Thus to Axioms he adds another class of Propofitions called Demonstrations, which, though less general, are of equal force, and which he applies, in the same way and by the same process, to the proof of relations which lie more distant and concealed. And, as it is the peculiar privilege of his science, that all its ideas are general. and these general ideas inexhaustible, in pursuing all their various and multiplex relations he can produce many Demonstrations : which Axioms and Demonstrations he can apply by the fame Syllogistic process, to the proof of theorem after theorem almost ad infini quod tam per fe evidens præsumitur, ut probatione non indigeat. Nimirum, Quicquid de Subjecto quopiam univerfalitur Affirmatur vel Negatur, id fimiliter vel Affirmatur vil Negatur de omni co de quo hoc fubjeétum dicitur. Utputa, Quicquid universaliter affirmatur aut negatur de Animali; fimiliter affirmatur vel negatur de quopiam Animali, seu de omni eo quod est Animal: puto de Homine, de Bruto, de Alexandro, de Bucephalo, alioque quopiam Animali. Wallis's Logic, B. iii. C. 5: And, if the reader would see at one short view the whole jet and force of all Syllogistic reasoning, he cannot do better than read this chapter De fundamento Syllogifmi ; et, Modis Figure Prima. H 2 tum; tum :' and which Syllogistic process is, (to express it in a few words) To reduce general Co The relations of quantity are so susceptible of exact • mensuration, that long trains of accurate reasoning on that subject may be formed, and conclusions drawn very remote from the first principles. It is in this science and those which depend upon it, that the power of reasoning triumphs ; in other matters its trophies are inconsidera"ble. If any man doubts this, let him produce, in any < subject unconnected with mathematics, a train of rea• loning of some length, leading to a conclusion, which without this train of reasoning would never have been • brought within human fight. Every man acquainted (with Mathematics can produce thousands of such trains • of reasoning. I do not say that none such can be pro• duced in other fciences.' Dr. Reid's Appendix to Lord Kaimss Sketches, p. 281. I think Dr. Reid might have pronounced that no such lengthened trains of Reasoning can be produced in other Sciences. And hence it is that SYLLOGISM, which is Mathematical and constitutes the Aristotelian Logic, is of very little use in other parts of learning. Upon this ground the following observation of the same author is very just. “The Ancients seem to have had too high no. • tions, both of the force of the reasoning power in man, cand of the art of syllogism as its guide. Mere reasoning *[fyllogistic) can carry us but a very little way in most • subjects. By observation and experiments properly con ducted, the stock of human knowledge may be enlarged without end; but the power of reasoning alone, applied with vigour through a long life, would only carry a man 'round, like a horse in a mill who labours hard, but makes no progress. Ibid. p. 381. truths truths under more general, till they terminate in Axioms, which are the most general.. Such is the Method of Science or DEMONSTRATION, (belonging, I think, to Quantity alone, 4) which has been justly celebrated and admired through every age, in which Reason advances, by a sublime intelle&ual motion, from the simplest Axioms to the most complicated speculations, and exhibits truth springing out of its first and purest elements, and rising from story to story in a most elegant progressive way, into a luminous and extensive fabric. The certainty of self-evidence attends it through every stage, and every link of the Mathematical chain is of equal, that is, the utmost, strength. * See chap. iv. §. 2. of this volume. · Here I am under the necessity of differing in opinion from Mr. Locke, who thinks that Demonstration is not confined to Quantity. See Essay B. IV. C. ii. 8.9. and B. IV. C. iii. §. 18. I shall have occasion to confider this opinion of this great man in some future part of these Lectures. H 3 From From the fingular elegance and precision of MATHEMATICAL REASONING, and the amazing feats which it has performed in its progressive career, and form its wonderful effects in its application to some parts of physical learning, philosophers ancient and modern have not only held it in a just refpect 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 mistake, which I shall re Po 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 equa'ly applicable in other Sciences. • They begin with Definitions. From these they deduce their Axioms and Poftulates, which serve as Principles of • Reasoning ; and having thus laid a firm Foundation, ad vance 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 compleat the • Reader's Information ; those, 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 Darknesses, that 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. serve myself to observe upon in some future stage of this work. For the present I shall only remark, that, in this Demonstrative Reasoning, not only the Middle Terms and Propositions are general, but that all other Terms and Propositions are general also : from which observation I beg leave to appeal to the judgment 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 catego"rical propositions, which have one subject and one predicate; and of these to such only as have a general term for their fubject,' s were not milled in their Logic by the Mathematics a 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 milled by the Mathematics, when they founded the new principle of their Logic : Dr. Reid in the Appendix to Lord Kaim’sizd vol, of Sketches, p. 328. Ibid, upon |