The natural logarithm was discovered decades before the number e!
Now this doesn't make sense to our modern way of thinking,
because today we define the natural logarithm as a logarithm with the base e.
So how come it was discovered that long before it?
Understanding this will give us a better understanding
of what the natural logarithm is.
But first what are logarithms?
See there's a difference between the way we define logarithms today
and the way they were defined when they were first invented.
Today we say that logarithms are inverse functions of exponentials.
I'll explain with an example:
We say that since 100 is equal 10 to the power of 2,
then log 100 with a base of 10 is equal to 2.
Another example is we say since 8 is equal 2 to the power of 3,
then log of 8 to the base of 2 is equal to 3.
The general rule is: if y is equal n to the power of x,
then log of y with a base of n is equal to x.
But when logarithms were first invented, the link between them and exponentials
wasn't recognized yet. In fact logarithms weren't seen as functions to start off with.
They were seen as calculation tables. Think of them as primitive calculators!
Logarithms were tables that transformed multiplication into addition,
division into subtraction
and finding the nth root into dividing by n.
And pre-calculator age, this was invaluable
because addition, subtraction and dividing by n are way faster than
multiplication, division and finding the nth root. I'll explain with an example:
Imagine you have the following list of random numbers:
If you want to add these numbers together using nothing but pen and paper
then it will probably take you no longer than two or three minutes.
However, if you want to multiply these numbers with each other,
then it will take you a considerably longer time.
Same applies to subtraction versus division.
And dividing by n versus finding the nth root.
And such kind of lengthy calculations became
more and more required towards the end of the 16th century
and the beginning of the 17th century, mainly due to astronomy and navigation,
where astronomers and navigators had to look at the stars
and do a lot of tedious calculations which were becoming a problem.
And that's when John Napier, an eccentric mathematician,
solved the problem with his invention of logarithms.
At the time logarithms were calculation tables
that pair between two types of progressions:
an arithmetic progression
and a geometric progression.
An arithmetic progression is a sequence of numbers like
0, 1, 2, 3, 4, 5, 6
What makes this an arithmetic progression
is that we get each term by adding a constant to the term before it.
In this case we add 1 but we don't have to add 1
we can choose any other number so long that it is constant.
A geometric progression is a sequence of numbers like
1, 2, 4, 8, 16, 32
What makes this a geometric progression is that we get each term in the sequence
by multiplying the term before it with a constant.
In this case we multiply by 2 but again we don't have to multiply by 2
we can multiply by any other number so long as it is constant.
Let's see how pairing these two progressions solves the calculation problem.
We create a table - In one column we put a geometric progression
for now we will choose 2 as the base of the progression
but we will have to change it later.
In the other column we put an arithmetic progression
we will choose 1 for the base of this progression
but again we will have to change it later.
Say we want to multiply two numbers from the geometric column: 8 x 32.
We look at the corresponding numbers in the arithmetic column
in this case they are 3 and 5.
We add these numbers together: so 3 + 5 is 8
Then we look back at the corresponding number in the geometric column
and we get 256. And 256 is the result of 8 x 32.
And in this way we transformed multiplication into addition.
The power of this method is that we can multiply more than two numbers together at the same time.
And same applies to division but with division we subtract instead of adding .
And to finding the nth root but with finding the nth root we divide by n.
Say we want to divide 256 by 32 by 8.
The corresponding numbers in the arithmetic column are: 8, 5 and 3.
We subtract these numbers from each other so: 8 minus 5 is 3 minus 3 is 0.
Then we look back at the corresponding number in the geometric column and we get 1
and 1 is the result of 256 divided by 32 divided by 8.
And finally with finding the nth root.
Let's say we want to find the square root of 256.
We look at the arithmetic column we get 8.
We divide it by 2 we get 4. We look back at the geometric column and we get 16.
And 16 is the square root of 256.
And this way Napier was able to solve the calculation problem.
At first Napier called the corresponding number in the arithmetic column the artificial number.
So 0 is the artificial number of 1. 1 is the artificial number of 2 and on and on.
But then he changed it to logarithm a word he derived from two greek words: Logos Airthmos
which translate to: proportional number.
But now we want to address a limitation in our current table.
See choosing 2 as the base for our geometric progression is good for demonstration purposes
but it is not very practical as it leaves gaps between our entries.
So for example between 2 and 4 we don't have the logarithm of
3 therefore we can't use this table to do any calculation with the number 3.
And the gaps keep getting bigger and bigger so between 128 and 256 we have a gap of 127 numbers.
And as our table grows, the gaps grow exponentially.
So to make this table of any practical use, we want to fill in the gaps
and the way to do it is to choose a base very close to 1.
What base shall we choose?
Napier thought about this a lot and eventually he settled on a number slightly less than 1
0.9999999
and he started his geometric progression with 10 million instead of 1.
He applied the base and he kept going on
and on
and on.
Now Napier didn't keep going on for the whole 10 million numbers
that would have been an impossible task.
He found a method, a shortcut if you will,
and he used it to find the logarithms of the sine tables that were available at his time.
But even with the shortcut it took him 20 years to compile a list of 90 pages of logarithms
and in 1614 he published them in his book:
Mirfici Logarithmorum Canonis Descriptio
Which translates to: Description of the Wonderful Canon of Logarithms.
The book was received enthusiastically by the scientific community
and many different translations and different versions of logarithms started popping up.
And what interests us is that in 1618, in one of the translations of Napier's book, in the appendix
we see the following intriguing table probably written by William Oughtred,
one of Napier's students. Now what's intriguing about this table is that these logarithms here,
apart from the decimal point, closely resemble the natural logarithms of these numbers.
Let's take a closer look. See the natural logarithm of 2 is 0.693147
The natural logarithm of 3 is 1.098 (it's believed that this is an error on Oughtred's behalf) 612.
Because all the other logarithms closely resemble the natural logarithms of these numbers.
And it doesn't stop here. In 1622 John Speidell published his own version of logarithms
and again these logarithms here, apart from the decimal point,
closely resemble the natural logarithms of these numbers.
So how come two mathematicians, independently, arrived at what looks like natural logarithms
without knowing any link between logarithms and exponentials let alone with the number e?
Now they arrived at this result differently
(link in the description). However I will be talking about the essence of what they did.
See Napier chose a number slightly less than 1 as the base for his geometric progression.
If we choose a number slightly bigger than 1, something like 1.0001
and we apply it to our geometric progression these will be our first 8 entries.
This will be the 9th, the 10th. And as we keep going all the way till we get 2 in our geometric
progression, then the corresponding number in the arithmetic progression will be 693148.
A number very close to both Oughtred's
and Speidell's and we can attribute the difference to rounding errors.
And when we continue all the way till we get 3 in the geometric progression
then the corresponding number in the arithmetic progression will be 1098613.
And this gives us the first what natural logarithms are.
One more step to get the natural logarithm.
See all this time we've been using 1 as the base for our arithmetic progression.
If however we choose a very small base something like 0.000001
then we arrive at natural logarithms!
So when we pair two types of progressions: a geometric progression with a very small base
and an arithmetic progression with a very small base
then we get natural logarithms!
But mathematicians at the time weren't aware of this fact
and even if they were it wouldn't have meant much to them,
because for them
logarithms were calculation tables and they were valued for their practicality and ease of use.
And for all practical purposes logarithm's job was complete: making calculations easier.
However a little over two decades later, in 1647,
logarithms showed up in a completely unrelated field: in the area under the hyperbola.
Now the hyperbola is one of the conic sections that were studied in antiquity
and we will be focusing on a specific kind of hyperbola what is called "the square hyperbola"
which is the graph we get from the equation y equals 1 over x.
Now we will be focusing on the positive side of the graph only.
Now what we mean by the area under the hyperbola: it's the area between the graph
and the x-axis from point x equal to 1 to a random point x equal to a.
Now we don't have to start with x equal to 1. We can start at any
other point so long as it is not 0 to avoid division by 0.
It's customary to start with x equal 1 because it makes calculations easier.
Finding the area of the simple looking shape proved to be very difficult
and mathematicians since antiquity tried unsuccessfully.
Most famously: Archimedes, Renee Descartes and Pierre de Fermat.
Fermat's work paved the way for one of his contemporaries:
Gregoire de Saint Vincent, to finally arrive at an equation for the area under the hyperbola.
Spoiler alert it's the natural logarithm.
Saint Vincent borrowed an idea from Fermat.
He divided the area under the hyperbola into rectangles and he drew them in a specific way:
Their tops touch the graph. And their bases form a geometric progression where rectangle 1 starts at
point a, rectangle 2 starts at point ar, rectangle 3 starts at point ar squared and on and on.
Where r is a number less than 1.
Now the idea is to find the area of each rectangle,
by multiplying its height times its width, and then summing up the areas of these rectangles.
Clearly we have errors which are these bits here as our rectangles don't cover them.
However these errors are a factor of r:
If r is a lot less than 1 then these errors become bigger.
If r is very close to 1 then these errors become smaller.
And Fermat reasoned that if r is infinitely close to 1 then these errors disappear
and we have a perfect fit between our rectangles and the area under the hyperbola.
Let's try to find the area of these rectangles and see if we can find any pattern.
Now the height of rectangle 1 is this point here which by definition it's 1/x,
and x in this case is a. So the height of rectangle 1 is 1/a.
The width of rectangle 1 is (a - ar).
We factor a out and we get a times (1 - r).
The area is simply the height times the width.
a cancels out and we are left with (1 - r).
The height of rectangle 2 is this point here which again is 1/x
but x in this case is ar. So the height of rectangle 2 is 1/ar.
The width of rectangle 2 is ar - ar squared.
We factor ar out and we get ar times (1 - r).
The area is again the height times the width.
ar cancels out and we're left with (1 - r) again.
And when we continue with the areas of rectangles 3, 4 and 5.
We notice that all of them have the same area: (1 - r)!
And this fact gave Saint Vincent an insight:
He realized that this leads to an interesting relation between the distance
and the area under the hyperbola.
Let's see what kind of relation this is.
We start with point ar to the power of 5 and the area there is still 0 because we haven't moved.
When we move to point ar to the power of 4, the area is the area of rectangle 5
which we already know that it is equal (1 - r).
When we move to point ar cube, the area is the sum of rectangles 5
and 4 and since each one of them is (1 - r) so the area is 2 times (1- r).
And when we continue the same way we get the following table.
Now when we take a closer look we notice that we get each entry in the distance column
by multiplying the entry before it with a constant (1/r)
And we get each entry in the area column by adding a constant (1 - r) to the entry before it.
This means that the distance is progressing geometrically,
while the area is progressing arithmetically.
And Saint Vincent concluded that the relation between the distance and the area is logarithmic.
And Alfonso de Sarasa, one of Saint Vincent's students, wrote this relation explicitly as:
The area of a is equal to log of a.
And this marks one of the first times, if not the first time, a logarithm was written as a function.
Whereas before logarithms were regarded merely as calculation tables.
And this logarithm was suitably called: the hyperbolic logarithm.
But Saint Vincent and de Sarasa didn't provide a way to calculate it.
Two decades later Nikolaus Mercator and Isaac Newton,
independently, provided an infinite series to calculate the hyperbolic logarithm.
And Mercator called it the natural logarithm
Now it's not clear why Mercator called it the natural logarithm!
Regardless of the reason the term natural logarithm stuck.
And by the beginning of the 18th century the connection between logarithms and exponentials
was finally recognized and the concept of a base of a logarithm was understood.
And in 1748, 101 years after the work of Saint Vincent and de Sarasa,
Leonhard Euler calculated the base of the natural logarithm and found it to
be what we call today the number e or Euler's number or in some countries Napier's number.
And this is the story of the natural logarithm.