French curves, which may be purchased at any Artist's Colourman's. The operation is greatly facilitated by employing the first method, as a considerable portion of the ellipse can be drawn by means of the bow-pen.

PROBLEM 25.

To determine by geometrical construction, lines which
shall represent √5, √6, or √5, v§.

From A, erect a perpendicular A D equal A B. Then (A D)2

=

6. By a similar construction, lines representing other quantities may be found.

2o. Make A B equal to of a given unit, and upon it

Fig. 44.

D

B

describe a semi-circle. From a, set off a c equal to the given unit, and from c, draw c D perpen

dicular to A B, cutting the semicircle in D. Then, by Prob. 12,

VACXCB=CD=V1X+=V7.

From the above, it will be readily seen how to determine a

line which shall represent v

TRANSFORMATION OR REDUCTION OF PLANE

FIGURES.

PROBLEM 26.

To make a rectangle equal to a given triangle.

Let A B C be the given triangle. Bisect B C in D, and through A, draw an indefinite straight line parallel to в C.

erect perpendiculars to в C,

From B, D,

cutting the

parallel, drawn from A, in E, F; BDEF is the rectangle required. For the reason of the construction here employed, see Euc. I., 41.

A

PROBLEM 27.

To make an isosceles triangle equal to any other triangle.

angle with AB, and equal in length to the undivided line. Join B to 5, and through the points in A B draw lines parallel to в 5; then will ▲ 5 be divided proportionally to a B.

PROBLEM 17.

From a given point to describe a circle which shall touch
a given circle.

Let A в be the given circle and c the given point. Join a, the centre of the given circle, to c, cutting the circumference in b. From centre c, and radius cb, describe a circle. This circle will touch the given circle at the point ō.

Second Case. Let D, the given point, be within the given circle.

Join a, D, and produce it to meet the circumference in b. With centre D, and radius D b, describe a circle.

A

PROBLEM 18.

From a given point, to describe a circle which shall touch a

Fig. 35.

given straight line.

This instrument is used to lay down angles. It is sometimes made of brass of a semi-circular form, and sometimes of ivory of a rectangular form, as shown in the figure.

If the circumference of a circle be divided into 360 equal parts, each part is called one degree (written 1°), and if from these points of division, lines be drawn to the centre of the circle, the opening or angle between any two consecutive lines, will be 1°. Again, the angle between the lines in·cluding 15 divisions, will be an angle of 15°, and so on. In the figure, the semi-circle is divided into 18 equal parts (each part containing 10°), and is numbered from left to right, from zero to 180, and in the same manner from right to left. In the centre of the protractor there is a mark or index, o. The manner of using the instrument is this:-Suppose it were required from a given point in a line, A B, to draw a line making with it an angle of 20°. Make ab or cd coincide with the given line, so that the index o be upon the given point, then make a mark on the paper at g, or the edge