And as in arithmetic words stand for indistinct ideas, in order to help us to reason upon them as accurately as if they were distinct; also cyphers for words, and letters for cyphers, both for the same purpose; so letters are put for geometrical quantities also, and the agreements of the first for those of the last. And thus we see the foundation upon which the whole doctrine of quantity is built; for all quantity is expounded either by number or extension, and their common and sole exponent is algebra. The coincidence of ideas is the foundation of the rational assent in simple cases; and that of ideas and terms together, or of terms alone, in complex ones. This is upon supposition that the quantities under consideration are to be proved equal. But if they are to be proved unequal, the want of coincidence answers the same purpose. If they are in any numeral ratio, this is only the introduction of a new coincidence. Thus, if, instead of proving A to be equal to B, we are to prove it equal to half B, the two parts of B must coincide with each other, either in idea or terms, and A with one.
And thus it appears, that the use of words is necessary for geometrical and algebraical reasonings, as well as for arithmetical.
We may see also that association prevails in every part of the processes hitherto described.
But these are not the only causes of giving rational assent to mathematical propositions, as this is defined above. The memory of having once examined and assented to each step of a demonstration, the authority of an approved writer, &c. are sufficient to gain our assent, though we understand no more than the import of the proposition; nay, even though we do not proceed so far as this. Now this is mere association again; this memory, authority, &c. being, in innumerable instances, associated with the before-mentioned coincidence of ideas and terms.
But here a new circumstance arises. For memory and authority are sometimes found to mislead; and this opposite coincidence of terms puts the mind into a state of doubt, so that sometimes truth may recur, and unite itself with the proposition under consideration, sometimes falsehood, according as the memory, authority, &c. in all their peculiar circumstances, have been associated with truth or falsehood. However, the foundation of assent is still the same. I here describe the fact only. And yet, since this fact must always follow from the fixed immutable laws of our frame, the obligation to assent (whatever be meant by this phrase) must coincide with the fact.
And thus a mathematical proposition, with the rational assent or dissent arising in the mind, as soon as it is presented to it, is nothing more than a group of ideas, united by association, i.e. than a very complex idea, as was affirmed above of propositions in general. And this idea is not merely the sum of the ideas belonging to the terms of the proposition, but also includes the ideas, or internal feelings, whatever they be, which belong to
