The Harmony of the World

by Nicholas Mee on October 3, 2017

Johannes Kepler transformed astronomy in the early years of the 17th century. His revolution was built on accurate observations of the planets compiled over many years by Tycho Brahe.

The New Astronomy

Kepler published his findings in 1609 in a book called Astronomia Nova (The New Astronomy), which includes two laws describing how planets travel around the sun. The first law states that planetary orbits are ellipses with the sun located at a focus of the ellipse. The second law states that each planet sweeps out equal areas of its orbit in equal periods of time, which means that planets move faster when close to the sun.

Kepler encapsulated the motion of the planets in these elegant laws after years spent grinding through almost endless tedious computations. He wrote in Astronomia Nova:

If this wearisome method has filled you with loathing, it should more properly fill you with compassion for me as I have gone through it at least seventy times at the expense of a great deal of time.

As Kepler was keen to point out, astronomers and mathematicians desperately needed new ways to take the labour out of calculation.

This table includes numbers that would have been seared into Kepler’s brain during the untold hours of calculation. There is a pattern to the numbers and you may already know what it is. Kepler did not spot the pattern when he published Astronomia Nova, which is hardly surprising as it is not at all obvious.

The 8th Laird of Merchiston

John Napier was a Scottish aristocrat who inherited the title 8th Laird of Merchiston. The Merchiston Tower in Edinburgh where he was born in 1550 is now part of Edinburgh Napier University. Napier lived during the great age of exploration when navigation was cutting edge applied astronomy and he is remembered for his pioneering work in easing the burden of calculation for navigators and astronomers.


Napier devised a method of converting multiplication sums into addition sums. If numbers are expressed as powers of a base number, when we multiply two numbers we add the indices, as described in a previous post The Googolplex. Napier realised this principle could be developed into a great labour-saving technique. For instance, 2 × 2 = 4 may be written as

81/3 x 81/3 = 82/3 = 4 .

In this case the sum is so simple that finding the indices and adding them offers no advantage. In general, however, a great deal of work may be saved.

Consider a slightly more difficult example: 27 × 24. How can we express this in terms of indices? We write 27 = 10a and 24 = 10b, and although, at the moment, we may not know what a is, we know for sure that there is some number that satisfies the equation 10a = 27 and similarly that some number b satisfies the equation 10b = 24. The multiplication sum then becomes:

27 × 24 = 10a × 10b = 10(a+b) .

All we have to do now is work out the value of 10(a+b) in ordinary numerals and we have achieved our goal.

An Arcane Procedure

Now we will actually do this calculation. With the assistance of a calculator, we find that if 27 = 10a and 24 = 10b, then a = 1.4314 and b = 1.3802 to four places of decimals, and a + b = 2.8116. We now calculate 102.8116 = 648.04. We round down to 648, as the answer must be a whole number. (The rounding error arose because we used indices to only four places of decimals.)

This arcane procedure is known as taking logarithms and it was the basis for performing calculations for several centuries. It only fell out of use in the 1970s with the arrival of the electronic pocket calculator.

a and b are the base 10 logarithms of the numbers 27 and 24, respectively. We add them to find a+b, which is the logarithm of the answer to the multiplication sum. To convert to ordinary numerals we must find the anti-logarithm of a+b, which is the same as calculating 10(a+b).

Turning the Tables

Simple! Well, not quite. Without a calculator each step, such as finding 102.8116, is much more difficult than calculating 27×24. The way round this conundrum was for large lists of base 10 indices to be calculated in advance and for the results to be published as tables of logarithms and anti-logarithms. When we want to know what value of a satisfies 27 = 10a, we take out our log book and find the entry in the log table corresponding to 27 and this is the logarithm of 27, in other words a. We write this as log10(x) = a, or more specifically log10(27) = 1.4314. Clearly, this procedure would not have been practical before the invention of printing.

Why have we used the number 10 for the base of the logarithms? Simply because we are familiar with counting in tens. We could have used 2, 8 or any other number.

There is one further point that makes log tables a viable proposition. The tables only need to contain logarithms of numbers between 1 and 10. When consulting the tables to find the logarithm of 27, we actually look up the logarithm of the number 2.7, which is 0.4314, and reason as follows:

10a = 27 = 10 × 2.7 = 10 × 10c = 10(1+c) ,

so a = 1 + c, where a is the logarithm of 27 and c is the logarithm of 2.7. In general, each time we multiply a number by ten we increase its logarithm by one. But this is no real surprise, afterall the logarithm is the base 10 index, so it just counts powers of 10.

Clearly log tables can ease the labour involved in multiplication. Taking logarithms converts multiplication into addition, so squaring a number is equivalent to multiplying the logarithm by 2 and cubing is equivalent to multiplying the logarithm by 3.

The advantages for division are even greater. Logarithms convert division sums into subtractions. To divide a by b, we subtract the logarithm of b from the logarithm of a, then find the anti-logarithm of the result.

The Wonderful Law of Logarithms

In 1614 Napier published his work as Mirifici Logarithmorum Canonis Descriptio (A Description of the Wonderful Law of Logarithms) containing 90 pages of tables and 57 pages explaining how they were compiled and how they could be used. One of the recipients of the book was the greatest astronomer of the age, the Imperial Mathematician of the Holy Roman Empire—Johannes Kepler. Kepler was clearly very pleased with Napier’s work as he dedicated his ephemerides for the year 1617 to Napier.

Kepler had good reason to be pleased, logarithms were almost certainly the key to one of his greatest discoveries. The table below expands the table shown earlier in this article. R is the radius, or more precisely the length of the semi-major axis, of the planet’s orbit in units where the radius of the Earth’s orbit is 10. T is the period of the planet’s orbit in months.

The Harmony of the World

When the logarithm of R is plotted against the logarithm of T the points lie on a straight line. The standard expression for a straight line is: y = mx + c, where m is the slope of the line and c is the value of the y intercept—the value of y when x equals zero.

When log(R) is plotted against log(T) the slope is 2/3 or equivalently if 3log(R) is plotted against 2log(T) the slope is 1, as shown here. This corresponds to the algebraic expression:

3 log(R) = 2 log(T) + c ,

where c is a constant. Using the addition rule of logarithms, this can be rewritten as:

log(R3) = log(T2) + log(k) ,

where we have defined a new more convenient constant log(k) = c. Using the addition rule again gives:

log(R3) = log(kT2) .

Taking anti-logarithms produces the result:

R3 = kT2.

This is Kepler’s 3rd Law, found in his Harmonices Mundi (The Harmony of the World) published in 1619. It relates the time taken for each planet to complete an orbit to its distance from the sun.

In the Name of the Law

Kepler made this remarkable discovery almost as soon as he had access to Napier’s logarithms. He would not have drawn a graph, but he was so familiar with the numbers that he didn’t need to. Kepler included his own set of log tables in Harmonices Mundi.

Kepler’s 3rd Law is the prototype for other laws in physics, which are often expressed as a power law relationship between two variables. Power laws are now found throughout physics.

Log-log plots, as they are known, are a standard tool used by physicists. When the logarithm of one variable is plotted against the logarithm of another variable, the slope of the graph gives a power law relationship between the variables and the y intercept is the logarithm of the constant of proportionality between them.

Artist’s impression of the RS Ophiuchi binary system consisting of a red giant (left) and a white dwarf undergoing a nova explosion (right). Credit: Casey Reed.

Kepler’s 3rd Law is still very useful today, it enables astronomers to measure the masses of stars in binary systems. This is a vital step in understanding the physics of stars as it calibrates the mass scale for all stars.

Historically, the importance of Kepler’s 3rd Law cannot be overemphasised as it led directly to Newton’s theory of gravity.

 

Further Information

There is much more about Tycho, Kepler and the birth of modern astronomy in my book Gravity: Cracking the Cosmic Code.

 

{ 1 comment… read it below or add one }

Carl Larsen November 14, 2017 at 8:52 am

This takes me back to my college days using logarithms and Antilogarithms, also cologarithms, which had their own applications. Of course, all of these tables were accompanied by the Sine, Cosine, and Tangent tables, which no student could do without, unless he had a scientific scaled Slide-Rule. Napier also produced a set of “Rods” called Napier’s Rods, and these could be used to calculate numbers of difficult magnitudes. I made a set of these rods for my own amusement, and they worked well if you had nothing else. What a brain Napier had, to conceive the idea of Logarithms. It takes a great deal of concentration to understand how these logarithms magically enabled simple maths for difficult sums, let alone work out a massive list of numbers in table form in order for us to do our calculations.

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