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22. 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


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.

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a. b


23. This scale is thus constructed.

unit of the scale, which may be one inch,

inch, in length. On ab describe the
On ab describe the square abed.
the sides ab and de each into ten equal parts.
and the other nine

Take ab for the


of an Divide

Draw af

parallels as in the figure.

Produce ba to the left, and lay off the unit of the scale any convenient number of times, and mark the points

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 applying it to the scale.

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24. This instrument is used to lay down, or protract angles. It may also be used to measure angles included between lines already drawn upon paper.

It consists of a brass semicircle, ABO, divided to half degrees. The degrees are numbered from 0 to 180, both ways; that is, from A to B and from B to A. The divisions, in the figure, are made only to degrees. There is a small notch at the middle of the diameter AB, which indicates the centre of the protractor.

To lay off an angle with a Protractor.

25. Place the diameter AB on the line, so that the centre shall fall on the angular point. Then count the degrees contained in the given angle from A towards B, or from B towards A, and mark the extremity of the arc with a pin. Remove the protractor, and draw a line through the point so marked and the angular point: this line will make with the given line the required angle.



1. In every plane triangle there are six parts: three sides and three angles. These parts are so related to each other, that when one side and any two other parts are given, the remaining ones can be obtained, either by geometrical construction or by trigonometrical computation.

2. Plane Trigonometry explains the methods of com puting the unknown parts of a plane triangle, when a sufficient number of the six parts is given.

3. For the purpose of trigonometrical calculation, the circumference of the circle is supposed to be divided into 360 equal parts, called degrees; each degree is supposed to be divided into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds.

Degrees, minutes, and seconds, are designated respectively, by the characters". For example, ten degrees, eighteen minutes, and fourteen seconds, would be written 10° 18' 14".

4. If two lines be drawn through the centre of the circle, at right angles to each other, they will divide the circumference into four equal parts, of 90° each. Every right angle then, as EOA, is measured by an arc of 90°; every acute angle, as BOA, by an are less than 90°; and every obtuse angle, as FOA, by an arc greater than 90°.



5. The complement of an arc is what remains after subtracting the arc from 90°. Thus, the arc EB is the complement of AB. The sum of an arc and its complement is equal to 90°.

6. The supplement of an arc is what remains after subtracting the arc from 180°. Thus, GF is the

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supplement of the arc AEF. The sum of an arc and its supplement. is equal to 180°.

7. The sine of an arc is the perpendicular let fall from one extremity of the arc on the diameter which passes through the other extremity. Thus, BD is the sine of the arc AB

8. The cosine of an arc is the part of the diameter intercepted between the foot of the sine and the centre. Thus, OD is the cosine of the arc AB.

9. The tangent of an arc is the line which touches it at one extremity, and is limited by a line drawn through the other extremity and the centre of the circle. Thus, AC is the tangent of the arc AB.

10. The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and limited by the tangent passing through the other extremity. Thus, OC is the secant of the arc AB.

11. The four lines, BD, OD, AC, OC, depend for their values on the arc AB and the radius OA; they are thus designated :

sin AB for BD
cos AB for OD
tan AB for AC
sec AB for OC

12. If ABE be equal to a quadrant, or 90°, then EB will be the complement of AB. Let the lines ET and IB be drawn perpendicular to OE.


ET, the tangent of EB, is called the cotangent of AB;
IB, the sine of EB, is equal to the cosine of AB;
OT, the secant of EB, is called the cosecant of AB.

In general, if A is any arc or angle, we have,
Asin (90° - A)
Atan (90° - A)

= sec (90° - A)



cosec A

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