The making of the logarithm

Marianne Freiberger

Remember the natural logarithm? It’s intimately related to one of the most beautiful constants of mathematics, the number

  \[ e = 2.71828182845904523536028747135266249775724709369995... . \]    

The logarithm $\ln {x}$ of a real number $x$ is the number to whose power you must raise $e$ to get $x:$

  \[ e^{\ln {x}} = x. \]    

Today we use a calculator or computer to find logarithms, but traditionally people looked them up in tables that for a long list of numbers $x$ gave you the corresponding value of $\ln {x}$.

John Napier

Portrait of John Napier (1550-1617), dated 1616.

Tables of numbers related in a very similar way were first published in 1614 by the mathematician, physicist and astronomer John Napier in a paper called The construction of the wonderful canon of logarithms. Surprisingly, though, Napier had never even heard of the number e, nobody had at the time, and he wasn't really thinking about exponentiation either. When he inadvertently defined something very similar to the logarithm to base e, he did so by imagining points moving along lines!

One problem that was plaguing people at the time, especially astronomers, was arithmetic. Astronomical calculations required the multiplication and division of very large numbers, something that’s pretty hard to do without a calculator. One way of making things easier is to think in terms of powers. As the rules of exponentiation tell us, to multiply two powers of 2, say $2^ a \times 2^ b$, you only need to add their exponents. To divide them, you simply subtract their exponents:

$2^ a \times 2^ b = 2^{a+b}$

$\frac{2^ a}{2^ b} = 2^{a-b}.$

So a table telling you how to express large numbers as powers of 2, or of any other number, would help you simplify your calculations considerably. Given a number $N$ you would be looking for the number $L$ so that

  \[ N=2^ L. \]    

In other words, what you'd want are tables of logarithms to the base 2, or some other number.

In Napier's time, however, people were not used to thinking in terms of exponentiation. They didn't have the concept of a base and they didn't have our handy way of writing powers, using a little number at the top.

What they were aware of, though, since the time of Archimedes, was an interesting link between the sequence you get by starting with 2 and successive doubling (which we today recognise as the sequence of powers of 2):

2, 4, 8, 16, 32, 64, 128, … .

and the sequence of natural numbers

1, 2, 3, 4, 5, 6, 7, … .

The first sequence is called a geometric progression because successive numbers have the same ratio; 2. The latter is called an arithmetic progression because successive numbers have the same difference; 1.

People noticed that multiplying (or dividing) two numbers in the geometric progression corresponds to adding (or subtracting) the corresponding numbers in the arithmetic progression. (To us, these are just the laws of exponentiation again, as numbers in the geometric progression are powers of 2, and the numbers in the arithmetic progression are the corresponding exponents.) This seemed to offer a way of making calculations easier, as you could replace the harder operations in the geometric progression with easier ones in the arithmetic progression.

Napier wanted to produce a table that related numbers in a useful geometric progression to numbers in a corresponding arithmetic progression so that, as he wrote, "All multiplications, divisions and [...] extraction of roots are avoided," and replaced by "most easy additions, subtractions and divisions by 2."

It’s the way he found the two sequences that is so intriguing. Imagine a point, call it $P$, moving along a finite line segment from a point $A$ to a point $B.$ It doesn’t move at uniform speed, however, but slows down continually: its speed at any given moment is proportional to the distance still left to travel to the point $B$. The closer to $B$ the point gets, the slower it gets, so it will never actually reach $B$. If you measured the distance still left to travel to $B$ at regular time intervals, say every second, then the numbers you’d get would form a decreasing geometric progression: ratios between successive numbers would be equal, but unlike on our example above, they’d be smaller than 1.

Point P

How do you link that to an arithmetic progression? Intuitively, imagine marking the locations of $P$ at each time interval on the line: $x_1$ is the location of $P$ after one second, $x_2$ the location after 2 seconds, and so on. Then since $P$ is continually slowing down, the intervals $[x_ i, x_{i+1}]$ become smaller as $i$ gets larger. And since $P$ never reaches its destination, there are infinitely many such intervals. Now imagine stretching these intervals out so they all have equal length, but also imagine that $P$ still takes a second to traverse each of them. This would move the point $B$ all the way out to infinity (there are infinitely many intervals of equal length), the average speed of $P$ in an interval would be the same for all intervals (all intervals have equal length, which $P$ traverses in one second), and the sequence of distances travelled after one second, two seconds, three seconds, and so on, would form an arithmetic progression.

Using, presumably, this sort of intuitive reasoning, Napier imagined a second point $Q$, which starts out at point $A$ at the same time as $P$ and with the same speed, but travelling at uniform speed towards and beyond $B$ and to infinity. At a given point in time, he defined the distance already travelled by $Q$ to be the logarithm of the distance yet to travel by $P$. This links every number in the geometric progression defined by $P$’s journey to a number in the arithmetic progression defined by $Q$’s journey.

Point P

Napier took the length of the line segment from $A$ to $B$ to be very large: $10,000,000 = 10^7$ units. He did this to ensure accuracy and presumably also because he had astronomical calculations in mind, so you’d want to find a logarithm for large numbers. He also assumed that the initial speed of $P$ is $10^7$.

Today we would use calculus to work out Napier's logarithm. If you do this (see here to find out how), you will find that

  \[ \frac{x}{10^7} = \frac{1}{e}^{\left(\frac{y}{10^7}\right)}, \]    

where $x$ is the distance still to travel by $P$ and $y$ is the distance already travelled by $Q.$

This means that $\frac{y}{10^7}$ is the logarithm to base $1/e$ of $\frac{x}{10^7}$ — essentially that’s what Napier’s construction defines. But since calculus had not been invented in Napier’s time, his table gave an approximation of this logarithm, relating $x$ and $y$ by the relationship

  \[ \frac{x}{10^7} =\left(1-\frac{1}{10^7}\right)^ y. \]    

To see that this really is a good approximation, rewrite the expression as

  \[ \frac{x}{10^7} = \left(1-\frac{1}{10^7}\right)^{10^7\left(\frac{y}{10^7}\right)}. \]    

If you are familiar with the many beautiful properties of the number $e$, then you will know that for any real number $x$, $e^{x}$ is the limit as $n$ goes to infinity of

  \[ \left(1+\frac{x}{n}\right)^{n}. \]    

Taking $x=-1$ gives

  \[ \lim _{n \rightarrow \infty }{\left(\left(1-\frac{1}{n}\right)^{n}\right)} = e^{-1}=\frac{1}{e}. \]    

And since $10^7$ is a very large number, the number

  \[ \left(1-\frac{1}{10^7}\right)^{10^7} \]    

that appears as the base in Napier’s logarithms is very close to the limit $\frac{1}{e}$. Therefore, since

  \[ \frac{x}{10^7} = \left(1-\frac{1}{10^7}\right)^{10^7\left(\frac{y}{10^7}\right)}, \]    

$\frac{y}{10^7}$ is very close to the logarithm to base $\frac{1}{e}$ of $\frac{x}{10^7}$. That’s why Napier’s work is often counted as the first, albeit implicit, appearance of the number $e$ in mathematical history. And Napier is today credited with inventing the natural logarithm — without ever having heard of $e!$

About the author

Marianne Freiberger is Editor of Plus.


Logarithms discovered by Napier in 1614 were based on sine tables with 0.9999999 just below sine 90 degrees as the base which is raised to successive powers. Initially the results are nearly equal to the shortfall from 1.0000000. It would be a very onerous task to raise these powers from sine 90 degrees down to sine 1 degree, but this would be helped by by sine 75 degrees equalling 0.9659258 being raised to the power of 10 and equalling sine 45 degrees which is 0.7070299. Without these tables of logarithms there would be no theory from Nicholas Mercator of the area under a symmetrical hyperbola equalling the log of the distance along the x axis, nor of Isaac Newton's reversion of the hyperbola formula to achieve the infinite series for the antilogarithm e. This year is the 400th anniversary of Napier's discovery which is not being properly commemorated largely because modern mathematicians have no idea how Napier achieved it. submitted by Peter L. Griffiths.

Napier and Regiomontanus before him knew the formulae for constructing sine and cosine tables. Basically this is sin2u equals 2sinu.cosu. This can be converted into sin2u equals 2sinu. (1-[sinu]^2)^0.5 ,so that if the sine of a particular angle is known then the cosine can be calculated, also the sine of half that angle can be calculated by quadratic equation. The sine of 75 degrees can therefore be calculated from bisecting sin30 degrees ( which is 1/2) to give the sine of 15 degrees which is the cosine of 75 degrees, from which the sine of 75 degrees can be calculated. submitted by Peter L. Griffiths.

Very few modern mathematicians have grasped that sine 75 degrees raised to the power of 10 equals sine 45 degrees. Napier's angles just below 90 degrees are incorrect. The arc sine of 0.9999999 is much closer to 89 39/40 degrees than it is to 89 59/60 degrees. This could explain why nobody seems to have attempted to list the corresponding arc cosines. submitted by Peter L. Griffiths. We can hardly blame school maths when the experts are so negligent.

Perhaps the reason that so few mathematicians "have grasped that sine 75 degrees raised to the power of 10 equals sine 45 degrees" is that sin(75 degrees) = (1 + sqrt(3))^10/32768, while sin(45 degrees) = sqrt(2)/2, which are different numbers (although they happen to agree to 3 decimal places).

In the Constructio paragraph 44, Napier rather obscurely states a formula for calculating the logarithms of sine 89 39/40 degrees down to log sine 75 degrees. This is 10^7 X (0.9999999)^5000s equals 10^7 - 5000s + 5/4 s^2. If s is 1, then we have an equality where the shortfall is about 3200. However if s is about 69, we have the log of sine 75 degrees. From log sine 75 degrees to log sine 45 degrees, sine 75 degrees raised to the power of 10 is sine 45 degrees. This power of 10 can be split up to measure the intervening logarithms. It is no use expecting modern mathematicians to know anything about this, submitted by Peter L. Griffiths.

Napier recognised that the formula for the hyperbola 1/(1+x) = 1-x +x^2 - x^3....
could be converted into measuring the area under the hyperbola by integration, similar
to the conversion of the circumference of a circle 2PIr intothe area of the circle PIr^2.
S(1/1+x) = x - x^2/2 + x^3/3 - x^4/4.....
Multiplying both sides by 1/2 gives
1/2S(1/1+x) = x/2 - x^2/4 + x^3/6 -x^3/8.....
Napier very cleverly recognised that S(1/1+x) could be represented log(1 + x),
so that 1/2S(1/[1+x]) could be represented as 1/2 log(1 +x) or log (1 +x)^1/2.
In this way with x = 1, log2 = 1 - 1/2 + 1/3 - 1/4....= 0.693148
and log 2^1/2 = 1/2 - 1/4 + 1/6 -1/8.......= 0.346574.
Napier mentions amounts close to 0.693148 and 0.346574 in paragraphs 46-53
of the Constructio.

Hi Peter,
Could you please post some images of the above derivations so that it would be clear. I am also interested to know the resources from which you have got this.
If you have any blog of your own let me know. Thank you

Hi could you let me know the resources to study so that I can understand your comment. I have gone through the book invention of napier logarithm by hobson but couldn't get it. Can you explain me how logarithm were invented at the beginning of 16th century on trigonometric functions. A video tutorial would help a lot for the further generations to learn these inventions.

Rigth after «you will find that», I'd put parentheses around 1/e instead of y/10^7.
Thanks for this very interesting article.

Further to my previous comments, I have come across an interesting approximate relationship which could have helped Napier in constructing his log tables. This is cos30 equals (cos15)^4. This seems approximately to apply to angles below 30 degrees, and in particular should be helpful in calculating the logarithms of cosines below 30 degrees and of sines of angles above 60 degrees that is sin60 equals (sin75)^4. submitted by Peter L. Griffiths.

One clue as to how Napier constructed his log tables is obscurely contained in paragraph 44 of the Constructio. It is
10^7 X (0.9999999)^5000s equals 10^7 - 5000s + (5/4) s^2.

One crucial question question about Napier's logarithms which needs to be answered is how he arrived at the values for
log2 and 0.5log2 apparently early on in his investigations according to the Constructio. It seems that he was aware early on
that log2 equals 1 - 1/2 + 1/3 - 1/4 etc, and that half this series measured log2^0.5. This could be his first recognition
half a logarithm measure the log of the square root.

Before embarking on the detailed calculation of his logarithms, according to paragraphs 46-53 of the Constructio,
Napier seemed to be able to apply the hyperbola and its expansion y = 1/(1+x) = 1 -x + x^2 - x^3 ....also
its integration for computing the area S1/(1+x) = x - x^2/2 + x^3/3 ...Napier could have obtained this integration idea from
the relationship between Pir^2 and 2Pir. If 1 +x =2 we have the formula for log2 from the integrated hyperbola
1 - 1/2 + 1/3 -1/4 ..... = log 2= 0.69. The log symbol seems to be negative integration.

The interesting question has arisen as to whether Napier was aware of the base e for his logarithms.
The answer is that he knew of 1/e as ( 1- 0.9999999) to the power of 10 million which would be multiplied
by his other powers such as 0.693147 to arrive at 1/2. I have recently come to the conclusion that there
is far more to the historical origin of logarithms than most mathematicians suspect. Madhava of Sangamagrama
(1425) applied calculus to tangent formulae which could have inspired Napier to follow a similar path to arrive
at the basic formula for log2.

Further to my comment of 13 August 2016, the formula I mention of (1-0.9999999) to the power of 10 million, shows how logarithms can be calculated more accurately. There are the same number of 9s in 0.9999999 as there are 0s in 10 million. For even greater accuracy you add on the same number of 9s to 0.9999999 as 0s to 10,000000. This I think probably explains how Henry Briggs and others soon after 1614 were able to calculate logarithms to a much higher accuracy than Napier was able to achieve.