tances AC, AD; from C and D, as centres, with any radius greater than AC or AD, describe two arcs intersecting each other in B; from A to B, draw the line AB, which will be the perpendicular required.

PROBLEM III.

To raise a perpendicular on the end B of a right line AB, Fig. 20.

Take any point D not in the line AB, and with the distance from D to B, describe a circle cutting AB in E; from E through D draw the right line EDC, cutting the periphery in C, and join CB, which will be perpendicular to AB.

PROBLEM IV.

To let fall a perpendicular upon a given line BC, from a given point A, without it, Fig. 21.

In the line BC take any point D, and with it as a centre and distant DA describe an arc AGE, cutting BC in G ; with G as centre, and distance GA, describe an arc cutting AGE in E, and from A to E draw the line AFE; then AF will be perpendicular to AB.

PROBLEM V.

Through a given point A to draw a right line AB, parallel to a given right line CD, Fig. 22.

From the point A to any point F, in the line CD, draw the right line AF; with F as a centre and distance FA, describe the arc AE, and with the same distance and centre A describe the arc FG; make FB equal to AE, and through A and B draw the line AB, and it will be parallel to CD.

PROBLEM VI.

At a given point B, in a given right line LG, to make an angle equal to a given angle A, Fig. 23.

With the centre A and any distance AE, describe the arc DE, and with the same distance and centre B describe the arc FG; make HG equal to DE, and through Band H draw the line BH; then will the angle HBG be equal to the angle A.

PROBLEM VII.

To bisect any right lined angle BAC, Fig. 24.

In the lines AB and AC, from the point A, set off equal distances, AD and AE; with the centres D and E and any distance more than half DE, describe two arcs cutting each other in F; from A through F draw the line AG, and it will bisect the angle BAC.

PROBLEM VIII.

To describe a triangle that shall have its sides respectively equal to three right lines, D, E, and F, of which any two must be together greater than the third, Fig. 25.

Make AB equal to D; with the centre A and distance equal to E, describe an arc, and with the centre B and distance equal to F describe another arc, cutting the former in C; draw AC and BC, and ABC is the triangle required.

PROBLEM IX.

Upon a given line AB to describe a square, Fig. 26.

At the end B of the line AB, by Problem III. erect the perpendicular BC, and make it equal to AB; with A and C as centres, and distance AB or BC describe two arcs cut

ting each other in D; draw AD, and CD, then will ABCD be the square required.

PROBLEM X.

To describe a circle that shall pass through the angular points, A, B, and C, of a triangle ABC, Fig. 27.

By Problem I. bisect any two of the sides, as AC, BC, by the perpendiculars DE and FG; the point H where they intersect each other will be the centre of the circle; with this centre, and the distance from it to either of the points A, B, or C, describe the circle.

PROBLEM XI.

To divide a given right line AB into any number of equal parts, Fig. 28.

Draw the indefinite right line AP, making an angle with AB, also draw BQ, parallel to AP, in each of which, take as many equal parts AM, MN, &c. Bo, on, &c. as the line AB is to be divided into ; then draw Mm, Nn, &c. intersecting AB in E, F, &c. which will divide the line as required.

PROBLEM XII.

To make a plane diagonal scale, Fig. 29.

Draw eleven lines parallel to, and equidistant from each other; cut them at right angles by the equidistant lines BC; EF; 1, 9; 2, 7 ; &c. then will BC, &c. be divided into ten equal parts; divide the lines EB, and FC, each into ten equal parts; and from the points of division on the line EB, draw diagonals to the points of division on the line FC: thus join E and the first division on FC, the first division on EB, and the second on FC, &c.

Note.-Diagonal scales serve to take off dimensions or numbers of three figures. If the first large divisions be units, the second set of divisions, along EB, will be

10th parts; and the divisions in the altitude, along BC, will be 100th parts. If HE be tens, EB, will be units, and BC will be tenth parts. If HE be hundreds, BE will be tens, and BC units. And so on, each set of divisions being tenth parts of the former ones.

For example, suppose it were required to take off 242 from the scale. Extend the dividers from E to 2 towards H; and with one leg fixed in the point 2, extend the other till it reaches 4 in the line EB; move one leg of the dividers along the line 2, 7, and the other along the line 4, till they come to the line marked 2, in the line BC, and that will give the extent required.

PROBLEM XIII.

To find a third proportional to two given right lines, A and B.

B

Draw two right lines, CD, CE containing any angle; make CF A equal A, and CG, CH, each equal B; join FG and draw HL parallel to it: then will CL be the third proportional required.

PROBLEM XIV.

'D

H

E

To find a fourth fourth proportional to three given right lines,

A, B and C.

PROBLEM XV.

To find a mean proportional between two given right lines A and B.

To divide a given right line AB into two parts that shall have the same ratio to each other as two given lines C and D.

Draw AE making an angle with AB; in AE take AF equal C and FE equal D: join EB and draw FG parallel to it; then AG will have to GB the same ratio that C has to D.

PROBLEM XVII.

F

E

B

To divide a given right line AB in two parts in the point D, so that AD may be to DB in the ratio of two given numbers m and n. For example, let m=3, and n=4.

Draw AC making any angle with AB; take the number m from any convenient scale of equal parts, and lay it on AC, from A to E; and take the number n from the same scale, and lay it from E to C; join CB and draw ED parallel to it; then AB will be divided as required.

E

A

D

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