15. Mathematics an instance of this.
That these two (and not the relying on maxims, and drawing consequences
from some general propositions) are the right methods of improving our knowledge in the ideas of other modes
besides those of quantity, the consideration of mathematical knowledge will easily inform us. Where first we shall
find that he that has not a perfect and clear idea of those angles or figures of which he desires to know anything, is
utterly thereby incapable of any knowledge about them. Suppose but a man not to have a perfect exact idea of a
right angle, a scalenum, or trapezium, and there is nothing more certain than that he will in vain seek any
demonstration about them. Further, it is evident that it was not the influence of those maxims which are taken for
principles in mathematics that hath led the masters of that science into those wonderful discoveries they have
made. Let a man of good parts know all the maxims generally made use of in mathematics ever so perfectly, and
contemplate their extent and consequences as much as he pleases, he will, by their assistance, I suppose, scarce
ever come to know that the square of the hypothenuse in a right-angled triangle is equal to the squares of the two
other sides. The knowledge that "the whole is equal to all its parts," and "if you take equals from equals, the
remainder will be equal," etc., helped him not, I presume, to this demonstration: and a man may, I think, pore long
enough on those axioms without ever seeing one jot the more of mathematical truths. They have been discovered
by the thoughts otherwise applied: the mind had other objects, other views before it, far different from those
maxims, when it first got the knowledge of such truths in mathematics, which men, well enough acquainted with
those received axioms, but ignorant of their method who first made these demonstrations, can never sufficiently
admire. And who knows what methods to enlarge our knowledge in other parts of science may hereafter be
invented, answering that of algebra in mathematics, which so readily finds out the ideas of quantities to measure
others by; whose equality or proportion we could otherwise very hardly, or, perhaps, never come to know?