Page:Essays on the active powers of the human mind; An inquiry into the human mind on the principles of common sense; and An essay on quantity.djvu/614

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the mind, for enabling us to conceive more easily and more distinctly to express and demonstrate the properties and relations of those things that have real quantity. The propositions contained in the two first books of Newton's "Principia," might, perhaps, be expressed and demonstrated without those various measures of motion, and of centripetal and impressed forces, which he uses. But this would occasion such intricate and perplexed circumlocutions, and such a tedious length of demonstrations, as would fright any sober person from attempting to read them.

Sec. 3. Corollary First.

From the nature of quantity we may see what it is that gives mathematics such advantage over other sciences in clearness and certainty; namely, that quantity admits of a much greater variety of relations than any other subject of human reasoning; and, at the same time, every relation or proportion of quantities may, by the help of lines and numbers, be so distinctly defined as to be easily distinguished from all others, without any danger of mistake. Hence it is that we are able to trace its relations through a long process of reasoning, and with a perspicuity and accuracy which we in vain expect in subjects not capable of mensuration.

Extended quantities, such as lines, surfaces and solids, besides what they have in common with all other quantities, have this peculiar, that their parts have a particular place and disposition among themselves: a line may not only bear any assignable proportion to another, in length or magnitude, but lines of the same length may vary in the disposition of their parts; one may be straight, another may be part of a curve of any kind or dimension, of which there is an endless variety. The like may be said of surfaces and solids. So that extended quantities admit of no less variety with regard to their form than with regard to their magnitude; and as their various forms may be exactly defined and measured, no less than their magnitudes, hence it is that geometry, which treats of extended quantity, leads us into a much greater compass and variety of reasoning than any other branch of mathematics. Long deductions in algebra for the most part are made, not so much by a train of reasoning in the mind, as by an artificial kind of operation, which is built on a few very simple principles: but in geometry, we may build one proposition upon another, a third upon that, and so on, without ever coming to a limit which we cannot exceed. The properties of the more simple figures can hardly be exhausted, much less those of the more complex ones.

Sec. 4. Coroll. 2.

It may I think be deduced from what hath been above said, that mathematical evidence is an evidence sui generis, not competent to any proposition which does not express