Another Method.

1. Place the given diameters, as before, and find the foci F 1, F 2 (Pr. 80).

2. Fix a pin at each of the foci, and another at one end of the conjugate diameter, as C.

3. Tie a piece of thread tightly around the three pins, forming a triangle C, F 1, F 2.

4. Take out the pin at C, and put a pencil within and against the string at C, and keeping the string perfectly tight, and close to the paper throughout, describe the curve of the required ellipse, which will pass through B, D, A, C. N.B.-This method is by mechanical means.

Problem 81,

To find the centre and axes of a given ellipse A.

1. Draw any two chords B and C parallel to each other (Pr. 8), and bisect them in D and E (Pr. 1).

2. Draw a diameter FG through D and E, and bisect it in A; then A is the centre of the ellipse.

3. From A, with AG as radius, mark the point K, and join GK and KF.

4. Through A draw LM and NO parallel to GK and FK (Pr. 8); then NO and LM are the axes required.

Problem 82.

To describe an elliptical figure, one diameter AB being given. 1. Divide AB into four equal parts, in points C, E, D (Pr. 15).

2. From C and D, with radius CA or DB, describe circles touching each other in E.

F

Α

M

K

B

3. From C and D, with radius CD, describe arcs cutting each other in F and G.

4. Draw lines FC, FD, GC, GD, and produce them until they cut the circles in H, K, L, M.

5. From F and G, with radius FH or GM, draw ares uniting H with K and L with M, which will complete the required elliptical figure.

Problem 83.

To construct an elliptical figure, two squares ABCD and CDEF being given.

1. Draw diagonals in each of the squares, intersecting each

other in G and H.

2. From C, with radius CA or CE, describe the arc AE.
3. From D, with the same radius, describe the arc BF.

E

4. From G, with radius GA, describe the arc BA.
5. From H, with the same radius, describe the arc EF, which
will complete the required elliptical figure.

Problem 84

To draw a perpendicular to the curve of a given ellipse from a given point 4.

B

1. Draw the transverse axis, and find the foci B and C (Pr. 80).

2. Draw the lines BA and CA, and produce them, making

the angle DAE.

3. Bisect angle DAE by line AF (Pr. 4); then AF is perpendicular to the curve of the given ellipse.

F

Problem 85.

To draw a tangent to the curve of a given ellipse at a given point of contact A.

C

1. Draw the transverse axis BC (Pr. 81), and obtain the foci D and E (Pr. 80).

2. From D and E draw lines DA, EA through the given point of contact A, producing one of them, as DA to F.

3. Bisect the external angle FAE in G (Pr. 4).

4. Draw line GA, and produce it; then GA is a tangent to the given ellipse, through the given point of contact A.

Problem 86.

To complete the curve of an ellipse which is partly constructed, one quarter ABC being given.

1. Produce CA beyond A, and make AD equal to AC; also produce BA beyond A, and make AE equal to AB.

2. Find the foci F1 and F 2 (Pr. 80), and proceed as in ProThe curve BDEC is the required completion.

blem 80.

Problem 87.

To complete the curve of an ellipse which is partly constructed, more than half of the curve ABCD being given.

1. At some portion of the given curve, not opposite the part which is incompleted, draw two parallel chords M, N, and find the centre E (Pr. 81).

2. From E, with any sufficient radius, describe an arc, cutting the curve in Fand G.

3. Bisect the arc FG in B (Pr. 1), and produce the line of bisection through E to H; then BH is the conjugate axis.

4. At E, draw a line EC at right angles to BH (Pr. 2). meeting the given curve in C.

5. Produce CE, and make EK equal to EC; then CK is the transverse axis.

6. Find the foci F 1, F 2 (Pr. 80), and complete the curve by means of intersecting arcs (Pr. 80). The curve drawn from A to D through K is the required completion.

Problem 88.

To describe an ellipse about a given rectangle ABCD.

1. Draw the diagonals AC, BD, meeting in E, the centre of the required ellipse, and draw the diameters indefinitely beyond the sides of the given rectangle.