COMPARATIVE SCALES. One scale is said to be comparative to another when the distances measured by the one can be measured by the other. For example, if we have a scale of miles to which a drawing is made, we can construct another scale by which the distances from place to place can be measured in furlongs. The scale of furlongs would then be comparative to the scale of miles. In making one scale comparative to another, therefore, we must of necessity have two units of measure, as an inch and a foot, a foot and a yard, a mile and a chain, etc. Suppose it is required to make a scale of yards comparative to No. 1. By proportion we have 4ft. : lin. :: 3ft. : x= 3-inch. That is to say, if 1 inch represents 4 feet, it will take three fourths of an inch to represent 1 yard or 3 feet: and, therefore, the number of inches to represent 60 feet (20 yards) will be 3X20 = 15 inches; because it will take 20 times the 4 length of scale to represent 60 feet (20 yards) that it takes to represent 3 feet; but it takes 3-inch to represent 3 feet, therefore it will take 20 times 4-inch 15in., to represent 60 feet. = Make the scale 15 inches long, divide it into 2 primary divisions, and the first primary division into 10 sub-divisions to measure single yards. The scale is completed in every respect as before. The proportion may also be stated thus: 12 : 5 :: 36 : X= 15 inches. That is to say, as 12, the number of inches in 1 foot, is to 5 inches, the length of the scale, so is 36, the number of inches in 1 yard, to x, the required length of the scale. 20 : 5 :: 20 x 3 : x= 15 inches. That is to say, as 20 feet is to 5 inches, the length of the scale, so is 20 yards brought to feet, to x, the required length of the scale. = 6. Draw a scale of kilomètres comparative to No. 3. The kilomètre 1093-63 English yards. (Scale No. 6.) The number of miles represented in Scale No. 3 is 20 miles. By proportion we have 20 × 1760 : 7.5 :: 20 x 1093.63 : X = 4.66 inches. That is to say, as the number of yards in 20 miles is to 7.5 inches, the length of the scale, so is the number of yards in 20 kilomètres to x, the required length of the scale. Make the scale 4.66 inches long; divide it into two equal parts, and the first primary division into 10 equal parts. Complete the scale as before, and write kilomètres on the right. (See No. 6.) The representative fraction is 7. In examining a French plan, I find only a scale of decimètres, 10 to 1 inch; supply a comparative scale of English feet, the decimètre being equal to 327 English foot. Show 20. Number of English feet in 10 decimètres =327 x 10 = 3.27. Then, by proportion, we have— 3.27ft. : lin. :: 20ft. : x= 6.11 inches. That is to say, as the number of feet in 10 decimètres is to 1 inch, the length which represents it, so is 20 feet to the required length of the scale. Therefore, make the scale 6.11 inches long, divide it into 2 equal parts, and the first primary division into 10 equal parts. Complete the scale as before, and write feet on the right. (See No 7.) Obs. It is not necessary that the comparative scale should represent the same number of units as the original scale. If this were the case, the scale would frequently be inconveniently long. For example, if the number assumed in the last question had been 80 instead of 20, the scale would have been more than 24 inches long. Perhaps we shall be more clearly understood by giving the question (No. 7) in another form. Thus :-In examining a French plan, I find that 100 decimètres are represented on a scale 10 inches long; supply a comparative scale of English feet, the decimètre being equal to 327 English foot. Number of feet in 10 decimètres feet. Therefore, as— = 327 x 10 = 3.27ft. : lin. :: 100ft. : X= 30.58 inches; or thus:-The number of feet in 100 decimètres 3.27 •327 X 32.7ft. : 10in. :: 100ft. : x= 30.58 inches. Now, 30.58 inches, the length of the scale, bears the same proportion to 100 feet, the number of units represented, as 6.11 inches, the length of the scale (see No. 7), bears to 20 feet, the number of units represented. The student, therefore, has only to bear in mind, that the number of units represented on the original scale need not necessarily be represented on the new scale. DIAGONAL SCALES. Diagonal Scales are used for taking off more minute distances than can be done by an ordinary scale. 8. Draw a Diagonal scale of 7 feet to 1 inch to show inches. Assume 40. (No. 8.) 7ft. : lin. :: 40ft. : x = 5.71 inches. Draw a line 5.71 inches long; divide it into 4 equal parts, and the first primary division into 10 equal parts. Each of these sub-divisions will represent 1 foot, as in the preceding scales. Parallel to the first line, draw 12 lines about the same distance apart as shown in the scale. Through the points of divisions 10, 0, 10, 20, 30, draw perpendiculars. (See Scale No. 8.) Number the points on the first primary 10, and those on the perpendicular drawn through 10, on the extreme left of the scale, 2, 4, 6, 8, 10, 12. Join 12 to the ninth sub-division, and through the points 8, 7, 6, 5, 4, 3, 2, 1, 0, draw parallels (diagonals). division 2, 4, 6, 8, Below the line first taken, draw another line at a greater distance apart than the others; make it a thick line and write feet and inches on the extreme right and left of the scale. We have thus constructed a diagonal scale by which we can measure feet and inches. The distance between the dots on the upper line shows 11 feet, while the distance between the second dots (on the parallel drawn through 9) shows 10 feet 9 inches. Suppose it is required to take off 15 feet 5 inches. Place one leg of the dividers at the point where the diagonal drawn through 5 intersects the parallel drawn through 5; and the other leg at the point where the same parallel intersects the perpendicular drawn through 10. The distance is shown by the dots on the parallel drawn through 5. The distance between the dots on the parallel drawn through 7 shows 24 feet 7 inches. Again, let it be required to construct a scale of 10 miles to 1 inch, showing furlongs diagonally. Take a line any convenient length, say 6 inches; divide into 6 equal parts, and the first inch to the left into 10 equal parts. Since the inch represents 10 miles, each of the subdivisions will represent 1 mile. Now, to show furlongs, we want the one-eighth part of one of these sub-divisions. Therefore, draw 8 lines parallel to the line first drawn, and about the same distance apart as shown in Scale No. 8. Complete the scale as shown at No. 8. Let it be required to construct a scale of 1 furlong to 1 inches, showing yards diagonally. Take a line any convenient length, say 6 inches long. Divide this line into 4 equal parts, each part of which will contain 1 inches and represent 1 furlong. Now, the number of yards in 1 furlong = 220, which resolved into factors, 11 x 20. Therefore, divide the first division into 20 sub-divisions, each of which will represent 11 yards (22); and to measure single yards, we have merely to draw 11 lines parallel to the first line-completing the scale as before. = It is hoped that the foregoing examples will render the principles, involved in the construction of a diagonal scale, sufficiently obvious without further explanation. EXAMPLES. Note. Those questions marked with an asterisk (*) are taken from the Reports on the Military Examinations. 1. Construct a scale of 30 feet to 1 inch, to show single feet. 2. Construct a scale of yards of T. Show 50 yards. 3. The distance between two places is 10.6 miles, and measures on the scale 2 inches; draw the scale. Show 30 miles. 4.* An Englishman wishing to examine a Spanish plan, finds only a scale of Spanish palms, 20 to 1 inch; supply him with a corresponding scale of English feet, taking the palm as 684 English foot. Show 50 feet. 5.* Make a diagonal scale of 10 feet to 1 inches, show |