Lay the triangle on the paper so that one of its sides will coincide with the given line to which the parallel is to he drawn; then, keeping the triangle steady, lay the ruler on the paper, with its edge applied to either of the other sides of the triangle; then, keeping the ruler firm, move the triangle along its edge, up or down, to the given point; the side of the triangle which was placed on the given line will always keep parallel to itself, and hence a parallel may be drawn through the given point.

31. To erect a perpendicular on a given line, and from any given point in that line, we have only to apply the ruler to the given line, and place the triangle so, that its right angle shall touch the given point in the line, and one of the sides about the right angle, placed to the edge of the ruler-the other side will give the perpendicular required.

32. If the given point be either above or below the line, the process is equally easy. Place one of the sides of the triangle about the right angle on the given line, and the ruler on the side opposite the right angle, then slide the triangle on the edge of the ruler till the given point from which the perpendicular is to be drawn is on the other side, then this side will give the perpendicular.

33. Other problems may be performed with these instruments, the method of doing which it will be easy for the reader to contrive for himself.

34. When arcs of circles of great diameter are to be drawn, the use of a compass may be substituted by a very simple contrivance. Draw the chord of the arc to be described, and place a pin at each ex

tremity, A and B, then place two rulers jointed at C, and forming an angle, ACB the supplement of half the given number of degrees; that is

=

B

to say, the number of degrees which the arc whose chord given is to contain, is to be halved, and this half being subtracted from 180 degrees, will give the degrees which form the angle at which the rulers are placed, that is, the angle ACB. This being done, the edges of the rulers are moved along against the pins, and a pencil at C will describe the arc required.

35. Large circles may be described by a contrivance equally simple. On an axle, a foot or a foot and a half

F

long, there are placed two wheels, M and F, of which one is fixed to the axle, namely, M, and the other is capable of being shifted to different parts of the axle, and, by means of a thumb-screw, made capable of being fixed at any point on the axle. These wheels are of different diameters, say of 3 and 8 inches, the fixed wheel F being the largest. This instrument being moved on the paper, the circles M and F will roll, and describe circles of different radi. the axle will always point to the centre of these circles, and there will be this proportion:

As the diameter of the large wheel

Is to the difference of the diameters of the two wheels, So is the radius of the circle to be described by the large wheel

To the distance of the two wheels on the axle.

36. If the diameters of the wheels are as above stated, and it is required to describe a circle of 3 feet radius, then from the above proportion we have 6: 6-3 :: 3 feet or 36 inches 18 inches : the distance of the two wheels, to

=

describe a circle 6 feet in diameter.

37. It may be observed, that it will be best to make the difference of the wheels greater if large circles are to be described, as then a shorter instrument will serve the purpose.

38. We will conclude this appendix, by making a few remarks on the Diagonal Scale and Sector, the great use of the latter of which, especially, is seldom explained to the young mechanic.

39. The diagonal scale to be found on the plain scale in common pocket-cases of instruments, is a contrivance for measuring very small divisions of lines; as, for instance, hundredth parts of an inch.

1

2

3

4

5

B E

A

40. Suppose the accompanying cut to represent an enlarged view of two divisions of the diagonal scale, and the bottom and top lines to be divided into two parts, each representing the tenth part of an inch. Now, the perpendicular lines BC, AD, are each divided into ten equal parts, which are joined by the crossing lines, 1, 2, 3, 4, &c., and the diagonals BF, DE,

6

7

8

9

10

are drawn as in the figure. Now, as the division FC is the tenth part of an inch, and as the line FB continually approaches nearer and nearer to BC, till it meets it in B, it will follow, that the part of the line 1 cut off by this diagonal will be a tenth part of FC, because B1 is only onetenth part of BC; so, likewise, 2 will represent two-tenth parts, 3, three-tenth parts, and so on to 9, which is ninetenth parts, and 10, ten-tenth parts, or the whole tenth of an inch; so that, by means of this diagonal, we arrive at divisions equal to tenth parts of tenth parts of an inch, or hundredths of an inch. With this consideration, an examination of the scale itself will easily show the whole matter. It may be observed, that if half an inch and the quarter of an inch be divided, in the same manner, into tenths and tenths of tenths, we may get thus two-hundredth and fourhundredth parts of an inch.

THE SECTOR.

41. THIS very useful instrument consists of two equal rulers each six inches long, joined together by a brass folding joint. These rulers are generally made of boxwood or ivory; and on the face of the instrument, several lines or scales are engraven. Some of these lines or scales proceed from the centre of the joint, and are called sectorial lines, to distinguish them from others which are drawn parallel to the edge of the instrument, similar to those on the common Gunter's scale.

42. The sectorial lines are drawn twice on the same face of the instrument; that is to say, each line is drawn on both legs. Those on each face are,

A scale of equal parts, marked L,

A line of chords, marked

A line of secants, marked

C,

S,

A line of polygons, marked P, or Pol.

These sectorial lines are marked on one face of the instru

ment; and on the other there are the following:

A line of sines, marked

A line of tangents, marked

S,

T.

A line of tangents to a less radius, marked t.

This last line is intended to supply the defect of the former and extends from about 45 to 75 degrees.

43. The lines of chords, sines, tangents, and secants, but not the line of polygons, are numbered from the centre, and are so disposed as to form equal angles at the centre; and it follows from this, that at whatever distance the sector is opened, the angles which the lines form, will always be respectively equal. The distance, therefore, between 10 and 10, on the two lines marked L, will be equal to the distance of 60 and 60 on the two lines of chords, and also to 90 and 90 on the two lines of sines, &c., at any particular opening of the sector.

44. Any extent measured with a pair of compasses, from the centre of the joint to any division on the sectorial lines, is called a lateral distance; and any extent taken from a point in a line on the one leg, to the like point on the similar line on the other leg, is called a transverse or parallel distance.

With these remarks, we shall now proceed to explain the use of the sector, in so far as it is likely to be serviceable to mechanics.

USE OF THE LINE OF LINES.

45. This line, as was before observed, is marked L, and its uses are,

To Divide a line into any number of equal parts: Take the length of the line by the compasses, and placing one of the points on that number in the line of lines which denotes the number of parts into which the given line is to be divided, open the sector till the other point of the compasses touches the same division on the line of lines marked on the other leg; then, the sector being kept at the same width, the distance from 1 on the line L on the one leg, to 1 on the line L on the other, will give the length of one of the equal divisions of the given line to be divided. Thus, to divide a given line into seven equal parts :—take the length of the given line with the compasses, and setting one point on 7, on the line L of one of the legs, move the other leg out until the other point of the compasses touch 7 on the line L of that leg; this may be called the transverse distance of 7 on the line of lines. Now, keeping the sector at the same opening, the transverse distance of 1 will be the length of one of the 7 equal divisions of the

given line; the transverse distance of 2 will be two of these divisions, &c.

46. It will sometimes happen, that the line to be divided will be too long for the largest opening of the sector; and in this case we take the half, or third, or fourth of the line, as the case may be; then the transverse distance of 1 to 1, will be a half, a third, or a fourth of the required equal part. 47. To divide a given line into any number of parts that shall have a certain relation or proportion to each other: Take the length of the whole line to be divided, and placing one point of the compasses at that division on the line of lines on one leg of the instrument which expresses the sum of all the parts into which the given line is to be divided, and open the sector till the other point of the compasses is on the corresponding division on the line of lines of the other leg. This is evidently making the sum of the parts into which the given line is to be divided a transverse distance; and when this is done, the proportional parts will be found by taking, with the same opening of the sector, the transverse distances of the parts required.To divide a given line into three parts, in the proportion of 2, 3, 4: The sum of these is 9; make the given line a transverse distance between 9 and 9 on the two lines of lines; then the transverse distances of the several numbers 2, 3, 4, will give the proportional parts required.

48. To find a fourth proportional to three given lines: Take the lateral distance of the second, and make it the transverse distance of the first, then will the transverse distance of the third be the lateral distance of the fourth; then, let there be given 6 : 3 :: 8,-make the lateral distance of 3 the transverse distance of 6; then will the transverse distance of 8 be the lateral distance of 4, the fourth proportional required.

49. This sector will be found highly serviceable in drawing plans. For instance, if it is wished to reduce the drawing of a steam engine from a scale of 11⁄2 inches to the foot, to another of to the foot. Now, in 11⁄2 inches there are 12 parts; so that the drawing will be reduced in the proportion of 12 to 5. Take the lateral distance of 5, and keep the compasses at this opening; then open the sector till the points of the compasses mark the transverse distance of 12; keep now the sector at this opening, and any measure taken on the drawing, to be copied and laid off on the sector as a