# Difference between revisions of "The Logarithms, Its Discovery and Development"

Two Pages from John Napier's Logarithmic Table
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Two Pages from John Napier's Logarithmic Table

These are the two pages from John Napier's Canon. As you can see, the logarithms given in the tables are those of the sines of asgles from 0^\circ to 90^\circ at intervals of one minute, to seven or eight figures. The table is arranged semi-quadrantlly, so that the logarithms of the sine and the cosine of a n angle appear on the same line, their difference being given in the table of differentials which thus forms a table of logarithmic tangents. Why? Well \tan \theta = \frac {\sin \theta}{\cos \theta} and taking logarithms of both sides we will have log \tan \theta = log \sin \theta - log \cos \theta. So, the difference forms the logarithmic tangents. Therefore, it is safe to assume that he invented logarithms to aid calculation in astronomy and geometry (in Mirifici Logarithmorum Canonis Descriptio, he gave the application of logarithms in solution of plane and spherical triangles).

# Basic Description

As you can see, the logarithms given in the tables are those of the sines of asgles from 0^\circ to 90^\circ at intervals of one minute, to seven or eight figures. The table is arranged semi-quadrantlly, so that the logarithms of the sine and the cosine of a n angle appear on the same line, their difference being given in the table of differentials which thus forms a table of logarithmic tangents. Why? Well \tan \theta = \frac {\sin \theta}{\cos \theta} and taking logarithms of both sides we will have log \tan \theta = log \sin \theta - log \cos \theta. So, the difference forms the logarithmic tangents. Therefore, it is safe to assume that he invented logarithms to aid calculation in astronomy and geometry (in Mirifici Logarithmorum Canonis Descriptio, he gave the application of logarithms in solution of plane and spherical triangles).

# A More Mathematical Explanation

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

This section, though embedded in this page as a sub section, stands equal to an independent article a [...]

This section, though embedded in this page as a sub section, stands equal to an independent article and hence, I direct you to The Logarithms, Its Discovery and Development. Having found a very thin but immensely interesting volume, John Napier and the Invention of Logarithms, 1614 --- A Lecture, by Ernest William Hobson, I was fascinated by logarithm's history and its subsequent development into what we know today. The book is a very concise and succinct volume that presented how John Napier delivered his original ideas. It is an absolute a pleasure to read because it translated Napier's arguments and thoughts into relatively modern mathematical symbols and notations and at the same time preserved and revealed Napier's ingenuity. In the redirected page, most of the ideas are from the above mentioned book and another wonderful book by Lancelot Hogben, Mathematics for the Million: How to Master the Magic of Numbers. In addition, I have supplied some additional proofs and necessary information for understanding. Though a thorough understanding of the books requires some careful thoughts and deliberate ruminations, in the end, you will find that you will appreciate logarithms a lot more than you did before. However, you should not be intimidated and discouraged by the section, in which case you should continue to the next section.

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