774
THEORY AND PRACTICE OF ENGINEERING.
Book 11.
Fig. 842.
Fig. 843.
service to architects and others, enabling them to discover the entire form, when any portionof the figure has been destroyed.
To find the centre of a circle or any portion of it, takeany three points in the circumference of the circle, as A, B,and C: then from the points A, B, with the radius A B,describe four arcs intersecting at E and D, through whichpoints of intersection draw at pleasure the right lineE D; then from the points B and C with the radiusB C, describe four other arcs intersecting at F and G, andthrough these points draw the line F G cutting E D inII, which will be the centre of the circle ABC.
To find the centre of a portion of a circle, as that ofIKL, draw the right line IL, and divide it into twoequal parts in the point M; then raise a perpendicularfrom the point M, and prolong it towards the base untilit cuts the portion of the circle I K L in K; then measurethe distance I M, which here is found to be 6ij parts, andthe distance MK 4 parts ; then multiply 6^ by itself; theproduct 44£ must then be divided by 4, and the quotient11£ must be counted as the line M N from M to C; thendivide the length K O into two equal parts in the point
P, which will be the centre of the circle. The centres ofbasins or reservoirs of water may be thus found, evenwhen portions of them have been destroyed.
Method of describing Ellipses. —To draw the oval A, tracea right line C D of the length required for the oval, anddivide this line into four equal parts by the points E, F, G ;from the points E, F, and G, with one of these divisionsas radius, describe three equal circles: on the point F,the centre of the line C D, let fall the perpendicular H I,which will cut the circumference of the circle FG in Kand L; from the point K through the centre E, drawthe right line K E, until it cuts the circle C F in M, andin like manner from the point I, through the centre G,draw the right line L G until it cuts the circle F D inM; then from the point K, with the radius KM, drawthe line M O, and from the point L the line P N. Thecurved line C P N D O M is the oval.
When it is required to draw a more rounded figure, asB, whose diameter is the same as C D, it is only ne-cessary to divide the line C D into three equal parts in
Q, R, and from the two points Q, R, w'ith the radius of onepart, as QC, describe the circles C R and Q.D ; then attheir points of intersection, S, T, draw the perpendicularS T across C D ; then take two equal parts out of thethree, and with the compasses, from the point S describethe arc VX, and from the point T the arc Y Z : theline C Y Z D X V is the oval required.
To draw an Oval when the two diameters are given,as C D for the longer, and E F for the smaller, take halfthe smaller diameter, as G E, and set it off on the greaterdiameter CD from G to H, and from G to I: divideG H into three equal parts, and carry two parts on to thesmaller diameter E F from G to K, and from G to L.
From the two points K and L, draw four right lines ofany length through the points H and I: from the pointII as a centre, with the radius H C, describe the arcN C M, and from the point I, with the same radius, de-scribe the arc O D P; then from the point L, with theradius LM, describe the arc MEO, and from thepoint K, with the radius K N, describe the circle N F P;the line CMEODPFN will form the oval A on the twogiven diameters.
To draw an oval similar to B, the two diameters ofwhich, QR and ST, cross at right angles in the pointV, take with a thread or cord the length of the greatestdiameter Q R, and double the thread ; place its two ex- Fig. 846
Fig. 844.