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perfectly independent of each other : and a
farther advantage, by which they contribute
to the illumination of this science, is, that all
the differences between themselves are most
minutely and distinctly marked and ascertain
ed. It is the exclusive privilege of both the
kinds of Quantity to supply a certain mea.
fure or standard, not only to themselves, but
to every other subject in the universe capa-
ble of mensuration.° Extention or continu-
ous Quantity is measured by any fixed part
or division of itself, as by an Inch, a Foot, and
other stated lengths formed out of each o-
ther; and Number, which is discrete, is mea-
sured by stated portions or classes of Num-
ber, as by Tens, Hundreds and so on, as they
rise in due order above each other. So that
any part, form, collection, or relation of such
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 Mechanicæ universalis pars illa quæ artem men, furandi accurate proponit ac demonstrat. Præf. Newtoni in Princip,

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as the subject of other parts of knowledge, which are the qualities of things, as hot and cold, bard and soft, 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.

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Ρ Ιδίον δε μάλιςα τα σοσέ, το τσόν τε και άνισον λέγε, σθαι· έκαςον γαρ των ειρημένων σοσων, τσόν τε και άνισον λέγεται. Οιον σωμα, τσόν τε και άνισον λέγεται. Και αριθμός, και χρόνος, ίσος και άνισος λέγεται. Ωσαύτως δε και επί των άλλων των ρηθέντων, έκαςον ίσον τε και άνισον λέγεται. Των δε λοιπών, όσα μή έςι τοσα, και πάνω αν δόξαιεν ισά τε και άνισα λέγεσθαι. Οιον, η διάθεσις, ίση τε και άνισος και πάνυ λέγεται, αλλά μάλλον ομοία και ανομοία. Και το λευκόν, τσόν τε και άνισον και σάνυ, αλλα μάλλον όμοιον τι ανόμοιον. "Ωςε τα ΠΟΣΟΥ μάλιςα αν είη ίδιον, το ΙΣΟΝ τε και ΑΝΙΣΟΝ λέγεσθαι. Αriftot. Categ. cap. vi.

Natural philosophers have, indeed, invented with acute address and ingenuity various instruments for the mensu, ration of the Qualities of things. This is done by applying them, in some medium or other in which they are differently affected, to a graduated fcale: and thus they have availed themselves, as well as they can, of that exactness and precision which properly belong to Quantity alone.

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From such adequate Definitions of these general ideas thus aftfully and mechanically 2 ti expressed, which are so different and distinct from all other kinds, so absolute. and unchangeable in themselves, and which admit of having their equality, inequality and proportion exactly measured and ascertained, a a few simple 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 so strongly and palpably felt, that as soon as pronounced they irresistibly compel conviction.

Their truth is, indeed, so direct and obvious, that some philosophers assert that it results from an instinctive impulse of the mind which they call Intuition, without the exersise of any act of Reasoning at all ; whilst others have, perhaps, more truly and philosophically 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,

9. Under the word Reason I comprehend the Intuition of the truth of Axioms' (meaning Mathematical]: 'For

certainly

though the truth result immediately and be properly self-evident, though not intuitive.

These general Propositions fo formed are the Axioms of Mathematical science, which are the SECONDARY PRINCIPLES, from which Reason derives all its numerous and extensive operations, and into which they are ultimately to be resolved.

And now that I am upon the subject of Mathematical Principles, I beg leave to make a Distinction which, however new, may prove of great and general importance to the more easy discovery, and more successful cultivation, of all the different kinds of Truth,

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

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

necessarily true, is an act of Reason, though performed ' quick, or perhaps all at once.' Wollaston's Religion of Naturc, §. iii. Note,

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To Mathematical Axioms they have both been attributed with the fullest confidence, because their truth is so direct and palpable that mathematicians think they cannot do them more than fufficient honour by afford ing them too strong an appellative. And, as these Axioms are so obvious in formation and so easy in apprehension, no injury has been derived to this science from the mistake. But, when philosophers and logicians assert that all other Axioms are likewise both intuitive and self-evident, great misfortunes arise from this false idea ; as it precludes inquiry, and secures them by an invincible bar against all sorts of examination and reasoning, which they most faftidiously reject and spurn.

So far, however, from being Intuitive, the Axioms of all other parts of knowledge are the consequences and deductions of the most attentive reasoning and laborious investigation constituting 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 senses, of all men.

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