An ellipse is a figure bounded by a continuous curve, whose nature will be shown presently.

The dimensions of an ellipse are taken at its extreme length and narrowest width, and they are designated in three ways, as by the length and breadth, by the major and minor axis (the major axis meaning the length, and the minor the breadth of the figure), and the conjugate and transverse diameters, the transverse meaning the shortest, and the conjugate the longest diameter of the figure.

In this book the terms major and minor axis will be used to designate the dimensions.

The minor and major axes are at a right angle one to the other, and their point of intersection is termed the axis of the ellipse.

In an ellipse there are two points situated upon the line representing the major axis, and which are termed the foci when both are spoken of, and a focus when one only is referred to, foci simply being the plural of focus. These foci are equidistant from the centre of the ellipse, which is formed as follows: Two pins are driven in on the major axis to represent the foci A and B, Figure 75, and around these pins a loop of fine twine is passed; a pencil point, C, is then placed in the loop and pulled outwards, to take up the slack of the twine. The pencil is held vertical and moved around, tracing an ellipse as shown.

Fig. 75.

Now it is obvious, from this method of construction, that there will be at every point in the pencil's path a length of twine from the final point to each of the foci, and a length from one foci to the other, and the length of twine in the loop remaining constant, it is demonstrated that if in a true ellipse we take any number of points in its curve, and for each point add together its distance to each focus, and to this add the distance apart of the foci, the total sum obtained will be the same for each point taken.

Fig. 76.

Fig. 77.

In Figures 76 and 77 are a series of ellipses marked with pins and a piece of twine, as already described. The corresponding ellipses, as A in both figures, were marked with the same loop, the difference in the two forms being due to the difference in distance apart of the foci. Again, the same loop was used for ellipses B in both figures, as also for C and D. From these figures we perceive that -

1st. With a given width or distance apart of foci, the larger the dimensions are the nearer the form of the figure will approach to that of a circle.

2d. The nearer the foci are together in an ellipse, having any given dimensions, the nearer the form of the figure will approach that of a circle.

3d. That the proportion of length to width in an ellipse is determined by the distance apart of the foci.

4th. That the area enclosed within an ellipse of a given circumference is greater in proportion as the distance apart of the foci is diminished; and,

5th. That an ellipse may be given any required proportion of width to length by locating the foci at the requisite distance apart.

The form of a true ellipse may be very nearly approached by means of the arcs of circles, if the centres from which those arcs are struck are located in the most desirable positions for the form of ellipse to be drawn.

Fig. 78.

Thus in Figure 78 are three ellipses whose forms were pencilled in by means of pins and a loop of twine, as already described, but which were inked in by finding four arcs of circles of a radius that would most closely approach the pencilled line; a b are the foci of all three ellipses A, B, and C; the centre for the end curves of a are at c and d, and those for its side arcs are at e and f. For B the end centres are at g and h, and the side centres at i and j. For C the end centres are at k, l, and the side centres at m and n. It will be noted that, first, all the centres for the end curves fall on the line of the length or major axis, while all those for the sides fall on the line of width or the minor axis; and, second, that as the dimensions of the ellipses increase, the centres for the arcs fall nearer to the axis of the ellipse. Now in proportion as a greater number of arcs of circles are employed to form the figure, the nearer it will approach the form of a true ellipse; but in practice it is not usual to employ more than eight, while it is obvious that not less than four can be used. When four are used they will always fall somewhere on the lines on the major and minor axis; but if eight are used, two will fall on the line of the major axis, two on the line of the minor axis, and the remaining four elsewhere.

Fig. 79.

In Figure 79 is a construction wherein four arcs are used. Draw the line a b, the major axis, and at a right angle to it the line c d, the minor axis of the figure. Now find the difference between the length of half the two axes as shown below the figure, the length of line f (from g to i) representing half the length of the figure (as from a to e), and the length or radius from g to h equalling that from e to d; hence from h to i is the difference between half the major and half the minor axis. With the radius (h i), mark from e as a centre the arcs j k, and join j k by line l. Take half the length of line l and from j as a centre mark a line on a to the arc m. Now the radius of m from e will be the radius of all the centres from which to draw the figure; hence we may draw in the circle m and draw line s, cutting the circle. Then draw line o, passing through m, and giving the centre p. From p we draw the line q, cutting the intersection of the circle with line a and giving the centre r. From r we draw line s, meeting the circle and the line c, d, giving us the centre t. From t we draw line u, passing through the centre m. These four lines o, q, s, u are prolonged past the centres, because they define what part of the curve is to be drawn from each centre: thus from centre m the curve from v to w is drawn, from centre t the curve from w to x is drawn. From centre r the curve from x to y is drawn, and from centre p the curve from y to v is drawn. It is to be noted, however, that after the point m is found, the remaining lines may be drawn very quickly, because the line o from m to p may be drawn with the triangle of 45 degrees resting on the square blade. The triangle may be turned over, set to point p and line q drawn, and by turning the triangle again the line s may be drawn from point r; finally the triangle may be again turned over and line u drawn, which renders the drawing of the circle m unnecessary.