# The Logarithms, Its Discovery and Development

Two Pages from John Napier's Logarithmic Table
These are 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 translates 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.[1]

# 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 some if John Napier's original key ideas. The book 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 and like Napier said, "find that it gives as much as a thousand big books"

As you can see, the logarithms given in the tables are those of the sines of angles 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 Napier invented logarithms to aid calculation in astronomy and geometry (in Descriptio, he gave the application of logarithms in solution of plane and spherical triangles).

# 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 $a^x=b$. We learn how to take powers before learning how to operate with logarithms and how to differentiate before integrate because that is only natural. However, the order (which is best for understanding and learning) in which we learn mathematical knowledge is not necessarily the order in which they were discovered. That is why we still find JN's discovery as wonderful as it was some 396 years ago.

Consider the following questions: What if we don't have a calculator? How would you calculate $\log_57$? 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 popular use? Eventually, 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. 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.[2]

# 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, Geometric Progression and Napier Logarithms

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.

 Article 23. To increase arithmetically is, in equal times, to be augmented by a quantity always the same. Article 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. Article 25. Whence a geometrically moving point approaching a fixed one has its velocities proportionate to its distances from the fixed one. Article 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.

Article 23 means that if we have a point moving with constant velocity, $v$, then in equal time interval, $t$, it will move equal amount of distance. Refer to the diagram above. $AB=BC=CD=DE=EF=vt$ and hence, the lengths $AB,AC,AD,AE,AF...$ form an arithmetic series.

Article 24 means that if we have a distance of $TS=10^7$ and a point moving in such a way that in equal time interval, $t$, it will move a distance that is of a fixed portion of the distance the point has yet to travel. Say we choose that constant portion as $\frac {1}{8}$ and refer to the diagram above. $\frac {TG}{TS}=\frac {GH}{GS}=\frac {HI}{HS}=\frac {IJ}{IS}=\frac {1}{8}$. As a result, the lengths $TS,GS,HS,IS,JS...$form a decreasing geometric series.

Article 25 needs a leap in the understanding. JN claimed that the ratio between successive velocities of the particle at $T,G,H,I,J...$ are the same as the successive distances that the particle has yet to travel, i.e. $\frac {v_T}{v_G}=\frac {v_G}{v_H}=\frac {v_H}{v_I}=\frac {v_I}{v_J}=\frac {TS}{GS}=\frac {GS}{HS}=\frac {HS}{IS}=\frac {IS}{JS}$. How so? This is JN's argument.

For we observe that a moving point is declared more or less swift, according as it is seen to be borne over a greater or less space in equal times. Hence the ratio of the spaces traversed is necessarily the same as that of the velocities. But the ratio of the spaces traversed in equal times, T1,12,23,34,45,&c (we used TG,GH,HI,IJ), is that of the distances TS,1S,2S,3S,4S,&c(we used TS,GS,HS,IS,JS). Hence it follows that the ratio to one another of the distances of G (the particle) from S, namely TS,1S,2S,3S,4S,&c., is the same as that of the velocities of G at the points T,1,2,3,4,&c.,respectively.

In notation language, he means that

 since $\frac {v_T}{v_G}=\frac {v_G}{v_H}=\frac {v_H}{v_I}=\frac {v_I}{v_J}=\frac {TG}{GH}=\frac {GH}{HI}=\frac {HI}{IJ} \cdots \cdots$ Eqt. 1        , and also $\frac {TS}{GS}=\frac {GS}{HS}=\frac {HS}{IS}=\frac {IS}{JS}\cdots \cdots$ Eqt. 2        . Therefore, $\frac {v_T}{v_G}=\frac {v_G}{v_H}=\frac {v_H}{v_I}=\frac {v_I}{v_J} = \frac {TS}{GS}=\frac {GS}{HS}=\frac {HS}{IS}=\frac {IS}{JS}\cdots \cdots$ Eqt. 3        .

Eqt. 2 is an easy argument to buy. Using the results from Article 24, both sides of Eqt. 2 equals $\frac {8}{7}$. Eqt. 1 is an argument that is difficult to understanding. It is correct to say that in equal time interval, a particle with higher velocity will have a larger displacement compared to the particle with lower velocity and hence the ratio of the displacements will be equal to the ratio of the velocities. But the velocity of the particle on the line $TS$ changes constantly and how can we compare then? As a matter of fact, JN did not prove that because calculus was not invented yet and intuitively assumed the validity of his argument. How can we use calculus to do this?

Article 26 gives the definition of Napier's logarithm. Say we have two particles moving in the ways stipulated by Article 23 and Article 24. At point $A$ and point $T$, the two particles have the same velocities. In a certain time period, particle $\alpha$ has moved to point $B$ and $\beta$ has moved to point $G$. Hence $Nap \log GS=AB$. As a result, $Nap \log 10^7=0$ which is the result from Article 27. Using calculus, if $x=GS$, we have ${\operatorname{d}x\over\operatorname{d}t}=-\frac{vx}{10^7}$, where $v$ denotes the velocities of $\alpha$ and $\beta$ at $A$ and $T$; and if $y = AB$, ${\operatorname{d}y\over\operatorname{d}t}=v$; thus ${\operatorname{d}y\over\operatorname{d}x}=-\frac{10^7}{x}$. Hence, $y=-10^7 \ln x+c$ where $c$ is a constant. I have questions about this.[3] I got is from a book.

## The limits of a logarithm

Having defined what logarithm is, JN still have no method to approximate with accuracy the values them. However, with his great intuition and ingenuity, he proposed that,

Article 28. Whence also it follows that the logarithms of any given sine is greater than the difference between radius and the given sine, and less than the difference between radius and the quantity which exceeds it in the ratio of radius to the given sine. And these differences are therefore called the limits of the logarithms.

It is indeed a mouthful. This is how he thought about it. If the two particles have the same velocities at point $A$ and $T$, then a certain moment later, say $t$ seconds, then $AB>TG$ since $\alpha$ moves with constant velocity while $\beta$ is subjected to deceleration. What about $t$ seconds previously? Well, $\alpha$ will be at $M$ with $MA=AB$. On the other hand, $\beta$ will be at $N$ with $NT > MA$ I know this is obvious but why?

Given $GS=x$, we have $Nap \log x=AB$ by definition. We know that $AB>TG$ and $TG=10^7-x$. Therefore, $Nap \log x>10^7-x$. On the other hand we know that $AB=MA$ and as a result $Nap \log x< NT$. How do we express $NT$ in terms of $10^7$ and $x$? We go back to the original relation below.

 $\frac {NS}{TS}$ $=$ $\frac {TS}{GS}$ $\frac {NT+TS}{TS}$ $=$ $\frac {TS}{GS}$ $NT+TS$ $=$ $\frac {(TS)^2}{GS}$ $NT$ $=$ $\frac {(TS)^2}{GS}-TS$

Therefore $NT=\frac {(10^7)^2}{x}-10^7$

Now we have the limits that JN proposed $10^7-x. Furthermore, JN proposed that,

Article 39. The difference of the logarithms of two sines lies between two limits; the greater limit being to radius as the difference of the sines to the less sine, and the less limit being to radius as the difference of the sines to the greater sine.

Why is that? JN used the same geometric argument. Say we still have $TS$ as the radius of $10^7$ and let $BS=x$ and $CS=y$ where $x>y$. Now suppose we have a point, $V$, on the left of $T$ such that $\frac {TS}{TV}=\frac{CS}{BC} \cdots \cdots Eqt4$. Furthermore, let there be a point, $A$, on the right of T such that $\frac {TS}{TA}=\frac {BS}{BC}$.

Now, two simple conclusions follow which are $\frac {VS}{TS}=\frac {TS}{AS}$ and $\frac {TS}{AS}=\frac {BS}{CS}$. As a result, $Nap \log CS - Nap \log BS = Nap \log AS - Nap \log TS$. Since $Nap \log TS = 0$, then $Nap \log y - Nap \log x = Nap \log AS$. Then the argument comes full circle as we know that $TA < Nap \log AS < VT$ and what are $TA$ and $VT$ but $TS \frac {BC}{BS}$ and $TS \frac {BC}{CS}$! In all, $10^7\frac {x-y}{x}.

As of now, we have three important conclusions,

 1. The definition of Napier's Logarithms 2. $10^7-x Relation. 1 3. $10^7\frac {x-y}{x} where $x>y$ Relation. 2

## Numbers Which We Can Work With

Now we take a break and play with three tables that JN created. The first table is a 1 by 101 matrix, $A$, starting with $10^7$ and each entry below is $1-\frac{1}{10^7}$ of the previous entry. Thus we have the matrix below. The ratio of the last entry to the first, $10^7$, is approximately $1-\frac{1}{10^5}$ by binomial expansion

 [4] $i=1...101$ $10^7(1-\frac{1}{10^7})^{i-1}$ $i=1$ $10^7$ $i=2$ $10^7(1-\frac{1}{10^7})=9999999$ $i=3$ $10^7(1-\frac{1}{10^7})^2=9999998.0000001$ $i=4$ $10^7(1-\frac{1}{10^7})^3=9999997.00000029999999 \approx 9999997.0000003$ $i=5$ $10^7(1-\frac{1}{10^7})^4=9999996.000000599999960000001 \approx 9999996.0000006$ $i=...$ $...$ $i=101$ $10^7(1-\frac{1}{10^7})^{100} \approx 9999900.00049505$

The second table is a 1 by 51 matrix, $B$, with first entry $10^7$ and subsequent ones $1-\frac{1}{10^5}$ of the previous ones. The ratio of the last entry to the first,$10^7$, is $1-\frac {1}{2000}$ approximately by binomial expansion.

 [5] $i=1...51$ $10^7(1-\frac{1}{10^5})^{i-1}$ $i=1$ $10^7$ $i=2$ $10^7(1-\frac{1}{10^5})=9999900$ $i=3$ $10^7(1-\frac{1}{10^5})^2=9999800.001$ $i=4$ $10^7(1-\frac{1}{10^5})^3=9999700.00299999 \approx 9999700.00300000$ $i=5$ $10^7(1-\frac{1}{10^5})^4=9999600.0059999600001 \approx 9999600.00599996$ $i=...$ $...$ $i=51$ $10^7(1-\frac{1}{10^5})^{50} \approx 9995001.22480404$

The third table is a 21 by 69 matrix, $C$, with $C(1,1)=10^7$. Each subsequent entry in a row is $1-\frac {1}{100}$ of the previous one. Each subsequent entry in a column is $1-\frac {1}{2000}$ of the previous one. Thus the entry in $i^{th}$ row and $j^{th}$ column is $10^7(1-\frac {1}{100})^{j-1}(1-\frac {1}{2000})^{i-1}$. It is clear by now that Matrix $B$ is the result of subdividing the difference between $C(1,1)$ and $C(2,1)$, and Matrix $A$ is the result of subdividing the difference between $B(1,1)$ and $B(1,2)$.

 [6] $j=1$ $j=2$ $j=\cdots$ $j=69$ $i=1$ $10^7$ $9900000$ $\cdots$ $5048858.88787070$ $i=2$ $9995000$ $9895050$ $\cdots$ $5046334.45842676$ $i=3$ $9990002.5$ $9890102.475$ $\cdots$ $5043811.29119755$ $i=4$ $9985007.49875$ $9885157.4237625$ $\cdots$ $5041289.38555195$ $i=5$ $9980014.995000625$ $9880214.84505061875$ $\cdots$ $5038768.74085917$ $i=\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $i=21$ $9900473.57802330$ $9801468.84224306$ $\cdots$ $4998609.40185319$

The above three tables were generated by Matlab. To see the complete table, download the plaint text Image:Tables.txt here. For Matlab M-script to generate your own table, click here. How can I upload a regular file and link it?

## A Matter of Arithmetics

So now JN had the numbers in geometric progression and the two theorems on the limits of logarithms, he had to start making the table. It was an enormous undertaking considering there was no computer available. But he did!

Using Relation. 1, he managed to find the logarithms of the numbers in the Table 1 with sufficient approximation. For example, for the second number, $9999999$, in Table 1, its logarithms lies between $10^7-9999999 which is $1.0000000 . The difference between the limits are really "insensible" so Napier took the average of the two and thus $Nap \log 9999999 = 1.00000005$. Repeat the process 100 times, we will have the answer.

 $ $i=1...101$ $10^7-x$ $\frac {(10^7)^2}{x}-10^7$ $Nap \log x$ $i=1$ $1.0000000$ $1.0000001$ $1.00000005$ $i=2$ $1.99999990$ $2.00000030$ $2.00000010$ $i=3$ $2.99999970$ $3.00000060$ $3.00000015$ $i=4$ $3.99999940$ $4.00000100$ $4.00000020$ $i=5$ $4.99999900$ $5.00000150$ $5.00000025$ $i=...$ $...$ $...$ $...$ $i=101$ $99.99950495$ $100.00050495$ $100.00000495$

Next, he proceeded to Table 2. The nearest number to 9999900, the second number of Table 2, in Table 1 is 9999900.00049505 whose limits are 100.00050495 and 100.00000495. Using Relation. 2, he found that $Nap \log 9999900.00049505 - Nap \log 9999900 = 0.0004950569156743573$ and thus $100.00000495+0.0004950569156743573 which gives him $Nap \log 9999900.00049505 = 1.000005000050325$. The next logarithm has double the limits of the previous one. Thus using this method, he managed to get all the logarithms of the numbers in the second table.

there is a little problem here. JN used the 7 decimals for calculation for table 1 and Matlab used something different which I cannot change. so small deviation in table 1 to the 8th decimal place and that results in subsequent difference in the limits and logarithms A number, $y$, that is near to a number, $x$, of table 2 is found this way: say $y, find $z$ such that $\frac {z}{10^7}=\frac {y}{x}$ then we have $Nap \log z = Nap \log y - Nap \log x$

Relation. 3        . Find the limits of $Nap \log z$ from table 1 and $Nap \log x$ from table 2, adding them together gives the limits of $Nap \log y$ from where the mean can be calculated and JN took that as the true value of the logarithm. Then the logarithm of the next number in the column has double the limits of $Nap \log y$. In this way, logarithms of all the numbers in the first column of table 3 are found. The logarithms of the first number of the second column is found by using Relation. 3 with the last number of first column. Then all logarithms are found.

The new table with all logarithms of the numbers in third table is called the radical table. Now the first two tables can be discarded as they have served the purpose. For a number that is within the limits of the radical table, its logarithm can be obtained using Relation. 2 with a table number that is nearest to it. For a number beyond the limits of the table, there is a way to do it. JN created a table of radios. A number is multiplied by 2 repeated until it is with in limits of the table. Then the new number's logarithm is found as per normal but the original number's logarithm is found by adding that number and the difference.

## Back to Square One

What good is all those logarithms? I though he wanted logarithms of sines and cosines? Of course, let's not get distracted into the petty detail so much that we actually forget what we are doing here. Now the argument comes full swing.

## What is the point of all these?

Then with those tools, logarithms of all sines between 0 and 90 can be obtained.

## Henry Briggs and the Logarithms to the Base 10

I don't have time for this.

## Logarithms to the Base e

I don't have time for this.

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# Teaching Materials

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# Notes

1. The MacTutor History of Mathematics archive, 2006
2. Macdonald, 1888, p. xi-xix
3. Hobson, 1914, p. 26
4. Macdonald, 1888, p. 13
5. Macdonald, 1888, p. 13
6. Macdonald, 1888, p. 15-16
7. Macdonald, 1888, p. 34-35
8. Macdonald, 1888, p. 39

# References

1. Hobson, E. W. (1914). John Napier and the invention of logarithms, 1614; a lecture. Cambridge: Cambridge University Press.
2. Macdonald, W. R. (1888). The Construction of The Wonderful Canon of Logarithms And A Catalogue. Edinburgh and London: William Blackwood and Sons.
3. The MacTutor History of Mathematics archive. (2006, February). Quotations by John Napier. Retrieved from The MacTutor History of Mathematics archive, School of Mathematics, University of St Andrews Scotland: http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Napier.html

Finish up the last part and do the Henry Briggs part.

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