. The Canadian journal of industry, science and art. TRIANGLE AND THE SUM OF THE ANGLES. 319 We cau go ou thus iudeii- nitely, forming a series of pairs of equal triangles KMN and PNF, Trli and FL^, &c., to which there is no limit; and, if Sn be the sum of the angles of the first triangle in the w* pair, and s„ the sum of the angles of the second tri- angle in the t?**" pair, But, as the series of tri- angles, FPN, FL^, &c., is indefinitely increased in number, by a continued repetition of the construc- tion above described, the base (such as hh) of the triangle ultimately ob- tai
. The Canadian journal of industry, science and art. TRIANGLE AND THE SUM OF THE ANGLES. 319 We cau go ou thus iudeii- nitely, forming a series of pairs of equal triangles KMN and PNF, Trli and FL^, &c., to which there is no limit; and, if Sn be the sum of the angles of the first triangle in the w* pair, and s„ the sum of the angles of the second tri- angle in the t?**" pair, But, as the series of tri- angles, FPN, FL^, &c., is indefinitely increased in number, by a continued repetition of the construc- tion above described, the base (such as hh) of the triangle ultimately ob- tained becomes indefinitely small. For BC = CD + DE = CD + NP + MN ^ CD -t- 2NP + JiL + Th, and so on, without limit; so that, if the base (such as hL) of the triangle (such as FL/j) ultimately obtained did not become indefinitely small, the finite line BC would be greater than the sum of an indefi» nite number of lines, none of which was less than a given finite line : which is impossible. Since therefore the base (such as hL) of the triangle (such as FAL) ultimately obtained must become indefinitely small, the sum of the angles of the triangle (such as FL^) ultimately obtained cannot (Prop. III.) differ by any finite angle from two right angles. That is, S„ does not continue, as n is indefinitely increased,. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Canadian Institute. Toronto : Canadian Institute
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