ALIXKMliXT CHARTS IX I'oRKST MKXSUKATIOX 7?9 



tliat the line EIF is inverted. Here it will be seen that, as in figure I, 

 H] = y2 GB and D] = jA FI. but that now HD = HJ — DJ = j^ 

 (GB — FI). 

 It is obvious that such a chart may be reversed to perform addition. 



MUI/nPLICATION OF \ARIABLES 



J-^irst Method. — The equation r = .rv may be written in logarithmic 

 form: log. z^log. x -\- log. y. Multiplication can therefore be per- 

 formed by means of a chart similar in form to that for addition, ex- 

 cept that the axes are graduated to correspond with the logarithms 

 of successive values of .r, y, and rr, instead of with the values them- 

 selves. Figure 4 is a simple graph of this type. Since the logarithms 

 of I, 2. 3, 4, etc., are o.oooo, 0.3010, 0.4771, 0.6021, etc., the 

 graduations on the two initial axes are spaced at the latter number of 

 units from the origin. The final or central axis is similarly graduated, 

 but using a unit one-half as large. The broken line indicates the man- 

 ner in which this chart solves the multiplication 5 X 8 = 40. 



Second Method. — Multiplication can also be performed by what is 

 known, on accoimt of its shape, as a Z chart, in which logarithmic 

 graduation is not necessary. Its principle is indicated by figure 5. In 

 this AB and CD, 2 parallel straight lines, are the first initial and the 

 final axes. These are graduated in opposite directions, B and C being 

 the origins. BC is then the second initial axis. The broken line EFG 

 represents a single position of the straight-edge, connecting related 

 values. Since the triangles BEF and CGF are similar, their sides are 

 proportional and 



CG ^ FB 



BE ~ CF • 

 but 



FP, = CP> — CF: 

 therefore 



CG _ CF 



BE ~~CB— CF ' 

 \vhich mav be written 



CF 



CG=BE (ci._cf) 



An equation of the form.j = .vy can therefore be solved by a chart of 

 this type, if 



CG = .^. 



BE = .r, and 

 CF ^ 



CB — CP ~ ^'' 



