two and four are only different names for the same impression. And it is mere association which appropriates the word truth, its definition, or its internal feeling, to this coincidence.
Where the numbers are so large, that we are not able to form any distinct visible ideas of them, as when we say that 12 times 12 is equal to 144; a coincidence of the words arising from some method of reckoning up 12 times 12, so as to conclude with 144, and resembling the coincidence of words which attends the just-mentioned coincidence of ideas in the simpler numerical propositions, is the foundation of our rational assent. For we often do, and might always, verify the simplest numerical propositions, by reckoning up the numbers. The operations of addition, subtraction, multiplication, division, and extraction of roots, with all the most complex ones relating to algebraic quantities, considered as the exponents of numbers, are no more than methods of producing this coincidence of words, founded upon and rising above one another. And it is mere association again, which appropriates the word truth to the coincidence of the words, or symbols, that denote the numbers.
It is to be remarked, however, that this coincidence of words is by those who look deeper into things, supposed to be a certain argument, that the visible ideas of the numbers under consideration, as of 12 times 12, and 144, would coincide as much as the visible ideas of twice two and four, were they as clear and distinct. And thus the real and absolute truth is said by such persons to be as great in complex numerical propositions, as in the simplest. All this agrees with what Mr. Locke has observed concerning numbers, viz. that their names are necessary in order to our obtaining distinct ideas of them; for by distinct ideas he must be understood to mean proper methods of distinguishing them from one another, so as to reason justly upon them. He cannot mean distinct visible ideas.
In geometry there is a like coincidence of lines, angles, spaces, and solid contents, in order to prove them equal in simple cases. Afterwards in complex cases, we substitute the terms whereby equal things are denoted for each other, also the coincidence of the terms, for that of the visible ideas, except in the new step advanced in the proposition; and thus get a new equality, denoted by a new coincidence of terms. This resembles the addition of unity to any number, in order to make the next, as of 1 to 20, in order to make 21. We have no distinct visible idea, either of 20 or 21; but we have of the difference between them, by fancying to ourselves a confused heap of things supposed or called 20 in number; and then farther fancying 1 to be added to it. By a like process in geometry we arrive at the demonstration of the most complex propositions.
The properties of numbers are applied to geometry in many cases, as when we demonstrate a line or space to be half or double of any other, or in any other rational proportion to it.
