about universal notions, to which I fully agree: but then it doth not appear to me that those notions are formed by ABSTRACTION in the manner PREMISED--UNIVERSALITY, so far as I can comprehend, not consisting in the absolute, POSITIVE nature or conception of anything, but in the RELATION it bears to the particulars signified or represented by it; by virtue whereof it is that things, names, or notions, being in their own nature PARTICULAR, are rendered UNIVERSAL. Thus, when I demonstrate any proposition concerning triangles, it is to be supposed that I have in view the universal idea of a triangle; which ought not to be understood as if I could frame an idea of a triangle which was neither equilateral, nor scalenon, nor equicrural; but only that the particular triangle I consider, whether of this or that sort it matters not, doth equally stand for and represent all rectilinear triangles whatsoever, and is in that sense UNIVERSAL. All which seems very plain and not to include any difficulty in it.
16. OBJECTION.--ANSWER.--But here it will be demanded, HOW WE CAN KNOW ANY PROPOSITION TO BE TRUE OF ALL PARTICULAR TRIANGLES, EXCEPT we have first seen it DEMONSTRATED OF THE ABSTRACT IDEA OF A TRIANGLE which equally agrees to all? For, because a property may be demonstrated to agree to some one particular triangle, it will not thence follow that it equally belongs to any other triangle, which in all respects is not the same with it. For example, having demonstrated that the three angles of an isosceles rectangular triangle are equal to two right ones, I cannot therefore conclude this affection agrees to all other triangles which have neither a right angle nor two equal sides. It seems therefore that, to be certain this proposition is universally true, we must either make a particular demonstration for every particular triangle, which is impossible, or once for all demonstrate it of the ABSTRACT IDEA OF A TRIANGLE, in which all the particulars do indifferently partake and by which they are all equally represented. To which I answer, that, though the idea I have in view whilst I make the demonstration be, for instance, that of an isosceles rectangular triangle whose sides are of a determinate length, I may nevertheless be certain it extends to all other rectilinear triangles, of what sort or bigness soever. And that because neither the right angle, nor the equality, nor determinate length of the sides are at all concerned in the demonstration. It is true the diagram I have in view includes all these particulars, but then there is not the least mention made of them in the proof of the proposition. It is not said the three angles are equal to two right ones, because one of them is a right angle, or because the sides comprehending it are of the same length. Which sufficiently shows that the right angle might have been oblique, and the sides unequal, and for all that the demonstration have held good. And for this reason it is that I conclude that to be true of any obliquangular or scalenon which I had demonstrated of a particular right--angled equicrural triangle, and not because I demonstrated the proposition of the abstract idea of a triangle And here it must be acknowledged that a man may consider a figure merely as triangular, without attending to the particular qualities of the angles, or relations of the sides. So far he may abstract; but this will never prove that he can frame an abstract, general, inconsistent idea of a triangle. In like manner we may consider Peter so far forth as man, or so far forth as animal without framing the fore-mentioned abstract idea, either of man or of animal, inasmuch as all that is perceived is not considered.
17. ADVANTAGE OF INVESTIGATING THE DOCTRINE OF ABSTRACT GENERAL IDEAS.-- It were an endless as well as an useless thing to trace the SCHOOLMEN, those great masters of abstraction, through all the manifold inextricable labyrinths of error and dispute which their doctrine of abstract natures and notions seems to have led them into. What bickerings and controversies, and what a learned dust have been raised about those matters, and what mighty advantage has been from thence derived to mankind, are things at this day too clearly known to need being insisted on. And it had been well if the ill effects of that doctrine were confined to those only who make the most avowed profession of it. When men consider the great pains, industry, and parts that have for so many ages been laid out on the cultivation and advancement of the sciences, and that notwithstanding all this the far greater part of them remains full of darkness and uncertainty, and disputes that are like never to
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