or Ellipse Ellipsis

but the demonstration which accompanies the following method, is quite general, for every two diameters.

Method V.—Figure 6. A diameter, K n, and an ordinate, D L, of an ellipsis being given, to describe the curve by a con tinued motion.—B6ect S H at i, and through 1, draw A I A, parallel to D L; draw DEC. and KB, at right angles to A A; front L, with the distance x B. describe an are. cutting i A at F draw i F C ; through the points c and i, draw 14 v ; then if the point c be moved in ttt N. and the point r in A A, the point L will describe the curve elan ellipsis.

The method Gtr describing an ellipsis, having two vonju gate diameters given, may be tinind in the Marquis de Treeilese of Conic Sections, by Stone. But the author of the :Irrh iteetural Dietamary has chosen to give the de-ATiption and demonstration from a diameter and double ordinate instead of two conjugate diameters, as being more readily applied in perspective. It is stmilge, that this useful method has been neglected by all English writers that have fallen in our way. This property was dis covered by the author, and demonstrated by him, many years before he met with the above work, in to find out methods for describing the perspective representation by continued !notion.

Method VI.—Figure 7. No. 1. Let A a be the greater axis, bisected in c, by the lesser semi-axis C n ; take two rulers, c E and K F, of equal length, equal to the sum of the semi-axes c D and u B, moveable upon Cavil other at E, :111(1 the end c of the rule c e, 1110Veable upon the centre of the Make the part G of the ruler, E, equal to the ni-axis a D; now suppose C E and E F to ith each other, and with the axis c D ; then move the point r from c, in the direction c IL until the deseribent CI arrive at B : the point G then have traced the qinulrant a 11 of the ellipsis. The other quadrants hill be described in the same manner, by reversing and inverting the rulers.

Figure

S.—Another variation of this : Let A B be the greater, and c D the lesser semi-axis, as before; take the straight line n I, equal to the greater semi-axis, A c or c n ; from I it cut I K, equal to the lesser semi-axis, c D, and divide H K into two equal parts at I : then place the joint rule c E F in the following manner, viz., make c E K F each equal to n L, or L K; the part c E being moveable round the centre, c, of the ellipsis, and the two rules c E and E G being moveable round E ; now let c E and E G coincide with c D, and the point F to coincide with c, and consequently e with D ; then move the point F towards A, keeping it in the semi-axis c A ; and %%hen c E and z F coins. in the same straight line, the point o will have described the quadrant of an ellipsis ; the lesser axis (see Figures 7 and 8) is equal X F 0, and the greater =2X cz+2X z G.

Demonstration.

Figure 7, No. 2. I /raw G 1 parallel to c F, cutting c E at 1, and draw E L perpendicular to c F, meet ing C B at L ; produce C E to 11, and make C It = C B ; join 11 0, and produce it to K.

Now

by the general demonstration, aerompanying the article CYLINDER, we have CBs C :: AK X KB:K but by the property of the circle K there fore c C : : K : K ; vonsequently C 11 : C D : : K n : K G, a property of the ellipsis.

But c

E half slim of the two axis, and c r, or F G. equal to the lesser axis ; thou I E, or K 0, is equal to the (Inference between the half sum of the two semi-axes and the lesser seini-axis ; therefore I E = halt' the difference between the two seini-axes, and I n = the whole difference ; consequently I E = E U.

Then because F e is pantile' to c

F, we have mc:EF: : EI:EG; but EC= EV; therefore K I K j and because