# 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 original Mirifici logarithmorum cannonis descriptio (The Description of the Wonderful Canon of Logarithms) 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 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 was a very concise and succinct volume that presented how John Napier delivered his original ideas. It was absolutely 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.

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 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

## Introduction: Why is John Napier's Discovery so Extraordinary ?

Today, we regard taking logarith [...]

## Introduction: Why is John Napier's Discovery so Extraordinary ?

Today, we regard taking logarithms as nothing but the inverse of calculating exponential. i.e. obtaining $x=log_ab$ knowing $a^x=b$. It seems that taking logarithms is as natural as operations involving indices. Just punch the calculator and we can have the answer. What if we don't have a calculator? How would you calculate $log_57$? Apparently, the definition of logarithms is a lot easier than it real calculation. Taking the argument further, what if you are neither aware of the notion of index as we now accept as a basic algebraic operation nor the exponential notation? Not only that, what if the decimal notation was not even in use? How would you even come up with the definition and a table of logarithms. Well, it is impossible you would say because what we take to be facts and the basics were even available. It is like trying to calculus before we even know how to add and subtract. Well those were the narrowness of the means available to John Napier who predated Isaac Newton and Leibniz so calculus and subsequently calculation by means of infinite series were not available him as well. While these obstacles in mathematics were daunting enough, those in society were even greater. The period from 16th to early 17th century was full of transformation and turmoil in religion and politics of the British Islands. In spite of all those difficulties, Napier had accomplished what few even in better condition could not have accomplished.

Towards the end of the sixteenth century, further progress of science was greatly impeded by the continually increasing complexity and labor of numerical calculation. Thus it was Napier's intention that

Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers ... I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. --- Descriptio

Napier published the Descriptio 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, Constructio was published by his son Robert Napier in 1619. In the forward by him, it was mentioned that Constructio was actually written before the Descriptio.

## John Napier's Mirifici Logarithmorum Canonis Constructio and a Step-by-Step Explanation

### A Definition of Sine We Don't Know

Note that Sine was not defined as the ratio of opposite over hypotenuse 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. Refer to the diagram above, AB is the cord and AC is the semi-cord and we have the relation $AC = sin \frac {\theta}{2}$. Napier took the radius to be $10^7$ units and thus $\sin 0^\circ = 0$ and $\sin 90^\circ = 10^7$, a result we are all familiar with.

### Arithmetic Progression and Geometric Progression

As mentioned earlier, JN's conception of logarithms was not that of the clearly exact reserve process of taking exponential. However, people are familiar with two series, the arithmetic and geometric progressions. In an arithmetic progression, consecutive numbers differ by a constant amount. For example, 1,2,3,4,5....... In a geometric progression, consecutive numbers are of the same ratio. For example, 2,4,8,16,32....... The faster of you lot will realize that for a number, say 2, its consecutive exponentials are geometric while the powers are arithmetic.

Arithmetic 1 2 3 4 5 6 ... Logarithms
$2^1$ $2^2$ $2^3$ $2^4$ $2^5$ $2^6$ ...
Geometric 2 4 8 16 32 64 ... Antilogarithms

You should have realized that multiplication and division done in the geometric series can be translated into addition and subtraction in the arithmetic series. For example, we want to calculate $4 \times 16$. We go up to find the number that corresponds to 4 and that corresponds to 16 on the first row of the table above, which are 2 and 4, and add them together, which gives up 6, and then come down to find the answer directly below, which is 64. This operation is a direct result of $2^a + a^b = 2 ^{a+b}$ which JN did not explicitly know. Instead, he ingeniously defined logarithms in terms of the continuous motions of two particles. Below is how he defined it originally. He presented his argument in separate articles.

 23. To increase arithmetically is, in equal times, to be augmented by a quantity always the same. 24. To decrease geometrically is this, that in equal times, first the whole quantity then each of its successive remainders is diminished , always by a like proportional part. 25. Whence a geometrically moving point approaching a fixed one has its velocities proportionate to its distances from the fixed one. 26. The logarithms of a given sine is that number which has increased arithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine.

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