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What was Napier's original logic, leading to his invention of logarithms?
In other words, how did Napier, using the mathematics that was available at that time, derive them?

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The idea was to simplify multiplicaton of numbers. If you ever tried to multiply $10$-digit numbers by hand you will see what I am talking about.

The idea is this. We have $a^{m+n}=a^m a^n.$ On the right hand side we have a product of $a^m$ and $a^n$, while on the left hand side a sum $m+n$. So if you write two progressions, one arithmeric and one geometric in the parallel lines:

$1,2,3,4,...$

$a,a^2,a^3,a^4...$

it is very easy to multiply two numbers in the second row: you find the corresponding numbers in the first row, add them and look at the answer which stands below your sum.

The next idea was to choose the arithmetic progression with very small increment, so that the second row will become "dense" and you can approximate any number by some number of the second row. This is the idea.

Then one had to compute the table.

You may ask why people cared about multiplication of large numbers (with many digits). The reason is astronomy. As Kepler said, when he learned of the logarithms: this invention extended the life span of an astronomer many times.

Before Napier, there was another method called prosthaphaeresis (see Wikipedia). Instead of the simple formula $a^{m+n}=a^ma^n$ it used the more complicated formulas $$\cos m\cos n=(\cos(m+n)+\cos(m-n))/2$$ and the analogous formula for the $\sin m\sin n$. To find the product of two numbers $a=\cos m$ and $b=\cos n$ one used the tables of cosines (backwards) to find $m,n$ then compute $m+n$ and $m-n$, then use the tables of cosines again and then perform one more addition and divide by $2$. Division by 2 is easy.

This involves more additions/subtractions than the use of the logarithm tables.

But it was used because the tables of trigonometric functions existed long before Napier computed the tables of logarithms.

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    $\begingroup$ It can be interesting to note that one of the "possible inventors" of prostapheris, the german astronomer Paul Wittich, collaborator of Tycho Brahe, was teacher in Frankfurt during 1576 of the scottish physicians and astronomer John Craig which in turn was linked with John Napier. $\endgroup$ – Mauro ALLEGRANZA Dec 20 '14 at 19:53
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Multiplication is a lot of work; in numerical computing it is considered evil and many tricks are used to avoid it. So Napier created the table of logarithms. (Briggs worked with Napier to make the table more useful.) IIRC, Napier's logarithms were to base 0.9999999.

However, keep in mind that not all multiplication is evil. Particularly multiplication by 1+10^-n and 1-10^-n, which is merely a shift and add-- much less work than a full blown multiplication. Multiplication by 0.9999999 would be a shift and subtract. This is one of the tricks that Napier used. Similar trickery was used by Robert Flower in 1771, by Euler (year unknown), and in the CORDIC algorithm of Jack Volder (1959, eventually incorporated into most handheld calculators).

Robert Flower claimed that he could compute a logarithm to 20 decimal places in 7 or 8 minutes, with nothing more than a pencil and paper. * Back in the 1960's, before we had calculators, I would use an abacus. I wondered if there could be a way to compute logarithms on an abacus. I later found out that it is not possible. You need two of them. What about a recursive method that converges quadratically? You could get ten digit logs from your calculator, do one iteration, and have them to twenty digits. Yep, there is such an algorithm. *

Math teachers in elementary school dare not offer this knowledge to their students!

*** Left as an exercise for the student, of course!

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John Napier (1550-1617) published his table of logarithms Mirifici Logarithmorum Canonis Descriptio in 1614 after some twenty years of work and described his method of construction in Mirifici Logarithmorum Canonis Constructio, published posthumously in 1619 (Edinburgh) by his son Robert, with appendices by Napier and Henry Briggs (1561-1630). Briggs worked with Napier on improving the methods of calculation in the summers following the publication of the table until Napier's death and published his own method in 1624 (Vlacq's 1628 edition). The following is based on the Constructio, Macdonald's translation, and Goldstine's summary in A History of Numerical Analysis from the 16th through the 19th Century (1977), 2-13.

Properties of geometric and arithmetic progressions were well-known by Napier's time, and the connection between a sequence of powers and its corresponding sequence of exponents that we call the law of exponents has roots in ancient mathematics (cf. Euclid IX.11 and Archimedes, The Sand Reckoner). I do not know what enabled Napier to make the key connection that led to logarithms (or on the other hand, what prevented the discovery earlier). My take on Napier is that calculating was thought primarily in terms of integers or ratios of integers (or fractions), and Napier is at pains to make his calculation of logarithms in terms of integers accurate. His break-through idea is quite ingenious and surprising (to me, at least). It is to consider two points moving continuously, one (representing the logarithm) increasing "arithmetically" while the other (representing what we would now call the argument) decreasing "geometrically." Here arithmetic motion is the same as uniform motion while for geometric motion, Napier proves that the point moving geometrically toward a fixed point "has its velocities proportional to its distances from the fixed one." (Prop. 25) The notion is in modern terms that of the continuous interpolation of arithmetic and geometric sequences.

Like many first tries, this led to some inconveniences. By the time his table was published, Napier had in mind some improvements, which are included in an appendix to the Constructio. One of them is that the logarithm of 1 should be 0 and the logarithm of 10 should be some nice number, like $10^{10}$. (It is large, it appears, so that one can express logarithms accurately in terms of integers.) Briggs too saw some room for improvement, and this led him to seek out Napier. Briggs published a table in Logarithmorum Chilias Prima (1617), the first published table of base-10 logarithms, and contributed some Notes in another appendix to the Constructio.

Napier deals with velocities and distances as easily as we deal with derivatives and integrals. In fact, from a modern point of view, there's is no real difference. In our modern terms, we can represent Napier's definition of a logarithm (Prop. 26):

The "artificial number" (logarithm) of a given sine is that which has increased Arithmetically always with the same velocity as the total sine [began] to decrease Geometrically and in the same time as the total sine decreased to that given sine.

Throughout the Constructio, Napier calls the logarithm numerus artificialis and the given number either sinus or numerus naturalis. The "total sine" is a fixed length through which the geometrically moving point can move. It can be translated as "radius" and a given sine will be less than or equal to this; however, trigonometry does not figure in the construction, so the names are somewhat meaningless. So a point begins moving along the total sine $r = TS$ from $T$ toward $S$. Let $x$ be the distance to $S$, so that $r-x$ is the distance moved. Then by Prop. 25, we have $${d \over dt} (r-x) = x$$ (where we can pick the unit of time so that the constant of proportionality is $1$.) If we let $y$ denote the distance the arithmetically moving point has covered, we have $dy/dt = r$. Thus Napier's definition in modern terms is equivalent to the initial value problem $${d \over dt} (r-x) = x,\ {d \over dt} y = r,\ x(0) = r,\ y(0)=0$$ since at $x = r$ the two velocities equal. (The notation here is the same as Goldstine's.) Napier takes $r=10^7$, and it follows that Napier's logarithm is $$y = 10^r \log 10^r/x = 10^7 \log 10^7/x\,.$$ To calculate his logarithm $y$, he uses his definition to prove some inequalities $$\frac{r (r-x)}{x}<y<r-x, \quad \frac{r (x_1-x_2)}{x_1}<y_2-y_1<\frac{r (x_1-x_2)}{x_2}$$ where $x_1>x_2$ and $y_1,y_2$ are the corresponding logarithms. It is easy to prove these inequalities from properties of the integral, but Napier did in terms of velocities and distances.

Napier's first step in constructing his table is to approximate the logarithm of $x=9\,999\,999$, one less than the total sine $10^7$. From the first inequality, he has $$1 < y < 1.00000010000001$$ He splits the difference and takes $y = 1.00000005$ (which represents a error of less than $10^{-14}$). With this he can fill out the logarithms of the geometric sequence with $r$ and this $x$ as the first terms. He does this down to $9999900.0004950$. The he starts a new geometric sequence with $r$ and $x=9999900$ down to nearly $9995001$. He then constructs a more extensive table with 69 columns so that the columns and rows were each in geometric progressions, starting with $r=10^7$ in one corner and ending with $x=4998609.4034$, or roughly $r/2$, in the opposite corner. With these, he constructed his final table, the "Canon of Logarithms."

Note that if $y=l(x)$ is Napier's logarithm, then $$l(ab) = l(a)+l(b)-l(1)$$ From this he realized that if $l(1)=0$, logarithms would be easier to use. But he had already constructed his table. Some ask about the "base" of Napier's logarithm. Given that it is not a true modern logarithm, that's a bit difficult to answer. If you think that $10^7 \log x$ is the basic function underlying $l(x)$, you can think of it either as a scaled natural logarithm or a logarithm with the odd base of $\exp(10^{-7})$. Wikipedia's Napierian logarithm effectively has the base as $10^7/(10^7 - 1)$; but that would imply the logarithm of $x=9\,999\,999$ is exactly $y=1$, not $y=1.00000005$ as Napier calculated. Briggs' base-10 logarithm mentioned above is really $10^9 \log_{10} x$.

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