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All Axioms, though not intuitive, may, however, be properly said to be Self-evident ; because, in their formation Reason judges by single comparisons without the help of a third idea or Middle Term ;' so that they are not indebted to any other for their evidence, but have it in themselves : and, though Inductively framed, they cannot be Syllogistically proved. Till they are either legitimately established or presumptively assumed, the Middle Term is, indeed, no where to be had ;' and, so far from deriving their evidence from it, it derives itself from them: they are, therefore, properly said to be Immediate. It is in this sense that all Axioms are pronounced and should be understood, to be Self-evident, because Immediate and incapable of proof.

INTUITION is, therefore, properly attributed, and should be carefully restricted, to those instinctive faculties and impulses external and internal, which act instantaneously and

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* See the 48th page of this volume,
• See the 49th page of this volume,

' Ai $ AMEEOI WOOTáres dexai. Aristot. Analyt, Poft. lib. i. cap. 29.

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irreîstably," which were given by nature as the first inlets of all knowledge, and which we have called the Primary Principles, whilft SELF-EVIDENCE may be justly and properly attributed to Axioms or the Secondary Principles, of Truth.

This Diftintion I am induced to make in the sanguine hope, that, if justly considered and attended to, it will effectually contribute to the improvement of all learning in the act of conftituting the Principles, that is of diftinguishing the Evidences and establishing the Axioms, of all the different parts of knowlege: a point which every philosopher will acknowlege to be of the last importance. • I appre-hend,' says one in the conclusion of his remarks on the organon of Aristotle, it is a subject of • such consequence, that if inquisitive men • can be brought to the same unanimity in • the first Principles of the other sciences, as

in those of mathematics and natural philo• sophy (and why should we despair of a ge• neral agreement in things that are self-evi

u See page 28 of this volume.

• dent?)

• dent?) this might be considered as the thire

grand æra in the progress of human rea. • fon."

SECT. II.

Of Mathematical REASONING.

THIS species of REASONING is

1 employed in investigating the relations of such abstract and general ideas, poffefsed of such other qualifications as have been noticed in the preceding pages, by reducing them to Axioms or Secondary Principles which are universal propofitions; and the METHOD it pursues is, of course, the most perfectly and purely SYLLOGISTIC."

Dr. Reid in the Appendix to Lord Kaim's 3d vol. of Sketches.

* Aristotle says that all Mathematical Reasoning is reducible to Syllogisms in the first of the three Figures which is the most pure and perfect, and by which all other kinds of Syllogisms that are sound and legitimate are finally to be tried. See Analyt. Poft. lib. i. cap. 14.

.As

As Mathematic has a subject, so its Reasoning has a language, peculiar and appropriated to itself; but, when analysed, it is reducible to the following process.

The Mathematician may be considered as taking his ideas from the beginning in their general form. Every Propohtion composed of such ideas is, therefore, general; and those which are theoretic are reducible to two parts or Terms, a Predicate and a Subje&t, with a Copula affirmative or negative, but generally the former. If the agreement or the relation between the two Terms be not immediate and self-evident, he has recourse to an Axiom which is still more general, and which supplies him with a third or Middle Term. This he compares first with the Predicate and then with the Subject, or vice versa. These two comparisons, when drawn out in form, make two Propositions which are called the Premises; and, if they happen to be immediate and self-evident, the Conclu

· The Middle Term is the Subject of a more general Proposition than that of the Question, and the Predicate the same in both. See page 49 of this volume.

H

hon,

fron, consisting of the Terms of the question proposed, is said to be demonstrated :2 Which Method of Reasoning is conducted exactly in the Syllogistic form* delivered by Aristotle with so much labour and particularity in his Analytics.

z 'Aváyun two anodextinn freisimpen & annIw i civas, και πρωτων, και αμεσων, και γνωριμότερων, και προτέρων xai altowy tã cuu Triçãou.&tos. Aristot. Analyt. Poft. ' lib. i.

• Luracy.ouos di isi réyos, šv Q TEDévtWv tivūv, étsρόν τι των κειμένων, εξ ανάγκης συμβαίνει, τω ταύτα είναι, Aristot. Analyt. Prior. lib. i. cap. 2.

Every kind of Syllogism is reducible to a Categoric and every categorical Syllogism to one of the First Figure; and in the Premises of a Syllogism of the First Figure this is done-- In the Major Proposition, or the Axiom, the Predicate of the Question or Conclusion (which is the same thing) is universally affirmed or denied of some general idea, which is the Middle Term : In the Minor Proposition the Subject of the Question or Conclusion is always affirmed or asserted to be a PART of that MORE GENERAL idea or Middle term. And the ground of this reasoning is this, Whatever may be afirmed universally of any idea, may be affirmed of any species or number of particu. lars comprehended under it, ard vice versa; upon the great Logical maxim, Dictum de omni, et Diftum de nullo.

Fundamentum quo nititur Modorum omnium jam memoratorum vis, (unde probetur Conclusivos effe) est Poftelatum illud quod dici folet Diétum de Omni et de Nullo :

quod

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