— 649 —
Mathematicks an instance of it.
      §15. That these two (and not the relying on Maxims, and draw-
ing Consequences from some general Propositions) are the right
Method 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 any thing, is utterly thereby uncapable
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. Farther it is evident, that it was not
the influence of those Maxims, which are taken for Principles in
Mathematicks, 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 Mathematicks
never 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 Hypotenuse 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 otherways
applied: The Mind had other Objects, other Views before it, far
different from those Maxims, when it first got the Knowledge of
such kind of Truths in Mathematicks, which Men well enough ac-
quainted with those received Axioms, but ignorant of their Method,
who first made these Demonstrations, can never sufficiently ad-
mire. And who knows what Methods, to enlarge our Knowledge
in other parts of Science, may hereafter be invented, answering
that of Algebra in Mathematicks, which so readily finds out Ideas of
Quantities to measure others by, whose Equality or Proportion we
could otherwise very hardly, or, perhaps, never come to know?
Locke Hum IV, 12, §15, p. 649