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CHAP. I.

MATHEMATICS.

SECT. I.

Of the Logic of Mathematics.

VERY thing which is the subject of human knowledge belongs either to mind or body.

The two parts of learning metaphysic and logic which have been touched in a summary way in the preceding pages, treat more immediately of mind, its powers and operations, its acts and energies. The former, producing by speculation the general principles of all other parts of knowledge, or at least affecting to produce them, is the universal science. The latter, descending more practically to the particular investigation and establishment of the principles of

each as they exist in nature and affect the mind, and then pursuing them in a just and rational way into all their different effects, is the universal art. In universality this as well as in many other respects they have a close connection and affinity with each other'. All the other sciences and arts, as has been before observed, apply to particular subjects which are of different kinds and various extent.

Physics form the science which treats more immediately of body or matter organized, its properties and affections, its motions and operations, its qualities and internal struc

ture.

Between these sciences of mind and body, lies one which is intermediate and which partakes of both, taking its subject from the sensible qualities of body, but having it perfectly separated therefrom and made abstract

1 Aristot. Metaph. lib. iv. cap. 2.

Μόνη δὲ ἡ πρώτη φιλοσοφία καὶ ἡ διαλεκτικὴ ὑποκειμένον Exεi návтa тà övтa.—Philoponus in 1 Post. Analyt.

• Εκάστη μὲν ἐπιστημη περὶ ἕν τι γένος καταγίνεται.-Philoponus in 1 Post. Analyt.

by an act of mind3. This intermediate science is mathematics, which as it is related to both, becomes the connecting link by which they are united in the grand system of knowledge. The transition from metaphysics and general logic, to the particular logic of physics and the other parts of learning, will be made aptly and advantageously through the mathematics.

'Aristotle distinguishes the three sciences thus, pèv γὰρ φυσικὴ περὶ ἀχώρισα μὲν, ἀλλ' ἐκ ἀκίνητα τῆς δὲ μαθηματικῆς ἔνια περὶ ἀκίνητα μὲν, ὦ χωρισὰ δὲ ἴσως, ἀλλ' ὡς ἐν ὕλῃ· ἡ δὲ πρώτη καὶ περὶ χωριςὰ καὶ ἀκίνητα Aristot. Metaph. lib. vi. cap. i. Which is thus explained by Du Val-Physica quidem versatur circa substantiam mobilem et materialem : Mathematicæ puræ agunt de rebus reipsa mobilibus, et a materia sensibili re inseparabilibus, sed tamen ea ratione qua sunt immobiles, et cogitatione separatæ; vel, quod idem est, prout in sui consideratione, materiam sensibilem non includunt. Ut ergo physica, mobilium et inseparabilium; mathematica vero, velut immobilium et separabilium; sic metaphysica est revera immobilium, æternorum, separabilium, et divinorum contemplatrix. Doct. Peripat. Synop. p. 22. And again Aristotle distinguishes mathematics both. from physics and metaphysical forms, ἔτι δὴ παρὰ τὰ αἰσθητὰ καὶ τὰ εἴδη, τὰ μαθηματικὰ τῶν πραγμάτων εἶναί φασι μεταξύ, διαφέροντα τῶν μὲν αἰσθητῶν, τῷ ἀΐδια καὶ ἀκίνητα εἶναι· τῶν ὃ εἰδῶν, τῷ τὰ μὲν πόλλ' ἅττα ὅμοια εἶναι, τὸ δὲ εἶδος αὐτὸ, ἓν Exasov μóvov.-Metaph. lib. i. cap. vi, which is thus explained by another commentator.-Indicat Aristoteles Platonem aliud adhuc genus rerum posuisse [principalium] a rebus sensibilibus et ab ipsis ideis diversum. Nam, præter sensibilia et suas formas, res mathematicas constituit, quas

This science is confined to the predicament of quantity, which being of two kinds magnitude and multitude, multitude, that is, quantity continuous and quantity discrete, the first bounded and defined by figure, the second bounded and defined by number, is accordingly divided by these different subjects into two collateral correspondent branches

geometry and arithmetic. And as they are the simplest in their principles, the clearest in their reasoning, and the most convincing in their truth, the logic of both will be properly introductory to that of the other parts of learning, which are more complicated in their nature, and more involved in their construction.

medias esse dixit inter res sensibiles et inter ideas; et differunt a sensibilibus, quod sempiterna sunt et immobilia entia mathematicæ, sicut ideæ quoque sunt; a formis autem et ideis distant, quod pleraque mathematicæ similia sunt inter se, hoc est, quod plura sint ejusdem speciei individua, ut plures trianguli æquum laterum, plura quadrata, et sic deinceps. Forma autem ipsa et idea unaquæque unum quoddam sit tantum. Ita ut res mathematicæ sint inter res sensibiles et inter ideas, quia de utrisque aliquid commune habent, et tamen ab utrisque rursus differunt. Joan. Ludov. Havenruterius Comment. in locum.

• Aristot. Categ. cap. vi.

SECT. II.

Of Mathematical Principles.

EXTERNAL nature is the archetype and

original of all our sensations and of many of our ideas; and the evidence of the external senses, exercised upon the superficial properties of innumerable bodies with which they are familiarly and perpetually conversant, viz. their length, breadth, and depth', and other exterior qualities; and again, as familiarly and incessantly employed upon many different objects, which they cannot avoid distinguishing as individuals or monads, is undoubtedly the primary principle of mathematical learning in both its branches.

This is every where the doctrine of Aris

1 Καὶ γὰρ ἐπίπεδα καὶ τερεὰ ἔχει τὰ φυσικὰ σώματα, καὶ μήκη, καὶ σιγμὰς, περὶ ὧν σκοπεῖ ὁ μαθηματικός.—ἀλλ' οὐχ ᾗ φυσικά σώματος πέρας ἕκατον· οὐδὲ τὰ συμβεβηκότα θεωρεῖ ᾖ τοιύτοις ἔσι συμβέβηκε· Διὸ καὶ χωρίζει· χωριςὰ γὰρ τῇ νοήσει, κινήσεώς ἐτι· καὶ ὐδὲν διαφέρει, ἐδὲ γίνεται ψεῦδος χωριζόντων.-Aristot. Nat. Ausc. lib. ii. cap. 2.