of mathematical reasoning, and the amazing feats which it has performed in its progressive career, from the vast extent to which it can be carried, and its wonderful effects in its application to some parts of physical learning, philosophers ancient and modern have not only held it in a just respect and veneration, but have been so enamoured of its beauty as to embrace and adopt it as the praxis and exemplar of universal logic'. This is a mis

9" Thus we have taken a short view of the so much celebrated method of the mathematicians; which, to any one who considers it with proper attention, must needs appear universal, and equally applicable in other sciences. They begin with definitions. From these they deduce their axioms and postulates, which serve as principles of reasoning; and having thus laid a firm foundation advance to theorems and problems, establishing all by the strictest rules of demonstration. The corollaries flow naturally and of themselves. And if any particulars are still wanting, to illustrate a subject, or complete the reader's information, these that the series of reasoning may not be interrupted or broken are generally thrown into scholia. In a system of knowledge so uniform and well connected, no wonder if we meet with certainty, and if those clouds and darkness which deface other parts of human science and bring discredit even upon reason itself are here scattered and disappear."-Duncan's Logic, p. 188. See also p. 224. It was the great error of Aristotle's logic, that on this sole foundation he laboured to erect a universal instrument or organon for the investigation of truth in all other parts of learning, though springing from foundations very different and distinct from mathematical axioms.

And

take, fatal to the success of all other parts of knowledge, upon which I shall reserve myself to remark more particularly in some future stage of this work. For the present I shall only observe, that in this demonstrative reasoning, not only the middle terms and propositions are general, but that all other terms and propositions are also general. here likewise I beg leave to appeal to the authority of Dr. Reid, who allows both the ancient and modern logic to be defective as an universal art, whether-" the ancients, who attended only to categorical propositions which have one subject and one predicate, and of these, to such only as have a general term for their subject","—were not misled in their logic by the mathematics? And also whether" the moderns, who have been led to attend only to relative propositions, which express a relation between two subjects, and these subjects always general ideas","—were not likewise misled by the mathematics, when they founded the principle of their new

"Dr. Reid in the Appendix to Lord Kaims's third volume of Sketches p. 328.

" Ibid.

12

logic upon the axiom, "Things that agree with one and the same, agree between themselves "?" Hence they have confined their reasoning to general relations, and to the agreement and disagreement of ideas of quality as well as of quantity, measured by a third, just as a carpenter measures a piece of timber by the application of his rule ".

12 Nonnulli autem logici (nostri seculi, aut superioris) posthabita veterum probatione per "Dictum de omni et de nullo;" aliud substituunt illius loco postulatum, nimirum, "Quæ conveniunt in eodem tertio, conveniunt inter se." Atque ad hanc regulam exigentes singulos syllogismorum modos, inde conclusum eunt justam eorum consecutionem. Quique sic procedunt, negligere possunt eam distinctionem modorum perfectorum et imperfectorum; ut quæ ortum ducit ab ea methodo qua usi si sunt veteres, in probatione sua ab illo dicto.-Wallis's Logic, book iii. chap. 5.

13 Mr. Locke is the great advocate for the perception of the agreement and disagreement of ideas being the criterion of all truth, and in exemplifying this great logical maxim he uses the following words: “When a man has in his mind the idea of two lines, viz. the side and diagonal of a square, whereof the diagonal is an inch long, he may have the idea also of the division of that line into a certain number of equal parts; v. g. into five, ten, a hundred, a thousand, or any other number; and may have the idea of that inch line being divisible or not divisible into such equal parts, as a certain number of them will be equal to the side line. Now, whenever he perceives, believes or supposes such a kind of divisibility to agree or disagree to his idea of that line, he as it were joins or separates those two ideas, viz. the idea of that line, and the idea of that kind of divisibility, and so makes

Of these two logics, both of which are partial and imperfect, the former is entitled to the preference; because when the general principles are once established, it is the guide to truth in all parts of knowledge; whereas, out of mathematics pure or mixed the latter can usefully apply to none. Hence the Aristotelian logic, with all its defects, has been rendered still more deficient by the moderns, from its more extensive misapplication 14.

a mental proposition, which is true or false, according as such a kind of divisibility, a divisibility into such aliquot parts, does really agree to that line or no. When ideas are so put together or separated in the mind, as they or the things they stand for, do agree or not, that is as I may call it, mental truth. But truth of words is something more, and that is the affirming or denying of words one of another, as the ideas they stand for agree or disagree."-Essay, book iv. chap. v. sect. 6.

"On the general subject of this chapter, consult Reid'sAnalysis of Aristotle's Logic; Brown's Philosophy of the Human Mind, lect. 50; Stewart's Elements, vol. ii. chap. 3; Campbell's Philosophy of Rhetoric, &c.-Editor.

SECT. IV.

Of Mathematical Truth.

WHEN such abstract and general ideas

as are appropriated to mathematics in both its branches, which besides the exclusive privileges that have been enumerated are permanent and eternal, are thus syllogistically compared in their numerous relations, and ultimately brought to the test of a few simple axioms or universal propositions which are palpably and self-evidently certain, the truths that result from such an operation of reason must be eminently clear and luminous, bearing down all possibility of doubt, and carrying the most absolute and irresistible conviction. The reason of this greater certainty of mathematical truth is, that all mathematical propositions are acts of mind abstracted from the things themselves, and that the abstract evidence is clearer than that of things whose evidence depends merely on the senses.