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but if CB is much less than CA, S and H are near the extremities of the major-axis, and at a distance from the

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centre; consequently the ratio of CS to CA is called the eccentricity of the ellipse, which increases in quantity with its flatness, and decreases as it approaches nearer and nearer to a circle in form. A tangent is a straight line as PT, which touches the curve without cutting it; a normal is a right-line as PN, perpendicular to the tangent at the point of its contact with the curve.

104. To describe an ellipse of given length of axes by continuous motion.-Let A CD, BCE, be the given axes (fig. 87). From B as a centre, with AC as radius, draw an arc cutting AD in S and H the foci of the proposed ellipse. Fix pins at S and H, and attach thereto a string whose length, when tightened, is equal to AD. Tighten the string by means of a pencil round which it can move freely, and as the pencil moves from A towards B or E it will trace out the ellipse. The arc from A to B, or A to

E, is one-fourth of the ellipse, and is exactly equal and similar to the part from D to B, or from D to E; the axes dividing the ellipse into four equal and similar parts.

105. To find any number of points to the circumference of an ellipse of given axes.-Let S and H (fig. 87) be the foci of the ellipse, determined as described above (103). Take any point X between C and S, and from S as a centre, with AX as radius, describe a circle, and from H as a centre, with DX as radius, draw an arc cutting the circle in the point P, which will be a point upon the circumference of the required ellipse. Then by taking any number of points on the axis between C and S, we can find a corresponding number of points on the curve, in the same way as P is found. The curve can then be approximately drawn by hand through the points thus found, or by means of a straight-edge bent through the points.

106. To draw an ellipse by help of the circumscribed circle, the length of the axes being given.-Let A CD, BCE, be the axes (fig. 87) of the ellipse. From C as a centre, with CA as radius, describe a circle AFD. Take any point M on the major-axis and erect the ordinate MR, perpendicular to the axis, cutting the circle at R. Draw the chord of the circle FR and produce it to meet the axis produced in O; draw OE cutting MR at Q, then Q is a point on the required ellipse. MQ is also a fourth-proportional (16) to CF, CE, and MR. In this manner we can find any number of points on the ellipse by drawing ordinates from the axis to the circle and finding a fourth-proportional to CF, CE, and the ordinate to the circle.

The points on the ellipse can also be practically found by the use of two different scales. Let CD or C F represent 100 on any scale, and let CE measure 100 on a smaller

scale; then whatever M R, the ordinate of the circle, measures on the first scale, the same will the ordinate MQ of the ellipse measure on the second scale. Any number of points can be thus determined with tolerable accuracy, and the ellipse may be drawn through them by hand. The points on the ellipse can also be found from those on the circle by help of a pair of proportional compasses set in the ratio of AC to BC.

107. To draw a tangent at any given point on an ellipse, by help of the foci.-Let S and H (fig. 87) be the foci of the ellipse, P the given point on the circumference. Draw SP, HP, and produce HP to Z, making PZ equal to PS; and join SZ. Bisect SZ at K, and draw PKT, which is the tangent to the ellipse at the point P.

The tangents at A and D are perpendicular to the majoraxis AD and parallel to the minor-axis BCE; those at B and E are parallel to the major-axis A CD, and at right angles to the minor-axis BE.

108. To draw a tangent at any point of an ellipse, by help of the circumscribed circle.-Let A FD be the circumscribed circle (fig. 87) struck from the centre C; Q the given point on the circumference of the ellipse at which the tangent is to be drawn. Draw the ordinate MQR at right angles to the axis A D, and cutting the circle at R. Draw RV the tangent to the circle at R (48), cutting the axis produced at V. Draw VQ which will be the tangent to

the ellipse at Q.

109. To draw a tangent to an ellipse from a given point on the major-axis produced.—Let V (fig. 87) be the given point from which the tangent is to be drawn. Draw any line V W, making an acute angle with VA, and take V U equal to the half-major-axis DC. Join CU, and draw

AW parallel to CU. Take CM equal to UW and erect the ordinate MQ at right angles to CM; then the line VQ will be the tangent at the point Q. CM is the third proportional (15) to CV and CA, or CV: CA: CA: CM.

110. To draw the normal, or perpendicular to the tangent at any given point on an ellipse.-Let P be the given point (fig. 87) on the ellipse, S and H the foci. Draw SP, HP, and produce HP to Z, making PZ equal to PS; join ZS, and draw PN parallel to SZ; then PN is the normal at P and is at right angles to the tangent PT (107) at that point. The normal can also be drawn by bisecting the angle SPH by the line PN.

At the points A, B, D, E, the axes themselves are the normals to the ellipse.

Whenever the ellipse is employed in architecture as the contour of an arch, the directions of the joints of the voussoirs must be in the normals to the curve.

111. To draw tangents to an ellipse from any given point

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outside of it.-Let P be the given point (fig. 88) outside the ellipse. From P draw three right-lines PGC, PFD,

PHE, cutting the ellipse in G, F, H, C, D, E. Draw the diagonals EF, DH, DG, CF, intersecting at X and Y. Draw the right-line A B passing through the points X and Y, and cutting the curve at A and B. Then the lines PA, PB will be tangents to the ellipse. In this way two tangents can be drawn from any given point outside the curve.

112. About a given ellipse to circumscribe a trapezium of which two opposite vertices are given.-Let P and Q (fig. 88) be the given vertices of the proposed trapezium, ABCD the given ellipse. Draw from P by the last proposition (111) PR, PS, tangents to the ellipse; also from Q draw QR, QS, tangents to the ellipse intersecting the former pair of tangents in the points R and S; then the figure PRQS will be the trapezium required.

Fig. 89.

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113. To approximate to the outline of an ellipse by means of arcs of circles. —Let AD (fig. 89) be the major-axis,

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