and B C, D E, any two transverse distances taken on this pair of lines; then, from the construction of the instrument, we have A B equal to a C, and A D equal to a E, so that AB: AC:: A D A E, and the triangles A B C and A D E have the angle at A common, and the sides about this common angle proportional (Euc. vi. prop. 6); they are, therefore, similar and—

AB BC::AD: D E.

In the case of a compound solution, the angles at a are equal, but not common, and the reasoning is, in all other respects, exactly the same.

USES OF THE LINE OF LINES.

To find a Fourth Proportional to three given Lines.-Set off from the center a lateral distance equal to the first term, and open the sector till the transverse distance at the division thus found, expressing the first term, is equal to the second term; again, extend to a point whose lateral distance from the center is equal to the third term, and the transverse distance at this point will be the fourth term required.

If the legs of the sector will not open far enough to make the lateral distance of the second term a transverse distance at the division expressing the first term, take any aliquot part of the second term, which can conveniently be made such transverse distance, and the transverse distance at the third term will be the same aliquot part of the fourth proportional required.

A third proportional to two given lines is found by taking a third line equal to the second, and finding the fourth proportional to the three lines.

Example.-To find a fourth proportional to the numbers 2, 5, and 10. Open the sector till the lateral distance of the second term 5 becomes the transverse distance at 2, the first term; then the transverse distance at 10 will extend, as a lateral distance, from the center to 25, the fourth proportional required.

To bisect a given Straight Line.-Take the extent of the line in the compasses, and open the sector till this extent is a transverse between 10 and 10 on the line of lines: then the transverse distance from 5 to 5, on the same pair of sectoral lines, gives the half of the line, and this extent set off from either end will bisect it.

To divide a Straight Line into any Number of equal Parts.— 1. When the number of parts are a power of 2, the operations

are best performed by continual bisection. Thus, let it be required to divide the line A B into sixteen equal parts. 1. Make A B a transverse distance between 10 and 10 on the

line of lines; then take off the transverse distance of 5 and 5, and set it off from A or* в to 8, and A в will be divided into two equal parts at the division 8. 2. Make A Sa transverse distance at 10, and then the transverse distance at 5, set off from A or* 8 at 4, and from в or* 8 at 12, will divide the line into four equal parts at the divisions 4, 8, and 12. 3. Make the extent A 4 a transverse distance at 10, and the transverse distance at 5 will again bisect each of the parts last set off, and divide the whole line into eight equal parts at the divisions 2, 4, 6, 8, 10, 12, and 14. Each of these may be again bisected by taking the transverse distance at 21 or 2.5, that is, at the middle division between the 2 and the 3 upon the line of lines, and the line will be divided as required.

When the divisions become smaller than can be conveniently bisected by the method just explained, the operation may still be continued to any required extent by taking the extent of an odd number of the divisions already obtained as the transverse distance of 10 and 10, and setting off the half of this extent, or the transverse distance at 5, from the several divisions already obtained. Thus, in the preceding example, by making the extent of three of the divisions, or five, or seven, a transverse distance at 10, the transverse distance at 5, set off from the several divisions already obtained, will divide a B into 32 equal parts.

2. When the number of parts is not a power of 2, the operations cannot all be performed by bisections; but still, by a judicious selection of the parts into which the line is first divided, many of the after operations may be performed by bisections. Example.-Let it be required to divide the line H

A B into seven equal parts. 1. Make the whole extent, A B, & transverse distance between 7 and 7 on the line of lines; then

* Greater accuracy is obtained by setting off the distance from both ends of the extent to be bisected, and then, in case the two points so found do not accurately coincide, taking the middle point between them, as near as the eye can judge, for the true point of bisection.

take off the transverse distance of 4 and 4, and set it off from A and B to 4 and 3. 2. Make this extent from A to 4 a transverse distance at 10; then the transverse distance at 5 bisects A 4 and 3 B in 2 and 5; set off from 3, gives 1, and from 4 gives 6; and thus the line A B is divided into seven equal parts as required.

To open the Sector so that the Line of Lines may answer for any required Scale of equal Parts.-Take one inch in the compasses, and open the sector, till this extent becomes a transverse distance at the division indicating the number of parts in an inch of the required scale; or, if there be not an integral number of parts in one inch, it will be better to take such a number of inches as will contain an integral number of parts, and make the extent of this number of inches, if it be not too great, a transverse distance at the division indicating the number of parts of the required scale in this extent.

Example.-To adjust the Sector as a Scale of One Inch to Four Chains.-Make one inch the transverse distance of 4 and 4; then the transverse distances of the other corresponding divisions and subdivisions will represent the number of chains and links indicated by these divisions: thus, the transverse distance from 3 to 3 will represent three chains; the transverse distance at 47, or the seventh principal subdivision after the primary division marked 4, will represent 4 chains 70 links, and so on.

To construct a Scale of Feet and Inches in such a manner that an extent of Three Inches shall represent Twenty Inches.1. Make three inches a transverse distance between 10 and 10, and the transverse distance of 8 and 8 will represent 16 inches. 2. Set off this extent from A to B, divide it by continual bisection into 16 equal parts, and place permanent strokes to mark the first 12 of these divisions, which will represent inches. 3. Place the figure 1 at the twelfth stroke, and set off again the extent of the whole 12 parts, from 1 to 2, 2 to 3, &c., to represent the feet.

As an Example of the Use of the Line of Lines in reducing Lines, let it be required to reduce a drawing in the Proportion of 5 to 8.-Take in the compasses the distance between two points of the drawing, and make it a transverse distance at 8 and 8; then the transverse distance of 5 and 5 will be the distance between the two corresponding points of the copy. 2. These two points having been laid down, make the distance between one of them and a third point a transverse distance at 8, and with the transverse distance at 5 describe, from

that point as center, a small arc. 3. Repeat the operation
with the other point, and the intersection of the two
small arcs will give the required position of the third
point in the copy.
In the same manner all the other
points of the reduced copy may be set off, each one from
two points previously laid down.

LINE OF CHORDS.

The double scales of chords upon the sector are more generally useful than the single line of chords described on the plain scale; for, on the sector, the radius with which the arc is to be described may be of any length between the transverse distance of 60 and 60 when the legs are close, and that of the transverse of 60 and 60 when the legs are opened as far as the instrument will admit of: but, with the chords on the plain scale, the arc described must be always of the same radius.

To protract or lay down a right-lined Angle в A C, which shall contain a given number of Degrees, suppose 46°.-Case 1. When the angle contains less than 60°, make the transverse distance of 60 and 60 equal to the length of the radius of the circle, and with that opening describe the arc в C. (Fig. at page 40.) Take the transverse distance of the given degrees 46, and lay this distance on the arc from the point в to c. From the center A of the arc draw two lines A C, A B, each passing through one extremity of the distance в c laid. on the arc; and these two lines will contain the angle required. Case 2. When the angle contains more than 60°. Suppose, for example, we wish to form an angle containing 148°. Describe the arc B C D, and make the transverse distance of 60 and 60 equal the radius as before. Take the transverse distance ofor, &c., of the given number of degrees, and lay this distance on the arc twice or thrice, as from в to a, a to b, and from b to D. Draw two lines connecting B to A, and a to D, and they will form the angle required.

sup

When the required angle contains less than 5°, pose 3, it will be better to proceed thus. With the given radius, and from the center A, describe the arc D G ; and from some point, D, lay off the chord of 60°, which suppose to give the point &, and also from the same point D lay off in the same direction the chord of 56° (60° 34°), which would give the point E Then through these two points E and &, draw

lines to the point A, and they will represent the angle of 3° as required.

From what has been said about the protracting of an angle to contain a given number of degrees, it will easily be seen how to find the degrees (or measure) of an angle already laid down.

E

B

LINE OF POLYGONS.

The line of polygons is chiefly useful for the ready division of the circumference of a circle into any number of equal parts from 4 to 12; that is, as a ready means to inscribe regular polygons of any given number of sides, from 4 to 12, within a given circle. To do which, set off the radius of the given circle (which is always equal to the side of an inscribed hexagon) as the transverse distance of 6 and 6, upon the line of polygons. Then the transverse distance of 4 and 4 will be the side of a square; the transverse distance between 5 and 5, the side of a pentagon; between 7 and 7, the side of a heptagon; between 8 and 8, the side of an octagon; between 9 and 9, the side of a nonagon, &c., all of which is too plain to require an example.

If it be required to form a polygon, upon a given right line set off the extent of the given line, as a transverse distance between the points upon the line of polygons, answering to the number of sides of which the polygon is to consist; as for a pentagon between 5 and 5; or for an octagon between 8 and 8; then the transverse distance between 6 and 6 will be the radius of a circle whose circumference would be divided by the given line into the number of sides required.

The line of polygons may likewise be used in describing, upon a given line, an isosceles triangle, whose angles at the base are each double that at the vertex. For, taking the given line between the compasses, open the sector till that extent becomes the transverse distance of 10 and 10, then the transverse distance of 6 and 6 will be the length of each of the two equal sides of the isosceles triangle.

All regular polygons, whose number of sides will exactly divide 360 (the number of degrees into which all circles are supposed to be divided) without a remainder, may likewise be set off upon the circumference of a circle by the line of chords.