proposition aa, when considered as the representative of such a formula as the binomial theorem of Sir Isaac Newton? When applied to the equation 2 + 2 = 4, (which, from its extreme simplicity and familiarity, is apt to be regarded in the light of an axiom,) the paradox does not appear to be so manifestly extravagant; but, in the other case, it seems quite impossible to annex to it any meaning whatever.1

I should scarcely have been induced to dwell so long on this theory of Leibnitz concerning mathematical evidence, if I had not observed among some late logicians (particularly among the followers of Condillac) a growing disposition to extend it to all the different sorts of evidence resulting from the various employments of our reasoning powers. Condillac himself states his own opinion on this point with the most perfect confidence. "L'évidence de raison consiste uniquement dans l'identité: c'est ce que nous avons démontré. Il faut que cette vérité soit bien simple pour avoir échappé à tous les philosophes, quoiqu'ils eussent tant d'intérêt à s'assurer de l'évidence, dont ils avoient continuellement le mot dans la bouche."2

The foregoing reasonings are not meant as a refutation of the arguments urged by any one author in support of the doctrine in question, but merely as an examination of those by which I have either heard it defended, or from which I conceived that it might possibly derive its verisimilitude in the judgment of those who have adopted it. The arguments which I have supposed to be alleged by its advocates, are so completely independent of each other, that instead of being regarded as different premises leading to the same conclusion, they amount only to so many different interpretations of the same verbal proposition; -a circumstance which, I cannot help thinking, affords of itself no slight proof, that this proposition has been commonly stated in terms too general and too ambiguous for a logical principle. What a strange

inference has been drawn from it by no less a philosopher than Diderot! "Interrogez des mathématiciens de bonne foi, et ils vous avoueront que leurs propositions sont toutes identiques, et que tant de volumes sur le cercle, par exemple, se réduisent à nous répéter en cent mille façons différentes, que c'est une figure où toutes les lignes tirées du centre à la circonférence sont égales. Nous ne savons donc presque rien.”Lettre sur les Aveugles.

2 La Logique, chap. ix.

On another occasion, Condiilac expresses himself thus: "Tout le système des connoissances humaines peut être rendu par une expression plus abrégée et tout-a-fait identique: les sensations sont des sensations. Si nous pouvions, dans toutes les sciences, suivre également la génération des idées, et saisir le vrai système des choses, nous verrions

The demonstration here alluded to is extremely concise; and if we grant the two data on which it proceeds, must be universally acknowledged to be irresistible. The first is, "That the evidence of every mathematical equation is that of identity:" The second, "That what are called, in the other sciences, propositions or judgments, are, at bottom, precisely of the same nature with equations." But it is proper, on this occasion, to let our author speak for himself.

"Mais, dira-t-on, c'est ainsi qu'on raisonne en mathématiques, où le raisonnement se fait avec des équations. En sera-t-il de même dans les autres sciences, où le raisonnement se fait avec des propositions? Je réponds qu' équations, propositions, jugemens, sont au fond la même chose, et que par conséquent on raisonne de la même manière dans toutes les sciences."1

Upon this demonstration I have no comment to offer. The truth of the first assumption has been already examined at sufficient length; and the second (which is only Locke's very erroneous account of judgment, stated in terms incomparably more exceptionable) is too puerile to admit of refutation. It is melancholy to reflect, that a writer who, in his earlier years, had so admirably unfolded the mighty influence of language upon our speculative conclusions, should have left behind him, in one of his latest publications, so memorable an illustration of his own favourite doctrine.

It was manifestly with a view to the more complete establishment of the same theory, that Condillac undertook a work, which has appeared since his death, under the title of La Langue des Calculs; and which, we are told by the editors, was only meant as a prelude to other labours, more interesting and more difficult. From the circumstances which they have stated, it would seem that the intention of the author was to extend to all the other branches of knowledge, inferences similar to those which he has here endeavoured to establish with respect to

d'une vérité naître toutes les autres, et nous trouverions l'expression abrégée de tout ce que nous saurions, dans cette

proposition identique: le même est le même."

La Logique, chap. viii.

mathematical calculations; and much regret is expressed by his friends, that he had not lived to accomplish a design of such incalculable importance to human happiness. I believe I may safely venture to assert, that it was fortunate for his reputation he proceeded no farther; as the sequel must, from the nature of the subject, have afforded, to every competent judge, an experimental and palpable proof of the vagueness and fallaciousness of those views by which the undertaking was suggested. In his posthumous volume, the mathematical precision and perspicuity of his details appear to a superficial reader to reflect some part of their own light on the general reasonings with which they are blended; while, to better judges, these reasonings come recommended with many advantages, and with much additional authority, from their coincidence with the doctrines of the Leibnitian school.

It would probably have been not a little mortifying to this most ingenious and respectable philosopher, to have discovered, that, in attempting to generalize a very celebrated theory of Leibnitz, he had stumbled upon an obsolete conceit, started in this island upwards of a century before. "When a man reasoneth," says Hobbes, "he does nothing else but conceive a sum total, from addition of parcels; or conceive a remainder, from subtraction of one sum from another; which (if it be done by words) is conceiving of the consequence of the names of all the parts, to the name of the whole; or from the names of the whole and one part, to the name of the other part. These operations are not incident to numbers only, but to all manner of things that can be added together, and taken one out of another. In sum, in what matter soever there is place for addition and subtraction, there also is place for reason; and where these have no place, there reason has nothing at all to do.

"Out of all which we may define what that is which is meant by the word reason, when we reckon it amongst the faculties of the mind. For reason, in this sense, is nothing but reckoning (that is, adding and subtracting) of the consequences of general names agreed upon, for the marking and signifying of our thoughts;—I say marking them, when we reckon by ourselves;

and signifying, when we demonstrate or approve our reckonings to other men."1

Agreeably to this definition, Hobbes has given to the first part of his Elements of Philosophy the title of COMPUTATIO, sive LOGICA;' evidently employing these two words as precisely synonymous. From this tract I shall quote a short paragraph, not certainly on account of its intrinsic value, but in consequence of the interest which it derives from its coincidence with the speculations of some of our contemporaries. I transcribe it from the Latin edition, as the antiquated English of the author is apt to puzzle readers not familiarized to the peculiarities of his philosophical diction.

"Per ratiocinationem autem intelligo computationem. Computare vero est plurium rerum simul additarum summam colligere, vel unâ re ab aliâ detractâ, cognoscere residuum. Ratiocinari igitur idem est quod addere et subtrahere, vel si quis adjungat his multiplicare et dividere, non abnuam, cum multiplicatio idem sit quod æqualium additio, divisio quod æqualium quoties fieri potest subtractio. Recidit itaque ratiocinatio omnis ad duas operationes animi, additionem et subtractionem."2 How wonderfully does this jargon agree with the assertion of Condillac, that all equations are propositions, and all propositions equations!

These speculations, however, of Condillac and of Hobbes relate to reasoning in general, and it is with mathematical reasoning alone that we are immediately concerned at present. That the peculiar evidence with which this is accompanied is not resolvable into the perception of identity, has, I flatter myself, been sufficiently proved in the beginning of this article; and the plausible extension by Condillac of the very same

1 Leviathan, chap. v.

2 [Logica, cap. i. 2.]-The Logica of Hobbes has been lately translated into French, under the title of Calcul, ou Logique, by M. Destutt-Tracy. It is annexed to the third volume of his Elémens d'Idéologie, [1805,] where it is honoured with the highest eulogies

by the ingenious translator. "L'ouvrage en masse," he observes in one passage, "mérite d'être regardé comme un produit précieux des méditations de Bacon et de Descartes sur le système d'Aristote, et comme le germe des progrès ultérieures de la science."-Disc. Prél. p. 117.

theory to our reasonings in all the different branches of moral science, affords a strong additional presumption in favour of our conclusion.

From this long digression into which I have been insensibly led by the errors of some illustrious foreigners concerning the nature of mathematical demonstration, I now return to a further examination of the distinction between sciences which rest ultimately on facts, and those in which definitions or hypotheses are the sole principles of our reasonings.

[SUBSECTION] III.-Continuation of the Subject.-Evidence of the Mechanical Philosophy, not to be confounded with that which is properly called Demonstrative or Mathematical.-Opposite Error of some late Writers.

Next to geometry and arithmetic, in point of evidence and certainty, is that branch of general physics which is now called mechanical philosophy;-a science in which the progress of discovery has been astonishingly rapid, during the course of the last century; and which, in the systematical concatenation and filiation of its elementary principles, exhibits every day more and more of that logical simplicity and elegance which we admire in the works of the Greek mathematicians. It may, I think, be fairly questioned, whether in this department of knowledge, the affectation of mathematical method has not been already carried to an excess; the essential distinction between mechanical and mathematical truths being, in many of the physical systems which have lately appeared on the Continent, studiously kept out of the reader's view, by exhibiting both, as nearly as possible, in the same form. A variety of circumstances, indeed, conspire to identify in the imagination, and, of consequence, to assimilate in the mode of their statement, these two very different classes of propositions; but as this assimilation (besides its obvious tendency to involve experimental facts in metaphysical mystery) is apt occasionally to lead to very erroneous logical conclusions, it becomes the more necessary, in proportion as it arises from a natural bias, to