In geometry there is a like coincidence of lines, angles, spaces, and solid con tents, to prove them equal in simple ca ses. Afterwards, in complex cases, we substitute the terms whereby equal things are denoted for each other, and then the coincidence of the terms to de note the coincidence of the visible ideas, except in the new step advanced in the proposition : and thus we get a new qua lity, denoted by a new coincidence of terms; and this in like manner we em ploy in order to obtain a new eqeality. 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 ides of 20 or of 21; but we have of the difference be tween them, by fancying to ourselves a confused heap of things, supposed or call ed twenty in number, and then further fancying one thing to be added to it. By a like process in geometry we arrive at the demonstration of the most complex propositions.—The properties of num bers 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 ratio to it.—And as in arithmetic words stand forindistincti leas, in order to help us to reason about them as accurately as if they were distinct ; as also cyphers stand for words, for the same purpose ; and letters for cyphers, to render the conclusions less particular; so letters are put for geometrical quanti ties also, and the agreements of the let ters for those of the quantities.
Thus we seethe foundation upon which the whole doctrince of quantity is built ; for all quantity is denoted either by num bers, or by extension, or by letters de noting either one or the other. The co incidence of ideas is the foundation of ra tional assent in simple cases ; and that of ideas and of terms, or of terms alone, in complex cases. This is upon the suppo sition that the quantities are to be proved equal ; but if they are to be proved un equal, the want of coincidence answers the same purpose. If they are in any numerical ratio, this is only introducing a new coincidenee.—Thus it appears that the use of words, (either as visible or as audible symbols,) is necessary for geo metrical and algebraic reasonings, as well as for arithmetical. 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. The recollection of having once examined and assented to each step of a demonstration, the authority of an approved writer, lke. are often sufficient to gain our assent, though we understand no more than the import of the proposi tion; nay, even though we do not pro ceed so far as this. Now this again is a mere transfer of association ; the recol lection, authority, &c. being :n a great number of cases associated WItii the be fore-mentioned coincidence of ideas and terms.--But here a new circumstance arises ; for memary and authority arc sometimes found to mislead ; and the re collection of such exncrieoie pus the mind into a state of so that some times truth, sotrtefhacs Uselioncb will re cur, and unite itself with the proposition ender consideration, according as the re collection, authority, &c. in all their pe culiar circumstances, have been associat ed with truth or with ildsehood.
Thus the idea belonging to a mathema tical 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, and forming a very complex idea (§ 53).
And this idea is not merely the sum of the ideas belonging to the terms of the proposition, but also includes the notions or feelings, whatever they be, which be long to the words ,equality, coincidence, and truth, and, in some cases those of utility, importance, &c.—For mathemati cal propositions are, in some cases, at tended with a practical assent, in the pro per sense of these words; as when a person takes this or that method of exe cuting a projected design, in conse quence of some mathematical proposition assented to from his own examination, or from the authority of others. Now the train of voluntary actions denoting the practical assent, is produced by the fre quent recurrency of ideas of utility and importance. These operate by associa tion, and though the rational assent be a previous requisite, yet the degree of the practical assent is proportional to the vi vidness of those ideas ; and in most cases they strengthen the rational assent by reaction.
II. Propositions concerning natural bo dies are of two kinds, vulgar and scienti fical. Of the first kind are, " milk is white," "gold is yellow,"" a dog barks," &c. These are evidently nothing more than forming the terms denoting the whole or some component parts of the complex idea, into a proposition, or em. ploying those denoting some of its com mon adjuncts in the same way. The as sent given to such propositions arises from the associations of the terms as well as of the ideas denoted by them.
In scientifical propositions concerning natural bodies, a definition having been made of the body from its properties, another property or power is joined to them as a constant or common associate. Thus gold is said to be soluble in the nitro-muriatic acid. Now to persons who have made the proper experiments a suffi cient number of times, these words sug gest the ideas which occur in those experi ments, and conversely are suggested by them, in the same manner as the vulgar propositions above mentioned suggest, and are suggested by, common appearan. ces. But then, if they be scientific per sons, their readiness to affirm that gold is soluble in this acid universally, arises also from the experiments of others, and from their own and other persons' obser vations on the constancy and tenor aria tore. They find it to be a general truth, that almost any two or three remarkable qualities of a natural body, infer the rest, being never found without them ; and hence arises a readiness to affirm respect ing all bodies possessing those two or three leading qualities, whatever may be affirmed of one, The propositions formed respecting na tural bodies are often •attended with a high degree of practical assent, arising chiefly from some supposed utility and importance, and which is no way propor tioned to the foregoing or similar acknow ledged causes of rational assent. And in some cases the practical assent takes place before the rational ; but then, after some time, the rational assent is generat ed and cemented most firmly by the pre valence of the practical. This process is particularly observable in the regards paid to medicines ; that is, in the rational and practical assent to the propositions con cerning their virtues.