GEOMETRY.

775

Chap. VIII.

tremities X and Y on the greatest diameter QR , equally distant from the centre V, sothat the fold or angle of the thread exactly reaches the point S and the point T; thenplace a pencil-point at the end of the doubled thread, and move it round until theextremities Q R and ST have been passed through, which will form the oval required.

To find the Centres of an Oval — Draw in any part of theoval the right line C D, and at any distance its parallelEF; divide these two parallel lines into two parts inG and II, and trace through these points the linesIGIIK; then bisect this right line in the point L,which will be the centre required.

To find the two Diameters of an Oval when the centre isknown, it is required to find the lengths of the longerand shorter diameters of an oval whose centre is at L.

Describe from the centre L a circle which shall exceedthe oval both above and below, and note where the

Fig. 847.

circle cuts the oval, as in the points M N O P, in orderto draw through those points M and P the right lineMP, to which a parallel line must be drawn, passingthrough the centre L, to the circumference of the oval,as QR , which will be the lesser diameter of the oval;then cut the lesser diameter Q R at right angles in thecentre L by the right line ST, which will be the greaterdiameter of the ovaL

Met/tod of drawing Parabolas , &c. — Trace the right lineA B, of the length required for its base, and divide itinto two equal parts in the point C ; from this pointraise a perpendicular CD, of the length required for theaxis of the parabola; divide this axis into several equalparts in the points E, F, G, II, I, and D: through thesepoints of the axis CD, draw transverse lines parallel withthe base A B ; prolong the axis C D to infinity, as to K.Divide the first space I D into two equal parts, as at L;take the length D I, and set it off from D towards M;then take the distance M I, and set it off from L to theextremities of the first transverse line, at the points N andO; then take the length M H, and set it off on thesecond transverse line, as at P and A: take also thedistance M G, and set it off from L, to the extremitiesof the transverse line C, as at the points R and S, and dothe same with all the other lines: then through these ex-tremities so marked off, trace the line A XT, &c., whichwill give the parabola.

Method of describing Spiral Lines, which are eithersimple or compound; the first are those which areformed by a single line, and the latter those which havea double one. A simple spiral is drawn by tracing the lineC D, and making upon it the position of the eye of thespiral G; then from this point G, with the raclius G E,describe the semicircle EHF; and from the point E asa centre, with the radius E F, describe the semicircleFIK; then from the point G, taken again as a centre,with the radius G K, describe the semicircle K L M;then from E as a centre, with a radius E M, describe thesemicircle M N O. The same process must be repeatedalternately from the centres G and E; and thus semi-circles must be traced at the interval where the precedingcircle ceased, until the spiral is of the size required.

To draw a compound spiral, a simple one must befirst drawn; then set off from the point E to P, the ex-tent to be given to the width of the band, as E P ; thenfrom the point G, with the radius GP, describe the semi-circle PQR, and from the point E as a centre, with theradius E R, describe the semicircle RST: in like manneralso, from the point G as a centre, with the radius G T,describe that of TVX, which must be continued alter-nately from the centres E and G, until the several spiralsanswer to the former.

Fig. 848.

Fig. 849.

Fig. 850.

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