= bring 100 yards to inches. Number of inches in 100 yards = 100 × 36 3600 inches. Therefore, every line, in the original object, is 3600 times the line which represents it in the drawing; or, every line in the drawing is th part of the corresponding line of the object. Again, if a length of 60 feet be represented by 3 inches, 1 inch will represent 20 feet, and every line in the drawing will be th of the corresponding line of the object. The fractions, o, are called the representative fractions of the scales. The representative fraction, then, shows the ratio of 1 inch to the number of units represented by 1 inch, whether of feet, yards, miles, etc.; or in other words, it expresses the relation between the drawing and the original object. When the scale is constructed, the fraction should be written above it, or placed on the drawing when there is no scale. From what has been said, it is hoped that the student will clearly understand what is meant by the representative fraction, to find which, he has only to reduce the number of units represented by 1 inch to inches. He is recommended to make an attempt to answer the following questions. without referring to the book. (1) A drawing is made to the scale of 50 yards to 1 inch; required the representative fraction, or the relation between the drawing and the original object. Here, the number of inches in 50 yards 50 × 36 1800; therefore, the representative fraction is 1850, .e., every line in the drawing is th part of the corresponding line of the object. (2) A drawing is made to the scale of 1 mile to 1 inch ; required the representative fraction. 1 x 1760 x 36 Here, the number of inches in 1 mile E (3) A drawing is made to the scale of 3 leagues to 1 inch; required the representative fraction. Here, the number of inches in 3 leagues = = 3 x 3 x 1760 X 36 570,240; therefore, the representative fraction is 370210. EXAMPLES. 1. Construct a scale to show 20 feet, when 1 inch represents 4 feet. Give the representative fraction. Here, 1 inch represents 4 feet; therefore, to represent 20 feet it will take 5 times 1 inch 5 inches; or, by proportion, thus:: = ift. lin. 20ft. (the length required) :: : x = 5 inches. Draw three parallel lines 5 inches long at equal distances apart (about the same as in the drawing). Divide this length (5 inches) into 2 equal parts (Prob. 6, Prac. Geom.) and the left hand part or primary division into 10 equal sub-divisions. From the primary divisions on the lower line, draw perpendiculars to the top line; and from each sub-division, draw perpendiculars to the second line, except the centre one, which should be drawn half-way between the second line and the top line. (See Scale 1). Write 0 (zero) at the first primary division, 10 at the end of the sub-divisions (shown at the left of the scale), and 10 at the primary division (shown at the right of the scale); 5 may also be written at the centre of the sub-divisions. The representative fraction should be placed above the centre of the scale, and the units of measure on the right. In inking in the scale, it will be observed that the top line is omitted, while the bottom line is drawn thick. Since No. 1 is, in every respect, a model of the scales which follow, the student will see the importance of thoroughly understanding it. It may be well to point out here, that in making a drawing to scale, the length of the scale should be such as to enable us to set off considerable distances. The number assumed (as 20 in the present scale) should be a number divisible by 10; and the quotient, resulting from the division of this number by 10, is the number of primary divisions into which the length of the scale should be divided (as 2 in the present scale). The manner of using the scale is as follows:-Suppose it were required to take off 15 feet. Place one leg of the compasses at 10 on the right of the scale, and the other leg at 5, one of the sub-divisions, and the space included between these two points will contain 15 feet. In the same manner any number of feet from 1 to 20 may be taken off. 2. Construct a scale to show 70 yards, when 1 inch represents 9 yards. (No. 2.) 9yds. : lin. 70yds. : X= 7.77 inches. That is, if it takes 1 inch to represent 9 yards, it will take 7-77 inches to represent 70 yards. Therefore, make the scale 7.77 inches long, and divide it into 7 equal parts (the quotient of 70÷10). Divide the first primary division into 10 sub-divisions, and complete the scale as before, i.e., by writing 10, 20, 30, 40, 50, 60, to the right of the scale, etc. The representative fraction is 17. 3. The distance between two places is 4 miles, and is represented on a plan by 1.5 inches; construct the scale to show furlongs, when 20 miles is the number assumed. (No. 3.) Here, we have 4 miles : 1.5in. :: 20 miles : x=7.5 inches. That is to say, if it takes 1.5 inches to represent 4 miles, it will take 7.5 inches to represent 20 miles. Make the scale 7.5 inches long, divide it into 2 equal parts, and complete it as before. Divide the tenth sub-division (on the left hand) into 8 equal parts for furlongs; for since each subdivision represents 1 mile, the eighth part of one of these will represent 1 furlong. The representative fraction of the scale is 4. Construct a scale of 10 miles to 1 inch, to measure spaces of 1000 yards. Assume 70 miles. (No. 4.) 10 miles : lin. :: 70 miles : X = 7 inches. Make the scale 7 inches long, and divide it into 7 equal parts. Each primary division represents 10 miles, which, divided by 1000, will give 17% parts. Therefore, divide the first primary division into this number of parts, and complete the scale as shown in the figure. All distances less than 1000 yards must be taken by the judgment only. The distance between the dots shows 10 miles 1500 yards. 5. Draw a scale of 1 mile to 1 inch, to show furlongs. (No. 5.) Make the scale the required length, suppose 5 inches. Divide it into 5 equal parts, and the first primary division into 8 sub-divisions for furlongs. Write miles on the right, and furlongs on the left, as shown on the scale. (No. 5.) It will be observed that the first primary division in Scales 4 and 5, is not divided into 10 equal parts as in the preceding scales. Though, as a rule, the decimal notation should be adopted, there are special cases where an exception. must be made, as in the examples referred to. In fact, the construction of scales is a subject which demands both judgment and ingenuity. Suppose it were required to construct a scale of 40 feet to 1 inch to measure single feet, and that 240 feet is to be the number shown. We should make the scale 6 inches long; divide it into 6 equal parts, and the first primary division into 4 sub-divisions, to measure spaces of 10 feet. We should then sub-divide the first subdivision into 10 equal parts. In figuring the scale, we should write 0 (zero) on the extreme left, and then the numbers 10, 20, 30, 40, which will include one primary division. From this point, the numbers will be 80, 120, 160, 200, and 240. EXAMPLES. (a) Construct a scale of 1 foot to 1 inch to measure inches. Show 9 feet. (b) Draw a scale of 1 league to 1 inch to measure miles. Show 8 leagues. Since in the representative fraction is expressed the number of inches represented by 1 inch, the scale may be constructed when this fraction is given. Suppose a scale of feet is required, the representative fraction being, as in No. 1. By proportion we have— 48 : 1 :: 20 × 12 : x = 5 inches, the length of the scale. That is to say, as 48 inches (the denominator of the fraction) is to 1 inch (the numerator), so is the number of inches in the number of units assumed to x, the required length of the scale. Again, in No. 2 we have 324 : 1 :: 70 × 36 the length of the scale. : x= 7.77 inches, |