Lists!

People appear to love list. The Internet is full of lists. The 10 most popular dog breeds, the 10 biggest waves ever ridden by a surfer, the 10… you get the idea. The lists very often have ten entries, it’s a shame that we all have the same number of fingers otherwise we could a bit more variation, the 7 biggest… or the 11 smartest… Science and its history are far from immune from this cyber cancer, lists of all sorts being produced and posted with gay abandon. We recently even had the Top 10 scientists of the 13^th century! Apart from the fact that the use of the word scientist here is highly anachronistic any such selection is of course subjective and disputable. However the subject of this post is not medieval scholars, tempting though it is, but a list of “17 Equations That Changed the World”

17 Equations that Changed the World

Although it claims to be by Ian Stewart I have no idea of the original source of this list but I have stumbled across it several times in the last few months. Now when I was a young mathematical acolyte and budding historian of maths I devoured Ian Stewart’s books at the same rate as those of Martin Gardner and Isaac Asimov. Put another way Ian Stewart was a major influence on my development. As I got older, but probably not wiser, I came to realise that Stewart, a mathematician and populariser, wasn’t very accurate in his historical attributions, in fact he is down right sloppy. This list is no exception.

Don’t worry I’m not going to go through all seventeen entries but the first time I read it I immediately noticed that the first five all have significant problems and I thought it would make an interesting exercise to explain why.

We start off with what is possibly the most well known theorem in the whole of mathematics Proposition 47 from Book I of Euclid’s Elements. The correct attribution of this theorem is actually an exercise in history of mathematics 101.

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

Now Euclid is thought to have written his Elements around 300 BCE and he doesn’t attribute this theorem to anybody. The first to putatively attach Pythagoras’ name to Euclid’s Proposition 47 was Proclus in his commentary on the Elements written in the fifth century CE. However Proclus doesn’t sound very convinced by his own attribution.

If we listen to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discovery. But for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the Elements, not only because he made it fast by a more lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in the sixth Book. For in that Book he proves generally that, in right-angled triangles, the figure on the side subtending the right angle is equal to the similar and similarly situated figures described on the sides about the right angle.

Proclus would seem to want to award the credits to Euclid not Pythagoras. Those who wished to recount ancient history were Cicero writing in the first century BCE and Plutarch writing in the first century CE. One thing that makes this anecdote from antiquity somewhat dubious is the fact that the Pythagoreans rejected animal sacrifice. There is no actual contemporary evidence that associates either the Pythagoreans or Pythagoras to the theorem that we name after him. However all of this is rather academic, as the theorem existed more than a thousand years before the Pythagoreans.

There is clear evidence that the Babylonians knew of the theorem in the Old Babylonian period around 1700 BCE. However although we have several instances of them using the theorem we don’t have a Babylonia proof of the theorem Maybe they didn’t have one but there are still literally tons of Babylonian clay tablets that have never been transcribed let alone translated. It could well be that somewhere the Babylonian Pythagoras is still waiting to be discovered.

The Babylonians were not the only ones to have the theorem independently of the Greeks. A clear example of the theorem can be found in the Indian Sulba Sutras. Unfortunately the dating of early Indian texts is very problematic and the best we can do is to say the Sulba Sutras date from between 800 and 200 BCE, so if the Indian Pythagoras predated the Greek one is almost impossible to determine.

Never to be left out when it comes to ancient invention and discovery the Chinese also had their own Pythagoras. The greatest Chinese mathematical classic The Nine Chapters of the Mathematical Arts contains problems that require use of the theorem in Chapter 9. In Chinese it is known as the Gougu rule. Once again dating is a major problem, the earliest existing manuscript dates from 179 CE but the contents are probably much earlier in origin, currently thought to date to 300 to 200 BCE. A simple and elegant pictorial proof of the theorem turns up in another Chinese classic the Zhou Bi Suan

Chinese Pythagoras

Jing. Also very difficult to date but probably originating around 300BCE. As can be seen this theorem doesn’t have a simple history.

Stewart now takes a massive leap into the seventeenth century CE and the invention of logarithms. Once again his simple attribution to John Napier is exactly that, simplistic and historically misleading. We can find the principle on which logarithms are based in the work of several earlier mathematicians. We can find forms of proto-logarithms in both Babylonian and Indian mathematics and also in the system that Archimedes invented to describe very large numbers. In the fifteenth century Triparty, of the French mathematician Nicolas Chuquet we find the comparison between the arithmetical and geometrical progressions that underlay the concept of logarithms but if Chuquet ever took the next step is not clear. In the sixteenth century the German mathematician Michael Stifel studied the same comparison of progressions in his Arithmetica integra and did take the next step outlining the principle of logarithms but doesn’t seem to have developed the idea further.

It was in fact John Napier who took the final step and published the first set of logarithmic tables in his book Mirifici Logarithmorum Canonis Descriptio in 1614. However the Swiss clockmaker and mathematician, Jost Bürgi developed logarithms independently of Napier during the same period although his book of tables, Arithmetische und Geometrische Progress Tabulen, was first published in 1620.

We stay in the seventeenth century for Stewart’s next equation, which is the production of a first derivative using the so-called h-method confusingly labelled calculus, confusing that is because calculus is a branch of mathematics and not an equation, and attributed to Newton 1668. To say that this line has a lot of issues would be a mild understatement. I will try to keep it relatively short. Anybody with half an idea of the history of calculus will already be asking themselves, what about Leibniz? Newton and Leibniz both developed their ideas of the calculus independently in the same period with Newton probably developing his ideas first but Leibniz being the first in print. This situation led to what is probably the most notorious priority dispute in the whole of the history of mathematics and science. What makes Stewart’s statement even more piquant is that he attributes the discovery to Newton but his equation for the first derivative is written in Leibniz’ notations. Of course there is an about two thousand year long history to the development of the calculus that I outlined in an earlier post, so I won’t repeat it now. I will however point out that the h-method to determine the first derivative is not from either Newton or Leibniz but Pierre Fermat.

Newton gets a second bite of the cherry, this time, with the equation for gravity. I’ve lost count of the number of time that I’ve pointed out that the basics of the law of gravity, the inverse square relationship, does not originate with Newton. A very quick rundown.

The first to suggest that the planets were kept in their courses by a force was Kepler who suggested a directly proportional relationship based on Gilbert’s investigations of the magnet. Borelli also speculated on forces driving the planets in his Theoricae Mediceorum Planetarum ex Causius Physicus Deductae published in 1666 and known to Newton. The first to suggest an inverse square relationship was Ismael Boulliau, a story that I’ve already told here, although I there claim erroneously that Newton admits his knowledge of Boulliau’s priority in Principia, he doesn’t, it’s in the letters he exchanged with Halley in his dispute with Hooke. In the middle of the seventeenth century Wren, Halley, Hooke and Newton all independently came to the conclusion that the force governing the planetary orbits was probably inversely proportional to the square of the distance, i.e. the law of gravity. Newton’s achievement was to show that this law was equivalent to Kepler’s third law of planetary motion and that it also allowed the deduction of Kepler’s first two laws.

Stewart’s fifth equation is his simplest i = √-1, which he attributes to Euler. Now whilst it is probably true that Euler introduced the letter “i” as the symbol for the square root of minus one, by the time he did so mathematicians had been playing with and cursing the concept for a couple of hundred years.

The first person to consciously use imaginary or complex numbers was the sixteenth century polymath Girolamo Cardano in his Ars magna, the first systematic study of the solution of polynomials published by Petreius in Nürnberg in 1545. Cardano solved cubic equation in which during the solution so-called conjugate pairs of complex numbers turned up, which when multiplied together lost their imaginary parts thus delivering real solutions. (Conjugate pairs of complex numbers are ones of the form a + b√-c and a – b√-c which when multiplied together become a² +b²c.) Cardano thought the complex numbers were nonsensical but the solutions worked so he left them in.

Later in the century the Italian mathematician Rafael Bombelli worked quite rationally with complex numbers developing the rule for their manipulation in his Algebra published in 1572. If any one name should be attached to this equation then it’s Bombelli’s.

Of course not everybody was as happy with these very strange entities as Bombelli and it was Descartes who gave them the name imaginary in 1637. It was intended to be derogatory. Euler did much to develop the theory of complex numbers in the eighteenth century but it was first the development, independently by three mathematicians, Caspar Wessel in 1799, Jean-Robert Argand in 1806 and of course Gauss, of the geometrical interpretation of complex numbers that they became finally universally accepted.

Having ploughed your way through the historical thickets of this post some of you might be thinking that I’m just nit picking, but there is a deeper point that I’m trying to make. It is very rare in the history of mathematics or science that a theorem, theory, invention, discovery, idea, concept or hypothesis emerges in its final form, like Athena born fully armed from Zeus’ forehead. Almost always there is an, often long, period of evolution involving many thinkers and often taking long and devious routes. Very often they occur as multiple discoveries with more than one progenitor, frequently leading to priority disputes. The idea of a simple list of discoveries with one date and one name whilst superficially attractive leads inevitably to a false concept of the evolution of science and of scientific methodology. Let us get away from such lists and let students of science and mathematics really learn just how messy, complex and, I think, fascinating the histories of their disciplines really are.