RECENT ADVANCES IN SCIENCE 



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graduated similarly, and x graduated with half the unit used 

 in a, b. Setting them all parallel with x equidistant from 

 a and b, we get a nomogram for x = a and b. For take 

 (fig. i), e.g., the point + 6 on {b), + 2 on {a) ; join them ; 

 then the hne cuts {x) in the point 8, giving the value of 6 + 2. 

 Take, again, the point — 6 on (^) +4 on {a) ; join them. 

 This time the line cuts {x) in the point — 2, giving the value 

 of — 6 + 4. Joining up, then, pairs of points, we may get the 

 value oi a -^-b for various values of a and b shown on the x scale. 

 To get a nomographic chart iov x = a -\- 2b, take scales (a) 

 and (b), as before, graduated with the same unit, but let the 

 third scale (x) be twice as far from a as it is from b, and let its 

 unit of graduation be one-third of the unit used in a, b (Fig. 2). 



Then the collinearity of a set of points — a, b, x — on the three 

 scales means that three times distance x = distance a + twice 

 distance b ; and, since the :!k:-unit is one-third of the a and b units, 

 it follows that graduation x = graduation a + twice gradu- 

 ation b. The lines drawn show the particular results 



8+2 (^6) = - 4 



-4 + 2.5 =6 



and the line perpendicular to the scales the result 7 + 2.7 = 21 . 

 Obviously, very accurately drawn scales would be of value, 

 even in such simple matters as multiplication, for by adjusting 

 the distances between the scales, various multiplication tables 

 could be constructed. 



The generalised nomogram giving x = la -{- mb, when / and m 

 are rational numbers, easily follows, but is, of course, only 



