PROBLEM XIV.

An ellipsis A DBC being given, to draw a tangent through a given point H in the curve.

1. Find the foci F and G, join F H and G H.

2. Produce G H to I, upon H, with any radius, describe the arc K L I, cutting G I, and F H, at K and I.

3. Bisect the arc K L I, at L; through L and H draw L H, it will be the tangent required.

PROBLEM XV.

To draw two tangents to an ellipsis from a given point E, without it, having any two conjugate diameters A B, and CD, given, without drawing any part of the ellipsis.

1. Let the point E be in the diameter D C produced. 2. From the centre H, make H I equal to H C, and join I E.

3. Through C, draw C K parallel to I E, cutting H B in K.

4. Make H L equal to H K, through L draw F G parallel to A B, find the extreme points F and G of the ordinate F G; by problem XI. From E, through the points F and G, draw E F and E G, they will be the tangents required.

If the point E, is in neither of the given diameters A B, or CD, when produced; draw a line from the given point E, through the centre; by problem XII. find a conjugate to that line, and the extremities of both, then the construction will be the same as in this.

PROBLEM

PROBLEM XVI.

To describe an ellipsis similar to a given one AD BC, to any given length IK, or to a given width M L.

1. Let A B, and C D, be the two axes of the given ellipsis. 2. Through the points of contact A, D, B, C, complete. the rectangle G E HF, draw the diagonals E F, and G H, they will pass through the centre at R.

3. Through I, and K, draw P N, and O Q, parallel to CD, cutting the diagonals E F, and G H, at P, N, Q, O. 4. Join P O, and N Q, cutting C D at L, and M, then IK, is the transverse, M L the conjugate axes of an ellipsis that will be similiar to the given one A D B C, which may be described by some of the foregoing methods.

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Given the rectangle ABCD, to circumscribe an ellipsis, which shall have its two axes in the same ratio as the sides of the rectangle.

1. Draw the diagonals A C, and B D, cutting each other at S, the centre.

2. Through S, draw E F, and G H, parallel to AB, and

A D.

3. Upon S, with a radius S I, equal to half A D, or B C, describe the quadrant I K L, cutting E F, at L.

4. Bisect the arc I K L, at K; through K, draw M N parallel to E F, cutting the diagonal B D, at N.

5. Join IN; through B, draw B G, parallel to it, cutting G H at G, and make S H equal to S G.

6. Join NO; through B, draw B F, parallel to it, cutting E F, at F; make S E equal to S F, then E F, and G H, are the two axes which may be described by some of the methods which are shewn in the foregoing problems.