“The logarithm 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.”

-John Napier, 1614.

John Napier, of Scotland, published the first account of his concept of a “logarithm” in 1614, after working on the development of his computations for roughly 20 years. Today we think of logarithms as exponents which are applied to a base number, raising the base to a “power”, as is described in our Review of the Logarithm. The standard for the common logarithm is to use a base of 10, where \(\log 1\) = 0, \(\log 10\) = 1, \(\log 100\) = 2, and so on. This is not how Napier started things out, however. In his time the notion of raising a number to an arbitrary power (like taking \(4.9^{3.8}\), for example) did not yet exist, nor did the mathematical construct of the calculus. In this regard he was ahead of his time and so thought about things a little differently. He knew about arithmetic series and geometric series of numbers, and realized that there had to be a relationship between the two. The key to that relationship would turn out to be the logarithm, which he was able to compute through an iterative process of comparing the ratios of certain series of numbers.

Detailed tables of trigonometric sines and tangents had been developed in the 1500s, though the definitions being used were slightly different than those used today. It was common for fractions to be defined as ratios, but the use of a “decimal point” to define fractional portions of a number in units of tenths and hundredths and so on had only been introduced by John Napier himself, during his development of the logarithm concept. The sine, for instance, had basically the same geometrical definition as today, but was not referenced to unity. Prior to Napier, published tables were created that used a variety of “amplitudes”, or “radii” as their maximum values. For example, a maximum radius of 100,000 might be used for an angle of 90 degrees. All numbers in the table were “whole” and were gauged to a maximum whole number in order to gain “accuracy”, in this case in parts per 100 thousand. Today we would say that the sine of an angle of 30 degrees has a value of 0.5, but in Napier’s day, they might say that the sine of 30 degrees is 50,000 units for a radius of 100,000 units. The sine then was not thought of as a function, but rather a table of whole numbers from which ratios could be formed.

In the same sense, Napier used a “radius”, or maximum value, of 10,000,000 as the starting point of his logarithms, such that the logarithm of 10,000,000 was defined to be “zero” when the sine was at its maximum. From the very beginning he wanted to develop logarithms of the natural numbers and of the sines and tangents of angles, such that these logarithms could be added together to compute a multiplication; in astronomy and in navigation, one often multiplied large distances by the sines and tangents of angles when determining orbits and plotting courses.

To help himself and others to visualize what was going on, in his attempts to develop his logarithms Napier envisaged a physical system of objects moving along straight trajectories, and made correspondences between them. The concept of a geometric series of numbers was in vogue during this time, and he recognized that the geometric progression of sequences such as “1, 2, 4, 8, 16, 32, 64, …” showed a particular correspondence to an arithmetic series such as “0, 1, 2, 3, 4, 5, 6, …”. He recognized the fact that multiplying two numbers in the first series had a correspondence to the addition of the associated numbers in the second series. Below we describe Napier’s model for finding what he called logarithms, but will relate its aspects using more modern mathematical descriptions than were available to him in the late 1500s. We’ll see that he was, indeed, describing exponential behavior.

A.2.1 Napier’s Model

Lines with arithmetic (a) and geometric (g) motion.

Suppose an object moves along a line a at a constant rate \({\cal R}\) of distance per unit of time. From the starting point, the distance after time \(t\) is \(x_a ={\cal R}\cdot t\). If we examine its location at equal time intervals separated by time \(T\), where the integer \(n = t/T\), then the development of \(n\) is an arithmetic progression. That is, after the \(n\)-th time step,

\[
(x_a)_n = {\cal R} T\cdot \;n.
\]

For integer values of \(n\) starting at 0, the vector of values of \(x_a\) would be

\[
x_a = {\cal R} T \cdot \{ 0, 1, 2, 3, \ldots , n, \ldots \}
\]

where the set \(\{0,1,2,3,\ldots\}\) is an arithmetic series. Note that

Next, suppose a second, corresponding object moves along a line g. This particular line has a total length of \(X_0\). But along this line, as \(n\) evolves, we let the distances from the object to the end point form a geometrical sequence. In other words, if \(r\) is a constant and \(r<1\), and if \(x_g\) is the distance from the beginning of the line to the object, then after each time interval the remaining distance to the end, \(y_g = X_0 - x_g\), gets smaller and smaller by the constant ratio \(r\) according to

\[
\frac{(y_g)_{n+1}}{(y_g)_n} \; = \; r
\]

as the particle gets closer and closer to \(X_0\).

If we start out with an initial value of \(y_g\), say \(y_0\), then after \(n\) time intervals we see that

\[
(y_g)_n = r\cdot r \cdot \ldots \cdot r \cdot y_0
\]

with \(n\) factors of \(r\) in our expression above. Again, written as a vector, the series of values which \(y_g\) takes on will be

since at the beginning when \(n\)=0 we should have \(y_g = X_0\). In the above we introduce the notation \(r^n\) meaning that \(n\) instances of \(r\) are to be multiplied together. For increasing \(n\) with \(r<1\), \(y_g\) will get smaller and smaller and hence the “particle”’s position \(x_g\) will get closer and closer to but never quite reaching \(X_0\).

So Napier envisioned two associated series of numbers, one based upon an arithmetic series and one based upon a geometric series. His task, then, was to find an appropriate value for the ratio \(r\) along with appropriate values of \({\cal R}\), \(T\), and \(X_0\) such that adding values \((x_a)_i + (x_a)_j\), which he called logarithms, would give the logarithm of the product of \((y_g)_i \cdot (y_g)_j\). From

we can see that \((y_g)_i \cdot (y_g)_j\) = \(X_0^2\cdot r^{i+j}\) and hence we see the relationship between his logarithms and the exponents of \(r\).

But again, we remind the reader that Napier’s efforts all took place before the concepts and notation of “exponents” and “raising numbers to powers” – particularly numbers and powers that are not whole – had been established. His intense and complicated efforts involved the computation of sets of ratios and their evolution, using the above physical description as his conceptual model. For now, though, let’s continue to discuss this topic in more modern terms. First, let’s drop the subscripts for simplicity and use \(x \equiv x_a\) and \(y \equiv y_g\). We see that along the line g,

\[
y_{n+1}= r\; y_{n} = r^n y_{0}, \\
\]

or, for continuous time where \(nT \rightarrow t\), we have

\[
y(t)= r^{t/T}\; y_0
\]

where again, \(T\) is the time interval between any \(n\) and \(n+1\). Since at \(t=0\) we start with \(x=0\) and \(y=X_0\), then we see that \(x\) is given by

\[
x(t) = X_0-y(t) = X_0\cdot(1-r^{t/T}).
\]

As time progresses, the smaller and smaller numbers are subtracted from 1, until at large times the value of \(x\) tends more and more quickly toward \(X_0\). The rate of change of the particle’s distance \(x(t)\) is accelerating. Napier realized this fact, and he wanted this property for his logarithms. In modern terms, using calculus, we can see that

In other words, \[
y(t) = y_0 ~e^{\ln r\cdot (t/T)}~.
\]

Notice that for \(r<1\), then \(\ln r <0\), and so \(dy/dt<0\) and hence \(dx/dt>0\). Napier’s “particle” moving along g is accelerating exponentially.

A.2.2 Napier’s Definition of a Logarithm

The above physical description was used by Napier (though not in modern mathematical language) to introduce the concept of logarithms. The physical description of particles moving along lines, but which were “connected” using arithmetical and geometrical sequences, led Napier to develop a complicated process to produce a table of numerical values for his logarithms.

In the late 1500s tables of values for the “sine” of an angle tended to have a maximum value of 1,000,000. In those days, as the use of fractions was purposefully avoided, the sine of 90 degrees was taken to be a large number such as this, and then the sines of smaller angles were compared to it. For example, the sine of 30 degrees would have been quoted as 500,000 in such a table. So, in his work Napier defined \(X_0\) = 10,000,000 as his starting point, or “zero” point, of his logarithms to provide even more accuracy. For an arbitrary whole number \(z\) let us call its logarithm \(``\lambda(z)"\). Then, according to Napier,

Napier also used a value of \({\cal R} = 10^7\) as the uniform “speed” of his particle moving along the line a. In essence, he was setting his unit of time as \(T=1\).

Now if at some time \(t\) we find that the particle on a is at \(x\) and the particle moving along line g is at \(x_g\), then Napier set his logarithm for the number \(y\) as

For instance, at the beginning where \(x_g=0\), then \(\lambda(10^7) = 0\) as required. He then developed a very large series of ratios of small differences of the values of \(x_g\) in order to iteratively compute values of his logarithms. The whole process took him roughly 20 years to complete, resulting in the tabulation of logarithms for the first 1000 natural whole numbers.

From Napier’s model, he essentially makes an association between distance \(x\) (from zero on a) with the distance \(y\) (distance to the end of the line g). In modern terms: to \(x =10^7 n \equiv \lambda\), Napier associates \(y=10^7e^{n\cdot \ln r}\).

But notice that \(\ln y = \ln 10^7 + n\cdot \ln r\), and hence his number \(n\) is essentially \(n = (\ln y - \ln 10^7)/\ln r\), while his logarithm is defined as \(\lambda = 10^7 n\). Substituting for \(n\), Napier’s original logarithm is related to the more modern “natural logarithm” according to

Again, we see that when \(z=10^7\) then \(\lambda\) = 0. Since both \(z/10^7\) and \(r\) are less than one, then \(\lambda\) will be positive.

In Napier’s two decades of effort he arrived at an optimal value to use for his ratio, which turned out to be \(r \approx\) 0.3678. One might note that this is very close to the value of \(1/e\) = \(1\ldots\). And, in fact, if we plug \(r=1/e\) into our last equation and note that \(\ln e^{-1}\) = -1, then

Remember that to change logarithms from one base to another, \(\log_b x = \log_a x / \log_a b\). So, \(\log_{1/e} z = \ln z/\ln(1/e) = -\ln z\)

In other words, Napier’s original logarithms were essentially logarithms to base \(1/e\), but with an offset and a scaling that were due to the choice of making \(10^7\) a fixed upper limit. If instead of \(10^7\) the zero of the logarithms would have been chosen to be at 1, and since \(\ln 1 = 0\), then we would find that Napier’s original logarithm and the natural logarithm are related by

\[
\lambda (z) = -\ln z = \log_{1/e} z ~~~~~~~~~~ ({\rm for} ~~ r=1/e, ~~{\rm and}~~ X_0 = 1).
\]

The constant \(e\) is sometimes called “Napier’s constant”, though it is most commonly referred to as Euler’s number.^{164}

Once Napier’s definitions and procedures were established, the next step was to actually compute the values of his logarithms. But, as pointed out by Roegel, “although the logarithms are precisely defined by Napier’s procedure, their computation is not obvious. This, per se, is already particularly interesting, as Napier managed to define a numerical concept, and to separate this concept from its actual computation.” Napier’s computations of logarithms were actually quite involved and laborious. He developed intermediate tables of numbers, which allowed him to compute upper and lower bounds for each logarithm, and through a highly iterative process he was able to arrive at fairly accurate results. Taking 20 years to complete, it is best left to Roegel’s fine description in his report to explain Napier’s process to compute the first complete table of logarithms, as well as to Napier’s Canon of Logarithms, both of which are listed below.

After finishing his first complete table in 1614, Napier died three years later in 1617 at the age of 67. The detailed descriptions of logarithms and the methods for computing them, as written in the publications the Desriptio and Constructio, were complete by this time, but were reprinted in 1619 and 1620 with publications being finished by Henry Briggs. It was Briggs who first brought to Napier’s attention during the previous decade or so that a change to a base 10 system would be beneficial, which was quickly recognized by Napier. While this possible change of base was mentioned in the early writings of Napier and Briggs, it was Briggs and Edward Wright who finished the base 10 efforts in time for the final 1619 publications. These publications contained the logarithms in base 10 of the first 1000 integers to 14 places.

Napier started out in a quest to find a set of numbers that could be used to reduce multiplication and division to a mere addition or subtraction operation. He was not, at first, concerned about the most appropriate choice of a “base” number for his calculations. He no doubt wanted only to advance the mathematical procedure and sought any “base” system that would do the job. However, Briggs and Napier had both seen early on that a base 10 system was much more useful in general calculations. And, eventually, the use of Napier’s “decimal point” to monitor the fractional parts of numbers made the new base 10 system much more attractive. And hence, by 1620 or so, tables of logarithms using base 10 were constructed and published, wherein the logarithm of zero was minus infinity, the logarithm of 1 was zero, the logarithm of 10 was 1, and so on. In these early books only the logarithms of whole numbers were compiled, although the logarithms themselves were presented in decimal form.

Bibliography

Denis Roegel, “Napier’s ideal construction of the logarithms”, [Research Report] 2010. inria-00543934.

John Napier, The Construction of the Wonderful Canon of Logarithms, translated From Latin Into English with Notes and a Catalogue of the Various Editions Of Napier’s Works, by William Rae Macdonald, William Blackwood And Sons, Edinburgh And London (1889).

## A.2 Comments on Napier’s Logarithms

John Napier, of Scotland, published the first account of his concept of a “logarithm” in 1614, after working on the development of his computations for roughly 20 years. Today we think of logarithms as

exponentswhich are applied to a base number, raising the base to a “power”, as is described in our Review of the Logarithm. The standard for the common logarithm is to use a base of 10, where \(\log 1\) = 0, \(\log 10\) = 1, \(\log 100\) = 2, and so on. This is not how Napier started things out, however. In his time the notion of raising a number to an arbitrary power (like taking \(4.9^{3.8}\), for example) did not yet exist, nor did the mathematical construct of thecalculus. In this regard he was ahead of his time and so thought about things a little differently. He knew about arithmetic series and geometric series of numbers, and realized that there had to be a relationship between the two. The key to that relationship would turn out to be thelogarithm, which he was able to compute through an iterative process of comparing the ratios of certain series of numbers.Detailed tables of trigonometric sines and tangents had been developed in the 1500s, though the definitions being used were slightly different than those used today. It was common for fractions to be defined as ratios, but the use of a “decimal point” to define fractional portions of a number in units of tenths and hundredths and so on had only been introduced by John Napier himself, during his development of the logarithm concept. The sine, for instance, had basically the same geometrical definition as today, but was not referenced to unity. Prior to Napier, published tables were created that used a variety of “amplitudes”, or “radii” as their maximum values. For example, a maximum

radiusof 100,000 might be used for an angle of 90 degrees. All numbers in the table were “whole” and were gauged to a maximum whole number in order to gain “accuracy”, in this case in parts per 100 thousand. Today we would say that the sine of an angle of 30 degrees has a value of 0.5, but in Napier’s day, they might say that the sine of 30 degrees is 50,000 units for a radius of 100,000 units. The sine then was not thought of as a function, but rather a table of whole numbers from whichratioscould be formed.In the same sense, Napier used a “radius”, or maximum value, of 10,000,000 as the starting point of his logarithms, such that the logarithm of 10,000,000 was defined to be “zero” when the sine was at its maximum. From the very beginning he wanted to develop logarithms of the natural numbers and of the sines and tangents of angles, such that these logarithms could be added together to compute a multiplication; in astronomy and in navigation, one often multiplied large distances by the sines and tangents of angles when determining orbits and plotting courses.

To help himself and others to visualize what was going on, in his attempts to develop his logarithms Napier envisaged a physical system of objects moving along straight trajectories, and made correspondences between them. The concept of a geometric series of numbers was in vogue during this time, and he recognized that the geometric progression of sequences such as “1, 2, 4, 8, 16, 32, 64, …” showed a particular correspondence to an arithmetic series such as “0, 1, 2, 3, 4, 5, 6, …”. He recognized the fact that multiplying two numbers in the first series had a correspondence to the addition of the associated numbers in the second series. Below we describe Napier’s model for finding what he called

logarithms, but will relate its aspects using more modern mathematical descriptions than were available to him in the late 1500s. We’ll see that he was, indeed, describing exponential behavior.## A.2.1 Napier’s Model

Lines with arithmetic (a) and geometric (g) motion.Suppose an object moves along a line

at a constant rate \({\cal R}\) of distance per unit of time. From the starting point, the distance after time \(t\) is \(x_a ={\cal R}\cdot t\). If we examine its location at equal time intervals separated by time \(T\), where the integer \(n = t/T\), then the development of \(n\) is anaarithmeticprogression. That is, after the \(n\)-th time step,\[ (x_a)_n = {\cal R} T\cdot \;n. \]

For integer values of \(n\) starting at 0, the vector of values of \(x_a\) would be

\[ x_a = {\cal R} T \cdot \{ 0, 1, 2, 3, \ldots , n, \ldots \} \]

where the set \(\{0,1,2,3,\ldots\}\) is an arithmetic series. Note that

\[ \frac{ (x_a)_{n+1} }{ (x_a)_n} = \frac{n+1}{n}, ~~~~~{\rm for} ~~ n>0 \]

for which the ratio approaches 1 for large \(n\).

Next, suppose a second, corresponding object moves along a line

. This particular line has a total length of \(X_0\). But along this line, as \(n\) evolves, we let the distances from the object to the end point form aggeometricalsequence. In other words, if \(r\) is a constant and \(r<1\), and if \(x_g\) is the distance from the beginning of the line to the object, then after each time interval the remaining distance to the end, \(y_g = X_0 - x_g\), gets smaller and smaller by the constant ratio \(r\) according to\[ \frac{(y_g)_{n+1}}{(y_g)_n} \; = \; r \]

as the particle gets closer and closer to \(X_0\).

If we start out with an initial value of \(y_g\), say \(y_0\), then after \(n\) time intervals we see that

\[ (y_g)_n = r\cdot r \cdot \ldots \cdot r \cdot y_0 \]

with \(n\) factors of \(r\) in our expression above. Again, written as a vector, the series of values which \(y_g\) takes on will be

\[ y_g = X_0 \cdot \{ 1, r, r^2, r^3, \ldots , r^n, \ldots \} \]

since at the beginning when \(n\)=0 we should have \(y_g = X_0\). In the above we introduce the notation \(r^n\) meaning that \(n\) instances of \(r\) are to be multiplied together. For increasing \(n\) with \(r<1\), \(y_g\) will get smaller and smaller and hence the “particle”’s position \(x_g\) will get closer and closer to but never quite reaching \(X_0\).

So Napier envisioned two associated series of numbers, one based upon an arithmetic series and one based upon a geometric series. His task, then, was to find an appropriate value for the ratio \(r\) along with appropriate values of \({\cal R}\), \(T\), and \(X_0\) such that adding values \((x_a)_i + (x_a)_j\), which he called

logarithms, would give the logarithm of the product of \((y_g)_i \cdot (y_g)_j\). From\[ x_a = {\cal R} \cdot \{ 0, 1, 2, 3, \ldots , n, \ldots \} ~~~~\\ y_g = X_0 \cdot \{ 1, r, r^2, r^3, \ldots , r^n, \ldots \} \]

we can see that \((y_g)_i \cdot (y_g)_j\) = \(X_0^2\cdot r^{i+j}\) and hence we see the relationship between his

logarithmsand theexponentsof \(r\).But again, we remind the reader that Napier’s efforts all took place before the concepts and notation of “exponents” and “raising numbers to powers” – particularly numbers and powers that are not whole – had been established. His intense and complicated efforts involved the computation of sets of

ratiosand their evolution, using the above physical description as his conceptual model. For now, though, let’s continue to discuss this topic in more modern terms. First, let’s drop the subscripts for simplicity and use \(x \equiv x_a\) and \(y \equiv y_g\). We see that along the line,g\[ y_{n+1}= r\; y_{n} = r^n y_{0}, \\ \]

or, for continuous time where \(nT \rightarrow t\), we have

\[ y(t)= r^{t/T}\; y_0 \]

where again, \(T\) is the time interval between any \(n\) and \(n+1\). Since at \(t=0\) we start with \(x=0\) and \(y=X_0\), then we see that \(x\) is given by

\[ x(t) = X_0-y(t) = X_0\cdot(1-r^{t/T}). \]

As time progresses, the smaller and smaller numbers are subtracted from 1, until at large times the value of \(x\) tends more and more quickly toward \(X_0\). The rate of change of the particle’s distance \(x(t)\) is accelerating. Napier realized this fact, and he wanted this property for his logarithms. In modern terms, using

calculus, we can see that\[ \frac{d\;y}{dt} = -\frac{d\;x}{dt} = X_0\;\frac{d}{dt}~r^{t/T} = X_0\;\frac{d}{dt}~e^{(t/T)\cdot\ln r} = X_0\;(\ln r/T)e^{(t/T)\cdot\ln r} = \frac{\ln r}{T} \;\;X_0 \; r^{t/T} \\ ~\\ \frac{d\;y}{dt} = \left(\frac{\ln r}{T}\right)\cdot y. \]

In other words,

\[ y(t) = y_0 ~e^{\ln r\cdot (t/T)}~. \]

Notice that for \(r<1\), then \(\ln r <0\), and so \(dy/dt<0\) and hence \(dx/dt>0\). Napier’s “particle” moving along

is accelerating exponentially.g## A.2.2 Napier’s Definition of a Logarithm

The above physical description was used by Napier (though not in modern mathematical language) to introduce the concept of logarithms. The physical description of particles moving along lines, but which were “connected” using arithmetical and geometrical sequences, led Napier to develop a complicated process to produce a table of numerical values for his logarithms.

In the late 1500s tables of values for the “sine” of an angle tended to have a maximum value of 1,000,000. In those days, as the use of fractions was purposefully avoided, the sine of 90 degrees was taken to be a large number such as this, and then the sines of smaller angles were compared to it. For example, the sine of 30 degrees would have been quoted as 500,000 in such a table. So, in his work Napier defined \(X_0\) = 10,000,000 as his starting point, or “zero” point, of his logarithms to provide even more accuracy. For an arbitrary whole number \(z\) let us call its logarithm \(``\lambda(z)"\). Then, according to Napier,

\[ \lambda(X_0) = \lambda(10,000,000) \equiv 0. \]

Napier also used a value of \({\cal R} = 10^7\) as the uniform “speed” of his particle moving along the line

. In essence, he was setting his unit of time as \(T=1\).aNow if at some time \(t\) we find that the particle on

is at \(x\) and the particle moving along lineais at \(x_g\), then Napier set his logarithm for the number \(y\) asg\[ \lambda(y) \equiv \lambda(10^7 - x_g) \equiv x. \]

For instance, at the beginning where \(x_g=0\), then \(\lambda(10^7) = 0\) as required. He then developed a very large series of ratios of small differences of the values of \(x_g\) in order to iteratively compute values of his logarithms. The whole process took him roughly 20 years to complete, resulting in the tabulation of logarithms for the first 1000 natural whole numbers.

From Napier’s model, he essentially makes an association between distance \(x\) (from zero on

) with the distance \(y\) (distance to the end of the linea). In modern terms: to \(x =10^7 n \equiv \lambda\), Napier associates \(y=10^7e^{n\cdot \ln r}\).gBut notice that \(\ln y = \ln 10^7 + n\cdot \ln r\), and hence his number \(n\) is essentially \(n = (\ln y - \ln 10^7)/\ln r\), while his logarithm is defined as \(\lambda = 10^7 n\). Substituting for \(n\), Napier’s

original logarithmis related to the more modern “natural logarithm” according to\[ \lambda (z) = 10^7\cdot \frac{\ln z -\ln 10^7}{\ln r} = \frac{10^7}{\ln r}\cdot \ln\left( \frac{z}{10^7} \right). \]

Again, we see that when \(z=10^7\) then \(\lambda\) = 0. Since both \(z/10^7\) and \(r\) are less than one, then \(\lambda\) will be positive.

In Napier’s two decades of effort he arrived at an optimal value to use for his ratio, which turned out to be \(r \approx\) 0.3678. One might note that this is very close to the value of \(1/e\) = \(1\ldots\). And, in fact, if we plug \(r=1/e\) into our last equation and note that \(\ln e^{-1}\) = -1, then

\[ \ln 10^7 - \frac{\lambda(z)}{10^7} = \ln(z). \]

Remember that to change logarithms from one base to another, \(\log_b x = \log_a x / \log_a b\). So, \(\log_{1/e} z = \ln z/\ln(1/e) = -\ln z\)

In other words, Napier’s original logarithms were essentially logarithms to base \(1/e\), but with an offset and a scaling that were due to the choice of making \(10^7\) a fixed upper limit. If instead of \(10^7\) the zero of the logarithms would have been chosen to be at 1, and since \(\ln 1 = 0\), then we would find that Napier’s original logarithm and the

natural logarithmare related by\[ \lambda (z) = -\ln z = \log_{1/e} z ~~~~~~~~~~ ({\rm for} ~~ r=1/e, ~~{\rm and}~~ X_0 = 1). \]

The constant \(e\) is sometimes called “Napier’s constant”, though it is most commonly referred to as Euler’s number.

^{164}Once Napier’s definitions and procedures were established, the next step was to actually compute the values of his logarithms. But, as pointed out by Roegel, “although the logarithms are precisely defined by Napier’s procedure, their computation is not obvious. This,

per se, is already particularly interesting, as Napier managed to define a numerical concept, and to separate this concept from its actual computation.” Napier’s computations of logarithms were actually quite involved and laborious. He developed intermediate tables of numbers, which allowed him to compute upper and lower bounds for each logarithm, and through a highly iterative process he was able to arrive at fairly accurate results. Taking 20 years to complete, it is best left to Roegel’s fine description in his report to explain Napier’s process to compute the first complete table of logarithms, as well as to Napier’sCanon of Logarithms, both of which are listed below.After finishing his first complete table in 1614, Napier died three years later in 1617 at the age of 67. The detailed descriptions of logarithms and the methods for computing them, as written in the publications the

DesriptioandConstructio, were complete by this time, but were reprinted in 1619 and 1620 with publications being finished by Henry Briggs. It was Briggs who first brought to Napier’s attention during the previous decade or so that a change to a base 10 system would be beneficial, which was quickly recognized by Napier. While this possible change of base was mentioned in the early writings of Napier and Briggs, it was Briggs and Edward Wright who finished the base 10 efforts in time for the final 1619 publications. These publications contained the logarithms in base 10 of the first 1000 integers to 14 places.Napier started out in a quest to find a set of numbers that could be used to reduce multiplication and division to a mere addition or subtraction operation. He was not, at first, concerned about the most appropriate choice of a “base” number for his calculations. He no doubt wanted only to advance the mathematical procedure and sought any “base” system that would do the job. However, Briggs and Napier had both seen early on that a base 10 system was much more useful in general calculations. And, eventually, the use of Napier’s “decimal point” to monitor the fractional parts of numbers made the new base 10 system much more attractive. And hence, by 1620 or so, tables of logarithms using base 10 were constructed and published, wherein the logarithm of zero was minus infinity, the logarithm of 1 was zero, the logarithm of 10 was 1, and so on. In these early books only the logarithms of whole numbers were compiled, although the logarithms themselves were presented in decimal form.

BibliographyDenis Roegel, “Napier’s ideal construction of the logarithms”, [Research Report] 2010. inria-00543934.

John Napier,

The Construction of the Wonderful Canon of Logarithms, translated From Latin Into English with Notes and a Catalogue of the Various Editions Of Napier’s Works, by William Rae Macdonald, William Blackwood And Sons, Edinburgh And London (1889).