. School: a monthly record of educational thought and progress. ults applicable to ourpurpose. A chart of graphs representing on a large scale thegrowth of the amount of /i at compound interest at various ratesis easily constructed. Such a chart might with advantage formone of the ordinary wall diagrams of the mathematical class-roomand be used to illustrate the march of any magnitude whose rate ofincrease is proportional to itself. For useful hints see LodgesDifferential Calculus for Beginners, pp. 63-72. The property just proved was taken by Napier as the-fundamental one in explaining how th
. School: a monthly record of educational thought and progress. ults applicable to ourpurpose. A chart of graphs representing on a large scale thegrowth of the amount of /i at compound interest at various ratesis easily constructed. Such a chart might with advantage formone of the ordinary wall diagrams of the mathematical class-roomand be used to illustrate the march of any magnitude whose rate ofincrease is proportional to itself. For useful hints see LodgesDifferential Calculus for Beginners, pp. 63-72. The property just proved was taken by Napier as the-fundamental one in explaining how the correspondence oftwo series of numbers, one in , the other in , ledto the conception of logarithms. He did not use rectangularaxes, as we have done, but took the two lines of motionparallel to one another and made N move towards O insteadof away from O. Neglecting these modifications as unim-portant, we may say that, according to Napiers definition,if M moves uniformlv and N with a velocity proportionalto ON, then OM is the logarithm of ON.* Fig. J L M6 7 a 9 10 X. Let HP, KQ, LR, MS be ordinates of an exponentialcurve, and let HK = LM. Then HK, LM are made upof the sums of the same number of equal differences in an , —^ and are compounded of the same number of HP LR equal ratios in the corresponding _ MS?? HP ~ LR Hence, if an exponential curve has been plotted, itaffords a means of finding graphically a fourth proportionalto three given numbers. For, taking HP, KQ, LR tOrepresent the three given numbers, we have only to markoff LM = HK to find the ordinate MS, which gives therequired fourth proportional. But the same result may be obtained in a more con-venient manner. Let OX be graduated, not as an ordinaryruler (in which the graduations would give the values ofthe abscissa:), but in such a way that the graduations(i, 2, 3 ... 10) at each point show the value of the ordinatefor that point to the exponential curve. We should then have a logarithmic
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