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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1849 Excerpt: ...This process supplies the want of a theorem with which the student ought to be acquainted in its general form. Prove that if an equation be homogeneous with respect to a set of letters p, q, r, &c, that equation remains true if p, q, r, &c. be erased, and p', q, r', &c. substituted, provided that p' is to p as q' to q, and as / to r, &c. Show that this proposition is the arithmetical representative of Euclid II. 12, 13; and that the introduction of the distinction of positive and negative quantity prevents our needing two propositions. As in page 39, we may express the above thus: (, iA-1/tyab.coa$C f, ., 2ab.wC c = (a-f b) cossm l-i--)-(a-6) sec tan l--r-...(3). The formula (2) may be proved thus: --From the vertex of A draw a perpendicular upon a. In all cases it will be seen that each side of a triangle is the sum of the projections of the other two upon it, provided each projection be called positive or negative, according as the angle of projection is acute or obtuse. Thus a-b cos C+ c cos B, b = c cos A + a cos C, c = a cos B + b cos A. Now c2 = (c cos Af + (c sin t)2 = (&-a cos C)2 + (a sin 7)2 = 62-2ab cos C + a2. a2 + h2 o2 The form cos C= 0, (4) 2ab v y is often useful. From it we have l + cosC+f-cV g, (jL 4( - ), 2a6 2 4a6 1-cos C = c8-(0g:5)2, sin = d+ - )( + - ). 2ab ' 2 46 Let a + & + c = 2$, then a + 6-c = 2 0-c), & + c-a = 2 (s-a), c + a-b 2(s-b). By substitution we have C0S2=-'SmV ab ' tan2=W 2 ca 2 ca 2 s(s-&) _ g(ga) A _ M) sin(s-c) _ (s-b) (s-c) cos------= j om----, xan-----2 be 2 be 2 s (s-a). lis-a) (s-b) (s-c).., #J_..... Let P-i----, which it will presently be shown is the radius of the inscribed circle (Euc. IV. 4). Show that A p B p Co tan--=-&...

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