The Logarithms, Its Discovery and Development
|Two Pages from John Napier's Logarithmic Table|
Two Pages from John Napier's Logarithmic Table
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A More Mathematical Explanation
- Note: understanding of this explanation requires: *A little Algebra
Since people knew how to do where , [...]
Since people knew how to do where , it was natural for people to come up with an operation that tells us the power, knowing the base and the result, i.e. obtain in . The solution, of course, is as we know today.
Today, we regard taking logarithms as nothing but the inverse of calculating exponential. i.e. knowing . It seems that taking logarithms is as natural as operations involving indices. Then it should come at a huge surprise that at the time of Napier, the notion of index, in its generality, was no part of the stock of ideas of a mathematician, and that the exponential notation was not yet in use. In addition to that, Napier predated Isaac Newton and Leibniz so calculus and subsequently calculation by means of infinite series was not available him as well. It was with these difficulties that Napier invented and calculated the logarithms.
Napier published the Mirifici Logarithmorum Canonis Descriptio (The Description of the Wonderful Canon of Logarithms) in 1614 which did not contain an account of the methods by which the "wonderful canon" was constructed. It was not until after his death, Mirifici Logarithmorum Canonis Constructio (The Construction of the Wonderful Canon of Logarithms) was published in 1619. It was later found out that the "Constructio" was written before the "Descriptio".
It should be noted that Napier did not try to obtain the logarithms of natural numbers. Instead, he was trying to obtain the logarithms of sines of angles. Note that Sine was not defined as the ratio as we know it today. It was defined as the length of that semi-chord of a circle of given radius which subtends the angle at the center. Hence we have the relation . Napier took the radius to be units. Therefore, Napier was actually looking for the logarithms of the numbers between and , not for equidistant numbers, but for the numbers corresponding to equidistant angles. It also should be observed that the logarithms in Napier's table are not what we know under the name of natural logarithms. Therefore, to
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