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1. Things which are equal to the same thing, are equal to each other.
2 If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal.
4. If equals be added to unequals, the wholes will be unequal.
5. If equals be taken from unequals, the remainders will be unequal.
6. Things which are double of the same thing, are equal to each other.
7. Things which are halves of the same thing, are equal to each other.
8. The whole is greater than any of its parts.
9. The whole is equal to the sum of all its parts.
10. All right angles are equal to each other.
11. From one point to another, only one straight line can be drawn.
12. Through the same point, only one straight line can be drawn which shall be parallel to a given line.
13. Magnitudes, which being applied to each other, coincide throughout their whole extent, are equal.
Description of the Instruments used for Delineating or Drawing Lines and Angles on paper. Construction of Problems.
18. Drawings, or delineations on paper, are the copies of things which they are intended to represent.
In order that these copies may be exact, their different parts must bear the same proportion to each other that exists between the corresponding parts of the things themselves.
To enable us to delineate lines and angles correctly, upon paper, certain instruments are necessary; these we will now
19. 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 dis
tance equal to AB.
For this purpose, place the forefinger C on the joint of the dividers, and set one
foot at A: then extend, with the thumb and other fingers, the 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.
20. A Ruler of a convenient size, is about twenty inches in length, two inches wide, and a fifth of an inch in thickness. It should be made of a hard material, perfectly straight and smooth.
The hypothenuse of the right-angled triangle, which is
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 parallel to a given line.
Let C 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.
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.
II. To draw through a given point a line which shall be perpendicular to a given line.
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, and it will be perpendicular to AB.
SCALE OF EQUAL PARTS.
21. 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
.22.214.171.124.5 .6 .7 .8 .910
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.
is, in general, either an If, for example, ab the
The unit of scales of equal parts, inch, or an exact part of an inch. 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.
22. This scale is thus constructed.
of the scale, which may be one inch,,
Take ab for the unit
in length. On ab describe the square abcd.
or of an inch,
Divide the sides
ab and dc each into ten equal parts. Draw af and the other
nine 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 point 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 one-tenth (.1) of ab; they are therefore .1 of ad, or .1 of ag or gh.
If we consider the triangle adf, 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,
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 farther 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 1, and extend the other to e: and to take off the distance 2.58, we place one foot of the dividers and extend the other to q.
REMARK 1. If a line is so long that the whole of it can not 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.
23. If, with any radius, as AC, we describe the quadrant CD, and then divide it into 90 equal parts, each part is called a degree.