Some simple arithmetic…

This blog is dedicated to the memory of the singular John Machin, professor of astronomy at Gresham College. He calculated 𝜋 to 100 places of decimals, and, in The Principia, Newton quotes in full an article by Machin on the motion of the nodes of the moon. Cajouri, in his revision of Motte’s translation of The Principia, turned him into two professors, Professor Machin and Professor Gresham. This so enraged Bernard Cohen that he, with Anne Whitman, embarked on their translation, which inspired me to produce my own.

Various mechanical devices have been constructed to aid computation. This blog is headed by an illustration of Napier’s bones, as recommended by Newton, and later there was Babbage’s difference engine 2, designed in 1847-49, and built by the Science Museum in 1991. This engine inspired Ada Lovelace (née Byron) to become the first computer programmer, devising algorithms for a machine that was yet to be built, as modern mathematicians do for quantum computers. The historical context of The Principia consists of what came after as much as what came before.

It is very easy if you have Newton’s arithmetical skills, or some mechanical aid, to turn a fraction into a decimal. It is not obvious how to go the other way. Given a number as a decimal, correct to so many places, how do you find the original fraction? How do you find good fractional approximations to an irrational real number such as 𝜋?

Let us see what we can do with 5.26984127. The integral part of this number is 5, and the decimal part is 0.26984127. The idea is to iterate the following step. We have a positive real number. Remember and subtract the integral part, and invert the decimal part. This gives another positive real number, so we can iterate until something happens. Following this recipe, with the remembered integer part in bold, we get:
5.26984127
3.70588235
1.41666667
2.4
2.5
2
Now something has happened. The decimal part is now 0, and you can’t invert 0. So
we stop.

Any sequence a₀,a₁,…,a_n of integers, where a₀ ≥ 0, and a_i > 0 for i > 0, defines a
continued fraction. If n = 4 the continued fraction is:

$a_0+\frac{1}{\displaystyle a_1 + \frac{\strut 1}{\displaystyle a_2 + \frac{\strut 1}{\displaystyle a_3 + \frac{\strut 1}{a_4}}}}$

In our case, n = 5 and the sequence of integers is 5,3,1,2,2,2. If we truncate this sequence we get an approximation to our number. The more steps you take the better the approximation. So we get 6 approximations to our number, taking the continued fractions that rise from the sequences (5), (5,3), (5,3,1), (5,3,1,2), (5,3,1,2,2) (5,3,1,2,2,2), and the corresponding approximations are:

$5$

$5 \frac{1}{3}=5{\cdot}33333333$

$5+\displaystyle\frac{1}{\displaystyle 3+ \frac{\strut 1}{1}}=5\frac{1}{4} = 5{\cdot}25$

$5+\displaystyle\frac{1}{\displaystyle 3+ \frac{\strut 1}{\displaystyle 1 + \frac{\strut 1}{2}}}=5\frac{3}{11}=5{\cdot}272727$

$5+\displaystyle\frac{1}{\displaystyle 3+ \frac{\strut 1}{\displaystyle 1 + \frac{\strut 1}{\displaystyle 2 + \frac{\strut 1}{2}}}} = 5\frac{7}{26}=5{\cdot}26923077$

The final step gives $5\frac{17}{63}$ . And this is indeed 5.26984127 to 8 places of decimals. Of course we were lucky that the calculator came up with exactly 2.4 and 2.5 and 2. One should expect rounding errors. I suspect the calculator I used of being a little smart.

See how quickly the successive approximations approach the limiting value.

Every positive real number has a continued fraction expansion, and if the number is irrational the continued fraction expansion, like the decimal expansion, goes on for ever.

Play around with this idea. Start with 𝜋. So the first approximation is 3, the second is $3\frac{1}{7}$ . What is the third? Look up the history of 𝜋. Explore the continued fraction expansion of the golden ratio $(1+\sqrt{5})/2$ , and of $\sqrt{2}$ and $\sqrt{3}$ . Does anything strike you?

How do I know that Newton used continued fractions? In Corollary 1 to Proposition 45 of Book 1 he has the fraction $\frac{29523}{14641}$ which he approximates by $2\frac{4}{243}$ which you can work out as the third approximation by continued fractions. How else would he have found this excellent approximation? Similarly, in Corollary 3 to Proposition 37, Book 3, he approximates his estimate $\frac{4891}{4000}$ of the density of the moon to the density of the earth by $\frac{11}{9}$ which is a continued fraction estimate, though one might select this approximation by eye.

Newton knew that his density approximation was very rough, and it varied considerably between editions of The Principia. The Italian mathematician Pietro Cataldi published a notation for continued fractions in 1613. I do not know how widespread the theory of continued fractions was in Newton’s time. I expect that, if he had never come across the idea before, he would soon have discovered it when need arose.

Of course a continued fraction can be defined when the numbers a_i are not constrained to be positive integers. If they are so constrained then their construction, from the input, such as 𝜋, is uniquely determined by the above recipe. Otherwise, you can be inventive. For example, Thomas J. Picket and Ann Coleman, in the American Mathematical Monthly of December 2008, produced the following amazing continued fraction.

$\pi/2=1+\frac{1}{\displaystyle 1 + \frac{\strut 1}{\displaystyle \frac{1}{2} + \frac{\strut 1}{\displaystyle \frac{1}{3} + \frac{\strut 1}{\displaystyle \frac {1}{4} + \frac{\strut 1}{\cdots}}}}}$

So the numbers a₁,a₂,… are no longer integers, but the successive terms of the harmonic series 1, 1/2, 1/3, 1/4, 1/5, . . .

For my next two blogs we will look at the amazing way in which Newton and Halley (using Newton’s method) computed the orbits of comets.

Charles Leedham-Green
C.R.Leedham-Green@qmul.ac.uk

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My new translation of The Principia is published by Cambridge University Press

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