and a

perfectly independent of each other farther advantage, by which they contribute to the illumination of this science, is, that all the differences between themselves are most minutely and diftinctly marked and ascertained. It is the exclufive privilege of both the kinds of Quantity to fupply a certain meafure or standard, not only to themselves, but to every other subject in the universe capable of menfuration. Extention or continuous Quantity is measured by any fixed part or divifion of itself, as by an Inch, a Foot, and other stated lengths formed out of each other; and Number, which is discrete, is meafured by stated portions or claffes of Number, as by Tens, Hundreds and fo on, as they rife in due order above each other. So that any part, form, collection, or relation of fuch ideas, can be easily compared with others, and pronounced to be exactly equal or unequal, greater or less, or in a certain ratio: where

Fundatur Geometria in praxi Mechanica, et eft nihil aliud quam Mechanica univerfalis pars illa quæ artem menfurandi accurate proponit ac demonftrat. Præf. Newtoni in Princip

as

as the subject of other parts of knowledge, which are the Qualities of things, as hot and cold, hard and foft, good and evil, with innumerable others, have no certain measure or criterion to determine the judgment; and, after the best and most exact comparisons that can be made, one quality can only be pronounced to be more or less than another, like or unlike in different degrees, by a more vague and uncertain determination.

ΤΟ

* Ιδίον δὲ μάλισα τα ποσέ, τὸ ἴσόν τε και άνισον λέγετ σθαι ἕκασον γὰρ τῶν εἰρημένων ποσῶν, ἴσόν τε καὶ ἄνισον λέγεται. Οἷον σῶμα, ἴσόν τε καὶ ἄνισον λέγεται. Καὶ ἀριθμὸς, καὶ χρόνος, ἴσος καὶ ἄνισος λέγεται. Ωσαύτως δὲ καὶ ἐπὶ τῶν ἄλλων τῶν ῥηθέντων, ἕκασον ἶσόν τε καὶ ἄνισον λέγεται. Τῶν δὲ λοιπῶν, ὅσα μή ἐςι ποσά, ε πάνυ ἂν δόξαιεν ἶσά τε καὶ ἄνισα λέγεσθαι. Οἷον, ἡ διάθεσις, ἴση τε καὶ ἄνισος & πάνυ λέγεται, ἀλλὰ μᾶλλον ὁμοία καὶ ἀνομοία. Καὶ τὸ λευκὸν, ἶσόν τε καὶ ἄνισον ἐ πάνυ, ἀλλὰ μᾶλλον ὅμοιον τὶ ἀνόμοιον. Ωςε τῷ ΠΟΣΟΥ μάλιςα ἂν εἴη ἴδιον, τὸ ΙΣΟΝ τε καὶ ΑΝΙΣΟΝ λέγεσθαι. Ariftot. Categ. cap. vi.

Natural philofophers have, indeed, invented with acute addrefs and ingenuity various inftruments for the menfuration of the Qualities of things. This is done by applying them, in fome medium or other in which they are differently affected, to a graduated scale: and thus they have availed themselves, as well as they can, of that exactness and precision which properly belong to Quantity alone.

From

From fuch adequate Definitions of these general ideas thus artfully and mechanically expreffed, which are so different and distinct from all other kinds, so abfolute and unchangeable in themselves, and which admit of having their equality, inequality and proportion exactly measured and ascertained, a a few fimple PROPOSITIONS are formed, to which they apply, which are the most general that can be made; the truth and certainty of which, upon comparing their ideas, strike so forcibly upon the understanding, and are fo ftrongly and palpably felt, that as foon as pronounced they irresistibly compel conviction.

Their truth is, indeed, fo direct and obvious, that fome philofophers affert that it refults from an inftinctive impulfe of the mind which they call Intuition, without the exercise of any act of Reasoning at all; whilft others have, perhaps, more truly and philofophically determined that where there is an act of Comparison, there is an act of Judgment, and where there is an act of Judgment there is an act of Reasoning,

"Under the word Reafon I comprehend the Intuition f of the truth of Axioms' [meaning Mathematical]: For ⚫ certainly

though the truth refult immediately and be properly felf-evident, though not intuitive.

These general Propofitions fo formed are the AXIOMS of Mathematical science, which are the SECONDARY PRINCIPLES, from which Reafon derives all its numerous and extenfive operations, and into which they are ultimately to be refolved.

AND now that I am upon the fubject of Mathematical Principles, I beg leave to make a DISTINCTION which, however new, may prove of great and general importance to the more eafy discovery, and more fuccessful cultivation, of all the different kinds of Truth,

Intuitive and Self-evident are terms used promiscuously by philofophers and logicians as perfectly fynonimous; which has, I apprehend, been the cause of introducing much error and obftruction into general science,

⚫ certainly to discern the respect which one term bears to another, and from these to conclude the propofition ne

neceffarily true, is an act of Reason, though performed quick, or perhaps all at once.' Wollafton's Religion of Nature, §. iii. Note.

Το

To Mathematical Axioms they have both been attributed with the fulleft confidence, because their truth is fo direct and palpable that mathematicians think they cannot do them more than fufficient honour by affording them too strong an appellative. And, as these Axioms are fo obvious in formation and fo eafy in apprehenfion, no injury has been derived to this fcience from the mistake. But, when philofophers and logicians affert that all other Axioms are likewife both intuitive and felf-evident, great misfortunes arise from this falfe idea; as it precludes inquiry, and fecures them by an invincible bar against all forts of examination and reasoning, which they most faftidiously reject and fpurn.

So far, however, from being Intuitive, the Axioms of all other parts of knowledge are the confequences and deductions of the most attentive reasoning and laborious investigation conftituting the most useful and honourable part of human learning; whereas, if they were intuitive, they would flush direct conviction on the minds, as external objects do on the fenfes, of all men.

All