2. From E, with any radius, cut the produced long diameter both ways in F 1, F 2, the foci, and join them to one angle of the rectangle, as C.

3. Draw any straight line, as FG, equal to the sum of CF 1, CF 2, and bisect it in H (Pr. 1).

4. Make EK and EL equal to HF or HG; then KL is the transverse aris.

5. From F 1, with half the transverse axis, as EK, as radius, cut the line perpendicular to KL in M and N; then MN is the conjugate axis.

6. Complete the required ellipse by Problem 80.

Problem 89.

To describe an oval by arcs of circles.

1. Draw any straight line AB, and describe a semicircle CD equal in diameter to the proposed oval.

2. From C and D, with the radius of the semicircle, cut the straight line in A and B.

3. From A and B, with radius BC, describe the arcs DF and CE

4. From A or B, draw a straight line through the transverse diameter, cutting it in G, and meeting the opposite arc in E or F.

5. From G, with radius GE, describe arc EF, which will complete the required oval CDFE.

NOTE.-The oval may be made longer or shorter by increasing or diminishing the transverse diameter.

Problem 90.

To construct a parabola, its ordinate AB and abscissa BC being given.

6B

1. Through C draw a line CD, parallel and equal to AB (Pr. 8), and join AD.

2. Pride AB and AD into the same number of equal parts (say six).

3. From C, draw lines to the points of division in AD.

4. From the points of division in AB, draw lines parallel to BC, till each meets the corresponding line from AD. The points of intersection will be in the curve of the required parabola.

NOTE.-By repeating the process in the other half, the curve of the required parabola will be completed.

Problem 91.

To construct a hyperbola, its diameter AB, abscissa BC, and ordinate CD being given.

4C

1. Through B draw a line BE, parallel and equal to CD (Pr. 8), and join DE.

2. Divide DC and DE into the same number of equal parts (say four).

3. From B, draw lines to the points of division in DE.

4. From A, draw lines to the points of division in DC. The points of intersection will be in the curve of the required hyperbola.

NOTE.-By repeating the process in the other half, the curve of the required hyperbola will be completed.

SECTION VII.-INSCRIBED FIGURES.

DEFINITIONS.

1. Inscribed figures. Inscribed figures are either rectilineal or circular.

(a.) A rectilineal figure is said to be inscribed in another rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is inscribed. Ex. ABCD

A

B

C

(b.) A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle. Ex. ABCD

A

B

NOTE.-A circle is said to be inscribed in a rectilineal figure,

when the circumference of the circle touches each side of the figure. Ex. A—

2. A sector of a circle is a figure contained by two radii and the intercepted arc. Ex. A

Problem 92.

To inscribe an equilateral triangle within a given circle A.

B

A

C

1. Find the centre of the circle A (Pr. 45), and draw a diameter BC.

2. From C as centre, with radius CA, describe an arc cutting the circumference in D and E.

3. Join DE, EB, BD; then DEB is the required equilateral triangle inscribed within the given circle A.