through the point d, in the middle of this line, and the centre O, draw the line A C, which will be the transverse diameter of the ellipse.

From the points e, e, taken at equal distances from the centre O, describe arcs intersecting each other in g, g; through these points, and the centre O, draw the conju gate diameter B D, and it is done.

PROBLEM XXX.

To determine whether a given oval figure be greater or less than a true ellipse.

By the last Problem, draw the transverse and conjugate diameters AC and B D. With the radius AO in the compasses, and D as a centre, find the two focuses F, F. Take a string equal in length to the transverse diameter A C, and make its ends fast, with two pins, at the focuses. Draw the string tight, so as to form the triangle Fm F; move the point m towards C, always keeping both parts of the string stretched, and if it every where coincide with the curve, the figure is a true ellipse; if it extend beyond the curve as at n, it is less than an ellipse; and if it fall short of the curve as at r, it is evidently greater than an ellipse. (See Prob. 15.)

Note. The last two Problems are frequently of considerable utility in Practical Gauging.

PROBLEMS

IN

PRACTICAL GEOMETRY,

FOR THE EXERCISE OF THE LEARNER.

1. The length of a given line A B, is 15 inches; it is required to lay it down by a scale of equal parts, and bisect it geometrically.

2. It is required to bisect a given angle A B C.

3. Draw a line CD, parallel to a given line A B, at the distance of eight-tenths of an inch.

4. A given line measures 18 inches; it is required to erect a perpendicular to this line at the distance of 8 inches from one end.

5. The three sides of a triangle are 16, 20, and 24 inches respectively; it is required to lay it down by a scale of equal parts.

6. The base of a triangle measures 24, the distance of the perpendicular from one end of the base 14, and the perpendicular itself 12 inches; it is required to lay down the triangle.

7. Lay down a square, whose side is 16 inches.

8. The length of a rectangle is 24, and its breadth 12 inches; it is required to construct the figure.

9. Construct a regular rhombus whose side is 24 inches.

10. The base of an irregular rhombus measures 16, and the perpendicular breadth 12 inches; it is required to construct the figure.

11. Lay down a regular rhomboid whose sides measure 36 and 18 inches respectively.

12. The transverse diameter of an ellipse, measures

28, and the conjugate 16 inches; it is required to construct the figure.

13. The side of a regular hexagon measures 14 inches; it is required to construct the figure.

14. It is required to make an acute angle that shall contain 42° 30'.

15. It is required to make an obtuse angle that shall contain 136° 45',

GEOMETRICAL THEOREMS,

THE

DEMONSTRATIONS

OF WHICH

May be seen in the Elements of Euclid, Simpson,

and Emerson.

THEOREM I.

If two straight lines A B, CD, cut point E, the angle A E C will be equal and CEB to AED. (Euclid I. 15. I. 2.)

C

to

each other in the the angle D EB, Simp. I. 3. Em.

The greatest side of every triangle is opposite to the

greatest angle, (Euc. I. 18. Simp. I. 13.

Em. II. 4.)

THEOREM III.

Let the right line E F fall upon the parallel right lines A B, CD; the alternate angles A G H, G H D, are equal to each other; and the exterior angle E G B, is equal to the interior and opposite, upon the same side, GHD; and the two interior angles BGH, GHD, upon the same side, are together equal to two right angles. (Euc, I. 29. Simp. I. 7. Em. I. 4.)

Let A B C be a triangle, and let one of its sides B C be produced to D; the exterior angle A CD is equal to the two interior and opposite angles CA B, A BC; also the three interior angles of every triangle are together equal to two right angles. (Euc. I. 32. Simp. I. 9, 10. Em. II. 1, 2.)

Let the parallelograms A B C D, DBCE be upon the same base B C, and between the same parallels A E, BC; the parallelogram A B C D is equal to the parallelogram DBCE. (Euo. I. 35, Simp. II. 2. Em. III. 6.

A DE

B

THEOREM VI.

Let the triangles A BC, DBC be upon the same base BC, and between the same parallels AD, BC; the triangle A B C is equal to the triangle DBC. (Euc. I. 37. Simp. II. 2. Em. II. 10.)

A D

THEOREM VII.

Let A B C be a right angled triangle, having the right angle B A C; the square of the side B C is equal to the sum of the squares of the sides AB, A C. (Euc. I. 47. Simp. II. 8. Em. II. 21.)

Note. Pythagoras, who was born about 2400 years ago, discovered this celebrated and useful theorem; in consequence of which, it is said, he offered a hecatomb to the gods.

THEOREM VIII.

Let A B C be a circle, and B DC an angle at the cen