PART THE FIRST.

PRACTICAL GEOMETRY.

PROBLEM 1.

To divide a line A B into any number of equal parts.

E

Draw A F, in any direction, to an unlimited length, take any convenient arbitrary length A C, and repeat it along the line A F, at the points C, D, &c., as often as the number of equal parts into which the line A B is proposed to be di- A vided. In this case it is four; join F B, parallel to which draw EI, DH, CG, then the line AB is divided in the points G, H, I.

PROBLEM 2.

G H I B

To divide a line A B in the same proportion as a given line CD.

Draw AH, in any direction, equal to CD, and place upon it AI, IK, KL, &c., equal to CE, a EF, FG, &c.; join HB, parallel to which draw L O, KN, IM; then A B is divided in the points M, N, O, in the same proportion as CD.

E F

G

LH

K

M

N

B

PROBLEM 3.

To bisect a given angle, A B C.

With the centre, B, and any radius, B E, describe the arc D E; and with the same radius, or less if more conve- B nient, and centres D and E, describe arcs cutting each other in F, join FB; then FB bisects the angle ABC.

B

PROBLEM 4.

From a given point A, to draw a line perpendicular to B C.

With centre B and radius B A describe the arc A D, with the centre C and ra

dius C A describe the arc A D, join A D Ē cutting B C in E; then E A is the perpendicular required.

E

If the point A be in the line B C the above construction fails.

Take any point, E, and with the radius E A describe the circle D A F, join F E and produce it to meet the circle in D ; join D A, which is the perpendicular required.

E

B

PROBLEM 5.

To find a third proportional to the lines A B and A C.

A

Place A B and A C making an angle at A, Ajoin C B, and with the centre A and radius A C describe the circle C E, draw EF parallel to B C; cutting A C in F, then A F is the third proportional required.

PROBLEM 6.

EB

To find the fourth proportional to the lines A B, A C, and A D.

Place A B, A C, making an angle at A, join C B and draw D E parallel to B C; then A E is the fourth proportional required.

4

PROBLEM 7.

To find a mean proportional to two lines, A B, C D.

Place A B, CD in the same straight line and bisect it in O, with the centre O and radius O C describe the semicircle A E C, draw B E perpendicular to A C, then BE is the mean proportional required.

A

E

PROBLEM 8.

OB

To make a triangle with the given straight lines A B, C D, E F.

E

Take A B as the base, and with the centre A and radius CD C describe the arc at H, with the centre B and radius E F describe the arc at H, join A H, B H, and the triangle A B H is the triangle required.

H

PROBLEM 9.

To make a parallelogram equal to a triangle A B C.

Bisect AB in D, draw DE making any given angle with AB, draw C E F parallel to A B, and BF parallel to DE; then DEFB is the parallelogram required.

A

CE

PROBLEM 10.

To make a triangle equal to the trapezium A B C D.

Produce A B to E, join D B, and draw C E paralled to D B, join DE, and the triangle A DE is the triangle required.

D

A

B

-B

B

PROBLEM 11.

To make an angle of any number of degrees.

From a scale of chords take A B the chord of 60 degrees, and with the centre A, and radius A B describe the circle B C; with the centre B, and a radius equal to the chord of the given number of degrees describe the arc at C, join A A C, and the angle B A C will be the angle required.

PROBLEM 12.

To find the centre of the circle A B C.

Take any chord, A B, and bisect it in D; draw CDE perpendicular to A B, and bisect it in O, which is the centre required.