But it were no hard matter, did I think
it necessary to my present purpose, to show that propositions and
demonstrations in geometry might be universal, though they who make them
never think of abstract general ideas of triangles or circles.
125. After reiterated endeavours to apprehend the general idea a
triangle, I have found it altogether incomprehensible. And surely if
anyone were able to introduce that idea into my mind, it must be the
author of the ESSAY CONCERNING HUMAN UNDERSTANDING; he who has so far
distinguished himself from the generality of writers by the clearness
and significancy of what he says. Let us therefore see how this
celebrated author describes the general or abstract idea of a triangle.
'It must be (says he) neither oblique nor rectangular, neither
equilateral, equicrural, nor scalenum; but all and none of these at once.
In effect, it is somewhat imperfect that cannot exist; an idea, wherein
some parts of several different and inconsistent ideas are put together'
ESSAY ON HUM. UNDERSTAND. B. iv. C. 7. S.9. This is the idea which he
thinks needful for the enlargement of knowledge, which is the subject of
mathematical demonstration, and without which we could never come to know
any general proposition concerning triangles. That author acknowledges it
doth 'require some pains and skill to form this general idea of a
triangle.
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