30. The sign is called minus; it indicates subtraction: Thus, AB, represents the sum of the quantities A and B; A-B represents their difference, or what remains. after B is taken from A; and A−B+C, or A + C − B, signifies that A and C are to be added together, and that B is to be subtracted from their sum.

31. The sign × indicates multiplication: thus A× B represents the product of A and B.

The expression A× (B+ C − D) represents the product of A by the quantity B+C-D. If A+D were to be multiplied by A- B+ C, the product would be indicated thus;

whatever is enclosed within the curved lines, being considered as a single quantity. The same thing may also be indicated by a bar: thus,

A+B+CX D,

denotes that the sum of A, B and C, is to be multiplied by D.

32. A figure placed before a line, or quantity, serves as a multiplier to that line or quantity; thus, 3AB signi fies that the line AB is taken three times; A signifies the half of the angle A.

33. The square of the line AB is designated by AB3 ; its cube by AB3. What is meant by the square and cube of a line is fully explained in Geometry.

34. The sign √ indicates a root to be extracted; thus, √2 means the square-root of 2; √AX B means the square

GEOMETRICAL CONSTRUCTIONS.

35. Before explaining the method of constructing geometrical problems, we shall describe some of the simpler instruments and their uses.

DIVIDERS.

C

α

36. The dividers is the most simple and useful of the instruments used for drawing.

It consists of two legs ba

bc, which may be easily turned around a joint at b.

One of the principal uses of this instrument is to lay

off on a line, a distance equal to a given line.

For example, to lay off on CD a distance equal to AB.

For this purpose, place the forefin

ger on the joint of the dividers, and A

set one foot at A: then extend, with
the thumb and other fingers, the CL

B

D

E

other leg of the dividers, until its foot reaches the point B. Then raise the dividers, place one foot at C, and mark with the other the distance CE: this will evidently be equal to AB.

RULER AND TRIANGLE.

о

37. A Ruler of convenient size, is about twenty inches

ness. It should be made of a hard material, perfectly straight and smooth.

The hypothenuse of the right-angled triangle, which is used in connection with it, should be about ten inches in length, and it is most convenient to have one of the sides considerably longer than the other. We can solve, with the ruler and triangle, the two following problems.

I. To draw through a given point a line which shall be paral· lel to a given line.

C

38. Let be the given point, and AB the given line. Place the hypothenuse of the triangle against the edge of the ruler, and then place the ruler and triangle on the paper, so that one of the sides of the triangle shall coincide exactly with AB: the triangle being below the line.

+

A

B

Then placing the thumb and fingers of the left hand firmly on the ruler, slide the triangle with the other hand along the ruler until the side which coincided with AB reaches the point C. Leaving the thumb of the left hand on the ruler, extend the fingers upon the triangle and hold it firmly, and with the right hand, mark with a pen or pencil, a line through C: this line will be parallel to AB.

I. To draw through a given point a line which shall be per pendicular to a given line.

A

D

B

39. Let AB be the given line, and D the given point. Place the hypothenuse of the triangle against the edge of the ruler, as before. Then place the ruler and triangle so that one of the sides of the triangle shall coincide exactly with the line AB. Then slide the triangle along the ruler until the other side reaches the point D: draw through D a right line,

40. A scale of equal parts is formed by dividing a line of a given length into equal portions.

If, for example, the line ab of a given length, say one inch, be divided into any number of equal parts, as 10, the scale thus formed, is called a scale of ten parts to the inch. The line ab, which is divided, is called the unit of the scale. This unit is laid off several times on the left of the divided line, and the points marked 1, 2, 3, &c.

The unit of scales of equal parts, is, in general, either an inch, or an exact part of an inch. If, for example, ab, the unit of the scale, were half an inch, the scale would be one of 10 parts to half an inch, or of 20 parts to the inch.

If it were required to take from the scale a line equal to two inches and six-tenths, place one foot of the dividers at 2 on the left, and extend the other to .6, which marks the sixth of the small divisions: the dividers will then embrace the required distance.

of an Divide

41. This scale is thus constructed. Take ab for the unit of the scale, which may be one inch,, or inch, in length. On ab describe the square abcd. the sides ab and dc each into ten equal parts. and the other nine parallels as in the figure.

Draw af

Produce ba to the left, and lay off the unit of the

1, 2, 3, &c. Then, divide the line ad into ten equal parts, and through the points of division draw parallels to ab, as in the figure.

Now, the small divisions of the line ab are each onetenth (.1) of ab; they are therefore .1 of ad, or 1 of ag or gh.

If we consider the triangle adf, we see that the base df is one-tenth of ad, the unit of the scale. Since the distance from a to the first horizontal line above ab, is one-tenth of the distance ad, it follows that the distance measured on that line between ad and af is one-tenth of df: but since one-tenth of a tenth is a hundredth, it follows that this distance is one hundredth (.01) of the unit of the scale. A like distance measured on the second line will be two hundredths (.02) of the unit of the scale; on the third, .03; on the fourth, .04, &c.

If it were required to take, in the dividers, the unit of the scale, and any number of tenths, place one foot of the dividers at 1, and extend the other to that figure between a and b which designates the tenths. If two or more units are required, the dividers must be placed on a point of division further to the left.

When units, tenths, and hundredths, are required, place one foot of the dividers where the vertical line through the point which designates the units, intersects the line which designates the hundredths: then, extend the dividers to that line between ad and be which designates the tenths: the distance so determined will be the one required.

For example, to take off the distance 2.34, we place one foot of the dividers at 7, and extend the other to e and to take off the distance 2.58, we place one foot of the dividers at p and extend the other to q

REMARK I. If a line is so long that the whole of it cannot be taken from the scale, it must be divided, and the parts of it taken from the scale in succession.

REMARK II. If a line be given upon the paper, its length can be found by taking it in the dividers and ap