many curious constructions. There are many questions of no easy nature, belonging to this class, in Mr. Buckley's remains. I may select the following as not unworthy of notice : QUESTION I. It is required to describe a circle to pass through the centres of two given circles, to cut them in C and D, so that C D shall be of a given length. Construction. Let A B (fig. 1), be the centres of the given circles: join A B, and produce this line to a, making Ba= the given distance CD; produce B a to M so that BM: Ba BC: A D. From the points M, A, draw the lines A C, M C, to meet the circle (B) in C, so that AC : MCAD: BC: the method of doing which is well known; then through the points A, B, C, describe the circle A B C, and C D joining the points where this cuts (A) and (B) will be the line required. : = : = Demonstration. Join A D and CD. Then ◄ A D C <C B M ; and if a c be supposed drawn parallel to C M, we have B M B a :: B C : B c ;-and by cons. B M B a : BC AD. Hence A D C B. Again, by parallels c a: CM BC = AD: BC; and by cons. AC: MC:: A D = BC: B C. Consequently AC a c. Lastly, because <AD C <c Ba; AD BC; and A C =ca; CD Ba= the given line by construction. Q. E. D. = QUESTION II. It is required to describe a circle to touch two right lines given in position, so that if a line be drawn from one of the points of contact, parallel to another right line given in position, to meet a given circle, the intercepted chord may be of a given length. = Construction. In the given circle (Q), (fig. 2) apply D E the given chord, and demit QR, perpendicular to D E, meeting it in R. Then with centre Q, and radius Q R, describe the circle QR I;-draw the tangent GIF C parallel to the right line given in position, meeting A C in C. At the point C erect the perpendicular C P, meeting the line A B, which bisects the < C A B in P ;-then P is the centre of the required circle. Demonstration. Describe the circle C P B. This evidently touches the lines given in position; and since CFIG is drawn from one of the points of contact parallel to the other line given in position, and D E and F G being at equal distances from Q,.. F G D E, the required intercept. Q. E. D. = By a similar combination of Tangencies with the different cases of the Section of Space, Mr. Buckley has produced a series of very interesting results, but the complexity of the requisite diagrams precludes the possibility of inserting more than the following neat example: QUESTION III. To describe a circle to touch two right lines given in position, so that if a right line be drawn to touch this circle, parallel to another right line given in position, the rectangle of the segments made by the point of contact may be equal to a given rectangle. Construction. Let A B, A C, be the lines given in position (fig. 3); describe any circle (O1) to touch them, and draw DF parallel to LM, the other line given in position, and touching it at E. Join A and the centre Oi, upon which take A O, a fourth proportional to D E, E F, A O,2, and the given rectangle. With centre O thus obtained, describe the circle O G to touch the lines A C, A B, and it will be that required. Demonstration. Draw B G C parallel to L M, touching the circle (O), and meeting the right lines in B and C. Then by similar figures A O: E D. DF:: A O2 B G. G C. But by cons. A O2: E D. D F :: A O2: given rectangle. .. BG. GC given rectangle, and B G C is drawn parallel to L M, the right line given in position. But the circle (0) touches A B and A C; and hence it is the one required. Q. E. D. Problems respecting Ratio are very numerous. Many of them embrace the properties of two or more circles and appear well worthy of transcription, did space permit. The following will suffice for illustrating his method of treating one of the more difficult combinations : QUESTION IV. : Three points being given, it is required to find a fourth, such that, if a circle be described with it as a centre, and a given radius, tangents drawn to it from the three given points may have given ratios to one another. = = Analysis. Suppose the circle described (fig. 4), with the given radius and centre O, so that the tangents A D, BE, C F, drawn from the given points A, B, C, may have the given ratio. Join A O, B O, C O, and draw the radii, D O, E O, FO. Then, per conditions, m:n::AD: BE; or, m2: n2:: A D2 (A O3—0 D3) : E B2 (B 02—E 02). Upon O E take O I, a fourth proportional to m2, n2, and O D2 or O E2; with centre B, and radius the side of a square O E2O I, describe a circle, to which draw the tangent O G, and radius B G. Then, since m2: n2:: A 0-0 12 B 02: O E H 02: B 02 (0 E2-0 I2) = B O2 — B G2 = G O2. Hence m2 n2:: A 03: G 02; or A O GO :: mn. Therefore, A O has to G O a given ratio, and .. the locus of the points of intersection of A O and G O is a circle by Prop. 11, Book II, Dr. Simson's "Locis Planis." In the same manner it may be shown that A O has a given ratio to a tangent to a circle whose centre is C. These loci being described, will evidently intersect in O, the centre of the circle required. Hence the Construction and Demonstration are obvious. It may be remarked that the loci will intersect in two points, either of which may be taken for centre. |