for Engineering and other Purposes. 



S9 



From the name selected for the instrument, namely, the 

 scale for geometrical equivalents, it may be seen it is princi- 

 pally designed for the alteration of known forms and quan- 

 tities; it therefore remains to show this application to any 

 solid of known contents, in any ratio, and in any prescribed 

 manner. 



1. When one dimension only is altered, it will be in the 

 direct proportion of the ratio, the original figure being con- 

 sidered as unity. 



2. When two dimensions are altered, it will be in the du- 

 plicate ratio of the two, or as the square root of the ratio. 



3. When all three dimensions are altered, it will be in the 

 triplicate ratio of the sides, or as the cube root of the ratio. 



For example, a given vessel measures 6 feet long, 4 feet wide, and 

 3 deep, its contents are therefore 6 X 4 X 3 = 72 cubic 

 feet. Required the dimensions of other vessels to con- 

 tain 180 cubic feet, according to each of the modes. The 

 new vessels will be to the original or unity as 180 to 72 

 or '^J> times the size, which reduced to its lowest terms 

 is ^ or 2| times as great. 



1. To enlarge the one dimension only, say the depth, multiply it by 

 the ratio |, 3xf = Ih, the measure of the new depth, ^ ^ 

 and 6x4x7'5 »= 180, the new contents. 



2. A new vessel to be of the same depth, but to have 

 its area enlarged 2^ times. The new sides will be found 

 by multiplying the given sides by the square root of the 



ratio, or V I or f !f|-| ; this by any of the modes of 

 arithmetic, or by logarithms, would take some time; 

 whereas by the scales, the 5 quadratic scale having for 

 its unit 2-236 inches, and the 2 scale 1'4142 inches, 

 (the square roots of 5 and 2,) they represent the above fraction, and the 

 application is precisely the same as seeking |ths of 8 already explained : 

 set the index to 6, the given side on the numerator scale 5, and the new 

 side is found in the denominator scale 2, namely, 9*48 ; and do the same 

 for the other side 4 feet, which comes out 6'32. 

 I have never taken the trouble to obtain these 

 quantities otherwise, and the multiplication of 

 the 3 sides one into the other gives the new con- 

 tents, 1 79*74, whereas it ought really to be 180, 

 so that the error only amounts to ^ about the 

 720th part of the whole quantity. 



3. The vessel to contain 21 times as much as 

 the given one 6x4x3, enlarged in each of its di- 

 mensions, will be found by multiplying each of the sides by the cube root 

 of the ratio, or V f- Therefore, seek 6, 5, 3 in 

 the 5 cubic scale, and the new sides will appear 

 respectively over against them in the denomina- 

 tor scale 2, namely, 8"16, 5*43, 4*08, which multi- 

 plied into one another for the new contents give 

 180-779, about the 240th part too much, a result 

 sufficiently near for most practical cases. 



Of course, what is true of the two scales em- 



