Fig. 25 b.
To draw an Ellipse.
—I. By means of a piece of string and pins. Place the diameters A B and C D ([Fig. 23]) at right angles to each other, and set off from C half the major axis at E and F; then will E and F be the two foci in the ellipse. Fix a pin at E and another at F; take an endless string equal in length to the three sides of the triangle E F C and pass it round the pins, stretch the string with a pencil G, which will then describe the required ellipse. II. From the centre O ([Fig. 24]) describe a circle of the diameter of the minor axis of the required ellipse. From the same centre, describe another circle with a diameter equal to its major axis. Divide the inner circle into any number of equal parts as 1, 2, &c., and through these points draw radii cutting the outer circle in 4, 3, &c. From 1, 2, &c., draw horizontals, and from 3, 4, &c., draw perpendiculars cutting each other in E F, &c.; the curve traced from C through the points C E F A, &c., will complete the curve of the required ellipse. III. Let A B ([Fig. 25 a]) be the major and C D the minor axis of the required ellipse. On any convenient part of the paper draw two lines F G, F H ([Fig. 25 b]) at any angle with each other. From F with the distance E C or E D, the semi-axis minor, describe an arc cutting the lines F G, F H, in I and K; and from F with the distance E A or E B, the semi-axis major, describe the arc L M. Join I M, and from L and K draw lines parallel to I M, cutting F G, F H, in N and O. From A and B ([Fig. 25 a]) set off the distance F N ([Fig. 25 b]) in points N′, and from these points as centres, with F N as radius, describe an arc of about 15° on each side of the major axis. From C and D ([Fig. 25 a]) set off on the minor axial line the distance FO ([Fig. 25 b]) in points O′, and from these points as centres, with radius FO, describe arcs of about 15° on each side of the axis C D. To obtain any number of intermediate points take a slip of paper ([Fig. 25 a]) and mark upon one edge, with a sharp-pointed pencil, 1, 3, equal to the semi-axis major, and 2, 3, equal to the semi-axis minor. If the slip of paper be now applied to the figure and moved over it in such a manner that the point 2 is always in contact with the major axis, and the point 1 in contact with the minor axis, the outer point 3 will describe a perfect ellipse, any number of points in which can be marked off as the “trammel” is moved into successive positions.
For this last method, which in practice is by far the best, we are indebted to Binns’ ‘Orthographic Projection.’
Fig. 26.
To construct a Semi-Elliptical Arch.
—The span A B ([Fig. 26]) and rise C D being given, divide C A and C B into any number of equal parts. Through the point D, draw E F parallel to A B, and from the points A and B erect the perpendiculars A E and B F. Divide A E and B F similarly to C A and C B. Produce C D and make C G equal C D. From D draw lines to the points 1, 2, 3, &c., in the lines A E and B F; also from G draw lines through the points 1, 2, 3, &c., in the line A B, and produce these lines until they cut those drawn from D to the corresponding numbers in A E and B F. Through the points thus obtained draw the curve of the ellipse.