Page:Essays on the Principles of Human Action (1835).djvu/174

 equally belongs to every other 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 argues 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, not a regular 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 neither because 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 proofs of the proposition. It is not said the three angles are equal to two right ones, because one of these is a right angle, or because the sides comprehending it are of the same length. Which sufficiently shews that the right angle might have been oblique and the sides unequal, and for all the others the demonstrations have held good. And for this reason it is that I conclude that to be true of any oblique angular, 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. The author then adds some further remarks on the use of abstract terms, and concludes—"May we not, for example, be affected