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been so fully qualified as he was to pronounce on the characteristical merits of both.
"I have often said, that, after the writings of geometricians, "there exists nothing which, in point of force and of subtilty, can "be compared to the works of the Roman lawyers. And, as it "would be scarcely possible, from mere intrinsic evidence, to dis"tinguish a demonstration of Euclid's from one of Archimedes or
(the style of of
"than if reason herself was speaking through their organs,) so also in all "the Roman lawyers all resemble each other like twin-brothers; in "so much that, from the style alone of any particular opinion or "argument, hardly any conjecture could be formed with respect to "the author. Nor are the traces of a refined and deeply meditat"ed system of natural jurisprudence any where to be found more "visible, or in greater abundance. And, even in those cases where "its principles are departed from, either in compliance with the "language consecrated by technical forms, or in consequence of "new statutes, or of ancient traditions, the conclusions which the "assumed hypothesis renders it necessary to incorporate with the "eternal dictates of right reason, are deduced with the soundest "logic, and with an ingenuity which excites admiration. Nor are "these deviations from the law of nature so frequent as is common"ly imagined." 11*
I have quoted this passage merely as an illustration of the analogy already alluded to, ween the systematical unity of mathematical science, and that which is conceivable in a system of municipal law. How far this unity is exemplified in the Roman code, I leave to be determined by more competent judges.†
As something analogous to the hypothetical or conditional conclusions of mathematics may thus be fancied to take place in speculations concerning moral or political subjects, and actually does take place in theoretical mechanics; so, on the other hand, if a A general
mathematician should affirm of a general property of the circle, that it applies to a particular figure described on paper, he would at once degrade a geometrical theorem to the level of a fact resting start mas ultimately on the evidence of our imperfect senses. The accuracy ong noi of his reasoning could never bestow on his proposition that peculiar applicable evidence which is properly called mathematical, as long as the fact to a parti remained uncertain, whether all the straight lines drawn from the law dia. centre to the circumference of the figure were mathematically equal.
⚫Leibnitz, Op. Tom. IV. p. 254.
It is not a little curious that the same code which furnished to this very learned and philosophical jurist, the subject of the eulogium quoted above, should have been lately stigmatized by an English lawyer, eminently distinguished for his acuteness and originality, as "an enormous mass of confusion and inconsistency." Making all due allowances for the exaggerations of Leibnitz, it is difficult to conceive that his opinion, on a subject which he had so profoundly studied, should be so very widely at variance with the truth. 12
These observations lead me to remark a very common misconI common ception concerning mathematical definitions; which are of a nature different from definitions in of the
misappre-essentially diff. It is usual for writers on logic, after taking notice
ension ret of the errors to which we are liable in consequence of the ambiguity tive to of words, to appeal to the example of mathematicians, as a proof of nath: defin: the infinite advantage of using, in our reasonings, such expressions only as have been carefully defined. Various remarks to this purpose occur in the writings both of Mr. Locke and of Dr. Reid. But the example of mathematicians is by no means applicable to the sciences in which these eminent philosophers propose that it should be followed; and, indeed, if it were copied as a model in any other branch of human knowledge, it would lead to errors fully as dangerous as any which result from the imperfections of language. The real fact is, that it has been copied much more than it ought to have been, or than would have been attempted, if the peculiarities of mathematical evidence had been attentively considered.
That in mathematics, there is no such thing as an ambiguous word, and that it is to the proper use of definitions we are indebted for this advantage, must unquestionably be granted. But this is an advantage easily secured, in consequence of the very limited vocabumoth lary of mathematicians, and the distinctness of the ideas about which their reasonings are employed. The difference, besides, in this wn word, their reasonings are employed. respect, between mathematics and the other sciences, however mone great, is yet only a difference in degree; and is by no means suffiistinct cient to account for the essential distinction which every person ideos must perceive between the irresistible cogency of a mathematical demonstration, and that of any other process of reasoning.
From the foregoing considerations it appears, that, in mathemation, tics, definitions answer two purposes; first, To prevent ambiguities of language; and, secondly, To serve as the principles of our two reasoning. It appears further, that it is to the latter of these circumstances (I mean to the employment of hypotheses instead of facts, as the data on which we proceed) that the peculiar force of demonstrative evidence is to be ascribed. It is however only in the former use of definitions, that any parallel can be drawn between mathematics and those branches of knowledge which relate to facts; and, therefore, it is not a fair argument in proof of their general utility, to appeal to the unrivalled certainty of mathematical science, -a pre-eminence which that science derives from a source altogether different, though comprehended under the same name, and which she will forever claim as her own exclusive prerogative.*
dly. yin finitions.
Nor ought it to be forgotten, that it is in pure mathematics alone, that definitions can be attempted with propriety at the outset ale com of our investigations. In most other instances, some previous discuswith
• These two classes of definitions are very generally confounded by logicians; among others, by the Abbé de Condillac. See La Logique, ou les premiers developpemens de 'Art de Penser, Chap. VI.
sion is necessary to shew, that the definitions which we lay down correspond with facts; and, in many cases, the formation of a just definition is the end to which our inquiries are directed. It is very Burke's judiciously observed by Mr. Burke, in his Essay on Taste, that "when we define, we are in danger of circumscribing nature within remarks "the bounds of our own notions, which we often take up by hazard, "or embrace on trust, or form out of a limited and partial consider"ation of the object before us, instead of extending our ideas to take "in all that nature comprehends, according to her manner of com"bining. We are limited in our inquiry by the strict laws to which "we have submitted at our setting out."
The same author adds, that " a definition may be very exact, and "yet go but a very little way towards informing us of the nature of "the thing defined ;" and that, "in the order of things, a definition "(let its virtue be what it will) ought rather to follow than to "precede our inquiries, of which it ought to be considered as the "result."
From a want of attention to these circumstances, and from a blind imitation of the mathematical arrangement, in speculations where facts are involved among the principles of our reasonings, numberless errors in t writings of philosophers might be easily traced. The subject is too great extent to be pursued any farther here; but it is well entitled to the examination of all who may turn their thoughts to the reformation of logic. That the ideas of Aristotle himself, with respect to it, were not very precise, must, I think, be granted, if the following statement of his ingenious commentator be admitted as correct.
"Every general term (says Dr. Gillies) is considered by Aris- Amstette "totle as the abridgment of a definition; and every definition is "denominated by him a collection, because it is the result always "of observation and comparison, and often of many observations "and of many comparisons."*
These two propositions will be found, upon examination, not very tritiuen consistent with each other. The first, "That every general term "is the abridgment of a definition," applies indeed admirably to mathematics; and touches with singular precision on the very circumstance which constitutes (in my opinion) the peculiar cogency of mathematical reasoning. But it is to mathematics that it applies exclusively. If adopted as a logical maxim in other branches of knowledge, it would prove an endless source of sophistry and error. -The second proposition, on the other hand, "That every defini❝tion is the result of observation and comparison, and often of many "observations and many comparisons;" however applicable to the definitions of natural history, and of other sciences which relate to facts, cannot, in one single instance, apply to the definitions of geometry; in as much as these definitions are neither the result of observations nor of comparisons, but the hypotheses, or first principles, on which the whole science rests.
Gillies's Aristotle, Vol. I. p. 92. 2d edit.
If the foregoing account of demonstrative evidence be just, it follows, that no chain of reasoning whatever can deserve the name very dle of a demonstration, (at least in the mathematical sense of that word) monstration which is not ultimately resolvable into hypotheses or definitions. been already shown, that this is the case with geometry: and solvable it is also manifestly the case with arithmetic, another science to to hypoth which, in common with geometry, we apply the word mathematical. The simple arithmetical equations 2 + 2 = 4; 2 + 3 = 5, and other def elementary propositions of the same sort, are (as was formerly nitions observed) mere definitions ;t perfectly analogous, in this respect to
those at the beginning of Euclid; and it is from a few fundamental principles of this sort, or at least from principles which are essentially of the same description, that all the more complicated results in the science are derived.
To this general conclusion, with respect to the nature of mathematical demonstration, an exception may perhaps be, at first sight, apprehended to occur, in our reasonings concerning geometrical problems; all of these reasonings (as is well known) resting ultimately upon a particular class of principles called postulates, which are commonly understood to be so very nearly akin to axioms, that both might, without impropriety, be comprehended under the same name. "The definition of a postulate (says the learned and ingenious Dr. Hutton) will nearly agree also to an axiom, which is 66 a self-evident theorem, as a postulate is a self-evident problem."‡ The same author, in another part of his work, quotes a remark from Dr. Barrow, that "there is the same affinity between postu"lates and problems, as between axioms and theorems."§ Dr. Wallis, too, appears, from the following passage, to have had a decided leaning to this opinion: "According to some, the difference "between axioms and postulates is analogous to that between theo"rems and problems; the former expressing truths which are self“evident, and from which other propositions may be deduced; the "latter, operations which may be easily performed, and by the help "of which more difficult constructions may be effected." He afterwards adds, "This account of the distinction between postulates "and axioms seems not ill adapted to the division of mathematical "propositions into problems and theorems. And indeed, if both "postulates and axioms were to be comprehended under either of
* Although the account given by Locke of what constitutes a demonstration, be different from that which I have here proposed, he admits the converse of this doctrine as manifest; viz. That if we reason accurately from our own definitions, our conclusions will possess demonstrative evidence; and “hence (he observes with great truth) it comes to pass, that one may often meet with very clear and coherent discourses, that amount yet to nothing." He afterwards remarks, that" one may make demonstrations and undoubted propositions in words, and yet thereby advance not one jot in the knowledge of the truth of things." "Of this sort (he adds) a man may find an infinite number of propositions, reasonings, and conclusions, in books of metaphysics, school-divinity, and some sort of natural philosophy; and, after all, know as little of God, spirits, or bodies, as he did before he set out."-Essay on Human Understanding, Book IV. chap. viii.
+ See page 25.
Mathematical Dictionary, Art. Postulate.
"these names, the innovation would not, in my opinion, afford much
"ground for censure."*
In opposition to these very high authorities I have no hesitation Author's to assert, that it is with the definitions of Euclid, and not with the axioms, that the postulates ought to be compared, in respect of remark their logical character and importance ;-in as much as all the demonstrations in plane geometry are ultimately founded on the former, and all the constructions which it recognizes as legitimate, may be resolved ultimately into the latter. To this remark it may be added, that, according to Euclid's view of the subject, the problems of geometry are not less hypothetical and speculative than the theorems; the possibility of drawing a mathematical straight line, and of describing a mathematical circle, being assumed in the construction of every problem, in a way quite analogous to that in which the enunciation of a theorem assumes the existence of straight lines and of circles corresponding to their mathematical definitions. The reasoning, therefore, on which the solution of a problem rests, is not less demonstrative than that which is employed in proof of a theorem. Grant the possibility of the three operations described in the postulates, and the correctness of the solution is as mathematically certain, as the truth of any property of the triangle or of the circle. The three postulates of Euclid are, indeed, nothing more than the definitions of a circle and a straight line thrown into a form somewhat different; and a similar remark may be extended to the corresponding distribution of propositions into theorems and problems. Notwithstanding the many conveniences with which this distribution is attended, it was evidently a matter of choice rather than of necessity; all the truths of geometry easily admitting of being moulded into either shape, according to the fancy of the mathematician. As to the axioms, there cannot be a doubt 'whatever opinion may be entertained of their utility, or of their insignificance) that they stand precisely in the same relation to both classes of propositions.†
* Wallisii Opera, Vol. II. pp. 667, 668.
In further illustration of what is said above, on the subject of postulates and of problems, I transcribe, with pleasure, a short passage from a learned and interesting memoir, just published, by an author intimately and critically conversant with the classical remains of Greek geometry.
"The description of any geometrical line from the data by which it is defined, must always be assumed as possible, and is admitted as the legitimate means of a geometrical construction: it is therefore properly regarded as a postulate Thus, the description of a straight line and of a circle are the postulates of plane geometry assumed by Euclid The description of the three conic sections, according to the definitions of them, must also be regarded as postulates; and though not formally stated like those of Euclid, are in truth admitted as such by Apollonius, and all other writers on this branch of geometry. The same principle must be extended to all superior lines.
"It is true, however, that the properties of such superior lines may be treated of, and the description of them may be assumed in the solution of problems, without an actual delineation of them.-For it must be observed, that no lines whatever, not even the straight line or circle, can be truly represented to the senses according to the strict mathematical definitions; but this by no means affects the theoretical conclusions which