. An encyclopaedia of architecture, historical, theoretical, & practical. New ed., rev., portions rewritten, and with additions by Wyatt Papworth. x FB : FG2; But, by ]n-operty of parameter, CA2 ; CE2::CE2 : FG2. Therefore CE2= AF x FB= CF- CA ; Fig. i.^n. And, by transposition, CF2=CA2+ CE2. Coroll. 1. The two semi-axes, and the distance of the focus from the centre, are the sidesof a right-angled triangle CEA, of which the distance AEis the distance of the focus from the centre. Coroll. 2. The conjugate axis CE is a mean proportionalnetween FA and FB, or between fB and /A, for CE- =(•F2-CA =
. An encyclopaedia of architecture, historical, theoretical, & practical. New ed., rev., portions rewritten, and with additions by Wyatt Papworth. x FB : FG2; But, by ]n-operty of parameter, CA2 ; CE2::CE2 : FG2. Therefore CE2= AF x FB= CF- CA ; Fig. i.^n. And, by transposition, CF2=CA2+ CE2. Coroll. 1. The two semi-axes, and the distance of the focus from the centre, are the sidesof a right-angled triangle CEA, of which the distance AEis the distance of the focus from the centre. Coroll. 2. The conjugate axis CE is a mean proportionalnetween FA and FB, or between fB and /A, for CE- =(•F2-CA = (CF+ CA) x (CF-CA)= BF x AF. 1088. Theoreji V. The difference of the radius vectorsis equal to the transverse axis. (Jig. 435.) That is, /M-FM=AB = 2CA = 2CB. For CA2 : CE2;: CP2- CA2 : PM2 ; And CE2=CF2-CA2. Therefore CA2 : CF2-CA2:: CP2_ CA2 : PM2. And by taking the rectangle of the extremes and means, anddividing by CA2, PM2 = ^^^^^-CF2-CP2+ CA2; Hut FP2 = (CP-CF)2=CP2=2CPx CF+CF2, And FM2 = PM2+FP2. Therefore FM2 = ^^1^-2CP x CF+ CA2. Now each side of this equation is a complete square. Therefore, extracting the root of each number,. FxM = CF> CICA -CA. C I X i P In the same manner we find _/M= ~qj^— + CA ; And, subtracting the upper equation from the lower,/M— FM = 1. Ilence is derived the common method of describing the liyperbolic curvemechanically. Thus : — In the transverse axis AB produced (Jig. 435.), take the foci F,/,and any point I in the straight line AB so produced. Ilien, with the radii A I, HI, and the 312 THEORY OF ARCHltECTUllE. Book II centre F, /, describe arcs intersecting each other ; call the points of intersection E, then E willbe a point in the curve ; with the same distances another point on theother side of the axis may be found. In like manner, by taking anyother points I, we may find two more points, one on each side of theaxis, and thus continue till a suflficient number of points be found todescribe the curve by hand. By
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