sent some preliminary results to Tycho Brahe for approval ) and actually he dealt with logarithms of sines. Regiomontanus' tables of sines were based on a circle with radius 107 and, because of this, Napier chose 107 for the length | AB |, this being the largest number for which he needed the logarithm. To see how Napier's logarithms are related to functions used today, we can interpret Napier's definitions using the modern language of the calculus. The geometrical definitions are equivalent to the differential equations dX = -X, ^ = 107, dt dt with the condition that y = 0 when x = 107. Eliminating t leads to y = 107ln(107/x)

in terms that we now call the 'natural logarithm'. Napier's logarithms thus have the property that

NLog(x1 X2) = NLog(x1) + NLog(x2) - 107 ln 107 (7.1)

and so, with the help of a table of these logarithms, any multiplication can be reduced to two additions, a huge saving in time when the numbers to be multiplied had large numbers of digits.50

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