Thus to Axioms he adds another clafs of Propofitions called Demonftrations, which, though less general, are of equal force, and which he applies, in the fame way and by the fame procefs, to the proof of relations which lie more diftant and concealed. And, as it is the peculiar privilege of his fcience, that all its ideas are general and these general ideas inexhaustible, in purfuing all their various and multiplex relations he can produce many Demonstrations: which Axioms and Demonftrations he can apply by the fame Syllogiftic procefs, to the proof of theorem after theorem almost ad infini

quod tam per fe evidens præfumitur, ut probatione non indigeat. Nimirum, Quicquid de Subjećto quopiam univerfalitur Affirmatur vel Negatur, id fimiliter vel Affirmatur vil Negatur de omni eo de quo hoc fubjectum dicitur. Utputa, Quicquid univerfaliter affirmatur aut negatur de Animali; fimiliter affirmatur vel negatur de quopiam Animali, feu de omni eo quod eft 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 fee at one fhort view the whole jet and force of all Syllogiftic reasoning, he cannot do better than read this chapter De fundamento Syllogifmi; et, Modis Figura Prima.

tum and which Syllogistic process is, (to exprefs it in a few words) To reduce general

The relations of quantity are so susceptible of exact menfuration, that long trains of accurate reasoning on that fubject may be formed, and conclufions drawn very ' remote from the first principles. It is in this science and thofe which depend upon it, that the power of reasoning ⚫ triumphs; in other matters its trophies are inconfiderable. If any man doubts this, let him produce, in any 'fubject unconnected with mathematics, a train of reafoning of fome length, leading to a conclufion, which without this train of reafoning would never have been brought within human fight. Every man acquainted "with Mathematics can produce thoufands of fuch trains of reasoning. I do not fay that none such can be produced in other fciences.' Dr. Reid's Appendix to Lord Kaims's Sketches, p. 281.

I think Dr. Reid might have pronounced that no fuch lengthened trains of Reasoning can be produced in other Sciences. And hence it is that SYLLOGISM, which is Mathematical and conftitutes the Ariftotelian Logic, is of very little use in other parts of learning. Upon this ground the following obfervation of the fame author is very juft. The Ancients feem to have had too high notions, both of the force of the reasoning power in man, and of the art of fyllogifm as its guide. Mere reafoning [fyllogiftic] can carry us but a very little way in most fubjects. By observation and experiments properly conducted, 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 horfe 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,) which has been justly celebrated and admired through every age, in which Reason advances, by a fublime intellectual motion, from the fimpleft Axioms to the most complicated fpeculations, and exhibits truth fpringing out of its first and pureft elements, and rifing from story to story in a most elegant progreffive way, into a luminous and extenfive fabric. The certainty of felf-evidence attends it through every stage, and every link of the Mathematical chain is of equal, that is, the utmost, ftrength.

See chap. iv. §. 2. of this volume.

• Here I am under the neceffity of differing in opinion from Mr. Locke, who thinks that Demonftration is not confined to Quantity. See Effay B. IV. C. ii. §. 9. and B. IV. C. iii. §. 18. I fhall have occafion to confider this opinion of this great man in fome future part of thefe Lectures.

From the fingular elegance and precifion of MATHEMATICAL REASONING, and the amazing feats which it has performed in its progreffive career, and form its wonderful effects in its application to fome parts of phyfical learning, philofophers ancient and modern have not only held it in a juft refpect and veneration, but have been fo enamoured of its beauty, as to embrace and adopt it as the praxis and exemplar of univerfal Logic. This is a mistake, which I fhall re

THUS We have taken a fhort View, of the fo much celebrated Method of the Mathematicians; which to any one who confiders it with proper Attention, muft needs. appear universal, and equally applicable in other Sciences. They begin with Definitions. From these they deduce their Axioms and Poftulates, which ferve as Principles of Reasoning; and having thus laid a firm Foundation, advance to Theorems and Problems, establishing all by the ftrictest Rules of Demonftration. The Corollaries flow naturally and of themfelves. And if any Particulars are ftill wanting, to illuftrate a Subject, or compleat the • Reader's Information; thefe, that the Series of Reasoning may not be interrupted or broken, are generally thrown ' into Scholia. In a Syftem of Knowledge fo uniform and 'well connected, no wonder if we meet with Certainty;

and if thofe Clouds and Darkneffes, that deface other parts of human Science, and bring Difcredit even upon Reason itself, are here fcattered and difappear.' Duncan's Logic, p. 188. See alfo p. 224.

ferve myself to obferve upon in fome future ftage of this work. For the present I shall only remark, that, in this Demonstrative Reasoning, not only the Middle Terms and Propofitions are general, but that all other Terms and Propofitions are general also: from which obfervation I beg leave to appeal to the judgment of Dr. Reid, who allows both the Ancient and Modern Logic to be defective as an univerfal Art, Whether 'The ancients, who attended only to categorical propofitions, which have one subject and one predicate; and of these to fuch M. only as have a general term for their fub'ject,' were not misled in their Logic by the Mathematics and alfo Whether the 'moderns, who have been led to attend only 'to relative propofitions, which express a ' relation between two fubjects, and these 'fubjects always general ideas,' were not likewife mifled by the Mathematics, when they founded the new principle of their Logic

h

* Dr. Reid in the Appendix to Lord Kaim's3d vol. of Sketches, p. 328.

Ibid,

upon