The Logarithms, Its Discovery and Development

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Two Pages from John Napier's Logarithmic Table
Napier logtable.jpg
Field: Algebra
Image Created By: John Napier
Website: Milestones in the history of thematic cartography, statistical graphics, and data visualization

Two Pages from John Napier's Logarithmic Table

This page is under development and I am doing research on it. But I will make this my biggest project to undertake for the summer and I vow to make this page perfect and complete. Xingda 18/6


Basic Description

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A More Mathematical Explanation

Note: understanding of this explanation requires: *A little Algebra

Since people knew how to do x^y where x,y \in \mathbb{R}, [...]

Since people knew how to do x^y where x,y \in \mathbb{R}, 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 x in a^x=b. The solution, of course, is x=log_ab as we know today.


Today, we regard taking logarithms as nothing but the inverse of calculating exponential. i.e. x=log_ab knowing a^x=b. 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 crd \theta = 2sin \frac {\theta}{2}. Napier took the radius to be 10^7 units. Therefore, Napier was actually looking for the logarithms of the numbers between 0 and 10^7, 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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