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than 1". From it draw a second similar figure having its sides one and a-half times as long (see pp. 7, 8).

EX. 11.-Draw a line, A B, 4′′ long, and mark three points in it, CDE. Then draw a second line 3′′ long, and divide it proportionately to the divisions of the line A B.

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Scales. In most mechanical drawings, the objects represented are too large to be drawn full size, and are, therefore, drawn so that all parts are proportionately smaller. When this is done the drawing is said to be to scale. The ratio of the drawing to the object is decided beforehand, and generally varies with the size and nature of the object and the size of the paper. In machine drawings details of complicated parts are drawn to a larger scale than simple parts, while structures, such as roofs and bridges, plans of fields and buildings, are drawn to a small scale. The fraction which expresses the ratio of the drawing to the object it represents is called the "representative fraction." Thus, suppose a drawing be made where a length of 1" represents a length of 1 foot on the object. This is shown on the drawing by writing upon it, "Scale 1 inches = 1 foot," and as 11 1 11" 12", the ratio is therefore, the representative frac12 8' tion is. Hence the drawing might be marked, "Scale of full size." The former method is, however, generally adopted, but the student should notice that the results are the same, and that a scale is described when either its representative fraction is given, or when the number of inches representing l' is stated. In a scale whose ratio is a fraction-that is, one where the drawing is made smaller than the object, the scale is said to be a "reducing scale;" but in the case of physical apparatus, clocks and watches, and other small mechanisms, the drawing requires to be larger than the object that is, a length of 1" on the object, is shown in the drawing by a length of probably 3" or 6". Hence, in the latter case, the drawing would be marked, "Scale

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1 foot 2 inches," and the ratio would be 6, therefore the representative fraction is a whole number, and the scale is called an "increasing scale." It is important to notice the different ways of stating an increasing or a decreasing scale.

Scales are constructed for the draughtsman's use, by dividing the edges of boxwood or ivory strips in a machine capable of working with great accuracy, and any ordinary scale is easily obtained. But it is necessary to be able to construct a scale, as a drawing has sometimes to be made to an unusual scale, or a machine-made scale may be unobtainable.

Before constructing a scale, it is necessary to know-1st, its size, or representative fraction; 2nd, the longest length it has to represent; 3rd, the different units of length it must show, as feet and inches, yards and feet, miles and furlongs.

It does not follow that in drawing a field 300' long, say to a scale of the scale must be 3′ long, as that would be absurd. Scales are generally made 12" or 18" long, and longer lengths are taken off by marking off successive lengths.

The method of constructing a simple or plain scale is as follows:

PROBLEM VII. (Fig. 11). To construct a scale where l' long enough for 6', to show feet and inches.

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Draw a straight line upon the paper of indefinite length, and

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9",

from one end, A, mark off a length, A B, equal to 11" since 1" show l', and 6' are to be shown. Divide A B into six equal parts, then each part will represent l', as the whole length shows 6'.

Divide the first of these divisions, A C, into twelve equal parts, then each part will represent 1".

Complete the scale in the manner shown in the figure. Notice that the divisions representing feet are carried to near the top line, that the 6′′ division is somewhat shorter, the 3" and 9′′ divisions still shorter, the other inch divisions being shortest. This is done to better distinguish the different divisions and to make the important ones clearly seen. Notice also that a line is drawn through the top of the inch divisions, and repeated in alternate foot divisions. This is done to help in counting, a lined division and a plain division representing 2'. The bottom line A B is generally made dark as a finish.

Marking the scale is very important, and is generally wrongly done by beginners. What is desired is that the marking shall agree with the length taken off the scale, and this is only accomplished by marking as shown in the figure. The zero point is at C where the inch and foot divisions begin, and from that point inches are marked to the left, and feet to the right.

It is a common fault to mark the point C as 1', this means that a length on the scale marked 2′ 3′′ is really only 1' 3". An equally wrong result follows when the inch divisions are

marked from A to O, beginning at A, then a length marked as 2′ 3′′ is really only 1' 9".

It should be noticed that it is not necessary to further divide the scale. It is, therefore, waste labour to divide up the whole of the foot divisions into equal parts, although this is sometimes done in machine-made scales. The scale as drawn shows any length between 1" and 6'.

The importance of accuracy in constructing a scale cannot be too strongly insisted upon. The same length taken from different parts of the scale should agree, otherwise the drawing made with the scales will be wrong, and all scales should be tested in this way. In showing inch divisions for a small scale this is very difficult, and the student will find that such small divisions can be made quite as accurately with the eye, as by using dividers. But accuracy is only obtainable with great care, and by using good instruments and hard pencils with fine points.

In setting off the total length of the scale, do not take a distance of say 1" in the dividers and set this distance off repeatedly along the line until the right length is obtained. This cannot be accurate, as suppose the 1" to be taken off the rule" too short or too long, a very probable error, then the whole line in the example given would be 10 × 9 '09" short or long.

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EXAMPLES.

EX. 12.-A line 2-5′′ long is drawn to represent a length of 1'. What fraction is the line of the length it represents, and what length of line should be drawn to show lengths of 1" and 5′′? Divide the line to show inches and mark the divisions.

EX. 13.-Construct a scale, the representative fraction of which is reading yards and feet, long enough for 3 yards.

EX. 14. The plan of a room, 41′ long and 28′ wide outside, is to be drawn upon a sheet of paper 22′′ × 16′′, leaving about 1" border all round. Construct and mark the scale that should be used.

EX. 15.-Construct carefully the following scales, writing above each, its representative fraction, and marking clearly what the divisions represent:—

(a) Scale of 11" Î foot, long enough for 6 feet, showing feet

and inches.

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(b) Scale of " = 1 foot, long enough for 8 feet, showing feet

and inches.

(c) Scale of 1" yards and feet. (d) Scale of " chains and poles.

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1 chain, long enough for 10 chains, showing

(e) Scale of 1 cm. = 10 cms., long enough for 1 metre, showing divisions of 10 cms. and c.metres (1 cm. = = 0.39′′).

EX. 16.—On a map 2.5 chains is represented by 1.5 inches. Draw a scale of feet for the map showing 500 feet, and divide it to show distances of 20 feet. What is the representative fraction of the scale?

EX. 17.-Construct a scale of, long enough for 15 feet, showing feet and inches.

Diagonal Division and Diagonal Scales.-The number of equal parts into which it is possible to accurately divide a line by the methods previously described soon reaches a limit. It is, for example, difficult to show lengths of 1 inch on a scale where or inch 1 foot, yet much smaller divisions than these are constantly required in scales for land measure, and on rules designed for measuring very small fractions of an inch such as 100 or 200

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The principle of diagonal division by which such small divisions may be accurately obtained, is as follows:-Suppose we require to show lengths of of the line A B (Fig. 12).

At one end, A, of the line draw a perpendicular of indefinite length, and mark along the perpendicular any c

ten equal lengths, starting from A, and ending

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at C. Join the last point to B, mark the points 3
9 as shown, and through the points
1 9, draw lines parallel to A B.

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Then all

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of

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the small triangles, as C 3 F, 05 E, and C8D are similar; and, therefore, since C3 is CA, so also is 3 F of AB, and so on with each of the triangles. Consequently the distance, 1G, is 0·1 of A B, and if AB be 0.5′′ long, then the length of 1 G 0.05" or ".

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PROBLEM VIII. (Fig. 13).—To construct a diagonal full size scale, 6" long, to show inches, tenths of an inch, and hundredths of an inch.

Draw the line A B 6′′ long, and divide it into six equal parts to show inches, and divide the first of these divisions, A C, into ten equal parts to show inches. From the end, A, draw a perpendicular, and starting from A, mark off any ten equal lengths to the point E. Complete the oblong, E A B D, and through the points 1 . . . 9, draw lines parallel to A B, terminated by BD. From each of the points marking the inch division draw lines perpendicular to AB, and mark them 1", 2′′ . . . 5′′, the division at C being 0. Join the points E. 9, and draw parallels to E.9

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through the points C. 1, 2, 3, 8, in A C. The scale is now complete and can be used to measure any length between 0.01" and 6"; as, for example, the length between the two points, xy, is 4.26", and between the points, mn, is 3.14".

A scale of this construction is usually marked on one side of the common 6" boxwood protractor, and should form a part of every student's drawing outfit, as it is the only convenient and accurate method of measuring to the second place of decimals.

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Diagonal scales, which are chiefly required for drawings of land and buildings, are constructed in this way. For example, if 11" = 1 furlong, and a scale is required to show poles, the construction should be as follows:-(Fig. 13) Make AC 11′′ long, and erect an indefinite perpendicular, AE, from the point A. Now as 10 chains 1 furlong, and 4 poles = 1 chain, it will be best to divide AC into ten equal parts to show chains and to set four equal divisions up A E, and then finish as before.

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Comparative Scales.-Comparative scales are those which enable different standards of length to be compared. Suppose a drawing is made in France to a scale of 2 cm. 1 metre, or of full size, then it is convenient to be able to know what measure of yards and inches in the English standard corresponds to any given length on the French drawing, which has been drawn in metres and centimetres. To accomplish this a scale to English measure should be drawn having the same representative fraction." The usual method is to draw the English scale along one edge of the scale, and the French scale along the other edge. The comparison and conversion is then easily made.

EXAMPLES.

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EX. 18.-Construct a full size diagonal scale 6" long, showing inches, inches, and inches.

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EX. 19.--A distance of 11 miles 3 furlongs is shown on a map by 4". Draw a scale for the map, showing furlongs by

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