Early Mathematics Day

 6 May 2011

Early Mathematics Day

The Archimedes Codex 

Reviel Netz

The Archimedes Palimpsest really is a phenomenal thing, because Archimedes was a phenomenal man in a very precise and unique way. Studying the Palimpsest, you can appreciate a certain conceptual subtlety that becomes integral to the overall history of mathematics, and which makes every last detail that we tease out of the text extraordinarily meaningful. In this lecture, I intend to uncover the Palimpsest to you. I will race through a few of those subtle things that we have discovered that are of rich significance. I shall start with the two major things that are covered in my book – The Archimedes Codex – and then move onto two, more recent issues, not covered in the book, but which will give you a sense of the kind of project that I have been working on. 

Let me briefly acquaint you with the Palimpsest. This is what it looks like today. You can see the mould, the purple marks. You can see that it has suffered fire damage, though not significantly. You can see that it is harmed in all sorts of ways. What you cannot see here at all, by the way, is Archimedes. The writing that you see here is a Greek prayer book. 

In order to preserve a book like this, you take the book apart and keep each page separately. We can look at photos taken of the Palimpsest pages from 1906, which reveal information that can no longer be seen in photographs taken today. However, there are other techniques that we can use nowadays. We can use technology to bring out a powerful contrast between the overlaying and underlying texts, which allows us to see the latter in greater detail.

Perhaps the most spectacular technique that we have been using is x-ray fluorescence. We bombard the page with x-ray waves and then count the florescence in this way, because it reacts differently to different elements in the page’s surface. The ink includes iron, which the parchment, the animal skin, does not include in any significant amount at all, so this provides a certain contrast. The image produced is not a visual image at all but a physical image.  It is an iron count of the manuscript.  The technique is very interesting and powerful and gives you information that you cannot see otherwise. It also creates an extra complication because, these being x-rays, they also show us the other side.  So, instead of a normal palimpsest, where you see two layers, we now see four layers, which we need to separate as we are reading. That takes some work, but it can be done.

Indeed, this technique has allowed us to read the very first page of the manuscript, which is otherwise completely illegible. This was especially important because, being the very first page, it contains the prayer-book dedication. Thanks to this x-ray fluorescence, we found that the book was apparently dedicated in Jerusalem in 1229.

So, what are we looking at?  It is a manuscript of Archimedes.  It is one of the medieval copies made of the works of Archimedes.  It was made, apparently, in the 10^th Century AD.  It was not used much at all.  There are no marginalia or corrections to the various errors, so it fell quickly into disuse.  It was always just a collectors’ item, nothing else.  Quite naturally and appropriately, in 1229 it was recycled, the pages taken out, torn, rotated, scraped and written over as a prayer book.

There were other medieval copies of the works of Archimedes produced, most of them lost, but one of them made its way to Italy.  It was there from the 13^th century onwards, but it was available from the 15^th century. It was often copied in the 15^th Century, and you could say that, in a sense, it was the manuscript that sparked the scientific revolution.

With this manuscript, we have certain works that we do not have elsewhere. This is a manuscript that failed to spark the scientific revolution.  It was not there to spark the scientific revolution. The manuscript that did got lost, so this remains the only medieval copy we have of the works of Archimedes.

It was finally identified as containing works by Archimedes in 1906 by Johan Ludvig Heiberg, the great Danish philologer. Heiberg did an incredible job of editing the text from the material he had, from photographs of the manuscript when it was in a better shape.

It was promptly stolen, shortly after Heiberg worked on it, in the aftermath of the First World War.  It was once again hidden from view.  We did not know where it had been all this while.  And then, finally, in 1998, it was sold for a relative bargain, for $2 million in an auction in New York. Since then, it was given for preservation and for scholarship, housed at the Walters Art Museum at Baltimore where we have been working on it. It is now going back to its owner, much enhanced in value – a worthy investment, I would say.

There is going to be a facsimile and transcription volume, together with an entire volume of commentary and study, which will appear in the autumn. In a sense, however, all of this has already been made available online.

I would now like to race through some of the breakthroughs that we have achieved. Perhaps the most important discovery is that of The Method, a work that is preserved only in the Palimpsest. Archimedes measures, among other things, the volume of the object created by passing a slanted plane through a cylinder. 

It resembles a fingernail and it is shown here encased in a prism because, very remarkably, this very curved object is exactly one-sixth of the perfectly rectilinear prism that encloses it. That is a result that Archimedes really liked. It is a great achievement.  He does this by taking this figure and the prism that encloses it and passing essentially infinite arbitrary planes through it. We are concentrating on two-dimensional images.  There is a triangle set up by each of these arbitrary planes - a triangle which is within the prism, a triangle which is within the cylinder - and there are also lines at the base of the cylinder - lines associated with the rectangle, lines associated with the circle. Archimedes also constructs an ad hoc parabola for the sake of the construction and for the geometrical work that he is doing with it.

We discovered something completely unexpected about the geometrical work. Essentially, Archimedes relies on claims such as ‘the triangles in the prism are equal in number to the lines in the rectangle’.  It turns out that he finds certain proportions between triangles, lines, and then sums up the proportions to find that the proportion of the triangular prism to the cylindrical object is a certain ratio.  He does a certain summation from infinitely many ratios to one ratio that encompasses all the infinitely many ratios, and for this purpose, he relies on certain theorems of summations of ratios and certain equalities. The triangles in the prism are equal in number to the lines in the rectangle, the numbers in question obviously being infinite. There is no question at all that this is how Archimedes envisages it.  We are not talking about thin wedges, we are talking about actual two-dimensional planes. 

This is very exciting from the perspective of the history of mathematics, where we have all been taught, and indeed, all have been teaching (myself included) that the Greeks never used infinity, because they had all sorts of logical problems with the concept. On the basis of this discovery in the work of Archimedes, we should be very sceptical about saying that ‘The Greeks did so-and-so’.  I think when people say ‘the Greeks’, usually what they mean is Aristotle.  That is the way our understanding of antiquity works - we take certain economical authors, find certain tendencies in them, and then try to generalise from those views, to apply those views to the Greeks as a whole. It is important to see that the Greeks are really the Greeks – there are plenty of Greeks, and they did very different things, which was essential to their culture. It was a culture of polemic debate and not a culture of consensus, so we should not be surprised to see Archimedes doing something that Aristotle perhaps would not have approved of.

Here is all that remains of Archimedes’ Stomachion. The beginning is very difficult to read, and so is the end – it is the actual end of the book.  It is the end of the prayer book and it was the end of the original book that was produced in the 10^th Century AD, and ends do not fare well, generally speaking. Because you touch the ends of books with your hands, you expose them more and they tend to get ruined much faster.  The end is also closer to the cover, which is made of leather, a very hostile sort of organism that parchment does not like.  So the end really was in a terrible condition in 1229, which explains why the person making the Palimpsest made the choice to throw away the last few pages, seeing as they would not survive should he try scraping them again. Of the Stomachion, they used only this.  So what did he use?  He used the introduction.  That is all we have of the Stomachion and in very bad shape, so Heiberg really could not read much of it in 1906.

So, what is the situation?  The situation is that we do not really know what the Stomachion was, but we have a few more hints than Heiberg had because now we have read it.  Now we know what is written in this fragmentary text.  We also know, from other sources, that there was a game in antiquity called Stomachion, which involved various pieces that could be fitted together, a tangram puzzle. Archimedes speaks in a way which makes it clear that he is not inventing anything; he is talking about something that is already known. From what I read in the introduction, I think Archimedes is telling us that the point of doing the treatise is the fact that there are many ways of putting the puzzle together and the point is to count them. There many different types of substitutions and rotations, I think, so he might be aware of those things. 

I am not a mathematician myself, rather an historian of mathematics, so I did not try to calculate this myself. Mathematicians started working on this problem and arrived at this solution. So here is a solution that was arrived at:

That is actually just a set of canonical answers, out of which one can multiply more through basic symmetries. 536 basic solutions, which, through the obvious symmetries of a square, become 17,512 solutions to the problem of putting together the square of the “Stomachion.” This is momentarily surprising, but once you start thinking about it, it becomes obvious there are quite a few substitutions there, and they interact in interesting ways.  So some of them will cancel out each other and will rule out others, so it is not a matter of simply counting how many substitutions there are and multiplying them. There is an interesting puzzle there, but one that can be solved through very elementary techniques.  Essentially, it is something that you just count, rather than developing a theory for. Perhaps this is what Archimedes could have come up with, something such as this, expressed in a verbal form. Once again, that is very exciting. 

In the previous case, we had Archimedes doing something which we thought the Greeks were not doing.  Here, Archimedes is doing something that we thought Archimedes was not doing - we do not normally think of Archimedes as doing combinatorics.  But there is something very deeply playful, very deeply counting-based, in what Archimedes is doing here. Actually, when you start looking at the corpus of Archimedes, you find that, once again, it is a matter of our modern perceptions.  I think the modern perception is to take Archimedes as the person who did the measurement of the sphere and the cylinder, the quadrature of the parabola, the author of this tradition that gives rise to the calculus, the person thinking seriously about problems in the measurement of curved figures in a way that gives rise to integration.  That is how we think about him.  But really, he did a very great variety of things, and this actually seems to be the essence of what he was doing.  The Greeks in general engaged in this polemical society of debate, and Archimedes in particular comes from this culture of the Hellenistic world where one is especially interested in variety, in trying to do many things together, of trying to do things that, in a single work, brings together many different strands. I think we understand Archimedes and his culture a bit better by thinking about works such as the Stomachion and not just the canonical works.

I now want to talk about things which are not in the book. However, before I do so, I want to say a word about diagrams.  Diagrams are extremely important, obviously, for understanding the way Archimedes was thinking, for understanding science in general, but diagrams were not studied at all by people such as Heiberg. For one thing, they had so much to do in recovering the text, but there was also a certain bias perhaps against the use of diagrams for mathematics purposes.  It is clear to us today that diagrams were a real part of the logic of Greek mathematics, so it is very important to recover them in our work.

Let us turn to On Floating Bodies. I have mentioned that Archimedes leads to the calculus.  Archimedes also leads to mathematical physics. Somehow, the combination of the calculus and mathematical physics was really essential to the way in which the scientific revolution was shaped and modern science was born. Archimedes looked at the mathematics of objects in water.  What does he do primarily?  We tend to think of Archimedes’ hydrostatics as fundamentally a study in achieving the law of buoyancy, and that is certainly a very remarkable achievement, through using absolute principles – just a single postulate, and you get the law of buoyancy. That is it. But actually, for Archimedes, that is just a stepping stone. What he does primarily, in what is the most difficult and sophisticated part of his entire work, is to look at various objects, in particular segments of conoids of revolutions (objects you get by taking conic sections and turning them about their axes), and consider the conditions of their stability immersed in a liquid.  So, if you have a very long hyperboloid, it will lose its stability in different angles.  What is the stability when the base is inside the liquid?  What is the stability when the base just touches the liquid? He goes through all those cases, which give rise to very complicated applications of geometrical principles and physical principles at the same time, again demonstrating this interest in variety.

This begins with a study of the segment of the sphere immersed in liquid - this is where he sets the ground for this study. We have, as is typical, a planar and fairly schematic picture. We have a segment of a sphere, immersed in the water. It is immersed in the entire ocean.  For the sake of the exercise, we imagine that the entire Earth is made just of water, so the ocean is the Earth, and of course everyone knows that the Earth is spherical, so there is no problem describing the Earth as a circle as well. The object is rather big, relative to the Earth.  It does not really matter, but it makes the picture easier to draw.  So, we are looking at a segment of a sphere immersed in the Earth. This is the picture drawn by Heiberg in his modern edition, so it will be useful to see that that is not actually what we see in the ancient diagram. 

This is my version of the ancient diagram, which we can compare with the modern drawing: 

There is an immediate noticeable difference, involving the position of the point represented here by the lambda. In the ancient diagram, this is very obviously not at the centre. I must tell you that this point stands for the centre of the Earth. This is how the proposition works. We can begin to see what Heiberg is doing, but we begin to wonder why the ancient diagram is different. What function could it serve?

As I said, Archimedes studies stability.  He says that a segment of a sphere will be stable under certain conditions. How does he claim this? He considers alternative arrangements and argues for the instability in them.

Here is an example of a diagram from Euclid.  This is Euclid book three, proposition 10. At the top, you can see a drawing by Heiberg, and below it, six different ancient diagrams, all quite similar to each other, and all quite different from the one that Heiberg drew.  The ancient diagrams have much more elongated objects in the middle. 

What is this proposition about?  This is a proposition that proves that two circles do not intersect each other at more than two points. How does Euclid go about it?  Through a proof by contradiction. Assume that two circles can intersect each other at more than two points - in this case at four points - and we get an impossibility.  The ancient diagram makes a very different choice from Heiberg.  Heiberg draws it in the way that it appears most plausible. He draws two ‘circles’ that are as near to circles as one can draw them under the conditions.  Euclid obviously made the opposite choice.  He made a choice to draw circles that are very obviously not circles. The other ‘circle’, the one that is supposed to intersect the circle at four points, is very elongated.  We can see that two circles cannot intersect each other in more than two points. 

Here is another, very closely related, example, once again from Archimedes. This is proposition 13.  Archimedes makes a very similar claim: he claims a tangent line touches a spiral at no more than one point.  If a line is a tangent to a spiral, it will not touch it again after the point at which it has touched it in, and for the sake of this proposition, he does something which Heiberg does not reproduce.   Heiberg does not reproduce this diagram – he has his own diagrams.  He has a diagram where the line, the tangent, touches first at gamma and then at etta, being broken very powerfully in the middle.  You emphasise visually the impossibility of the impossible case, in a proof based on impossibility, and I would argue that this is what happens here.                                                 

When Archimedes wants to show us the instability of a certain case, he shows this by making the case appear impossible. This suggests something very profound about the way Archimedes thinks about mathematical physics.  Archimedes is actually misleading us. He says, “…for if possible, let the segment of a sphere be in that position.” But of course, the segment of sphere can be in that position.  The unstable position is not impossible.  It is merely unstable, and this has important consequences.  The logic of possibility and impossibility has certain rules that allow one to work with it.  The logic of impossibility leads us from one case to its complementary case in an instantaneous fashion. We tend to do the same in physics.  From a not stable disposition, we do not automatically claim that the complementary position is stable. We need a physical path.  Archimedes never provides us with a physical path and, for all we know, it cannot be achieved.  The statement that an object will reach its stability from its instability is essentially based on exporting the logic of possibility and impossibility and mapping it onto the logic of stability and instability.  That is not simply an example of what Archimedes is doing in floating bodies.  I think it is the key to the way in which he constructs his mathematical physics.  He constructs his mathematical physics by taking physical objects and putting them, not purely in the geometrical space of three-dimensions and lines and surfaces, but also putting them in a space defined by the logic of pure geometry.  That is how mathematical physics is constructed. I think that this diagram provides us with an important clue about the way Archimedes thinks about physics. That is an example of one thing that we can learn, or at least debate, when looking at the diagrams in the Palimpsest.

Finally, let me give you a small reading from the end of the Palimpsest, which is my favourite. It consists of three simple lines, all of which were previously proved by Euclid. That is very exciting because, if you a historian of mathematics, you care about the relationship between Archimedes and Euclid because we do not really know when Euclid lived.  We know when Archimedes lived because we have historical evidence of his death with an actual date, which essentially forms the foundation of all our chronology of Greek mathematics. If we could know that Euclid is before Archimedes, this would be very useful. 

This comes from Sphere and Cylinder, in which Archimedes, midway through, suddenly makes a few citations of previous results concerning cones and cylinders, and says that these were proved. Sphere and Cylinder exists in the other manuscript – the manuscript that got into Italy and sparked the scientific revolution and then got lost.  It is there, so we have evidence for this manuscript.  That manuscript has very similar text for Sphere and Cylinder in general. In this case, the other manuscript states that these problems were all proved by previous mathematicians. This gives us a different reading, and we want to know which is the correct reading.

Proclus, in his commentary to Euclid’s Elements, says “…and then came Euclid”.  We know that Euclid came about then because we know that he is earlier than Archimedes, because Archimedes refers to him. That is interesting.

There is another moment in Sphere and Cylinder where both manuscripts state: “…and this can be shown the second proposition of the first book of Euclid’s Elements”. Maybe this is what Proclus is referring to and we have just added a small thing. However, that other place is very clearly wrong.  It is a very silly statement, certainly not in the original text. This gives us many different options. Perhaps this new reading, that all of this was proved by Euclid, is a mistake made by this scribe in the Middle Ages and that is all there is to it, and the correct reading is in the other manuscript, and the real text that Proclus also knew also said this was proved by past mathematicians and all Proclus referred to was the reading that we knew already - or perhaps the reading that we always thought Proclus was referring to.  Perhaps, ultimately, this could even be the correct reading and it could be the correct reading by Archimedes, in which case, not only does it tell us what Proclus was thinking about, but it also tells us that Euclid was earlier than Archimedes. The range of options here is very wide, and as with the diagrams, this opens up the sense of the possible debate that historical documents can lead to, and in general, what is involved in the study of historical documents. It should also give you a sense of why working with this manuscript was not only such an important piece of work, because of all the consequences it has, but also such a fun piece of work to do.

Thank you.

©Reviel Netz, Gresham College 2011

6 May 2011 

Money and Mathematics

Luke Hodgkin 

We all know that the Greeks invented money and then invented mathematics, and the application of money to mathematics has landed us all in our current economic mess. In a sense, the Greeks are to blame for all that.

“Where did Greek mathematics come from?” is my theme, and it is an old theme that a lot of people have talked about; it is a hopeless question, but there is no harm in posing it again.  As Mao said, “Where do correct ideas come from?” and he gave the three, obvious answers: scientific experiment; the struggle for production; and the class struggle.  What he did not mention was taking somebody else’s correct ideas, which is the fourth way that you can get a correct idea.

Where did Greek mathematics come from then? Well, there is any amount of speculation.  I want to talk about two heretical speculations which get currency from time to time, to be dismissed, and the first one is a very old one.  It was floated, in the first place, by Herodotus, and Aristotle, and more recently, by Martin Bernal.

What do you mean by Greek mathematics? The Greeks, or some Greeks, claimed that they had got it from Egypt. Martin Bernal writes:

“[I]t would seem difficult to argue that before the second half of the fourth century B.C. any aspect of Greek “science” - with the possible exception of axiomatic mathematics - was more advanced than that of Mesopotamia or Egypt.”

It is an interesting quote because he points out that Greek science, with the possible exception of axiomatic mathematics, was up there with the science of Egypt.  As one of his detractors pointed out, that is quite an exception. To except axiomatic mathematics is to except what the Greeks were seriously good at, so if you grant that exception, you give away a lot of ground. So, what are you talking about when you talk about Greek mathematics? As Reviel Netz and others have shown, Greek mathematics used to be thought of as one thing but is more diverse than we had thought.

The second heretical theory I want to raise is one which is much less often realised, but which I like. It goes back to the fact that, around the same time as they invented mathematics, the Greeks also invented currency. Alfred Sohn-Rethel, one of the Frankfurt School Marxists, thought about this and produced a very fine book called Intellectual and Manual Labour, in which he claimed that both Greek mathematics and Greek philosophy derived their abstract form from the idea that the introduction of currency forces you to lose the particularity of any object within exchange. As people, as the Greeks said, you stop talking about sheep numbers and ox numbers and you just talk about numbers. Sohn-Rethel wrote:

“We reason that this [abstraction] could result only through the generalisation intrinsic in the monetary commensuration of commodity values promoted by coinage.”

That is some rather heavy theory and there are pages and pages of it, so i cannot hope to summarise Sohn-Rethel’s arguments to you today. Even so, I shall just call your attention to the coincidence of time and the similarity of mental operations in constructing the market and in constructing mathematics. I shall try and move on to an example a little later.

Sources can be problematic, as Serafina Cuomo has already demonstrated in her lecture today. I have always been a great fan of the National Curriculum, which demands that students of History should “place events, people and changes into correct periods of time” and be taught how to find out about the events, people and changes “from an appropriate range of sources of information”.  Now, what is “an appropriate range of sources of information” in the case of Greek mathematics?  Our problem is that we have got one kind of source and we have not got another kind of source.

As a result of this, as well as other prejudices (coming from Aristotle or Plato or both of them), there is one thing that is seen as Greek mathematics, and it is essentially geometrical and deductive. I do not think that this idea is entirely wrong, but it is only part of the picture.  

Reviel’s excellent first book, The Shaping of Deduction in Greek Mathematics, set up a neat framework in which he showed that there was such a study as Greek mathematics, which was not like anything else.  It might be said to be more diverse than we thought, is based on the lettered diagram, and he gave a neat argument that dates it back to the time of Hippocrates of Chios. 

Here is our first source of real Greek mathematics. This is an Oxyrhynchus Papyrus with Euclid 2.5, and it is a very neat diagram. It is interesting that fairly straightforward people in Egypt – we do not know who – thought it was worth copying down a Euclid proposition on a piece of papyrus.  It is obvious that some serious Greek mathematics was in circulation, but because of the problems we face with papyrus as a source, we do not know a great deal about that.

This mathematics is not the mathematics of numeracy.  It is not what I would call demotic mathematics, the mathematics of surveyors and accountants, the mathematics of sums.  And that is good mathematics too, of course.  Greek mathematics includes sums.  Greek mathematics includes the work of surveyors. As Serafina pointed out, there were Greek surveyors and Greek accountants, as well as Roman ones.

This is a picture showing a bit of a survey, reproduced from David Fowler’s book The Mathematics of Plato’s Academy. It is a terrific book, which constantly gives me new insights into Greek mathematics.  A couple of days ago, I found that David was claiming that Socrates had a method of calculating the cube root of two, and that you could read that in his account of things in The Republic.  I really admire somebody who can say something like that! And it may be true.  How do we know? How do we know anything that we know about Greek mathematics? A lot of the time, we have to rely on ingenious reconstruction. What emerges from Fowler’s book is the idea that there is more of a connection between demotic mathematics and the serious theoretical mathematics of, say, Euclid or indeed Hippocrates than we think.

As we can now see, demotic mathematics fits easily into the thesis of Herodotus and Bernal.  You can see an almost seamless transition, from Egypt to Greece, of a kind of calculation of measuring fields and doing accounts. Herodotus does not say “We got the logical deductive method from the Egyptians”.  He says “We got how to measure fields from the Egyptians”.  

The existence of a break, in which some sort of abstract argument, some sort of logical, deductative argument comes, is crucial for Sohn-Rethel’s theory. Do we think that there was such a break? It looks as though there was, although I would again argue that it was quite varied.

Now, it is time to produce a few examples.  You may think of different examples.  I shall start with doubling because it lies at the beginning of what we think about generally. There are two well-known examples of doubling in Greek mathematics: one is Plato’s Meno, where you double a square.  Interestingly, you double a two-foot square – that is, you have got numbers in there.  Why are the numbers there? Why is it a two-foot square?  Why is it not just a square?  It is there so that the slave boy can make mistakes and say, “What about four?” and “What about three?” and so on.  But, is this logical deductive mathematics?  In a sense, yes, but in a sense, no. In a sense, it is an abstract argument, because you cannot actually arrive at a number.

But we know that the Babylonians knew what the diagonal of a square was, I think we can bet that Egyptians knew what the diagonal of a square was, and I think that we can be pretty sure that Socrates knew what the diagonal of a square was.  It is just that that was not what he was interested in when he was writing the Meno, or Plato was writing the Meno.

I think that the Egyptians, and the Delions, the people from Delos, could very likely have worked out the cube root of two as a fraction (go back to my remarks about David Fowler). I think that working out the cube root of two as a fraction is not beyond people, but that is not what many of the Greeks were doing.  That is not what Hippocrates did, and that is where Hippocrates comes in again as a founding figure. Hippocrates said that it was a question of constructing two mean proportionals, and I think it was Reviel who said, “What the hell is he on about?” When you are constructing two mean proportionals, what is equivalent to what? 

Let us return to Ancient Egyptian mathematics as it was in the sixth century BC. We have no idea what this mathematics was, but we can quite safely assume that it was not at very much like Ancient Greek mathematics. Hippocrates’ reduction and Menaechmus’ solution of the problem of doubling the cube are not the kind of thing that an Egyptian would have come up with, but I think an Egyptian might have come up with the following, or they would have understood what was being got at.

This is the well-known machine devised by Eratosthenes, who did away with constructions using conics and so on. Instead, he showed that you could double your cube by building a machine like this and fiddling around with triangles.   

What has this got to do with the currency or the coinage, or capitalism?  You might want, and Eratosthenes is quite explicit about this, to double the size of weights and measures. If you have a mould for a one drachma coin, you might want to make a mould for a two drachma coin, and you might want the two drachma coin to contain twice the amount of silver that the one drachma coin contains - how much bigger would it have to be? Eratosthenes would tell you how to do it.  I think that this is the kind of mathematics that some of the Greeks thought of, and which was also not outside the horizon of some Egyptians.  There is more interplay between those two ways of thinking about mathematics than we think.   

I am now going to come on to something that is seriously different, the trivial proposition of Euclid 1.35. You have two parallelograms with the same area.  If you read the proposition, from beginning to end – it is not very long – you will see that Euclid has not defined area.  You do not know what the area is, of course, and the reason why they have the same area is something to do with having added and subtracted congruent triangles.  Now, congruent triangles are the same in a totally different way from the way in which parallelograms of the same area are the same.  He is going back to the common notions and making (you could say) a pun on the word “equal”, or simply saying what he means by “equal” – I think that is probably a fairer way of looking at it.  He is saying “I mean by “equal” equal in the sense of the common notions”, that is, you can add or subtract equals and what I mean by “equal” is a) equal congruent triangles and b) something to which you can add or subtract congruent triangles.  But he has not said that, and you would need somebody to help you with that. I think that relates to the exchange of distraction, in a certain sense – that is that, in some way or other, you have produced an abstract idea of equality for parallelograms, which is illuminated by a remark quoted by David Fowler from Proclus:

“It may seem a great puzzle to those inexperienced in this science that the parallelograms constructed on the same base should be equal to one another.”

Fowler makes it clear that this is no longer a puzzle to us. It is a puzzle if you are a surveyor used to finding the areas of quadrilaterals by using a surveyor’s rule, which is a good way of cheating your taxpayers. In any case, in some sense or other, Proclus finds this is a surprising proposition.  Therefore Proclus, writing some 900 years later, still finds Euclid’s definition of area a puzzle.  That is interesting.

I think, therefore, that this is an example of a different kind of mathematics, while some of the examples of duplication that I gave are not. They are only “half examples” of a different kind of mathematics. Sometimes we get an example of something that is really different and sometimes we do not. 

What one might find surprising, I suppose, is the fact that if you are talking about exchange abstraction - currency, money and so on - why is Greek abstract not more concerned with numbers?  I shall leave that as a sort of question mark.  Of course, we know that numbers are involved. Heron’s theorem has been referred to, which is of course very interesting because you are simultaneously surveying and producing a Euclidian proof – that is, you prove that your surveying formula is accurate and then you give an approximation to how to work it out.  Heron is an extraordinarily interesting source, but he is unlike anybody else. 

The quadrature of lunes is another example of the exchange abstraction. It is always regarded as the point at which Greek deductive mathematics starts off, if you discount all the stories about Pythagoras and things like that.  It is the earliest reliable story, in some sense, of something that happened in Greek mathematics.  And what it says, in my interpretation, is that you can exchange two very different things for one another.  You can exchange a round area for a triangular area.  This explains the kind of problems that people had about squaring the circle and so on, but it must have seemed wonderful that you could exchange exactly a crescent for a triangle. 

There is more diversity in Greek mathematics than once thought. There is what I have called the demotic mathematics.  There is the mathematics of surveyors and accountants and so on.  There is the pure demonstrative mathematics, at the other extreme, wherever you like to locate that.  There are those peculiar kinds of construction that are later found to work, such as Eratosthenes’ construction. There are certain things that Reviel Netz calls ludic – I am thinking of the sound reckoner and the cattle problem as things that are dealt with in a demonstrative way by extraordinarily respectable mathematicians, but have nothing in particular to do with the geometric tradition.  There is Heron, and there is astronomy, and I have not finished there.

So, whose pursuit is mathematics? The following quotation from Aristotle is well-known:

“Hence when all such inventions were already established, the sciences which do not aim at giving pleasure or at the necessities of life were discovered, and first in the places where men first began to have leisure. This is why the mathematical arts were founded in Egypt; for there the priestly caste was allowed to be at leisure.” (Metaphysics, I,i.)

This is the other quote about getting mathematics from Egypt, and I like it because it is a complete lie.  Here is Aristotle saying that mathematics came from the priests because they had leisure time.  Now, it was not the priests who did mathematics in Egypt.  We know it was the scribes, and the scribes had very little leisure – they were kept scribing all the time.  Any psychoanalytic reading would see Aristotle projecting his own leisure onto the Egyptian priests and saying “You can do mathematics if you have leisure and I have it”, and you have to have a caste who have leisure, and we (the Greeks) have one.

My essential point is that the developments in Greek mathematics were varied, and we are still learning the skills to talk about them. I am not going to provide a conclusive answer, I rather wanted to raise the different possibilities, and there are of course others. Geoffrey Lloyd thought that it came from people having conversations with one another in Greek city states.  It might have come from that too. Its relation to abstraction is a difficult one because, of course, our sources are so propagandist and Platonic, and there is so much that we do not have, but we cannot spend our time lamenting that.  We have to work on what we do ahve, and I think that there are some very good people around doing work on this material, and I am very grateful to them for providing me and the rest of us with the material that we have to think about.

Thank you.

GreshamCollege, 6^th May 2011

Early Mathematics Day

Decoding the Heavens:
Solving the Mystery of the World’s First Computer

Jo Marchant

This is what I will be talking about – the biggest surviving part of it anyway. The Antikythera mechanism dates from the early 1^st or 2^nd century BC, and it is a clockwork mechanical calculator. Some people call it a computer. It is the most sophisticated scientific artefact that survives from antiquity – in terms of a physical object that we have in our hands there is nothing else like it. You can see gear wheels here – at least thirty survive. We do not have another single gear wheel that survives from the ancient world. Nothing as sophisticated as this appears for well over a thousand years afterwards, just to give you a sense of how unique this is.

First, I am going to give you a bit of background about how it was discovered. Then I shall talk you though what it did, what it was for, what it calculated. I shall then talk about where that knowledge and technology came from as far as we know, and what happened to it afterwards.

The Antikythera mechanism was discovered by a crew of sponge divers from the island of Syme, in the eastern Mediterranean. They spent the summer diving for sponges just off the coast of North Africa, but on their way home (around the autumn of 1900) they were blown off course by a storm and took shelter by the island of Antikythera. It is a tiny island, barely inhabited.

They found a shipwreck. The Greek government hired them to salvage it, which took about ten months. The site is at about 60 metres depth, which in the suits that they had was incredibly dangerous. One of them died from the bends, two of them were paralysed from the operation. They brought back the most incredible haul of treasure from the ancient world that had been found up until that point.

They mostly found bronze and marble statues, which can be seen in the National Archaeological Museum in Athens. The bronzes fared pretty well although they had to be reconstructed from pieces.

It was a Roman ship carrying Greek treasure, probably stolen Greek treasure from the eastern Mediterranean that was being carried back to Rome when the ship sank.

While all this stuff was coming back to the museum, staff were desperately trying to put the pieces back together. The finds were making headlines around the world. It was 1901. Anything that could not be identified was just thrown into a crate, and this particular piece of rock sat in an open courtyard for months before it cracked open. We do not know if someone hit it with a hammer or if it just dried and broke open.

Nevertheless, inside was this. Gear wheels, pointers, precisely marked scales, inscriptions. It was like nothing anyone had ever seen from the ancient world. People did not know what to make of it.

This is the largest surviving piece of the Antikythera mechanism. Here is another piece where you can see what looks like concentric dials. Here is a third piece where you can see two dials that are marked very precisely. It looks modern, much like a protractor that you might have used in school. You can see a lot of inscriptions here.

People were very excited about it. They assumed that it had something to do with astronomy (because it had the names of the months on it), that it was Greek (the writing is in Greek) and that it dated from around the time of the shipwreck, which was early first century BC (although this particular artefact is probably a bit older than that) Beyond that, however, no one really made much progress.

Unfortunately, I do not have time today to go into the hundred years of research that happened since its discovery, but one of the most important researchers was Derek Price, a British historian of science who worked at Yale. He was the first to X-ray the pieces and look at the gear wheels inside and how they fitted together.

This is what he said about it: “If it is genuine the Antikythera machine must entail a complete re-estimation of ancient Greek technology. Its discovery 55 years ago was as spectacular as if the opening of Tutankhamun’s tomb had revealed the decaying but recognisable parts of an internal combustion engine.” Price was one of the first to really get the significance of this find. Unfortunately, he got a lot right about what it did, but he also got a lot wrong.

Michael Wright, who lives in Hammersmith, was a curator at the Science Museum and he also X-rayed the pieces. Then, a big international team with sophisticated X-ray 3D scanning technology came along and looked at it as well.

Their combined efforts produced this diagram, which I think shows just how complicated the Antikythera mechanism was. This represents the thirty or so surviving gear wheels, but we think that there were probably many more which have been lost.

I am going to take you through some of this. First of all, what did it calculate?

This is Michael Wright’s reconstruction of the machine. We do not know exactly how all of the bits fitted together but this is probably the most accurate model that we have. It was not necessarily exactly like this but this is pretty good.

It was bronze – the gear wheels and mechanism inside, as well as the dials on the front and back – housed in a wooden box with a handle on the side. You would operate it by turning a handle on the side and it tells you everything about the sky at any particular moment in time. It is giving you the positions of the Sun, Moon and planets on the front here. There are two dials here. The zodiac dial shows the 12 signs of the zodiac and is divided into 360 degrees. And there is a calendar, showing the days and months of the year.

Then, there are pointers going around the main dial, which show you the positions of the Sun and the Moon, and possibly the planets. As you turn the handle you can turn forwards and backwards in time and it shows you all of those bodies in the sky and what they are doing.

These are the names of the Sun, Moon and planets here, you have got days of the month round here, and these are the dials. This is a star calendar or parapegma. As the date pointer reaches one of those letters, you refer down to the corresponding letter at the bottom and it tells you what stars are rising and setting at that particular moment in time. There is also this little ball here, which rotates and shows you the phase of the moon.

These are some of the inscriptions from the star calendar, which show the kind of information you are getting. There are other inscriptions as well, which have not been completely read, there are just fragments of it. But they look like instructions, explaining what is going on in the mechanisms.

This is what is going on in the back. These are two spiral dials. It was only recognised quite recently that they are spirals, they are not concentric circles. This top one is a 235-month repeating calendar. This pointer is like the stylus on a record player. It is quite clever. You start here, and as it goes round it extends, the arm gets longer as it goes around the spiral, until it gets to the end. When you have finished you just pick it up and put it back to the beginning, and it goes again.

This is a close-up of a little subsidiary dial, which was initially thought to be a way of multiplying this calendar. This is a 235-month calendar, which I shall return to in a minute, but the Greeks also used a calendar that was four times that, which was more accurate. So it was thought that this was enabling you to read four sets of these 235 months. But when researchers actually read the names on the dial they realised that they were the names of Greek athletic games, including the Olympics. That was completely unexpected! So this is a four-year dial, and it tells you the names of the games that were happening in that particular year.

It was thought that this was a completely scientific, astronomical instrument, but this discovery made researchers realise an added social importance to the mechanism. This tells you something about who this was used by and what it was used for.

On the bottom is an eclipse prediction dial. I shall come back to the details of how this works, but this is a 223-month cycle. Patterns of eclipses tend to repeat themselves after 223 months. Again, you have this extendable pointer that goes all the way around and when you get the end you lift it up and put it back to the beginning. A little subsidiary dial measures three sets of those cycles, which gives you a more accurate period.

So, the maths! Inside you find many different gearwheels, like the inside of a modern clock. How do you calculate with these?

Imagine that you have a pair of interlocking gears – one has 48 teeth, the other has 16 teeth. If you turn the 48-tooth wheel once, the 16-tooth wheel is going to turn three times. That is a function of the number of teeth of the gears. You can write that as a fraction. I doubt the Greeks would have written it in that way but you can understand it in that way.

And you can reverse that. If you turn the 16-tooth wheel once, the 48-tooth wheel will go a third of the way around, so you are multiplying or dividing depending on the number of teeth.

You can then pile different pairs of gears up onto each other. You can have an axle leading from the second gear of one pair that drives the first gear of another pair. You can multiply those fractions together.

We are going to look at a gear train within the mechanism that is involved with the Sun and the Moon. The Sun and Moon were very important to the ancient Greeks – the time of year, what the Sun was doing, was very important for agriculture; the Moon was very important for religious festivals. One thing the Greeks spent a lot of time doing was trying to come up with calendars that could tell you what both the Sun and the Moon were doing, because the number of months does not fit nicely into a number of years. You cannot just have a 12-month year and expect the Moon’s phases to repeat on the same days each year.

So they used something called the Metonic cycle. It was known to the Babylonians but this was named after a Greek astronomer, Meton, who lived in the 5^th century BC. It says that 235 synodic months – the time it takes for the Moon to come back to the same phase, full moon to full moon – is the same as 254 sidereal months – the time it takes for the Moon to orbit the Earth and come back to the same position with respect to the background stars; that is, nineteen years. After nineteen years the Sun, Moon and Earth all come back to the same position relative to each other, and you will once again have your full moons happening on the same day of each year. You can have a repeating  nineteen year cycle.

This is the speed of the Moon, going round the Earth 254 times, and in that time, the Earth goes around the Sun nineteen times. But of course the Greeks were thinking of it all from a geocentric perspective, so for them, it was as if the Moon was going through the sky 254 times, while the Sun was going through the sky nineteen times.

If you look in the Antikythera mechanism and follow the gearing through from the input that is driving the Sun pointer, you see three pairs of gears, and these are the number of teeth on those gears. What that adds up to is 254/19. What it is doing is converting the speed of the Sun into the speed of the Moon.

We can follow it here. This is the handle on the side. As you turn it you’re turning this wheel here, which has 64 teeth interlocked with 38 teeth. That drives the second pair here – 48 teeth and 24 teeth, and that drives the third pair here – 32 teeth and 127 teeth. So that is this equation here. You turn the handle and that turns the Sun pointer round once per year. Whatever speed you are turning it at, the pointer will go round and when it has been all the way round once, that gives you a year. You then have this chain of gears here, which converts that into the speed of the Moon.

It was originally thought that this went straight back up here and drove the Moon pointer around. So you have the Sun and Moon each going at their own constant speeds around the zodiac on the front dial. But it is actually more clever than that.

The Moon does not go around the Earth in a perfect circle. It goes round in an ellipse and it speeds up and slows down as it does so. The Greeks did not know about elliptical orbits but they did know that the Moon was speeding up and slowing down. It is only a tiny change, but they knew about it.

So the speed of the Moon feeds into this little collection of gears here. This is quite difficult to explain, but here you have got two wheels with slightly offset centres. This wheel here has got a pin sticking out of it that goes into a slot in the wheel below, driving it around. Because the two wheels are off centre, the pin on the first wheel moves towards and away from the centre of the second wheel as it turns. So as it is driving it around, the speed with which it is pushing that second wheel around speeds up and slows down in a cyclic way. It is giving you just the same speeding up and slowing down that you have when the Moon is going around the Earth.

But even that was not good enough for the Greeks who made this machine. The axis of the Moon’s orbit, or the ellipse, is actually shifting around the Earth once every nine years or so. So, this whole set of gears is mounted on another turntable. As it is speeding up and slowing down it is actually travelling around on this turntable as well, roughly once every nine years. And only then is it fed back to the Moon pointer.

So when the Moon pointer is going around, it is taking into account all of that different motion. Then you have got another set of gears up here which turns the Moon phase display – according to the relative movement of the Sun and the Moon, it turns that little ball to give you the phase.

And that is just the Sun and the Moon. I just wanted to give a sense of how clever this is!

The reason that I have coloured this wheel with stripes is that it has a double use. It is used in the Moon system, but it is used again here. These are the gears that lead to the eclipse prediction display, so I told you about the 223-month cycle. This has 223 teeth so it is key in determining that speed.

That is partly why it took so long to reconstruct all of this, because it is so cleverly put together. Whoever made this knew exactly what they were doing. It was really thought through and pre-planned.

Let us look at the eclipse prediction dial. This is the Saros cycle – 223 synodic months is equal to eighteen years and eleven and a third days. The Greeks knew about this pattern from Babylonian astronomers, who had been observing eclipses for centuries and marking down exactly when they happened. Eclipses repeat after 223 months but because of this third of a day, they are shifted by eight hours and occur a third of the way around the globe. So it is not massively useful. That is why the mechanism has this extra subsidiary dial to measure three of these cycles. After 223 months and after putting the main pointer back to the beginning of the spiral, this little dial will have moved round one, so that you know you are in the next phase and you have to shift everything by eight hours. Do it again and you are in the second phase and you have to move everything by sixteen hours, and then you are back to the beginning again.

Michael Wright who went to Athens and looked at the pieces of the Antikythera mechanism for himself and X-rayed them, told me something very intriguing about this. The pieces are now battered and crumbling, with corrosion products covering everything. When he looked beneath the overhang of corrosion products on this dial, he saw an eight and a sixteen in two of these little segments, and he says he saw a symbol in the other one as well, which he speculates might be a symbol for zero. That would be a long time before zero should have been around, but I just thought I would tell you that! The team that came after Wright, who took 3D X-ray slices through everything, did not see anything there. Wright counters that he looked at it with his own eyes whereas the later researchers were not allowed direct access to the pieces, they were always handled by museum staff. I am just throwing that in there.

It seems likely that the mechanism did show the planets, because several of them are named in the inscriptions. However, the relevant gears do not survive so it is not clear how this was done. Michael Wright thinks the mechanism would have shown the planets on the same dial as the Sun and Moon. This is not straightforward because as the planets are orbiting the Sun and not the Earth, they have even more irregular motion (as we look at them) than the Moon and Sun do.  Sometimes they stop, sometimes they appear to go backwards. Wright thinks they would have used a separate gear train for each planet. It is called epicyclic gearing, where you have gear wheels that ride around on other wheels. This is the same principle as I just described for the Moon. He thinks they would have done this for each of the planets. 

This is your epicyclic gear here that is riding around on this bigger wheel, and there is a pin sticking out of it. It is looping the loop as it goes round in the bigger circle. This is modelling the Greek theory of epicycles. If this is the pin here, you can see it will sometimes stop and change direction, modelling how the Greeks thought the planets were moving. Then you have got this slotted lever that translates that movement back to the pointer in the centre. They would have done a similar thing for the Sun as well.

That is one idea, which is quite impressive. However, another astronomer, James Evans, more recently came out with a paper where he measured all the spacings on the zodiac dial, and he concluded that they are split into a fast zone and a slow zone. Evans thinks that in half of the zodiac dial the divisions were closer together, and in half of the zodiac they were further apart. So you could have a pointer going at constant speed and the variation would be accounted for by your scale, not your pointer.

If you did that, you would not then be able to have your planets on the same dial, you would have to have them on separate dials. It is not clear at this point who is right about that.

So, where did the technology come from? That is really two questions in one. Where did the machine itself come from, and then where did the maths come from? I already said this was a Roman ship sailing from the eastern Mediterranean, we know that from the rest of the cargo on the ship. It probably would have stopped off at Rhodes, then it sank at Antikythera, on its way to Rome. It sank between 70 and 60 BC, which we know from coins found by Jacques Cousteau when he went to the wreck site in the 1970s.

There are some interesting clues from Cicero, who we have already heard about today. He wrote a couple of things about machines that sound a lot like the Antikythera mechanism. He wrote about “…an instrument recently constructed by our friend Posidonius, which at each revolution reproduces the same motions of the Sun, Moon and the five planets that take place in the heavens each day and night.” This sounds like a very similar sort of thing.

He also wrote about a device built by Archimedes, and other authors wrote about this as well. Cicero said: “The invention of Archimedes deserves special admiration because he thought out a way to represent accurately by a single device for turning the globe those various and divergent movements with their different rates of speed.”

Stories like this were never taken that seriously by historians because Cicero did not have any technical training and he did not explain how these things worked, so maybe he was just making it up to make a point. You want to have more evidence really, before you conclude that they had these machines.

However, now that we have the Antikythera mechanism, which fits the description so well that you start to think perhaps he really was writing about something that was around at the time.

Posidonius was a philosopher who lived on Rhodes at exactly the time that the Antikythera ship would have stopped off there, and Cicero was on Rhodes at this time as well. Posidonius had an interest in astronomy, so perhaps there is a connection there.

Hipparchus, one of the most prominent astronomers we know about in the Greek world, also lived on Rhodes just before, in the 2^nd century BC, and he was very interested in the motions of the planets and in the varying speed of the Moon, in particular, so a lot of the stuff that he was working on is reflected in the Antikythera mechanism.

Unfortunately, if you follow other lines of evidence they do not all tell you the same story. These are the month names from the calendar on the back of the mechanism. These are not month names that astronomers would have used particularly, they are social. Each city would have had its own calendar, so you can look at the names and ask where this calendar would have been used.  It comes down to Sicily or northwest Greece. Sicily is interesting because that is where Archimedes was working a long time before.

Then there is the Olympiad dial that tells you the timings of different games. It has six games named on it, five of which have been read so far. The first four are quite big games, which would have been important across the Greek world. But Naa was apparently a small local affair, where you would only really have been interested if you lived nearby. That was held in northwest Greece. So there are two things that match up there.

However, with Archimedes, several authors wrote that he had made something similar to this. He probably did not make the actual Antikythera mechanism because he was living quite a bit earlier. But we do know that he was working with gears, making machines to lift heavy weights and change the ratios of forces. These experiments were much simpler than the gears in the Antikythera mechanism, but worked on the same or a similar principle. His father was an astronomer. He also wrote a treatise called On Sphere-making, which we only have the title of. Sphere, or sphaera, was the word used for these kinds of astronomical models, so it is intriguing to think that he might have come up with this idea originally, and if we could ever find that treatise it might tell us about how they were made.

In terms of the knowledge, I have made it clear that a lot of the astronomy in the Antikythera mechanism was not Greek originally, but came from Babylonian astronomers. They had been observing the centuries and marking down very diligently exactly what was happening when, because for them it was all related to omens about the future and the possibilities of bad things happening to the king; it was very important to know what was going to happen when, and to try and predict it.

They saw the heavens in a very arithmetical way, things moving almost as if on a 2D screen. They had algorithms to be able to plot the motion of different things and to predict what was going to happen next, but they were not really interested in how things were arranged.

The Greeks, on the other hand, saw things much more geometrically. They were interested in how the cosmos was arranged but they did not really put numbers to it. It was more about coming up with models that were philosophically pleasing, incorporating perfect spheres, and what was where. They did not make much attempt to match that numerically with what was happening in the sky.

The Antikythera mechanism represents those two traditions coming together. You have got the numbers being put to it, but you have also got the geometric aspect, trying to model how these things are actually moving through space. Hipparchus is very interesting in this regard because he is thought to have been one of the astronomers who brought those two ways of thinking together. It is thought that he had direct access to the Babylonian astronomers’ records. It does feel as though he may have had a hand in this tradition of model-making.

In terms of significance for the history of astronomy, I think it is really significant that the Antikythera mechanism represents the bringing together of those two traditions. However, there is another aspect as well, which takes us back to the disagreement about the planets. If you go with Michael Wright’s interpretation, this is a representation of the Greeks’ epicyclic theory of the planets - the idea that the planets must have been moving in perfect circles. They saw circular orbits as perfect, and things in the heavens were divine, so these must have been moving in circles, nothing else was really conceivable. But to explain the irregular movement of the planets, the Greeks came up with the idea of epicycles, circles superimposed on other circles. In the Antikythera mechanism, you see gears riding around on other gears to model the motion of the Moon, and if Michael is right you would have had that for the planets as well, so this would be a lovely mechanical representation of that theory.

However, James Evans maintains that you do not need any epicyclic gearing in here at all. For the Sun, they could have modelled its motion by marking the scale differently – and that model comes directly from the Babylonians as well, so if they were doing that, that would be quite interesting. He believes that this is what is going on with the Moon mechanism, with gears riding on other gears. This suggests a Greek modelmaker who was taking the Babylonian arithmetic astronomical tradition and thinking about modelling that with gearwheels? He concludes that the best way to do that would be by putting gears to move round on other wheels. It is not that the astronomers came up with the idea and then the modelmakers copied it, it was actually the other way around. It may have been a modelmaker who came up with the idea to model the Babylonians’ astronomy, which in turn convinced Greek astronomers that that was how the cosmos was really arranged. Models like this could have been the original inspiration for the epicyclic theory in astronomy. It is an idea, an interesting thing to think about.

Where did the technology go? We do not see anything like this again until Medieval Europe, with the appearance of mechanical clocks. It is clear that when the Roman civilisation collapsed, a lot of this technology was lost. We do, however, see little kernels of this technology that did survive.

Chronologically, the next similar object to emerge is this sundial. This is from the Byzantine Empire, 6^th century AD, many hundreds of years later but still Greek-speaking. This is a reconstruction that was also made by Michael Wright, from battered pieces that are in the Science Museum.

Mostly this is a sundial, and there are quite a lot of these around. But here is what is really unusual – these represent the seven days of the week. It is a little calendar with a ratchet that you click around once for every day. That leads to a set of eight gear wheels that are like a mini geared calendar. You move it around once every day, and this is what it drives.

Here is a dial showing the Sun’s position in the zodiac, this is the Moon’s position in the zodiac, this is a rotating disc that gives you the phase of the Moon, and this is the day of the month. So it is exactly the same idea. It is much simpler, it is only eight wheels compared to 30 plus, but the concept of using gear wheels to model astronomical motions has survived into the Byzantine era.

This is a copy of a tenth century manuscript, in Arabic. It is thought to be by an astronomer called Nastulus, in the Islamic world. If you look at this, it is a diagram of exactly the same eight-wheel calendar. You can count the teeth on the gear wheels. Here are the Sun and the Moon in the zodiac – two little dials – the phase of the Moon and the day of the month. It is exactly the same calendar. Once again, there is a specific, written instruction to indicate that this is a calendar to be attached to a sundial. We are now into the Islamic world, several hundred years later.

This is an astrolabe that is held in the Museum for the History of Science in Oxford. This is from the 13^th century, and it is from Isfahan in Iran.  On the back, we find the same thing again. This time it is attached to an astrolabe instead of a sundial so it seems that these geared calendars could be attached to different kinds of instruments depending on what you wanted to do. Here you have concentric dials to show the Sun and Moon going around in the zodiac, phase of the Moon, day of the month. Again it has eight gear wheels.

You can see that the technology has survived through. What is particularly interesting about that sundial is that it is not a luxury gadget like the Antikythera mechanism would have been. It is actually an everyday piece of kit, suggesting that this could have been quite widespread.

Many historians think that the technology did then come back to Europe. One of the questions in the history of science is how mechanical clocks emerged. This is because they appeared very suddenly, all across Europe at the same time, big extravagant display pieces, predicting eclipses, showing the position of the Sun, Moon and planets and so on. Only later did they became smaller and simpler and more about telling the time. How did they appear everywhere, all at once, with such complicated technology? It seems that as a lot of other knowledge was coming back from the Islamic world at that time, this geared technology probably came back as well, and when somebody invented the mechanical escapement, everything got put together.

So when you look at the pieces of the Antikythera mechanism and you see how modern they look, it is not actually a coincidence. There may well be a consistent thread that runs all the way through to the gears that are in our watches, cars, cameras and everything around us today.

© Jo Marchant, Gresham College, 2011

6 May 2011 

Early Mathematics 

Crore Blimey: My trip to India to uncover the truth about vedic mathematics 

Alex Bellos 

My name is Alex Bellos, author of Alex’s Adventures in Numberland. I am not really a mathematician, though I have a degree in Maths and Philosophy from twenty years ago.  I am a journalist and a former foreign correspondent, which gives me rather a different attitude towards mathematics compared with today’s other speakers. I have written about Vedic Mathematics, which certainly qualifies as ‘early’ mathematics because the Vedas are probably the world’s most ancient sacred texts that are still around today.  They date from about 1,500 years ago. This afternoon I intend to tell you the story of how my interest in Vedic mathematics developed, what it is and what it revealed to me about mathematics and arithmetic more generally.

I first came across Vedic mathematics simply by browsing the internet. There was, at that time, a viral YouTube video calling itself ‘Vedic mathematic’, which had half a million hits - probably the most-watched piece of mathematics on the internet.  This was a couple of years ago, and as of yesterday, 600,000 people have watched it. Essentially, it is a way of doing simple multiplication by translating numbers into lines. For example, to work out something like 12 times 32, we turn each digit into a line - so 12 becomes one line, and two lines, and 32 becomes three lines and then two lines. We arrive at the answer by counting the sections, which gives us the answer – 384.

The first time watching this method is always amazing, and it is very rare that a piece of mathematics becomes a viral hit. I decided that I would discover where this comes from, so I flew to the place of its origin – a place called Puri, which is in Orissa, in the Bay of Bengal, India.

While I was waiting in my hotel when I got there, I picked up the paper, and the headline read: FIVE CRORE MORE INDIANS THAN GOVT THOUGHT. It was the first time I had ever been to India, so perhaps this is not so exciting to those who have been before, but I was confused by this word ‘crore.’ What on earth is a crore? I was then told that, in India, you never use the word ‘a million’ or ‘a billion’. When counting the powers of ten, they move through 10, 100, 1000, 10,000, lakh (100,000), 10 lakh (a million), and then crore, which is 10 million, and then 10 crore, and so on. I thought about the word ‘million’, and I asked, ‘Well, what about millionaire – do you have the word ‘millionaire’?’ and they said, ‘No’. I said, ‘Well, what about the film Slumdog Millionaire? That is about India!’  To which they replied, ‘Correct, though it is called Slumdog Crorepati in India!’ In fact, when you watch the film, you notice that the prize is not a million rupees at the end, it is 10 million rupees, because the guy speaking in Hindi asks, ‘Now, do you want to go for the crore?’ Already, on my first day in India, I was encountering differences in simple ways of dealing with numbers. When writing the powers of ten, the comma comes after every second number, so they count off in hundreds rather than in thousands, like we do. 

I carried out some research into the reasons behind this, and found that lakh and crore come from the Hindi words lakh and karod, which come from the Sanskrit, laksa and koti. A thousand, two thousand years ago, they had a word for 100,000 and 10 million. I know we are not supposed to make generalisations about the Greeks, but I do believe that the Greeks did not have a number word larger than a myriad, which is 10,000. So, when the Greeks did not get higher than a myriad, the Indians were way up into the crores. There is a wonderful section from the Lalitavistara Sutra, from the 4^th Century, where the Buddha is asked to show his prowess at mathematics, and he is told to ‘go up’ from lakh and crore, which would have been laksa and koti. He said that 100 koti (crore) are in an ayuta, 100 are in a niyuta, a 100 niyuta make a kangkara, and so on, until he gets to 10⁴²¹!  That is a big number!  In terms of all the atoms in the universe, there are probably only about 10⁸⁰, so this is beyond everything really, it is not really about numbers or calculating, there is something mystical is going on there. 

Then, interestingly, he is asked the opposite – if he is good at counting big numbers, how good is he at counting small numbers? The Buddha is asked to divide up a yoyana (about ten kilometres), and he says, ‘Four krosa are 1,000 arcs, each of which are the length of four cubits, each of a length of two spans, each of which is the length of twelve phalanges of fingers,’ so we are dealing with a few centimetres now, ‘each of which is the length of seven grains of barley, each of which is the length of seven mustard seeds, which was the length of seven poppy seeds, which is the length of seven particles of dust stirred up by a cow, each of which is the length of seven specks of dust disturbed by a ram, each of which is the length of seven specks of dust stirred up by a hare, each of which is the length of seven specks of dust carried away by the wind, each of which is the length of seven tiny specks of dust, each of which is the length of seven minute specks of dust, each of which is the length of seven particles of the first atoms.’

This is obviously poetic as much as anything, but a physicist or chemist calculated that, if the phalanges of the fingers are four centimetres, then the Buddha went as far as 4cm x 7^-10, which works out as 0.00000000001416, which is more or less the size of a carbon atom. Spooky! This is another reason why lots of people get very mystical and slightly lose a sense of reason when they are talking about Vedic mathematics and Vedic physics. Within these areas, however, there is a lot of serious stuff going on.

Because I am a journalist and not a mathematician, I did not have the brain power to work out some of the equations. I read a wonderful, enormous book called The Universal History of Numbers, which was something of a labour of love by a French schoolteacher who was asked by one of his kids, ‘Where do numbers come from?’ He basically spent the next 30 years going all round the world trying to work it out.  He discovers many seriously amazing things, and one of the things he writes about – and he is the only person that seems to write about these things that I could find – is the way in which, before the symbols for digits in India were secure, the Indians had a very interesting way of remembering large numbers.  Obviously, they knew about large numbers because they have got words for them, but within certain advanced areas like Indian astronomy, they needed some ways of remembering these really large numbers. This is what they would do. I am not a Sanskrit scholar, but Sanskrit is a sacred language andhas to be written according to certain rules. This produces something that looks like a poem:

The apsides of the moon in a yuga

Fire. Vacuum. Horsemen. Vasu. Serpent. Ocean,

and of its waning node

Vasu. Fire. Primordial Couple. Horsemen. Fire. Twins.

Which equates to:

[The number of revolutions] of the apsides of the moon in a cosmic cycle [yuga] is

Three. Zero. Two. Eight. Eight. Four.

and of its waning node

Eight. Three. Two. Two. Three. Two.

The numbers of revolutions of the aspects of the Moon in the cosmic cycle, the yuga, is just 302884, and its waning node is 832232. Actually, the number would be read the other way round – ‘Three’ means ‘three units’, ‘zero’ means ‘no 10s’, ‘two’ 100s, ‘eight’ 1,000s, ‘eight’ 10,000s, and ‘four’ 100,000s. To make it easier to remember these important numbers, you translate each one  into a word, because you are much more likely to remember something that sounds poetic.  So, if you had a zero in these Ancient poems, you could use either ‘dot hole’ or ‘the serpent of eternity’, amongst other things.  For one, you could use earth, moon, pole star, curdled milk – there is a 100 page dictionary of these different words that they used for numbers in The Universal History of Numbers. As you can see, you can be quite clever when finding different ways to adopt numbers and astronomy in that culture.

I went to Puri because of Shankara, who was the very first Shankaracharya, a sort of Hindu seer. The most reliable sources of information claim that he was around in the 8^th century AD. Hinduism is older than Buddhism, but then Buddhism came along and then India was Buddhist for a while. Following this, Hindu seers realised that they needed to adapt some of these Buddhist ideas, which led to India becoming Hindu again. In the great wave of Indian ‘re-Hinduisation’, Shankara was quite important. He set up four big temples, one at each of the four corners of India, which were called, auspiciously, ‘Mathas’. That is nothing to do with ‘maths’ – it is just the Sanskrit or Hindu word for ‘temple’. I was visiting the westernmost temple, in Puri. At each of these temples, there is a Shankaracharya, which is comparable to an Archbishop.  It is very difficult to compare Hindu leaders (of a system that is totally non-hierarchical) with the Judao-Christian idea of leaders, which is very hierarchical.  The Shankaracharyas are possibly some of the most revered spiritual figures in Hinduism. 

In 1925, the Shankaracharya at Puri was Jagadguru, which is a title meaning ‘big guru’. His full name was Shankaracharya Swami Sri Bharati Krishna Tirthaji, and really known as Tirthaji.  He was a fascinating character, both in terms of his interests and the time that he lived in.  He was born in 1876.  He was a child prodigy.  When he was twenty, he graduated in Sanskrit, Philosophy, English, Maths, History and Science, and people saw that that he was going to become a great thinker.  He was also very political and, at a time when Indians were fighting the English, he was imprisoned for a while for being so polemical. During this time he came up with what is called Vedic mathematics. In 1925, he was made Shankaracharya at Puri, and he was Shankaracharya up until 1960, when he died.

During this time, he would go round this part of India and speak to his flock, as it were, about faith but also about Vedic mathematics. He said that he had gone into the forest for several years, taking only the Vedas, which he studied. He discovered that, in the Vedas, there were sixteen hidden sutras, or formulae, which made mathematics much easier. That which would normally take you ten years to learn, could now be learnt over three months using these sutras.  

The first eight are as follows:

1. By one more than the previous one

2. All from 9 and the last from 10

3. Vertically and crosswise

4. Transpose and apply

5. Transpose and adjust (the coefficient)

6. If the Samuccaya is the same (on both sides of the equation, then) that Samuccaya is (equal to) zero

7. By the Paravartya rule

8. If one is in ratio, the other one is zero.

When you first looks at them, they seem completely bonkers! What do they mean?! His idea was to hold lectures and explain that these sixteen sutras were all in the Vedas and could help people do maths better. At the end of each lecture, he would give his audience a few helpful tricks.

He attracted quite a lot of fame in India, and in 1958 he was the first Shankaracharya ever to leave India, which was supposedly forbidden – no one has done it since.  He went to the East Coast of America, which was promoted as big news in The New York Times and The LA Times. This was one of the world’s most spiritual men, and accordingly he went on a number of chat shows and gave a number of lectures on world peace. This was the time of the Cold War, and he was very much into ‘Make love, not war’. By the way, this was before the 1960s image of the hippy Indian guru had established itself. This was 1958, and at that time, America had not seen anything like it. 

He also did a famous talk at Cal-Tech, to an audience comprising the smartest, brightest mathematicians in America, on the West Coast (apologies to Stanford!). The lecture notes are still around, showing exactly the things that he taught them. Firstly, he told the story of his journey into the forest and stressed that his audience members could not have discovered tehse formulae themselves – only he could have done that, as a spiritual man, well versed in the Vedas. He described the formulae like puns, like words that most people might read one way but that he could read another way. I am sure he said that with a smile on his face. 

One of the things that he showed was an interesting way to multiply two numbers together. I shall multiply 7 x 8:

You start by taking the difference of each number from 10. For 7, this is 3; for 8, this is 2. You then multiply these together, which gives you 6. Then, you just either add -3 to 8 or -2 to 7, and you get 5. 56 is the answer. As long as you multiply two numbers above 5, it always works. 

I shall now multiply 97 x 96:

The differences from 100 are 3 and 4 respectively, so we multiply these together to get 12. We then either add -3 to 96, or -4 to 97, to get 93. The answer is 9312.

Supposedly, all of these students at Cal-Tech were amazed and Vedic mathematics became immensely popular. However it still remained largely unknown in India, except to those who had attended Tirthaji’s lectures, because he never actually published anything.

He died in 1960, supposedly having written sixteen books on the subject, of which fifteen have never been found. The first one, the introduction, was published as Vedic Mathematics in 1965. Upon publication, there was little interest. This remained the case until the beginning of the 1970s, when India was ‘in vogue’, and a group of British visitors found the book at the back of the shelf in an ashram. They brought it back to the UK and began to promote it. There are lots of books about it, mostly from Britain and with a few pioneers in the field, and now India is ‘re-learning’ the concept from us. It is certainly a growing field in India nowadays.

So this is basically what Vedic mathematics is.  It gives you many fun tricks to do with arithmetic. I shall return to the problem I showed you at the start of the lecture:

Essentially what we are doing here is 2 x 2; here it is 2 x 1; here it is 3 x 2; here it is 3 x 1. The reason why long multiplication is called long multiplication is because it takes longer than what we could do, which is short multiplication – the method shown above – and which is also known as cross-multiplication.

While at Puri, I wanted to meet the current Shankaracharya. I wanted to ask if these formulae were really just derived from the Vedas. He was obviously a Brahmin and spoke perfect English, but in the temple you are not allowed to speak English, so I had to communicate via his interpreter - which is a brilliant way of not answering my questions! I would ask a question such as, ‘Can you give me a page number where I can find ‘vertically and cross-wise’ in the Vedas?’ This question would go through the disciple, who would spend some time translating what took me ten words to say into about twenty words in Hindi. He would then pose this to the Shankaracharya, who would think about it as if he had only just understood it (though he had obviously understood it before). He would spend about five minutes replying to the interpreter in Hindi, who would then spend about ten minutes replying to me. It was like a game of Chinese whispers – absolutely nothing at all to do with the questions I asked! I began to realise that if you ask a rational, scientific question to someone so concerned with religion, spirituality and metaphysics, you are never really going to get anywhere. I visited him four times, because each time the interview took two hours and I only managed to ask three questions.

As mentioned earlier, the place value system of numbers first occurred in China and India. The numbers that we use today are akin to Hindu-Arabic numbers, from India. One of the great mathematic advances from India was the invention of zero. (Usually, when I give talks, I make the old joke: ‘What did India give the world? Nothing!’) By just looking at this man, I realised that India is the obvious place for zero to come from. Hinduism is nothing, unless it is something. This man is the embodiment of having nothing. He has no possessions.  He has one thing that he wears all the time.  He apparently eats the same, bland curry every day. During my time with him, people would come in, prostrate themselves before him and offer gifts. As soon as he got a gift, he would simply pick it up and give it to someone else. To someone who was not aware of the culture, it looked like he was a bit annoyed with these incoming gifts coming, as if he had received too many presents and felt like redistributing them. I ended up with a whole bowl of fruit round me!

I returned from India with the realisation that the number system emerged there because of the metaphysical, spiritual way that nothingness meant something. Furthermore, why is the circle used as the digit for the number zero? When I was young, it seemed to me as if the hole of a ‘0’ was a visual symbol for nothing, but that completely misses the point. Rather, the circle represents the eternal cycle of the heavens – both the infinite and nothing at the same time. Each time we write a zero on a piece of paper, we maintain an ancient, Vedic way of looking at the world.

As well as giving the world zero, India has recently given the world the first zero currency notes – zero rupee! To me, this is very poetic. It was developed by an NGO as an anti-corruption tactic, so that when you bribe someone – which you are continually forced to do - you give them a zero currency note as a way of making the point ‘I am giving you nothing, but it is something.’

Now, do not suppose that everyone in India thinks that Vedic mathematics is great.  Actually, most serious mathematicians think that Vedic maths is mad. I went to see one of their top mathematicians, S. G. Dani, who is the top professor at the Tata Institute in Bombay. He has written quite a lot of savage attacks on Vedic mathematics, saying that it makes Indians seem stupid and silly, and reinforces prejudices of India being a third-world country. Would we believe it if someone said they read the Bible and discovered the quadratic formula in it? Dani argues that it is not good for the Indian self-image. However, as we got chatting, I found that he actually had a lot of sympathy for Tirthaji and what Tirthaji was trying to do. By calling it Vedic mathematics, the man was making a politically charged , anti-colonial statement at the turn of the century, when the Indians were turning against the British. The theory emerged out of a feeling that the English had even taken numbers from the Indians, which really belonged to the Indians, and Indians should feel proud of those numbers. So, though Dani does not approve of Vedic mathematics on an intellectual level, he can empathise with Tirthaji on an emotional level and see what he was trying to achieve.  

It is always quite easy to parody other cultures for doing something ‘a bit crazy’, and I would not want to be seen to be someone who is dismissing Vedic mathematics, because the actual arithmetical tricks that are there are all perfectly valid.  There is nothing silly about them at all. If, in order to try and educate a nation, you point out parts of a heritage and embellish them, I think that that is a perfectly acceptable means to do so and to popularise mathematics.

When I got home, I wanted to look into some of these tricks. I bought the Trachtenberg method of arithmetic, and found that there are huge overlaps. Then, I read the Liber Aberci, Fibonacci’s 13^th century book that introduced Hindu-Arabic numerals to Europe, and found that it is almost a recreational math book. It presents fun ways of counting and exciting number puzzles. In terms of multiplication, it shows methods that are vertical and crosswise! If the very first book has got the same tricks that are in Vedic mathematics, who is not to say that Indians were the first people to do that? 

I got in touch with David Sigmaster, who knows lots about recreational mathematics, and he told me that all this stuff was well-known by Victorians, and that Tirthaji was probably a schoolteacher, and he would have picked up all these things as tricks.  But actually, if you look at books from the 13^th, 14^th and 15^th centuries, they all use very similar methods. 

I was also reading Florian Cajori’s History of Mathematical Notations, a book that should be required reading for all those interested in the history of mathematics. It is an amazing book. It considers things like the origin of the ‘x’ in multiplication. The first person to use it was William Oughtred in 1631, but before he used it in the way we use it now, he said that there were eleven different ways that the ‘x’ had already appeared in multiplication literature. To me, the method that I have shown you today seems a likely source of this ‘x’ - multiplying two things by splitting them apart. 

Though my initial reaction to Vedic maths was one of distrust, I came to realise that all maths is essentially Vedic. We can trace our notions of maths to the Vedic, spiritual culture of zero, and the invention of number positions.  Even though the Greece had numbers, arithmetic only really  became part of mathematics when the Hindu-Arabic numerals superseded Roman numerals in the Renaissance.

Therefore, we are all Vedic mathematicians! 

Thank you very much.