From the other end, D, of the minor axis, draw a line through point 3' in A O, to meet the line C 3 in P.

Then P is a point in the curve, and by drawing other lines

from D to meet the lines from C, the remaining points can be found.

(Note D 4' meets C 4, D 5' meets C 5, and so on.)

The ellipse is best completed by repeating this construction for the portion in CO BF, and then using the ordinate method, as described at the end of Method II.

Method IV. by "trammels" (Fig. 22).—A very convenient way

G

Fig. 22.

(known as the trammel method) of finding points in the curve of an ellipse, is as follows:-Mark off along the edge of a strip of paper, card, or wood, a distance, EF, equal to half the minor axis, and from the same end, a distance, E G, equal to half the major axis.

Place the strip in such a way that the minor axis point, F, is always on the major axis, and the major axis point, G, always on the minor axis, then the end, E, of the strip will be a point in the curve.

The strip is shown in position for two different points.

This is a common drawing office and workshop method, and as it readily gives any number of points, is very useful.

Tangents and Normals.-It is necessary to be able to draw the tangent and normal to an ellipse. A tangent to a curve is often of use as showing the direction of the curve. In an elliptical arch, the ends of the stones, or the radial members, would be normals to the curve.

"The normal to an ellipse at any point in the curve bisects the angle between the lines joining that point to the foci. (These lines are called the focal distances.)

"The tangent at any point is at right angles to the normal at that point."

Thus to draw a normal and tangent at any point, P (Fig. 19), it is only necessary to join the point to the foci and bisect the angle FPF' between the joining lines. This bisecting line MN is the normal, and a line at right angles, TT, is the tangent.

Or, produce FP to Q, then the tangent bisects the angle QP' F', and the normal is perpendicular to it.

"If two tangents are drawn to an ellipse from a point outside the curve, and the contact points are joined to a focus, then the angles between these lines and the line joining the focus to the point are equal."

Thus in (Fig. 20) the angles P F G and P F H are equal.

Hence to draw a tangent from a point outside the ellipse, it is necessary to adopt a construction making these angles equal. This can be done as follows:

With the point P as centre, draw an arc passing through a focus, F. With the other focus, F', as centre, and the major axis as distance, describe an arc, cutting the first arc in a and b.

Join a and b to the focus F', cutting the curve in the points G and H. These are the contact points of the tangents from P. In the triangles Pa F' and PbF' the three similar sides are respectively equal, therefore the angles PF' G and PF' H are equal (Euclid i., 8), and, therefore, G and H must be the contact points of the tangents.

Parallels to an Ellipse.-A parallel to a curve is equidistant from it at all points. It has not necessarily the same mathematical properties as the curve to which it is parallel.

The curve used in constructing the arches of bridges is

frequently a parallel to an ellipse, as this gives greater vertical clearance near the abutments than the true ellipse.

The parallel to an ellipse is most conveniently drawn by describing a large number of small radii of the required distance from points on the curve as centre; or it can be obtained by drawing a number of normals to the curve of the required length, and drawing the parallel curve through the ends of the normals.

The parallel curve, the curve, N N, to an ellipse, is shown partly constructed in Fig. 20. It is drawn touching the small arcs, described from points on the ellipse as centre, but might have been drawn by constructing a number of equal normals and drawing the curve through the ends.

A convenient practical way of finding whether a given curve is a true ellipse, is to draw lines representing the two axes, and mark the foci, supposing the curve to be an ellipse. Then measure the focal distances of a number of points and find if the sum is constant; or partly construct a true ellipse about the assumed axes and foci, when its difference from the given curve will show the error of the curve.

EXAMPLES.

EX. 27.-Construct an ellipse, major axis 7", minor axis 4′′, by the following methods:- -(a) arcs of circles;" (b) "two circles;" (c) "oblong;" (d) "trammels." In each case draw

a tangent and normal to the curve, from a point in the curve and from a point outside the curve.

EX. 28. Work the following by drawing:-(a) major axis of an ellipse is 6′′, minor axis 21", find the foci; (b) major axis 61′′, foci are 1" from each end, find the minor axis; (c) minor axis is 31", foci are 4′′ from the ends of the minor axis, find the major

axis.

EX. 29.-Carefully draw an ellipse by two circles method, major axis 7", minor axis 4". Rub out all lines except the curve and the axis, and find the foci. Then take six different points in the curve and find the sum of the distances of each point from the foci.

EX. 30.-Construct a semi-ellipse, axis 4" and 2′′. Then draw a second parallel curve 21′′ away, and find if this curve is a true ellipse.

EX. 31.-Draw an ellipse, the distance between the foci being 21", and the major axis 3" long. (S. & A. E., 1891.)

EX. 32.-Two points, F and F', 2′′ apart, are the foci of, and P (2" from F and " from F') is a point on, an ellipse. Draw the curve. (S. & A. A., 1892.)

EX. 33.-Draw an ellipse inscribed in a parallelogram the sides of which are 4" and 5" long, and are inclined at 60° to one another. (V. U. Hon., 1889.)

SECTION V.

CIRCLES AND TANGENTS-AREAS-MISCEL-
LANEOUS PROBLEMS.

THE practical draughtsman does not usually adopt a geometrical construction to enable him to draw such circles and lines tangentially to each other as are required in ordinary mechanical drawings, since his own skill in draughtsmanship ensures sufficient accuracy. But instances often occur in solid geometry problems, as well as in mechanical drawings, where it is very advantageous, if not absolutely necessary, to be able to accurately determine the contact points of lines and circles, hence the student should render himself familiar with the chief geometrical constructions.

PROBLEM XIV. (Fig. 23).-To draw a tangent to a circle, (a)

P

Fig. 23.

from a point in the circumference, (b) from a point outside the circle.

A tangent to a circle is a line which touches the circle, without cutting it. A tangent is at right angles to the radius passing through the point of contact.-Euclid iii., 18.

(a) Let P be the point in the circumference of the circle. Join P to the centre C, then the required tangent, ST, is the line drawn at right angles to the radius, PC.

(6) Let P' be the point outside the circle. In order to draw the tangent correctly,

we require to find its contact point, and knowing that the tangent is at right angles to the radius, "the angle in a semicircle is a right angle."

we remember that Therefore join the

point P' to the centre C, and on P'O describe a semicircle cutting the given circle in D. Then P'D is the required tangent, for it touches the circle at D, and is perpendicular to the radius, DO.

Notice that a second tangent can be drawn from P' to touch the circle, as shown in the figure by dotted lines. It is easy to see that "the two tangents are equal in length," a fact which should be remembered.

PROBLEM XV. (Fig. 24).—In a given angle to inscribe a circle of given radius, and also to inscribe a second circle tangent to the first circle and to the angle.

Let B A C be the given angle. The circle must evidently have its centre on the line bisecting the angle, therefore first bisect the angle BAC by the line A D.

A

Fig. 24.

K

Draw a line parallel to A B, at a distance from it equal to the radius of the required circle cutting A D in E. (This is best done by first drawing a perpendicular to A B from any point in it, and making its length equal to the given radius.) E is the centre of the circle, and in order to accurately draw the circle, it is best to first draw a line from E perpendicular to A B or A C, to obtain the point of contact, F.

Next to draw a second circle touching the first and the sides of the angle. Draw the line GH from the point G, where the circle cuts A D, and perpendicular to A D. Then H G and H B will both be tangents to the required circle when it is drawn; therefore, if H B is made equal to H G, the point B so found will be the contact point of the required circle. A perpendicular, BK to A B, through the point B, will cut A D in K, which will be the centre of the required second circle.

PROBLEM XVI. (Fig. 25).—To draw three circles of given radius in contact with each other. (The method of this construction is useful in problems on spheres in contact.)

Let the circles be of 2′′, 11′′, and 1" radius.

Draw any straight line and draw any two of the circles (say of 2" and 11" radius) touching each other, having their centres at A and B in the line.

Set off the radius of the third circle along the line, beyond the two circles, A and B to C and D, as shown. With A as centre, draw an arc passing through C, and with B as centre, draw an arc