Let A (Fig. 402) be the line to be divided, and B the line with its divisions. Make ab equal to B with all its divisions, as at 1, 2, 3', etc.; from a draw ac at any angle with a b; make a c equal to A; join c and b; from the points 1, 2, 3, etc., draw lines parallel to cb; then these will divide the line a c in the same proportion as B is divided - as was required.
This problem will be found useful in proportioning the members of a proposed cornice, in the same proportion as those of a given cornice of another size. (See Art. 321.) So of a pilaster, architrave, etc.
Let A and B (Fig. 403) be the given lines. On the line ac make ab equal to A and bc equal to B; bisect ac in e; upon e, with ea for radius, describe the semi-circle adc; at b erect bd at right angles to ac; then bd will be the mean proportional between A and B. That is, ab is to bd as bd is to be. This is usually stated thus: ab: bd:: bd: be, and since the product of the means equals the product of the extremes, therefore, ab x be = bd2 This is shown geometrically at Art. 538.
If a cone, standing upon a base that is at right angles with its axis, be cut by a plane, perpendicular to its base and passing through its axis, the section will be an isosceles triangle (as a bc, Fig. 404); and the base will be a semi-circle. If a cone be cut by a plane in the direction ef the section will be an ellipsis; if in the direction ml, the section will be a parabola; and if in the direction ro, an hyperbola. (See Art. 499.) If the cutting planes be at right angles with the plane a bc, then -
545. - To Find the Axes of the Ellipsis: bisect ef (Fig. 404) in g; through g draw h i parallel to ab; bisect h i in j;
upon 7, with jh for radius, describe the semi-circle hki; from g draw gk at right angles to hi; then twice gk will be the conjugate axis and ef the transverse.
Let ml (Fig. 404), parallel to ac, be the direction of the cutting plane. From m draw m d at right angles to a b; then lm will be the axis and height, and md an ordinate and half the base, as at Figs. 417, 418.
Let o r (Fig. 404) be the direction of the cutting plane. Extend or and ac till they meet at n; from o draw op at right angles to a b; then r o will be the height, n r the transverse axis, and op half the base; as at Fig. 419.
Let ab (Fig. 405) and cd be the given axes. Upon c, with ae or be for radius, describe the arc ff; then f and f, the points at which the arc cuts the transverse axis, will be the foci. At f and f place two pins, and another at c; tie a string about the three pins, so as to form the triangle ffc; remove the pin from c and place a pencil in its stead; keeping the string taut, move the pencil in the direction cga; it will then describe the required ellipsis. The lines fg and gf show the position of the string when the pencil arrives at g.
This method, when performed correctly, is perfectly accurate; but the string is liable to stretch, and is, therefore, not so good to use as the trammel. In making an ellipse by a string or twine, that kind should be used which has the least tendency to elasticity. For this reason, a cotton cord, such as chalk-lines are commonly made of, is not proper for the purpose; a linen or flaxen cord is much better.