IV.

Continuation of the Subject.Peculiar and supereminent Advantages possessed by Mathematicians, in conse quence of their definite Phraseology.

If the remarks contained in the foregoing articles of this section be just, it will follow, that the various artifi cial aids to our reasoning powers which have been projected by Leibnitz and others, proceed on the supposition (a supposition which is also tacitly assumed in the syllogistic theory) that, in all the sciences, the words which we em ploy have, in the course of our previous studies, been brought to a sense as unequivocal as the phraseology of mathematicians. They proceed on the supposition, therefore, that by far the most difficult part of the logical problem has been already solved. Should the period ever arrive, when the language of moralists and politicians shall he rendered as perfect as that of geometers and algebraists, then, indeed, may such contrivances as the Ars Combinatoria and the Alphabet of human thoughts, become interesting subjects of philosophical discussion; although the probability is, that, even were that era to take place, they would be found nearly as useless in morals and politics, as the syllogistic art is acknowledged to be at present, in the investigations of pure geometry.

Of the peculiar and supereminent advantage possessed by mathematicians, in consequence of those fixed and de

finite relations which form the objects of their science, and the correspondent precision in their language and reasonings, I can think of no illustration more striking than what is afforded by Dr. Halley's Latin version from an Arabic manuscript, of the two books of Appollonius Pergæus de Sectione Rationis. The extraordinary circumstances under which this version was attempted and completed (which I presume are little known beyond the narrow circle of mathematical readers) appear to me so highly curious, considered as matter of literary history, that I shall copy a short detail of them from Halley's preface.

After mentioning the accidental discovery in the Bodleian library, by Dr. Bernard, Savilian Professor of astronomy, of the Arabic version of Appollonius, wepi 2078 añoτoμns, Dr. Halley proceeds thus :

"Delighted, therefore, with the discovery of such a treasure, BERNARD applied himself diligently to the task of a Latin translation. But before he had finished a tenth part of his undertaking, he abandoned it altogether, either from his experience of its growing difficulties, or from the pressure of other avocations. Afterwards, when, on the death of Dr. Wallis, the Savilian professorship was bestowed on me, I was seized with a strong desire of making a trial to complete what Bernard had begun; an attempt, of the boldness of which the reader may judge, when he is informed, that, in addition to my own entire ignorance of the Arabic language, I had to contend with the obscurities occasioned by innumerable passages which were either defaced or altogether obliterated. With the assistance, however, of the sheets

which Bernard had left, and which served me as a key for investigating the sense of the original, I began first with making a list of those words, the signification of which his version had clearly ascertained; and then proceeded, by comparing these words, wherever they occurred, with the train of reasoning in which they were involved, to decypher, by slow degrees, the import of the context; till at last I succeeded in mastering the whole work, and in bringing my translation (without the aid of any other person) to the form in which I now give it to the public."*

When a similar attempt shall be made with equal success, in decyphering a moral or a political treatise written in an unknown tongue, then, and not till then, may we think of comparing the phraseology of these two sciences with the simple and rigorous language of the Greek geometers; or with the more refined and abstract, but not less scrupulously logical system of signs, employed by modern mathematicians.

It must not, however, be imagined, that it is solely by the nature of the ideas which form the objects of its reasonings, even when combined with the precision and unambiguity of its phraseology, that mathematics is distinguished from the other branches of our knowledge. The truths about which it is conversant, are of an order altogether peculiar and singular; and the evidence of which they admit resembles nothing, either in degree or in kind, to which the same name is given, in any of our other in

* Appollon. Perg. de Sectione Rationis, &c. Opera et Studio Edm. Hal ley. Oxon. 1706. In Præfat.

tellectual pursuits. On these points also, Leibnitz and many other great men, have adopted very incorrect opinions; and, by the authority of their names, have given currency to some logical errors of fundamental importance. My reasons for so thinking, I shall state as clearly and fully as I can, in the following section.

SECTION III.

Of Mathematical Demonstration.

I.

Of the Circumstance on which Demonstrative Evidence essentially depends.

THE peculiarity of that species of evidence which is called demonstrative, and which so remarkably distin guishes our mathematical conclusions from those to which we are led in other branches of science, is a fact which must have arrested the attention of every person who possesses the slightest acquaintance with the elements of geometry. And yet I am doubtful if a satisfactory ac count has been hitherto given of the circumstance from which it arises. Mr. Locke tells us, that "what constitutes a demonstration is intuitive evidence at every step;" and I readily grant, that if in a single step such evidence should fail, the other parts of the demonstration would be of no value. It does not, however, seem to me that it is on this consideration that the demonstrative evidence of the conclusion depends,-not even when we add

to it another which is much insisted on by Dr. Reid,that, "in demonstrative evidence, our first principles must be intuitively certain." The inaccuracy of this remark I formerly pointed out when treating of the evidence of axioms; on which occasion I also observed, that the first principles of our reasonings in mathematics are not axioms, but definitions. It is in this last circumstance (I mean the peculiarity of reasoning from definitions) that the true theory of mathematical demonstration is to be found; and I shall accordingly endeavour to explain it at considerable length, and to state some of the more important consequences to which it leads.

That I may not, however, have the appearance of claiming in behalf of the following discussion, an undue share of originality, it is necessary for me to remark, that the leading idea which it contains has been repeatedly started, and even to a certain length prosecuted, by different writers, ancient as well as modern; but that, in all of them, it has been so blended with collateral considerations, altogether foreign to the point in question, as to divert the attention both of writer and reader, from that single principle on which the solution of the problem hinges. The advantages which mathematics derives from the peculiar nature of those relations about which it is conversant; from its simple and definite phraseology; and from the severe logic so admirably displayed in the concatenation of its innumerable theorems, are indeed immense, and well entitled to a separate and ample illustration; but they do not appear to have any necessary connexion with the subject of this section. How far I am right in this opinion, my readers will be enabled to judge by the sequel.

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