D. J. SCOURFIELD ON LOGARITHMIC PLOTTING. 421 



further useful peculiarity that it shows the same proportionate 

 changes in these numbers by lines having the same angle of 

 slope in whatever part of the chart they may be situated, and 

 this necessarily implies that similar-shaped curves denote the 

 same relative course of events. A very little consideration will 

 show that this must be so. As the horizontal lines are drawn at 

 distances equal to the logarithms of the numbers, similar pro- 

 portionate changes, whether increases or decreases, will always be 

 represented by the addition or subtraction of the logarithm of the 

 same number, i.e., of the number representing the change. Thus, 

 if two numbers, the one high and the other low, both become 

 doubled, the result is shown on the chart by a shifting up equal 

 to the logarithm of 2 in each case. But as the distances 

 representing equal periods of time are equal, it follows that, if 

 the doubling takes place in each case in the same time, the 

 lines showing the change must be inclined at the same angle. 

 The angle for any particular change, say per month, can be 

 easily obtained from its trigonometrical tangent, which is 

 evidently the logarithm of the number representing the change, 

 divided (in the present instance) by "5, i.e., the distance allotted to 

 each month. The following table gives the tangents and the cor- 

 responding approximate angles for various changes per month : — 



2-fold = log ; 2 = '602 = tan. 31° 3' 



•954 = „ 43° 39' 



= 1-204 = „ 50° 17' 



1-398= „ 54 25' 



= 1-556 = „ 57° 17' 



1-690= ., 59° 23' 



= 1-806 = „ 61 2' 



= 1-908 = „ 62 21' 



J =2-000= ., 63 26' 

 ■5 



100- ■„ = lo -- 100 = 4-000 = , 75° 58' 

 "6 



1000- .. = ] -°g- l00 ° = 6-000= .. SO 32 



