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

Two Pages from John Napier's Logarithmic Table
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

These are the two pages from John Napier's original Mirifici logarithmorum cannonis descriptio which started with the following

Hic liber est minimus, si spectes verba, sed usum. Sid spectes, Lector, maximus hic liber est. Disce, scies parvo tantum debere libello. Te, quantum magnis mille voluminibus.

which is translated into

The use of this book is quite large, my dear friend. No matter how modest it looks, You study it carefully and find that it gives As much as a thousand big books.

# Basic Description

During the initial creation of the page Logarithmic Scale and the Slide Rule, I have found a very thin but immensely interesting volume, John Napier and the Invention of Logarithms, 1614 --- A Lecture, by Ernest William Hobson. Fascinated by logarithm's history and its subsequent development into what we know today, I decided to have a separate page dedicated to explaining and imparting this knowledge, not only for my own learning but also for the learning of others. 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. Much of the ideas here are from the above mentioned book, another wonderful book by Lancelot Hogben, Mathematics for the Million: How to Master the Magic of Numbers and a relatively modern translation of the Mirifici Logarithmorum Canonis Constructio (The Construction of the Wonderful Canon of Logarithms). In addition, I have supplied some additional proofs and necessary information to aid understanding. Though a thorough understanding of the original publication 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.

# A More Mathematical Explanation

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

## An Interesting Introduction

As you can see, the logarithms given in the tables are those of the [...]

## An Interesting Introduction

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 an 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).

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

zzz