38 Mr. C. HoltzapfFel 07i a Scale of Geometrical Equivalents 



The above application of scales is true of all ratios ex- 

 pressed as vulgar fractions, decimal or others, as |, .y., S^-^., 

 ^•|36^ T-lff' ^^ being essential the scales should be 7 and 8, 



11 and 9, 28 and 4 2-236 and 1-414 times any 



unit, and that they are employed as explained. The three 

 first fractions are comprehended in the scales prepared for 

 lines or drawings, which are 1, 2, 3, 4 to '■25 inches long, si- 

 milarly divided and figured*. As we obtain from them all 

 the distinct proportions that may be expressed as vulgar 

 fractions in terms not exceeding 25, (which are given in the 

 pamphlet both as fractions and decimals, and arranged in the 

 order of magnitude) they offer great facility for making re- 

 duced or enlarged copies of drawings, models, &c. after va- 

 rious manners, many examples being given, and they likewise 

 serve for working numerical ratios of simple proportion. The 

 two latter or the decimal fractions would result from the em- 

 ployment, after the same manner, of two other series of scales 

 for the areas of superficies and the contents of solids, the units 

 of which are respectively the square and cube roots of the 

 numbers 1 to 10. 



These may need a little more explanation. Required |ths 

 the size of a given area A B C D. 



" The parallelogram A B C D is divided into 9 equal and 

 proportional parallelograms, by the 

 division of each of its sides into 

 three parts. If, however, we wnly 

 reckon two parts each way, we find 

 them to include but four out of the 

 nine equal areas. The square root 

 of 4 is 2, and the square root of 9 is 

 3, therefore the areas being as 4 to 9, the sides are as 2 or 3, 

 or as the square roots of the former, which was to be shown. 

 This would be equally true if the scales were applied in the 

 inverse order, and of any two ; also if the figures were com- 

 plex ; for a semicircle, or equilateral triangle on A B, would 

 have for the new diameter or side A E, and so with every 

 part of which the figure might be made up." 



The scales for solids admit of the same explanation. 



There are 24 lines of graduations on the other face of the 

 scale, (or it is somewhat more convenient to have them as 

 two distinct instruments,) namely, the scales for areas y^, 1, 

 2, 3, 4, 5, 6, 7, 8, 9, 10 and 100, and the same number for 

 solids. The first and last are for those cases in which the 

 ratio of enlargement or diminution cannot be expressed in 

 fractions not exceeding 10 in either term, which will be re- 

 ferred to hereafter. 



* A New System, &c. pp. 16—23. 



