As one fathom equals 6 feet, 6 lines parallel to the first will be required. = 4. A scale of miles 76 to 1.3 inches. 76: 1.3 :: 500: X 8.29 nearly. Five primary divisions will show hundreds of miles, 10 secondary divisions will show spaces of 10 miles, and 10 lines parallel will show single miles. COMPARATIVE SCALES Are scales representing precisely the same distance and consequently having the same representative fraction as other scales, but under different denominations; thus, whether we require scales of 2 fathoms to 1 inch, or of 12 feet to 1 inch, or of 4 yards to 1 inch, or of 144 inches to 1 inch, those scales will be comparative because their representative fractions, are the same: 2 fathoms being equal to 144 inches, 12 feet 144 inches, and 4 a scale of yards, 1760 yards to 1 inch; a scale of furlongs, 8 furlongs to 1 inch; the representative fraction 360 would be the same because all those different denominations or scales reduced to inches would give that number of inches represented by one inch, therefore all those scales would be comparative to each other. Having a scale on which a space of 2.7 inches represent 1 mile, and whose representative fraction will naturally be 1760 × 36 ÷ 2.7, I require a scale of yards comparative to 1 it; it is evident that the scale will only change its denomination, because 2.7 will represent 1760 yards instead of being called 1 mile. That scale, therefore, will be 1760 2.7: 4000: X = 6.18. It follows that in order to obtain a comparative scale we have only to change the denomination of the given scale into that of the scale required, which is performed by simple multiplication or division, as the case may be; thus, suppose that on a map a space of one inch represents 5 yards, show a comparative scale of feet. Here reduce the yards into feet, 5×3=15 feet, that scale will naturally become a scale of 15 feet to an inch, 1 with representative fraction 5×3×12, and conversely, should it be required to convert a scale of 15 feet to an 15 inch into yards, would give 5 yards, and the represen 3 15 × 12 would be the same as above. EXAMPLES. 1. Reduce 40 yards to Russian archines. 1 archine = .7777 of a yard. 2. Reduce 33 metres to Greek cubits. 1 cubit = .45 of a metre. 3. Reduce 72 feet to French metres. 1 metre 3.27 feet. 72 = 22.018 metres. 4. Reduce 73 Spanish palms to English feet. 1 palm = .684 English foot. .684 × 73 = 50.132 feet. 5. Reduce 57.61 metres into English yards. 1 yard 1.09363 metres. = 6. Reduce 12 feet to Roman palms. 1 palm of 1 foot. 12 = 16.32 palms. .735 7. Reduce 27 Roman palms to English feet. 27 × .735 = 19.845 feet. = .735 8. Reduce 168 yards to chains. 1 chain 22 yards. Reduce 64 Spanish palms to English feet. 1 palm .684 of English foot. = Reduce 2 miles to paces. A mile assumed to contain 2250 paces. Reduce 686 Neapolitan cannae to English yards. 1 canna = 2.3008 English yards. Reduce 17 Austrian miles to English miles. 1 Austrian mile 3.3312 English miles. Reduce 300 Russian sachines to English yards. chine 2.332 yards. 1 sa Reduce 8000 alners to English feet. 1 alner = .6493 of an English yard. EXAMPLES OF COMPARATIVE SCALES. 1. Construct a scale of 5 chains to 1.5 inches, and a comparative scale of yards to it. 1 chain = 66 feet, or 22 yards. For the first scale, 5 chains : 1.5 :: 20 : X = 6. 6 inches divided into two parts, and the first division subdivided into 10 parts, will give the scale of chains. For the second scale, reduce the chains to yards, 5 × 22 = 110 yards... 110: 1.5 :: 500 : X = 6.81. 2. On a Spanish plan, 47 palms are represented by a distance of 2.3 inches; construct a scale of English feet comparative to it. 1 palm.684 English feet. (.684 × 47 = 32.148 English feet) .. 32.148: 2.3 :: 3. Construct a scale of 2 inches to 1.5 mile, upon which hundreds of paces can be measured, assuming a mile to contain 2260 paces. 2260 × 1.5: 2 :: 10000: X = 5.9. 5.9 inches divided into 10 parts will show thousands of paces, the 10 secondary divisions will show hundreds, and the diagonal tertiary divisions will determine spaces of 10 paces. 4. On an Italian plan, 4 inches represent 325 cannae; show a comparative scale of English yards. 1 canna = 2.3008 English yards. (2.3008 × 325): 4 :: 1000: X = 5. The distance between two points three Austrian miles apart is represented on a map by a space of 6.67 inches. Construct a scale of English miles and chains. 1 Austrian mile = 3 3312 English miles. 1 English 3.3312 × 39.9936 English miles.. 9.9936: 6.67 ::10: X = 6. Construct a scale of 1.45 inches to a mile, and show furlongs; give a comparative scale of chains. = First scale-1 mile : 1.45 :: 5 : X 7.25. (Plain scale.) Second scale-1160 80 chains .. 80: 1.45: 400: 22 8. To the latter distance, construct a comparative scale of Turkish berri. 1 berri 1828.08 yards. Then 20 × 1760 1828.08 the number of berri in 20 miles. .. 19.239 berri: 6.66 :: 20 : X 9. On a Spanish plan, 46 palms are represented by a space of 2.3 inches; give a comparative scale of English yards. 1 palm .684 of an English foot. 46 = 10×.081) : 2.3: 30 yds. : X = 10. On a Swedish map, 6000 alners are represented by a space of 5 inches; give a comparative scale of English feet. 1 alner = .6493 of an English yard. (6000 .6493 × 3) 11687.4: 5: 15000 : X = 11. Required, a scale of 8 feet to 1 inch, and a comparative scale of French decimetres. 1 metre 1.0936 yards. First scale, 8 ft. 1: 50: X = 6.25. : Second scale, 1.0936 × 8 = 2.91626 metres : 1 :: 3 20: X 6.85. = 12. Construct a plain scale of yards, 290 yards to 1.9 inches, and also a corresponding scale of Russian sachines. 1 sachine 2.332 yards. = 13. Construct a scale of 30, 50 yards long, and a comparative scale of Russian archines. 1 archine = .7777 of a yard. 1 yard 300 yards.. 36 inches = 300 yards. = then 300 36:: 50: X = 6. Second scale, 300 4 387.1672 archines: 36 :: 60 : X 14. Draw scales of 70 to represent English feet, French metres, and Greek cubits. |