quantity, * cost as measured in number 
 of days, labour, and abstinence ' (p. 389), our e rather than 
 
 our / (e);* still he may seem both to have had the latter 
 
 quantity in view, and to have foregone some of the advantages 
 which would have been obtained by more clearly distinguishing it. 
 Professor Caimes's erposition of the bargain between em- 
 ployer and employed would probably have been enhanced by 
 the use of demand-curves, one representing the quantity of 
 work which the labourer is willing to give, and the other the 
 (total) amount of remuneration which the employer is willing 
 to give, at a certain rate of wages. It would have been sug- 
 gested that the Wage-Fund or -Offer, though for a given rate of 
 wages it have a determinate, has not necessarily a unique, value. 
 The demand-curves may intersect more than once. It would 
 not then, I think, be inconsistent with the premisses, though it 
 might be with the conclusions, of Caimes, that the effect of a 
 trades-imion might be to shift the position of the bargain from 
 the first to the third (or rather from third to first) intersection. 
 Also it would have been suggested as above, that, though the 
 labourer might have less total remuneration in consequence of 
 
 ' CJairnee, pp. 24, 25. ' Id. p. 21. 
 
 ' Cf. Cixnyngham, Notct on Exchange Value, p. 1. 
 * Cf. 62, 63, 79, &c. ' •=■ Appendix IV.
 
 122 APPENDICES. 
 
 a trades-union, yet he might have more utility, having less 
 labour. 
 
 Mr. Spencer. 
 
 iVIr. Spencer has * tried' the Utilitarianism of Mr. Sidgwick 
 (' Data of Ethics '), and condemned it ; but had the procedure 
 been according to the forms of quantitative science the verdict 
 might have been different. ' Everybody to count for one ' 
 is objected to Utilitarianism,' but this equation as interpreted 
 by Mr. Spencer does not enter into Mr. Sidgwick's definition of 
 the Utilitarian End, greatest possible product of number x 
 average happiness,* the definition symbolised above.' Equality 
 of distinction is no proprium of this definition; ato contraire.*^ 
 Not ' everybody to count for one,' but ' every just perceivable 
 increment of pleasure to count for one,' or some such definition 
 of the pleasure unit,* is the utilitarian principle of distri- 
 bution. 
 
 (S. 85.) The case of A B, C D, producers, among whom the 
 produce is to be distributed, presents no theoretical difficulty 
 to the * impartial spectator,' armed with the Calculus of Varia- 
 tions. The most capable of tvork shall do most work ; the 
 most capable of pleasure shall have most produce.* How could 
 the principle of equity be worked in the entangled case of co- 
 operative work ? ' But to the principle of greatest happiness all 
 is simple. Consider the whole produce as a given function of 
 the fatigues of the labourers, the pleasure of each as a given 
 function of his portion ; and determine the fatigues and the 
 portions so that the sum of the pleasures, ininus the sum of 
 the fatigues, should be the greatest possible, while the sum of 
 the portions equals the whole produce.* 
 
 (S. 86.) To insist that altruistic requires egoistic pleasure, 
 is open to the remarks above made (Appendix IV.). As to the 
 physical illustration (p. 228), grant that, in order that the 
 whole may be heated, the parts must be heated. What then ? 
 Is it not conceivable that to each part should be imparted just 
 
 ^ Data of Ethics, ch. xiii. 
 
 ' Book iv. ch. 1, § 2. ^ See above, p. 57. 
 
 * See Index gub voce Equality. * See above, p. 8. 
 
 • See above. ^ See above, p. 51. ' Se« above, p. 64-67.
 
 CKITICISM OF MR. SPENCER. 123 
 
 that amount of heat •which may conduce to anintegi'al tnaxirtium. 
 The illustration suggests a very different view from the author's, 
 viz., that there should not be ' equalness of treatment.' Let us 
 state, as the end to be realised, that the average temperature 
 of the entire cluster, multiplied by the number of the elements, 
 should be the greatest possible. Let us suppose that the 
 elements have different thermal capacities^ or that the same 
 amount of energy being imparted causes different increases of 
 temperatiu-e ; and (not troubling oiuselves about the conserva- 
 tion of energy) that each element, without diminishing its own 
 temperature, increases by radiation the temperature of its 
 neighbours. If thermal capacity (the received definition of the 
 term being inverted for the sake of the metaphor) ' and power 
 of radiation and absorption go together,^ then the larger por- 
 tions of a given fund of energy shall be assigned to higher 
 capacities. 
 
 The possibility of differences of capacity in the final state 
 of equilibrium does not seem to be entertained by the author. 
 But can we receive this ? Can we suppose that the Examina- 
 tion-list of the Future will consist of an all-comprehensive 
 bracket ? If capacities for work differ, possibly also capacities 
 for pleasure.^ If either or both species continue to differ, 
 Utilitarianism, it is submitted, will continue to have a function 
 not contemplated by the Data, unequal distribution. 
 
 A general agreement has been already ^ expressed with the 
 author's view that Pure Utilitarianism is not now absolutely 
 right. Some comment, however, may be made upon the 
 suggested comparison between 'absolute' rightness in the case 
 of an irregular imperfectly evolved society and mathematical 
 certainty in the case of ' crooked lines and broken-backed 
 curves.' Take a piece of string as crooked and broken-backed 
 as you please, and impart to its extremities given impulses. 
 Then it is mathematically deducible and accurately true * that 
 
 * See Clerk-Maxwell, Heid, p. 65. 
 
 ' Capacity for self-regarding and for sympathetic pleasures, each pro- 
 bably mcreaang with evolution. ' See above, p. 59, and below, p. 131. 
 
 * Appendix IV. 
 
 * Bertrand's Theorem, Thomwin & Tait. Cf. "Watson & Burbury, Gene- 
 ridised Co-ordinate$, Arts. 16, 17.
 
 124 APPENDICES. 
 
 the initial motion of each element is such that the whole initial 
 energy of the string shall be maximum. No doubt to actually 
 determine by the Calculus of Variations the motion for each 
 element, we must know the (original) form of the string. If 
 that form is broken-backed, a definite curve may be hypotheti- 
 cally assumed. So then it might be even now absolutely right 
 that each individual should act so that the general happiness, 
 as defined by Pure Utilitarianism, should be a maximum ; 
 though what that action is can only be approximately de- 
 termined. 
 
 Mr. Sidgwick. 
 
 i\Ir. Sidgwick's Economical reasonings have been already 
 noticed. Close and powerful as these reasonings are, it has 
 been impossible to conceal the impression that this distinguished 
 analyst would have taken the field in Economical speculation in 
 a manner more worthy of himself if he had not embraced the 
 unfortunate opinions of Cairnes • upon the application of 
 Mathematics to Political Economy. 
 
 Probably the only flaws in Mr. Sidgwick's ethical analysis 
 are where mathematical safeguards were required. 
 
 In the * Methods of Ethics,' « after defining the Utilitarian 
 End as the greatest simi of happiness, he supposes (as I under- 
 stand, but it is always very difficult to catch hold of those who 
 use ordinary language about mathematical subjects) that 
 happinesSj though not the means of happiness, should be 
 distributed equally. But this supposition is repugnant to his 
 definition. For, in general, either the capacities for happiness 
 (as defined above, p. 57) are, or are not, equal. If they are 
 equal, then both happiness and means should be distributed 
 equally ; if unequal, neither (p. 64). The supposition, then, 
 that happiness, though not the means of happiness, should be 
 distributed equally, is in general repugnant to the Utilitarian 
 End. 
 
 ■ Fortnightbj Jtevieto, February, 1879, p. 310. It is not for one whose 
 views about cbaDges in the * general purchasing power of gold ' are very 
 hazy to criticise a theory of that subject. It may be allowable, however, 
 to mention that the haze has not been removed by the theory of * aggregate 
 price,' &€., advanced in the article cited. 
 
 = Book iv. p. 386.
 
 CRITICISM OP MR. SIDGWICK. 125 
 
 In general ; for the beauty of mathematical analysis ' is that 
 it directs our attention not only to general rules but to excep- 
 tions. Suppose the two properties which constitute the defini- 
 tion of capacity for happiness not to go together, as in the 
 third imperfection of that definition noticed on the same page ; 
 then it is just possible that a given distribuend would be most 
 felicifically distributed among given distributees when the 
 happiness, though not the means of happiness, should be 
 distributed equally. 
 
 The interpretation that Mr. Sidgwick, in the passage just 
 discussed, has in view difierences of capacity for happiness, 
 is confirmed by explicit recognition of such (p. 256), * Some 
 require more and some less to be equally happy.' The pro- 
 blem raised in that context is not treated with mathematical 
 precision. *We should have to give less to cheerful, con- 
 tented, self-sacrificing people, than to the selfish, discontented, 
 and grasping, as the former can be made happy with less.' The 
 case would seem to be this : the minimum of means corre- 
 sponding to the zero of happiness (above, p. 64) is higher for 
 the discontented than the cheerful ; for values of means above 
 that minimum the cheerful have greater capacity for happiness. 
 If, then, the distribuend be sufficient to admit of all at least 
 reaching the zero of happiness, then the cheerful shall have a 
 larger portion of means. (See above, pp. 57, 65.) 
 
 These are shght steps of reasoning ; but they are at an 
 enormous height of generalisation, where a slip is ruin. 
 
 ^ I cannot refrain from illustrating this proposition by one more re- 
 ference to Principal Marshall's and Professor Walras's similar— doubtless 
 independent — theory of multiple intersection of demand-curve^ xmstable 
 equilibrium of trade.
 
 126 APPENDICES. 
 
 vn. 
 
 OiV THE PRESENT CRISIS IN IRELAND. 
 
 The consideration, however superficial, of a real case may serve 
 to put our method in a clearer light. Let us suppose, then, 
 that an intelligent reader, attracted by the heading of this 
 Appendix, inquires of what possible use can Psychical Mathe- 
 matics be in real life ? 
 
 First, it must be pointed out that deductive reasoning is not 
 to be too sharply pulled up with the demand, * What then do 
 you propose ? ' For, even if this highly deductive method 
 should prove more potent than the present tentative sketch 
 may warrant, it would have power only to give general instruc- 
 tions, not detailed regulations. From such a height of specula- 
 tion it might be possible to discern the outlines of a distant 
 country, but hardly the by-paths in the plain immediately below. 
 Mathematical Psychics would at best fiunish a sort of pattern- 
 idea to be roughly copied into human affairs ; ' in the language 
 of modem Logic hypothetical deductions to be corrected and 
 verified by comparison and consilience with experience. This 
 general character of deductive reasoning in Sociology has been 
 exhibited by Mill theoretically at length in his * Logic,' and 
 practically by repeated cautions in his 'Political Economy.' 
 The steps of Mill are followed by almost all considerable writers 
 upon method — Comewall Lewis, Caimes, Bain, Mr. Jevons in 
 the ' Principles of Science,' Mr. Sidgwick in behalf of ' Econo- 
 mic Method ' renouncing pretensions to precision of detail. 
 
 It cannot be expected that so terse a treatise as the present 
 should go over ground exhausted by such writers. We must 
 take for granted that our intelligent inquirer understands what 
 is intelligible to the intelligent. If he believe 'not the autho- 
 rities just cited, it would not be worth our while to resuscitate 
 considerations loug consecrated by universal acceptance. We 
 can only consider the position of one who, understanding in a 
 general way the nature and the need of deductive reasoning in 
 Sociology, draws the line at deductions couched in the language 
 of literature, refusing to employ as signs of general conceptions 
 
 ' Cfl n«k», Me^Mic, b. vi. & 501.
 
 POLITICAL UTILITAIIIAXISM. 127 
 
 mathematical symbols along ^ith ordinary words. The theo- 
 retical weakness of this position is that there is no logical 
 ground for drawing the line, other than the prejudice that ma- 
 theTnatical reasoning imports numerical data. Such, in fact, 
 appears to be the ground on which the objections against econo- 
 mical mathematics are based by Caimes ; Caimes, whose opinion 
 on this subject is shared by a stOl more distinguished analyst.' 
 This prejudice having been cleared away,* why should not 
 general reasonings about quantities be assisted by the letters 
 appropriate to the science of quantity, as well as by ordinary 
 words ? ' Ego cur, acquirere pauca si possum, invideor ? ' the 
 generalising genius of Mathematics unanswerably demands. 
 
 Practically, the objection solvitur ambula^ido, by the march 
 of science which walks more securely — over the ' flux and 
 through the intricate — in the clear beam of mathematical in- 
 tuition. The uses of this method may have been already 
 illustrated, at least by reference to the achievements of mathe- 
 matical economists. It will, however, be attempted here to 
 present some further illustration, introduced by the conspicuous 
 case of a country convulsed by political conspiracy and econo- 
 mical combination. 
 
 (I.) First as to the political aspect of the case has Calculus 
 anything to teach ? Nothing as to practical politics ; but as to 
 the first principles of political theory perhaps something. What 
 is the first principle of politics ? Utilitarianism, it would be 
 replied by most intelligent persons of the nineteenth century, 
 if in diflferent terminologies, yet virtually with one accord. Of 
 this basis what is the ground ? Here we leave the visible con- 
 structions of external action descending into a subterraneous 
 region of ultimate motives. 
 
 The motives to Political Utilitarianism are the same as in 
 the case of Ethical Utilitarianism, some would say; and they 
 would have to grope for a proof of utilitarianism, such as Mr. 
 Sidgwick grasps at with one hand, while with the other hand he 
 grasps the polar principle. His method proceeds by comparing 
 
 ' Fortnightly Reviero, 1879^ Economic Method. 
 
 ' See pp. 2-6, and Appendix I. 
 
 ' To treat variables as constants is the characteristic vice of the unmathe- 
 matical economist. Many of the errors criticised by M. Walras are of this 
 character. The predHerminate Wage-fund is a signal bstance.
 
 128 APPENDICES. 
 
 deductions from the utilitarian first principle with moral senti- 
 ments observed to exist ; * philosophical intuitionism ' does not 
 come to destroy common-sense, but to fulfil it, systematising it 
 and rendering it consistent with itself. Now this method may 
 be assisted, with regard to certain quantitative judgments of 
 common sense, by the science of quantity ; ' proving these moral 
 judgments to be consilient with deductions from Utilitarianism, 
 clipping off the rough edges of unmethodical thought. 
 
 But to others it appears that moral considerations are too 
 delicate to support the gross structure of political systems ; at 
 best a flying buttress, not the solid ground. It is divined that 
 the pressure of self-interest must be brought to bear. But by 
 what mechanism the force of self-love can be applied so as to 
 support the structure of utilitarian politics, neither Helvetius, nor 
 Bentham,^ nor any deductive egoist has made clear. To expect 
 to illuminate what Bentham has left obscure were presumptuous 
 indeed. Yet it does seem as if the theory of the contract- 
 curve ' is calculated to throw light upon the mysterious process 
 by which a crowd of jostling .egoists tends to settle down into 
 the utilitarian arrangement. 
 
 Thus the terms of the social contract are perhaps a little 
 more distinctly seen to be the conditions of ' Greatest Happi- 
 ness.' If the political contract between two classes of society, 
 the landlord and the tenant class for instance, is disturbed, 
 affected with the characteristic evil of contract * undecidable * 
 strife ' and deadlock, the remedy is utilitarian legislation ; as is 
 already felt by all enlightened statesmen. 
 
 Considerations so abstract it would of course be ridiculous 
 to fling upon the flood-tide of practical politics. But they are 
 not perhaps out of place when we remount to the little rills of 
 sentiment and secret springs of motive where every course of 
 action must be originated. It is at a height of abstraction in 
 the rarefied atmosphere of speculation that the secret springs 
 of action take their rise, and a direction is imparted to the pure^ 
 
 ' See above, pp. 76-80. And cf. the p-oof of utilitarianism in New and 
 Old Methods of Ethics (by the present writer). 
 
 ' I take the view which Mr. Sidgwick takes (^Fortnightly Revietc) of 
 Bent ham's aims, and of his success. 
 
 * CoroUaiy, p. 53. ■• Above, p. 29,
 
 EQUALITY. 129 
 
 fountains of youthful enthusiasm whose influence will ulti- 
 mately affect the broad current of events. 
 
 The province of ends is thus within the cognisance of 
 Mathematics. "WTiat shall we say of intermediate, or proxi- 
 mately final, principles ? The quantitative species of * Reason 
 is here no guide, but still a guard,' at present ; and might con- 
 ceivably be something more in some distant stage of evolution 
 related to the present (agreeably to the general description of 
 evolution) as the regularity of crystallization to the violent 
 irregular movements of heated gas. 
 
 Let us take a question suggested, however remotely, by our 
 heading. When ' peasant proprietorship,' ' expropriation of 
 landlords,' and even more communistic schemes, are talked 
 of, there are those whose way of thinking carries them on to 
 inquire whether the level of equality is a thing so much to be 
 desired 'per se, and abstracted from the expediencies of the 
 hour, and even the age. 
 
 The demagogue, of course, will make short work of the 
 matter, laying down some metaphysical ' rights of man.' Even 
 Mill never quite disentangled what may be a proximate from 
 what is the final end of utilita,rianism. And it is much to be 
 feared that a similar confusion between ends and means is en- 
 tertained by those well-meaning, generally working, members 
 of the social hive, who seem more concerned about the equi- 
 lateralness of the honeycomb than the abundance of the honey. 
 But the very essence of the Utilitarian is that he has put all 
 practical principles in subjection, under the supreme principle. 
 For, in that he has put all in subjection under it, he has left 
 none that is not put under it. 
 
 How then is it possible to deduce Equality from * Greatest 
 Happiness ; the symmetry of the Social Mechanism from the 
 maximum of pleasure-energy? By mathematical reasoning 
 such as that which was offered upon a pre\T[ous page,' or in an 
 earlier work,^ such as had already been given by Bentham and 
 the Benthamite William Thompson. Bentham, who ridicules the 
 metaphysical rights of man and suchlike * anarchical fallacies,' 
 
 ^ Above, p. 64. 
 
 ' New and Old Methods of Ethics. Tlie reasoning was offered in ignor- 
 ance of tbe analogous Benthamite reasoning.
 
 130 APPENDICES. 
 
 reasons down from Greatest Happiness * to Equality by a 
 method strictly mathematical ; even though he employ * repre- 
 sentative-particular ' numbers'* rather than general sjrmbols. 
 The argument might be tnade palpable by a parallel argument, 
 constructed upon another of the great arches of exact social 
 science, or those concave functions, as they might be called, in 
 virtue of which the Calculus of Variations becomes applicable 
 to human affairs — the law of diminishing returns. A given 
 quantity of labour (and capital) will be expended most produc- 
 tively on a given piece ofc land, when it is distributed uniformly, 
 equally, over the area ; by a parity of reasoning which makes 
 palpable the parity of proviso : provided that there be no dif- 
 ferences of quality in the ground. If, speaking both literally 
 and in parable, there is (indication and probability of) difference ; 
 if for the same seed and labour some ground brought forth a 
 hundredfold, some sixtyfold, some thirtyfold, the presumption 
 is that more should be given to the good ground. 
 
 Is there then any indication of such difference between 
 sentients ? We may not refuse once more to touch this ques- 
 tion, however unwelcome to the modem reader ; otiose to our 
 unphilosophical aristocrats, and odious to our democratical 
 philosophers. 
 
 (i.) First, then, it may be admitted that there is a difference 
 with respect to capacity for happiness between man and the 
 more lowly evolved animals ; and that therefore — among or 
 above other considerations— the interests of the lower creation 
 are neglectible in comparison with humanity, the privilege of 
 man is justified. Or if any so-called utilitarian, admitting the 
 practical, conclusion, refuses to admit its sequence from the 
 premiss, affirming some first principle in favour of the privi- 
 lege of his own species, he must be gently reminded that this 
 affirmation of first principles not subordinate to the Utilitarian 
 Principle is exactly what the great utilitarian 'called *ip8e- 
 dixitism ' ; and also — in case he protests against the ohgarchieal 
 
 ' Bentham apud Dumont, TraitSa de Ugitlation: Code Civil,eh. ti. ; 
 Pi-inciples of Pathology (Bowring's edition), vol. i. ; ib. vol. ii, 228, &c. ; 
 thua evincing a perfectly clear idea of the utilitarian end, more than might 
 have been inferred from some of his phraseology. 
 
 * Often a precarious method. Cf. Marshall, Foreign Trade, ch. L p. 4. 

 
 EQUALITY. 131 
 
 tendencies of our position — that he, not we, is the oligarch, the 
 ohgarchical demagogue levelling down to himself, and there 
 drawing the line. But the pure Utilitarian, drawing no hard 
 and fast line, according to the logical divisions of scholastic 
 genera or pre-Darwinian Real Kinds, and admitting no ultimate 
 ground of preference but quantity of pleasure, * takes every 
 creature in and every kind,' and * sees with equal eye,' though 
 he sees to be unequal, the happiness of every sentient in every 
 stage of evolution. 
 
 (n.) Again, it may be admitted that there are differences of 
 capacity for work, corresponding, for example, to differences of 
 age, of sex, and, as statistics about wages prove, of race. It 
 would be a strange sort of rational benevolence which in the 
 distribution of burdens would wish to equahse the objective 
 circumstance without regard to subjective differences. 
 
 (m.) Now (as aforesaid ') the admission of different relations 
 in different individuals between external circumstances and 
 internal feeling in the case of one species of (negative) pleasure 
 is favourable to the admission of such differences in the case of 
 other species of pleasure, or pleasure in general. Not only do 
 we see no reason why the latter difference, if agreeable to ob- 
 servation, ought not to be admitted ; but also we see a reason 
 why it has not been admitted or not observed. For in the 
 former case we have what in the latter case we have not, the 
 same quantity of feeling in different individuals corresponding 
 to different values of an ext,emal variable, namely the (neigh- 
 bourhood of) the infinite value of fatigue to different external 
 limits of work done. And everyone is acquainted with those 
 whose physical or intellectual power he himself could not equal, 
 *not even if he were to burst himself; ' whereas in the case of 
 pleasure in general — owing apparently to the rarity or irregu- 
 larity of the very high values of pleasure — we are reduced to 
 the observation of different increments of pleasure occasioned 
 by the same increment of means. 
 
 But is this observation insufficient ? Or can it be indifferent 
 to the utilitarian whether a given opportunity or increment of 
 means is bestowed where it occasions but a single simple sen- 
 suous impression of fiov6')(povos rjhovT), or a pleasure truly 
 
 * Above, p. 59,
 
 •|^22 '"*™^' APPE^^)ICES. 
 
 called * higher,' or * liberal,* or * refined ' — integrated by redin- 
 tegrating memory, multiplied by repeated reflection from the 
 * polished breast ' of sympathisers, in fine raised to all the 
 powers of a scientific and a romantic imagination? Can we 
 think it indifferent whether the former or the latter sort of 
 sentience shall be put into play ? 
 
 (iv.) Put into play, or brought into existence. For at 
 what point shall we stop short and refuse to follow Plato while, 
 inspired with an ' unconsciously implicit,' ' and sometimes an 
 explicit,'* utilitarianism, he provides for the happiness (it is 
 submitted, with due deference to Aristotle '), not only of the 
 present, but of succeeding generations ? Or should we be 
 affected by the authority of Mill, conveying an impression of 
 what other Benthamites have taught openly, that all men, if 
 not equal, are at least equipotetitial, in virtue of equal educa- 
 tabihty ? Or not connect this impression with the more transi- 
 tory parts of Mill's system : a theory of Eeal Kinds, more 
 Noachian than Darwinian, a theory of knowledge which, by 
 giving all to experience gives nothing to heredity, and, to come 
 nearer the mark, a theory of population, which, as pointed out 
 by Mr. Galton (insisting only on quantity of population) and, 
 taking no account of difference of quality, would probably re- 
 sult in the ruin of the race ? Shall we resign ourselves to the 
 authority of pre-Darwinian prejudice ? Or not draw for our- 
 selves very different consequences from the Darwinian law? 
 Or, rather, adopt the * laws and consequences ' of Mr. Galton ? * 
 
 To sum up the powers claimed for our method : if in 
 some distant stage of evolution there may be conceived as 
 practicable a distinction and selection, such as Plato adum- 
 brated in the * Republic, ' the selected characters perhaps not 
 so dissimilar from the Platonic ideal — wise and loving, with a 
 more modem spirit both of science and romance — but the 
 principle of Selection, not intellect so much as feeling, capacity 
 for happiness ; then the delicate ' reasoning about capacity 
 
 * Mr. Sidgwick's happy phrase. 
 
 * V.nWi(rTa yap bq tovto Xe'yrrat Kn\ XtXt^trm, ort ri fiiy af(f>eKifiov xoXov, 
 TO bi ffkafttftotf alaxpov. — Plato's Repvhlic. 
 
 » Poiifica, V. * Jfeteditnn/ Genntt: end of penultimate chapter. 
 
 * Above, p. iJo.
 
 EQUALITY. 133 
 
 would seem to stand in need of mathematical, if not symbols, 
 at least conceptions. And even at present it is well, at what- 
 ever distance, to contemplate the potentiality and shadow of 
 such reasoning. For though the abstract conclusions have no 
 direct bearing upon practical politics (for instance, extension or 
 redistribution of suffrage), determined by more proximate 
 utilities — just as Bentham protests that his abstract preference 
 for equality does not militate against the institution of property 
 — nevertheless it can hardly be doubted that the ideal reason- 
 ings would have some bearing upon the general drift and 
 tendency of our political proclivities. And at any rate the 
 history of all dogma shows that it is not unimportant whether 
 a faith is held by its essential substance, or some accidental 
 accretion. And the reasonings in question may have a use in 
 keeping the spirit open to generality and free from preposses- 
 sion, the pure ideal free from the accreting crust of dogma. 
 From semi-a-priori ' innate perceptions ' dictated by an ' ana- 
 lytic ' intelligence,' from * equity,^ and ' equalness of treatment,' 
 and * fairness of division ; ' ^ which, if they gave any distinct 
 direction at all (other, of course, than what is given by merely 
 utilitarian^ considerations), would be very likely to give a 
 wrong direction, meaning one which is opposed to the Univer- 
 salistic Hedonism or Principle of Utility established by the 
 more inductive methods of Sidgwick and of Hume. From 
 dictates indistinct and confusing, or, if distinct — at least about 
 a subject so amenable to prejudice as ' equalness ' and ' equity ' 
 — most likely to be wrong. To show which danger it is suffi- 
 cient (and it appears necessary, at a not unfelt sacrifice of 
 deference) to observe that the same semi-a-priori method, ap- 
 plied to Physics, in the course of a prolonged discussion of 
 * Force ' and its * Persistence,' never clearly distinguishes, nay, 
 rather confoimds, ' Conservation of Momentimi ' and ' Conserva- 
 tion of Energy ' ! while it is distinctly stated that the law of the 
 inverse square is ' not simply an empirical one, but one de- 
 ducible mathematically from the relations of space — one of 
 which the negation is inconceivable.'* Is it wise, is it safe, to 
 
 ' Herbert Spencer, Data of Ethics, b. 62. 
 
 « lb. 8. 60, p. 164. » lb. ss. 68, 69, elc. 
 
 * Id. First Principle, s. 18.
 
 134 APPENDICES. 
 
 weight and cramp science with a-priori dogmas such as this — 
 in view of the possibility of a Clerk-Maxwell after all discover- 
 ing, by the ordinary (Deductive) method of Inductive Logic, 
 that there is attraction between atoms according to a law of 
 inverse fifth power ? An inductively deductive method in 
 Sociology may have similar surprises for the dogmatic isocrat 
 forthcoming ; but they will certainly not come, there will not 
 come any development, if we resign ourselves with a Byzantine 
 sloth to a-priority or other authority more dear to the utili- 
 tarian ; not dissociating the faith of love from the dogma of 
 e([uity, from the accreted party-spirit and isocratic prejudice of 
 Benthamite utilitarianism, the * pure ethereal sense ' and un- 
 mixed flame of pleasure. 
 
 And lastly, * whether these things are so, or whether not ; * 
 about a subject so illusory, where the vanity and the very 
 virtues of our nature, oligarchical pride, democratical passion, 
 perturb the measurements of utility ; not slight the advantage 
 of approaching the inquiry in the calm spirit of mathematical 
 truth. 
 
 Thus it appears that the mathematical method makes no 
 ridiculous pretensions to authority in practical politics. There 
 is no room for the sarcasm of Napoleon complaining that La- 
 place wished to govern men according to the Differential 
 Calculus. The sense of practical genius need not take offence. 
 The mathematical method has no place in camps or cabinets ; 
 but in a philosophic sphere in which Napoleon had neither 
 part nor lot, and which he scouted as * Ideology.' ' 
 
 (II.) Let us turn now to the econoTuical aspect of the case 
 before us : combination of tenants against landlords, which the 
 present crisis in Ireland^ is thought to involve. Here also the 
 dry light may illuminate the troubled scene of dead-locked 
 unions ; and by an unobvious path lead up again to the prin- 
 ciple of utility as the basis ^ of arbitration. The /air rent is 
 seen to be the utilitarian rent.* 
 
 ' Bouirienne's Memoirs. 
 
 ^ The Pall Mall Gazette has persisted in regarding the agrarian as 
 Trades Unionist outrag^es. 
 
 ^ Read Mr. Crompton in Industrial Conciliation (cf. pp. 82, 83), and 
 realise the need of some principle of arbitration. 
 
 * Her Majeptj's CommisBioners of Inquiry into the working of the Land
 
 COMBINATION OF TENANTS. 135 
 
 Here it may be proper to indicate the relation which pre- 
 ceding considerations upon indetermiTiateness of contract are 
 supposed by their writer to bear to the considerations recently 
 adduced by others, in particular Mr. Cliffe Leslie ^ and ^Mr. 
 Frederick Harrison,'^ concerning the irregular and accidental 
 character of mercantile phenomena — as contrasted with what 
 may be called perhaps the old-Ricardian view. The two sets 
 of considerations, ours and theirs, may be mutually corrobora- 
 tive ; but they are for the most part distinct, though they 
 occasionally overlap. Thus Mr. C. Leslie's contention against 
 the equality of profits, &c., in different occupations, does not 
 form any part of these fragmentary studies ; while, on the other 
 hand, o\ir second and fourth ' imperfections have not perhaps 
 been noticed elsewhere. Again, the imperfection of the labour 
 naarket, due to the immobility of the labourer upon which IMr. 
 PVederick Harrison in a human spirit dwells is, analytically 
 considered, a case of out first imjperfection. 
 
 As there is a certain relation of alliance between these con- 
 siderations and those', so they may be all exposed to the same 
 attack, namely, that the irregularities in question, though 
 existent in fact, do not exist in tendency, tend to disappear, 
 and therefore may be neglected by abstract science. This is a 
 matter of fact upon which the present writer is ill-qualified to 
 offer an opinion. But he submits that the imperfections which 
 it has been in these pages attempted to point out in the case 
 of cooperative association and to trace in the case of trades- 
 unionism, do not tend to disappear, but rather to increase, in 
 the proximate future at least. The importance of the second 
 imperfection — affecting contract with regard to certain kinds of 
 
 Act of 1870, &c., having sanctioned and supposing settled a * fair rent,' 
 recommend that the ' unearned increment ' which may accrue should, in the 
 absence of first principles to determine the distribution between landlord and 
 tenant, be divided equally between them. Obser\"ing that the contrcut-curoe 
 in this case is the representation of all the possible rents (p. 342), we have 
 here a simple exemplification of the theory that the hasia of arbitration is a 
 point on the contract-curve, roughly and practically as here the quantitative 
 mean, the bisection of the indeterminate reach of contract-curve, tlieoreti- 
 cally the qualitative mean the utilitarian point (p. 55). 
 
 ' Fortnightly Reoicto, Hermatbena, &c. - Ibid. 1 8Go. 
 
 =« Pp. 4G, 47.
 
 136 APPENDICES. 
 
 service — might perhaps stand or fall with the importance o£ 
 Mr. Cliffe Leslie's considerations upon the inequality of re- 
 munerations.* 
 
 Lastly, if the argument attempted in these pages concerning 
 the indeterminateness of contract is as to the premisses some- 
 what similar to the Positivist argument, it would fain be also 
 as to the conclusion : the jiecessity of settling economical dif- 
 ferences by a moral principle — here clothed in the language more 
 of Mill than of Comte, and disfigured by the unfortunately ugly 
 term Utilitarianiain, which so imperfectly suggests what it 
 connotes. ' Fjw*e 'pour autruiJ' 
 
 Ketiirning from this digression, let us now sift a little more 
 accurately the light w^hich Mathematics may shed upon Combl- 
 nations. Compare the analysis suggested in a previous part of 
 this work with the general account of ' Monopolies and Combina- 
 tions ' in * Economics of Industry.' The conception of indeter- 
 iiiinateness increasing with the inci'easeof combination comes 
 out perhaps a little more clearly in the mathematical analysis. 
 To bring out the comparison, it is best to consider some par- 
 ticular species of combination. Here, however, occurs the 
 difficulty that the species as presented by the text of these 
 supplementary remarks upon method has not been nxuch, if at 
 all, treated by economists. Let us take, then, combinations 
 of v'orkmen against employers ; a deviation from our subject 
 for which the less apology is due as it is part of the purport 
 of some coming remarks to insist on the essential unity of the 
 dift'crent kinds of contract. 
 
 Let us consider the argument about Tnides Unions con- 
 tained in the ' Economics of Industry,' book iii. chapter 6, 
 §§ 1 and 2 ; or rather a certain popular argument against 
 Trades Unions strengthened by whatever it can borrow from 
 the passage under consideration. 
 
 It is submitted with great deference, ^rsi, that the conclu- 
 sion does not follow from the premisses, if the conclusion is 
 that trades unions tend to defeat their own object, the interest 
 of the unionists. The premiss is that the consequence of the 
 action of Trades Unions is a continually increasing * check to 
 the growth,' diminution from what it would have been, of the 
 
 ' Above, p. 47.
 
 COMBliVATIOX OF WORKliEN. 137 
 
 wages-and-profits fund, and so of the total Eemuneration of 
 operatives. But, since the utility of the operatives is a function 
 not only of their remuneration, but their labour; and, though 
 an increasing function of the remuneration, considered as 
 explicit, is a decreasing function of the same considered as 
 implicit in labour ; ^ it does not follow that there tends to de- 
 crease that quantity which it is the object of unions to increase 
 — the unionists* utility at each time, or rather time-integral of 
 utility. Rather, it appears from the general analysis of con- 
 tract that, if any effect is produced by unions, it is one bene- 
 ficial to the unionists (presupposed, of course, intelligence on 
 their part) ; and that, if combination is on a sufl&ciently large 
 scale, an effect is likely to be produced. 
 
 But, secondly, the premisses are not universally true, those 
 of the popular argument at least ; for the Marshall argument 
 keeps * intra spem veniae cautus.' For though it be true that 
 the action of imionists, if they * refuse to sell their labour 
 except at a reserve price,' would be to diminish ultimately the 
 Remuneration, this result would no longer hold if the unionists 
 were to insist, not on a rate of wages, leaving it to the em- 
 ployers to buy as much or as little work as they please at that 
 rate, but upon other terms of employment — a certain quantity 
 of remuneration in return for a certain quantity of work done. 
 If (in our terminology) they proceeded by way of contract- 
 
 ' Geometrically ; let an abscissa represent time. Let tlie remunerations 
 at each time, as they would have been, be represented by ordinat«s formings a 
 sort of byperbola-shaped curve as to the portion of time at least with whicb 
 we are concerned --from the present, far as human eye can t>ee (not to 
 trouble ourselves about the vertex and the asymptote). To fix the ideas, let 
 
 the approximate shape be given by ^ — •^^ — 1 = 0. Now let the series 
 
 d^ b' 
 
 of remunerations, as it is in consequence of the action of Unions, be 
 
 ^^ ^'^^ --^-1 = where b'< b.c is positive. Let the present time 
 cr b ' 
 
 correspond to the point where y' = i/; if y' be new ordinate at any point y 
 bein^ the old. We have then ^ ~ ^ the percentage of loss of remuneration 
 
 continually increasing. Bat the end of the unionists is not the ordinat<«8 
 nor the area, but the hedonic integral represented by the solid contents of a 
 certain gHoti-hiffterboloid described upon the quasi-hyperbola. From the 
 nature of tbe functions of this surface it appears that the solid contcutb may 
 be greater in the latter case Uuiu iji the former. — Q.E.D.
 
 138 APPENDICES. 
 
 cut^e,^ not by way of demand-curve, the presumption is that 
 their action would increase not only their utility but their 
 remuneration. 
 
 And, thirdly y even if the literary method by a sort of 
 intuition or guess-work apprehends the truth, it can hardly 
 comprehend the whole truth. For it appears from analysis 
 that the tendency of combinations is not only to make contract 
 more beneficial to the unionists, but also to make it indeter- 
 ininate ; a circumstance of some interest as bringing clearly 
 into view the necessity of a principle of arbitration where 
 combinations have entered in. 
 
 The Mathematical method does not, of course, show to 
 advantage measuring itself with the ungeometrical argmnents 
 of Mr. Marshall, himself among the first of mathematical 
 economists, and bearing, even under the garb of literature, the 
 arms of mathematics ; which peep out in this very place (* Eco- 
 nomics of Industry,' p. 201). A. much more favourable compa- 
 rison would be challenged with the popular economists, who 
 often express themselves rather confusedly, as JVIr. Morley, in 
 an eloquent address,^ points out. Mr. Morley's own opinion is 
 not very directly expressed, but is presumably opposed to 
 * those who deny that unions can raise wages.' Now, it is 
 submitted that this opinion, in face of the Caimes-lSIarshall 
 arguments, can only be defended by the unexpected aid of 
 mathematical analysis. The incident may suggest, what is the 
 burden of these pages, that human afiairs have now reached a 
 state of regular complexity necessitating the aid of mathematical 
 analysis ; and that the lights of unaided reason — though spark- 
 ling with eloquence and glowing with public spirit — are but a 
 precarious guide unless a sterner science fortify the way. 
 
 But what is all this to laridlords and tenants ? Or can 
 your scanty analysis of combination in general Jje securely ex- 
 tended to the peculiar case of rent ? The reply is : Yes ; the 
 reasoning about the tendency of combination to produce inde- 
 terminateness can with sufficient safety — by a sort of mathema- 
 tical reduction — be extended from the general to a particular 
 case. Symbols are not to be multiplied beyond necessity. 
 Bather the mathematical psychist should be on his guard to 
 
 » Sitt) pp. 48, IIG. ^ FuHnightly Review, 1877, p. 401.
 
 CONTRACT ABOUT RENT. 
 
 139 
 
 Deduct what is but vanity or dress, 
 
 Or learning's luxury, or idleness : 
 
 Mere tricks to show the stretch of human brain. 
 
 To show, however, this very thing, the substantial unity of 
 the theory of contract (whatever the articles), and also to fur- 
 ther illustrate the general theory, let us attempt an analysis of 
 the contract between landlord and cottier-tenant. We may ab- 
 stract all the complications of commerce, and suppose the 
 corapeiitive field to consist only of landlords and cottier-tenants. 
 
 Let us start, then, upon the lines of previous trains of 
 
 Fig! 6. 
 
 reasoning, and begin by imagining equal numbers of on the one 
 side equal-natured landlords, and on the other side equal- 
 natured tenants. The quantity and the quality, of the land 
 possessed by each landlord are supposed to be the same ; the 
 quantity limited, or more exactly less than a tenant if he had 
 to pay no rent would be willing to take into cultivation. The. 
 requirements and capacities of the tenants likewise are for the 
 moment supposed equal. Let us represent the portion of land 
 owned by the landlord as a portion of the abscissa o x, and the 
 corresponding rent paid by a length measured along the other 
 co-ordinate. And let us proceed to write down in this particu-
 
 140 APPENDICES. 
 
 Itir case the ftinctions wbose general character has been ahready 
 described. 
 
 P, the utility-function of X the landlord, is F {y) (subject 
 to a certain discontinuity which will be presently suggested). 
 n, the utility-function of Y the tenant, is 
 
 ^ (<^ (e) x^y)--^{xe) 
 
 subject to the condition I—— j = 0. Here ^ as before is a 
 
 pleasure-function, e is the amount of objective-labour (mus- 
 cular energy or other objective measure of labour) put forth by 
 Y, per unit of land. ^ (e) is the corresponding produce per 
 unit ; a function which, according to the law of diminishing 
 returiis, has its first differential continually positive, and its 
 second differential continually negative, x e is the total objec- 
 tive labour, yp- (x e) the corresponding subjective labour^ or dis- 
 utility ; a function which according to the law of increasing 
 fathjne has both its first and second differential continually 
 positive. Since e is variable at the pleasure of Y, he will vary it 
 (whatever x may be), so that his utility as far as in him lies may 
 
 be a maximum ; whence ( — y— ) = 0. Let us for convenience 
 
 \ deJ 
 
 designate the function which results from the indicated elimi- 
 nation of e by TT (x y). 
 
 The indijference-cu7'ves of the landlord if he have no other 
 use for his land are horizontal lines ; importing that it is in- 
 different to the landlord how much land he lets, provided he 
 gets the same (total) rent. Let us however for the sake of il- 
 lustration, and indeed as more real, suppose that the landlord can 
 always make sure of a certain minimum, by emplojdng his land 
 otherwise, e.g. not letting it to cottier cultivators, but to capitalist 
 graziers. If then the landlord's income from lands thus other- 
 wise employed be proportionate to the land thus "femployed at a 
 certain rate per unit of land, the landlord's indifference-curve 
 may be represented by y „ and parallel lines (Fig. 6). 
 
 The indifference-curves of the tenant are given by the 
 
 differential equation (-7- )<^^ + ( TT" ) ^ V — ^- ^^^ 
 f — j is by hypothesis positive in the neighbourhood of the
 
 CONTRACT ABOUT RENT. 141 
 
 origin, and negative ultimately ; since x has been assumed less 
 than the quantity of land which Y would be willing to take into 
 cultivation without rent, which quantity is given by the equation 
 
 (^J.(x,<,)- = 0. And (g = (^^^) = -*'(.^(e)-,, 
 
 is essentially negative. Thus the indifference curve ascends in 
 the neighbourhood of the origin and descends as indicated in 
 
 the figure to the point K where f— J tt (x, o)=0. Again, 
 dx^ \dxJ \dy'^J dx dy\dxdyJ \dy) Wa;V 
 
 \dyJ 
 
 where (%^^ = {^^^^^ f^M") f^ + (~) T^V + 
 Kdx^J \dxy \dx deJ dx \ d eV \dx) 
 
 f—-~ ^ -2-|, the last term being equal to zero in virtue of the 
 equation (^)=0. And^=(^)_ And simUarlj for 
 
 the other second differentials of little tt. Working out the 
 somewhat elephantine formula thus indicated, and attending to 
 the character of the functions <J> ^ i/r, we should find that ' the 
 
 curve is convex when -~^ is negative. The attention of the 
 
 student is directed to this, if expanded rather lengthy 
 Tnathematical reasoning^ for which never a numerical datum 
 is postulated, about a social subject. The curves may be (I 
 think) convex at starting. Thus in figure 6, o T t;^ s- is a fair 
 representation of Y's indifference-curve through the origin. 
 The curve through ym and {x' y') represents (part of) another 
 member of the same family. 
 
 The demand-curve of the landlord is the ordinate at the 
 point x from above the point y^. The landlord will be willing 
 to take any amount of rent for his land above that minimum ! 
 Or, in other terms, the quantity of land which he offers at any 
 
 ' Compare the reasoning at pp. 35, 36.
 
 142 APPENDICES. 
 
 rate of rent (indicated by the angle between a vector and the 
 abscissa) is o x. The demand-curve of the tenant is the 
 locus of points of contact between vectors dravm from the 
 origin and indifference-curves. In the figure it is supposed 
 to pass through T, 77, and R ; the last point indicating the 
 quantity of land demanded by the tenant at rate of rent zero. 
 
 So far as to what may be called personal or individualistic 
 functions. What of the mutual function, which plays so large 
 a part in our speculations, the contract-curve ? The available 
 portion of the contract-curve is 2/0 ''7o» ^he portion of the or- 
 dinate at X intercepted between the in difference-curves from 
 the origin. For it is easy to see that if the index be placed 
 anywhere to the left (it cannot by hypothesis be placed on the 
 right) of this line it will run down under the force of concur- 
 rent self-interests to the hue in question. For instance, at 
 the point T, the indifference-curve of Y is drawn in the figure, 
 and the indifference-curve of X is a line parallel to O^/o ; be- 
 tween which and the corresponding lines at each point the 
 index will continually move down to the line xrj^ (assuming at 
 least a certain limitation or relative smallness of ox). Here, 
 however, occurs the interesting difficulty that the general con- 
 
 ,.,. (IF dU dU dF . . ,. ^ , , .. r 
 
 dition , — -— — . -,- -J- = IS not satisfied by the une 
 dx dy dx dy 
 
 2/0 ^o* What is the rationale of this ? It may be thus stated. 
 The contract-curve expresses the condition of a certain he- 
 donic (relative) maximwm. Now the condition of this maxi- 
 mum is in general, according to the general principles of the 
 Calculus of Variations, the vanishing of a certain first term of 
 variation. But the general rule of the Calculus of Variations 
 is suspended in particular cases of imposed conditions ; accord- 
 ing to a principle discovered by Mr. Todhunter, which is pro- 
 bably of the greatest importance in the calculus as applied to 
 human affairs. Now the case before us of quantity of land 
 fixed and small constitutes such an imposed condition and 
 barrier as is presented in so many of Mr. Todhunter's pro- 
 blems. In the metaphorical language already employed,* we 
 might conceive the contractors' joint-team driven over the 
 plain up to the barrier 2/0 "^o J ready to move on to the right of 
 
 ' Above, p. 24.
 
 CONTRACT ABOUT RENT. 143 
 
 the line if the barrier were removed, but incapable of moving 
 either up or down the line. If the quantity of land were 
 fluent, as in general articles of contract are to be regarded, 
 then the ordinary form of the contract -curve will reappear. 
 That the quantity of land should be regarded as fluent it is 
 not necessary that it should be absolutely unlimited, as in 
 general articles of contract have a superior limit e.g., the quan- 
 tity of labour a man can offer. It suffices that the quantity of 
 land should be large ; more exactly that the angles made by 
 the indifference-curves of Y at each point of the ordinate with 
 the direction o x should be greater than the angles made by 
 the indifference-curves of X. 
 
 Let us now proceed to investigate the fiTial settlements in 
 the field of competition just described. The first condition' 
 of a final settlement is that the whole field be collected at a 
 point on the contract-curve. The second condition is that 
 recontract be impossible. What then are those points at 
 which the whole field being concentrated recontract is possible ? 
 Those at which p landlords can recontract^ with q tenants. 
 
 By definition of contractr-curve p and q are unequal. The 
 recontract, or at least the settlements to one of which it tends, 
 may be represented by a supplementary contract-system con- 
 structed on the analogy of that above ' indicated. A little at- 
 tention will show that p must be greater than q when the point 
 2/0 falls as in the figure below the point 77 to be presently de- 
 fined. The supplementary system then consists of the 
 original contract-curve and a perpendicular to the abscissa at 
 the point ccf such that p'KOX=iqxo x' ', and it imports that 
 the recontract-ors tend to the following arrangement: the p 
 landlords on a point, say x y, of the original contract-curve, 
 and the q tenants on a point x'y' determined by the intersec- 
 tion of a vector through x y, with the supplementary contract- 
 curve or perpendicular at x'. Accordingly, if as just supposed 
 the whole field is concentrated at a point xy on the contract- 
 curve p landlords can * recontract vrith q tenants so long as y 
 
 ' Above, p. 35. 
 
 " Each recontracting for himself, of course, the fourth imperfection being 
 not in general presupposed. ' P. 37. 
 
 * It may be a nice question how far, as a matter of fact, the process of
 
 144 APPENDICES. 
 
 is such that the corresponding point x' y' falls within the 
 tenant's indifference-curve drawn through x y. The recontract 
 will just be impossible when x' y' is on the intersection of the 
 indifference and supplementary curves. It will appear that 
 
 the larger is the fraction i- the longer, as we ascend the con- 
 tract-curve moving from 2/0. is impossibility of recontract de- 
 ferred. The last point, therefore, at which recontract is 
 possible, is y^, the (tenant's) indifference-curve through which 
 meets the vector from the origin on the ordinate at a;', where 
 {rtx—\)ox' — 'mox. The points beyond y^ are final settle- 
 ments. 
 
 By parity it may be shown that the points on the contract- 
 curve in the neighbourhood of tj^, are not Jinul settlements ; 
 but that the system if placed at any of them will move away 
 under the influence of competition between landlords ; on to 
 a point rjm^ the indifference-curve through which meets 
 the vector from the origin on the ordinate at a/' where mox:" 
 = (w,— 1) ox. 
 
 Between ijm and y^ there is a reach of contract-curve con- 
 sisting ofJiTial settlements. The larger m is the STnaUer^ is 
 the reach of indeterminate cmitract. 
 
 It is clear that similar reasoning will hold if we suppose 
 our landlords and tenants to be not individuals, but equal- 
 corporate competitive units, in short, equal combinations as in 
 these pages understood. Thus it is clearly seen how the in- 
 crease of combination tends to increase indeterminateness in a 
 sense favourable to the combiners. 
 
 Clearly seen in the abstract ; and what has been sighted in 
 the abstract will not be lost sight of as it becomes immersed 
 in the concrete : when we suppose the numbers of the parties 
 on each side, the natiures of the tenants, the quantities and 
 qualities of land, the size of combinations, &c., to be unequal. 
 
 recontract in imperfect competition will involve the conception of rate cf 
 exchange — the tenant for instance endeavouring? to vary any existing con- 
 tract — because at the rate presented by that contract, the ratio of the articles 
 exchanged, he would be willing to take, he demands, more land. It has 
 seemed best in treating of contract in general to keep clear of a conception 
 which is, it w submitted, essential only to one species of contract, that 
 determined by perfect competition.
 
 CONTRACT ABOUT RENT. 145 
 
 The treatment of different numbers on each side is suggested 
 by the theory of the supplementary contract-curve. The 
 treatment of different natures may be thus indicated in the 
 important instance when the numbers on each side are indefi- 
 nitely large. In this instance, it may be premised, upon the 
 supposition of equality the points r)^ and y^ coincide at the 
 point tjy where the vector from the origin touches the (tenant's) 
 indifference-curve on the contract-curve, and which is accord- 
 ingly on the tenant's demaTid-curve.^ And it is also on the 
 landlord's demand-curve.^ And thus contract is determined by 
 the intersection of the demand-curves. Here we suppose all 
 the tenants to have the same requirements, the same indiffer- 
 ence-curves. "We might conceive the perfectly similar curves 
 which are touched at rf coincidently heaped up. Now, the natures 
 varying, let the curves no longer identical slide away from 
 each other, still keeping in contact with the itself-moving 
 vector ; subject to the condition that the sum of the lands let 
 is equal to the sum of the lands rented. Or more precisely ; 
 subject to the said condition, draw a vector from the origin such 
 that it touches a member of every family of (tenant's indiffer- 
 ence) curves. It is clear that equilibrium is then attaitied. 
 No tenant wants any more land at the rate of rett indicated 
 by the vector, and therefore does not, as he otherwise would, 
 tend to raise the rate in order to obtain more land at the same, 
 or even a slightly increased, rate. And no landlord has an 
 effective demand for more rent, since he has no more 
 land. 
 
 The preceding investigation applies to the case of different 
 quantities of land. The case of different qualities is one which 
 has not been explicitly treated in these pages. But its treat- 
 ment is suggested by analogy. If, for instance, there are 
 two species of land, x and y, the rent being represented Z 
 ( = Z J + Z y), the contract-locus might be regarded as a cur^^e 
 of double curvature, down which — down from their maximum 
 utility — the tenants are worked by competition, the further as 
 they are less combined. It would be easy, were it relevant, to 
 contemplate from this point of view the Ricardo-Mill theory of 
 the * worst land paying no rent,' &c. 
 
 ' See Index $uh ptnf Demand-curve. ' A1>»ve, p. 141.
 
 146 APPENDICES. 
 
 With regard to combinations in the concrete, it may be 
 observed that, while in the abstract symmetrical case equality 
 of distribution between combiners might be taken for granted, 
 we must in case of unequal natures presuppose in general a 
 principle of distribution as an article of contract between 
 members of a co-mbinatimi ; presumably tending to the utili- 
 tarian distribution. 
 
 It was not promised that this final efflorescence of analysis 
 would yield much additional fruit, though perhaps one who 
 knew where to look might find some slight vintage. Attention 
 mav be directed to the possible initial convexity of the tenant's 
 indiflference-curve. It will depend upon the presence or absence 
 of this property whether or not the tenant can be deprived by 
 competition of the entire utility of his bargain in perfect com- 
 petition ; and the same property presents interesting peculiarities 
 in the case of imperfect competition. 
 
 What it has been sought to bring clearly into view is the 
 essential identity (in the midst of diversity oi fields and articles) 
 of contract ; a sort of unification likely to be distasteful to 
 those excellent persons who are always dividing the One into 
 the Many, but do not appear very ready to subsume the Many 
 under the One. 
 
 Mr. Cliffe Leslie is continually telling us that nothing is to be 
 got from such abstractions as the ' desire of wealth and aversion 
 for labour,' feelings different in different persons, and so forth. 
 Yet he would surely admit that there is a general theory of 
 contract, of the bargain between individuals actuated by those 
 abstract desires, irrespective of the diversity of their tastes,* and 
 all the information about particulars which Mr. Cliffe Leslie 
 desiderates. Thus confining our attention to the simple case 
 of two ^ sets of contractors, Xs and Ys — it may be Producers 
 and Consumers, Employers and Employed, Lenders and Bor- 
 rowers, Landlords and Tenants, International 'traders ; pre- 
 scinding this simple case for convenience of enunciation, we 
 micrht write down I think some such (not the most general, but 
 quite generalisable) laws of contract — contract qualified by 
 competition. 
 
 I. Where the numbers on both sides are indefinitely large, 
 
 • See p. 145. ' See above, p. 17.
 
 COTRACT Df GENERAL. 147 
 
 and there are no combhiathiis, and competition is in other 
 respects perfect, contract is determinate. 
 
 II. Where competition is imperfect, contract is indeter- 
 minate. 
 
 lu. Cceteris paribus, if the nmnbers on one side are de- 
 creased (or increased) each of the (original) members on that 
 side, in perfect competition gains in point of utility (or loses) ; 
 in imperfect competition stands ' to gain (or stands to lose). 
 
 TV. In perfect competition, if, cateris paribus, the supply on 
 on«^ side — meaning the amount of article offered at each price 
 — if this whole scale of offers is increased on one side, whether 
 froiD increase of numbers on that side or otherwise, then the 
 other side gains : and an analogous proposition is true of im- 
 perfect competition. 
 
 The last two theorems have important exceptions mostly 
 requiring mathematical analysis for their investigation ; those, 
 for instance, which may be presented by Mr. Marshall's second 
 class of curves (if the introduced change might cause a jump 
 from the neighbourhood of the first intersection of demand- 
 curves to that of the third). 
 
 The preceding and the many similar abstract theorems are im- 
 portant as well as those historical inquiries on which Mr. Leslie ^ 
 lays so much stress. It suffices to say that on a form of the 
 third theorem J. S. Mill propounded his counsels to the wage- 
 earning classes, and shaped and re-shaped the policy of millions 
 upon a theory of capital-supply, at first affected with what may 
 perhaps be called the special^ vice of unsjonbolical Economics, 
 at length * corrected, and after all ^ imperfectly because ungeo- 
 metrically apprehended. 
 
 It is easy with Caimes protesting against the identification 
 of Labour with commodities to say : ^ * Verbal generalizations 
 are of com^se easy,' and the equation of Demand to Supply is 
 * what any costermonger wiU tell you.' But the noblte coster- 
 monger would not perhaps find it so easy to tell us about Mr. 
 Marshall's Demand-curves Ckbss IL, or other exceptional cases, 
 
 ' Seep. 43. 
 
 ' There is room for all, as Prof. Jbtods points out in a temperate 
 article in the Fortnightly Review. 
 
 ' Above, p. 127. ♦ Revieio of Thornton. 
 
 * Above, p. 5. « Leadintf Principlff, Part II. ch. i. 5 2. 
 
 L 2
 
 148 A1'PE.\DIC£S. 
 
 such lis those which are presented by imperfect competition 
 (trades unions, &.C.). 
 
 Of course it is right to notice differences as well as simila- 
 rities. It is proper to attend to the differentia, as well as the 
 genus of Man ; in particiilar to dwell upon the high moral 
 attributes which distinguish him from other animals. But we 
 must not allow this distinction and the associated moral senti- 
 ments to oppose the unifications of science and our reception 
 of the Darwinian theory. It is very right and proper with Mr. 
 Frederick Harrison ' for high moral purposes to insist that 
 the labourer has not a thing to sell, that the kibour-market 
 is an unhappy figure ; to dwell upon the differentiae * of the 
 contract about labour. But we must not allow ourselves to 
 forget that there is a sense in which the labourer equally with 
 any other contractor Aos a thing to sell, an^ article; that there 
 is an abstract general mathematical theory of contract. 
 
 The need of this sort of generalisation is not imaginary, and 
 an example of the apparent deficiency in this respect of the 
 highest philosophical, without mathematical, analysis may im- 
 pressively conclude these somewhat unmethodical remarks upon 
 method. iMr. Sidgwick discussing the bargain between employer 
 and workman — with less than his usual clearness indeed, yet at 
 least by opposition to the, as it is here submitted, perfectly 
 correct statement of Walker upon wages — states that in un- 
 restricted competition (presumably in what is in these pages 
 called perfect competition) the bargain between employer and 
 workmen is as indeterminate in such a laboiu-market as the 
 bargain' between a single employer and a single workman (our 
 case a). Which is contrary to the first law of contract. 
 
 To have improved upon the statements of Mr. Sidgwick 
 would surely be a sufficient vindication of Mathematical 
 Psf/rhics. 
 
 ' Fortniijhth/ Reuieic. 
 
 '' Hut not to exagjrerate tlieiu, nf» Tliornton perhaps does when he speaks 
 lit" the contintinl perishinrr, the loss duririfr every moment that its 8ale is 
 (h'lftved, of labnrir. For is not tlie same true o^ capital and anything which 
 is for hire — of the use of a cab, -is well as the ljOx)ur of the cabman ? 
 
 ^ Fortnifjhthj Review, 1865.
 
 INDEX. 
 
 ACT 
 
 Action ^'momentum-potential), 
 
 11, 14,89,91 
 Airy, 86-7 
 Aristotle, 55, 75, 89 
 Article of contract, 1 7 
 
 Bain, 60, 62, 92, 99, 126 
 Barratt, 58, 65, 79, 80, 81 
 Beccaria, 117 
 Bentham, 52, 98, 117, 128, 129, 
 
 133 
 Buffon, 77 
 Burke, 78, 79 
 Butler, 82 
 
 C'AiBNES, 44, 94, 119, 126, 127, 
 
 133, 147 
 Capacity, 57-59 
 Capital, 31-33 
 Combination, 19, 43-48 
 Competition, 17 
 Comte, 85, 91, 109 
 Contract, 17, 21 
 Cooperative Association, 17, 49 
 Cost of Labour, 121 
 Courcelle-Seneuil, 30, 44, 109 
 Coumot, 40, 47, 83 
 Crompton, 134 
 Cunynghame, 96, 121 
 
 Dafwin (G. H.), 91 
 
 IND 
 
 Darwinian, 74, 132 
 Delboeuf, 60, 62 
 
 Demand-Curve, 38-42; Appen- 
 dix V. 
 Determinate, 19 
 Doubleday, 72 
 
 Equality, 81, 99 
 Euclid, 57 
 
 Fawcett, 44 
 Field of Competition, 1 7 
 Final (settlement), 19 
 Fourier, 59, 81 
 
 Galton, 70, 72, 132 
 Gossen, 94 
 Green, 76 
 Grote (John), 76 
 
 Hamilton (W. Rowan), 11, 94 
 Harrison (Frederick), 135, 148 
 Hegel, 97 
 Helvetius, 128 
 Holyoake, 52 
 Hume, 78 
 
 Indeterminate, 19 
 Jndifference-curvc, 21
 
 150 
 
 INDEX. 
 
 JEV 
 
 Jevons, 1, 7, 21, 30, 31. 33, 34, 
 39, 61, 83-87, Appendix III., 
 98; Appendix V., 118, 120, 
 126 
 
 Lagrange, 10, 13, 94 
 
 Laplace, 7, 62, 134 
 
 Leslie (Cliffe), 47, 135, 14G, 147 
 
 Lewes, 13 
 
 Lewis (Comewall), 126 
 
 Maltiiusian, 68, 73 
 
 Marshall, 5, 26, 30-33, 39, 42, 
 
 46, 92, 96, 105-109, 118, 125, 
 
 130, 137, 138, 147 
 Maximum-Principle, 9-15, 24 
 Maxwell (Clerk), 1 3, 76, 80, 90, 
 
 91, 100, 101, 123, 134 
 Means, 57 
 Mill, 5, 12, 52, 54, 75, 81, 82, 
 
 95, 98, 118, 126, 129, 132, 
 
 145, 147 
 Morley (John), 138 
 
 Napoleon, 134 
 Newton, 97 
 
 OVTEN, 81 
 
 WUN 
 
 Recontract, 17 
 Rent, 135-146 
 Ricardo, 135, 145 
 Rousseau, 78 
 
 Saturday Reviewer of Professor 
 Jevons, 83-86 ; Appendix V., 
 118, 120, 126 
 
 Settlement, 19 
 
 Sid^wiek, 16, 32, 33, 52, 62, 
 76-81, 93, 96, 98; Appendix 
 IV., 124, 126, 127, 148 
 
 Spencer (Herbert), 12, 51, 72, 
 75, 103-104, 122, 133-134 
 
 Stewart (Balfour), 14 
 
 Stewart (Dugald), 51 
 
 Stranch, 92 
 
 Sully, 58, 72, 100 
 
 Tait and Thomson, 4, 7, 85, 124 
 Thompson {on Wealth), 59, 129 
 Thornton, 5, 48, 148 
 Todhunter, 6, 55, 92, 93, 111, 
 
 112, 142 
 Ti-ades-Union. See *■ Gombina- 
 
 tion.' 
 
 Variations, Calculus of, 109. 
 
 Ste ' Todhunter ' 
 Venn, 61 
 
 Perfect (Competition), 18 
 riato, 4, 51, 94, 126 
 jPre/erence-Curve, 22 
 Price, 31, 48, 143 n. See De- 
 mand and Supply 
 Priestley, 117 
 
 Walker {on Wagas), 54 
 Walras, 5, 26, 30-32, 40, 42, 46, 
 
 47, 105, 119, 125, 127 
 Watson and Burbury, 6, 10, 90, 
 
 123 
 Wundt, 7, 60, 62. 75
 
 UNIVERSITY OF r« " ' "V 
 
 <^ t iL/fOKiVIA E7r- 
 ' Angeles 
 
 'ast datf 
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
 Los Angeles 
 
 This book is DUE on the last date stamped below. 
 
 Mt^f^. , 
 
 ■^^KW' 
 
 oeci ft^ 
 
 
 ifSW-lh" 
 
 
 FEB 2 7 1990 
 
 
 ,!f, MA«:'©s*99I 
 
 
 APR 29^ 
 
 
 APR 21 m 
 
 

 
 3 1158 00005 4444