149.  I do not design to trouble myself with drawing corollaries from the doctrine I have hitherto laid down.  If it bears the test others may, so far as they shall think convenient, employ their thoughts in extending it farther, and applying it to whatever purposes it may be subservient to:  only, I cannot forbear making some inquiry concerning the object of geometry, which the subject we have been upon doth naturally lead one to.  We have shown there is no such idea as that of extension in abstract, and that there are two kinds of sensible extension and figures which are entirely distinct and heterogeneous from each other.  Now, it is natural to inquire which of these is the object of geometry.

150.  Some things there are which at first sight incline one to think geometry conversant about visible extension.  The constant use of the eyes, both in the practical and speculative parts of that science, doth very much induce us thereto.  It would, without doubt, seem odd to a mathematician to go about to convince him the diagrams he saw upon paper were not the figures, or even the likeness of the figures, which make the subject of the demonstration.  The contrary being held an unquestionable truth, not only by mathematicians, but also by those who apply themselves more particularly to the study of logic; I mean, who consider the nature of science, certainty, and demonstration:  it being by them assigned as one reason of the extraordinary clearness and evidence of geometry that in this science the reasonings are free from those inconveniences which attend the use of arbitrary signs, the very ideas themselves being copied out and exposed to view upon paper.  But, by the bye, how well this agrees with what they likewise assert of abstract ideas being the object of geometrical demonstration I leave to be considered.

151.  To come to a resolution in this point we need only observe what hath been said in sect. 59, 60, 61, where it is shown that visible extensions in themselves are little regarded, and have no settled determinable greatness, and that men measure altogether, by the application of tangible extension to tangible extension.  All which makes it evident that visible extension and figures are not the object of geometry.

152.  It is therefore plain that visible figure are of the same use in geometry that words are:  and the one may as well be accounted the object of that science as the other, neither of them being otherwise concerned therein than as they represent or suggest to the mind the particular tangible figures connected with them.  There is indeed this difference between the signification of tangible figures by visible figures, and of ideas by words:  that whereas the latter is variable and uncertain, depending altogether on the arbitrary appointment of men, the former is fixed and immutably the same in all times and places.  A visible square, for instance, suggests to the mind the same tangible figure in Europe that it doth in America.  Hence it is that the voice of the Author of’ Nature which speaks to our eyes, is not liable to that misinterpretation and ambiguity that languages of human contrivance are unavoidably subject to.