Arthur Cayley

The Collected Mathematical Papers of Arthur Cayley.vol. 12

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. 1897 Excerpt: ...in a rhombus the sum of the squares of the diagonals is = sum of the squares of the four side. We have a rhombus ACEB, and therefore (AEf + (BGf = 2 (AB) + 2(ACf, that is, (AEf = 2 (ABf + rad.5. But by construction, EF = (AEf; therefore EF3 = 2 (ABf + rad.2, = 2 (GEf + rad.2. Hence in the right-angled triangle FCE, we have (GFf = (EFf-CE2, = (GEf + rad.a; and by construction, (EGf = CF)s; that is, (EGy-= (CE) + rR. = GEf + (GGy; viz. GEG is a right-angled triangle; that is, CG is at right angles to BCE, or the line CGF bisects the arc AB in G. For the second problem, (2), to find the intersection of two given lines, we require the solution of the problem to find a fourth proportional to three given distances, and this immediately depends on the following problem. Problem. In two concentric circles to place chords subtending equal angles. If AB (fig. 2) be a chord of the one circle, then with centres A and B respectively, and Fig. 2. G' an arbitrary radius AA' = BR, describing circles to cut the larger circle in the points A' and B1 (A' is either of the two intersections and B' is the intersection lying in regard to CB as A' lies in regard to GA) then clearly Z ACB = Z A'CB'. And hence also we have the following problem. Problem. To find a fourth proportional to three given distances. We have only to take as the given distances the two radii GA, GA' and the chord AB; and then from the similar triangles AGB, A'CB1, we have GA: GA':: AB: A'ff, viz. A'B is a fourth proportional to GA, GA', AB. We have now the solution of (2) To construct the intersection of two given lines AB and CD (fig. 3). Find Fig. 3. A. c the counter-point of G, and d the counter-point of D, in regard to the line AB: and find 7 such that the distances Cy, dy are = Dd, DC respectively; 7 is...