Archimedes of Syracuse (287? - 212 BC) pioneered highly original methods for working with irrational numbers. For reasons we will get to, this high level of Greek mathematics did not continue, but when the works of Archimedes were rediscovered in the Renaissance, they contributed very directly to the making of our modern technological world. Here is an example of Archimedes' work, from On the Equilibrium of Planes, showing how the distinction between rational and irrational actually operated.
Archimedes is creating a very careful theory of the balance. In doing this, he deals with two kinds of measurable things, lengths and weights. The balance consists of weights placed at various distances from the support (where it is supposed to balance). He also introduces the notion of center of gravity, which is just another name for the place where the support should be, to make the weights balance. He introduces several Postulates, of which the first three are [1]
1. Equal weights at equal distances balance, and equal weights at unequal distances do not balance, but incline towards the weight which is at the greater distance.
2. If, when weights at certain distances balance, something be added to one of the weights, they do not balance, but incline towards that weight to which the addition was made.
3. Similarly, if anything be taken away from one of the weights, they do not balance, but incline towards the weight from which nothing was taken.
As an example of things he then proves, here is his
Proposition 1: If two weights balance at equal distances, then they are equal.
If you are not thinking carefully, you might think this is just Postulate1, but if you look, you will see that Postulate 1 deals with the case of equal weights, whereas here we just have two weights, not known ahead of time to be equal. Proof of Proposition 1: suppose the two weights, balancing at equal distances, are unequal. Then remove weight from the heavier one to make them equal. The weights will now not balance, but will incline toward the other weight, by Postulate 3, but on the other hand they must balance, by Postulate 1. This contradiction shows they cannot be unequal, and so must be equal.
The next propositions establish that the center of gravity of various symmetrical arrangements of weights is in the middle.
This may seem merely qualitative, but in Proposition 6, Archimedes proves something surprisingly precise, namely that if weights A and B balance at distances a and b, then the ratios of weights and lengths satisfy A/B = b/a. He first proves it in case A and B can be measured in the same unit, so that A/B is a rational number. In Proposition 7 he proves it in case A /B is irrational.
Here is the proof in case A/B is rational. Let weight A be at point E and weight B be at point D. Divide the line connecting D and E as shown, at point C, so that so that A/B=b/a, where b is the distance CD and a is the distance EC. Now construct a new line as shown, twice as long as the line DE, containing two segments of length a and two of length b.
Since b/a is rational, there is some small length n which measures both b and a. Now take a weight N which is contained as many times in weight A as n is contained in 2b. This same weight N is contained in weight B as many times as n is contained in 2a, because B/N=(B/A)(A/N)=(B/A)(2b/n)=(a/b)(2b/n)=2a/n. Thus weights A and B can be split into equal pieces each of weight N and distributed on the line segments shown, with one weight N going into each length n. A is now redistributed over the segments of length 2b, directly beneath its original position, and B is redistributed over the segments of total length 2a directly beneath its original position. The centers of gravity of weights A and B do not change in this process, because the little weights N are symmetrically arranged around the original centers of gravity E and D. But now, by the same principle, the whole system balances at the midpoint of the long line, which is at C, as was to be proved.
That was the rational case. Now suppose A/B is irrational. Again choose C so that A/B=b/a. Suppose the weights do not balance when the support is at C. Then either A goes down or B goes down. Suppose first that A goes down. Then we can remove a weight Z from A, where Z is chosen so small that A still goes down, but so that (A-Z)/B is rational. Then, since (A-Z)/B<A/B=b/a, we can use the rational case to see that b is too large a distance for balance, and hence B must go down, contradicting that A goes down! The other case is handled similarly with the roles of A and B interchanged.
The key idea in Archimedes' argument is that an irrational number can be approximated arbitrarily well by a rational number -- that with the change Z you can move to a rational number and yet Z is so small that you do not change the imbalance. This kind of argument seems forbiddingly difficult, but it is apparently necessary. That is still the way we do it today, although the whole subject has become much easier to think about, as definitions and notations have been introduced to streamline our thoughts. The notation of decimal fractions is an example of that.
If you look at history with an eye particularly to what was happening in mathematics, the whole subject looks different and a little unfamiliar. Is this a distortion, by the "mathematical lens," or is there perhaps some new insight to be gained? We won't even attempt to answer that, but just note a few things.
Egypt, Babylon
One of the most striking things, to me, is how good Babylonian arithmetic was, and how bad Egyptian arithmetic was, in the same period, say 2000 BC - 300 BC, when neither system changed very much. These two civilizations were certainly in contact, at least through trade, and occasional military conflict, but superiority in mathematics did not translate into any general superiority for the Babylonian region. It doesn't seem to have mattered in any practical way.
There is a poignant look into the life of an Egyptian mathematician of the New Kingdom (~1200 BC?) in a papyrus which seems to be part of a textbook: the teacher is perhaps role-playing with a student, asking him to imagine what it will be like when he has attained a responsible position, and can no longer ask his teacher to help him [2]:
"You are given a lake to dig. You come to me to inquire concerning the rations for the soldiers, and you say, 'reckon it out.' You are deserting your office, and the task of teaching you to perform it falls on my shoulders. Come, that I may tell you more than you have said ... I disclose to you a command of your lord, you, who are his Royal Scribe. ... For see, you are the clever scribe who is at the head of the troops. A building-ramp is to be constructed, 730 cubits long, 55 cubits wide, containing 120 compartments, and filled with reeds and beams; 60 cubits high at its summit, 30 cubits in the middle, with a batter of twice 15 cubits and its pavement 5 cubits. The quantity of bricks needed for it is asked of the generals, and the scribes are all asked together, without one of them knowing anything. They all put their trust in you and say,'You are the clever scribe, my friend! Decide for us quickly! Behold your name is famous; let none be found in this place to magnify the other thirty(?)! Do not let it be said of you that there are things which even you do not know. Answer us how many bricks are needed for it? See, its measurements are before you. Each one of its compartments is 30 cubits and is 7 cubits broad!' "
What the scribe is agonizing over seems to be just a problem of multiplying several integers together, and perhaps dividing the product by the average volume of a brick. The astonishing building projects of Egypt seem to have been accomplished with minimal mathematics!
Greece
The Greeks believed they had gotten their mathematics from the Egyptians, but they certainly developed it in new directions, apparently creating the notion of proof, for example. Early Greek mathematics gives proofs, and the other traditions do not. In retrospect, this seems very important! Greek mathematics is much younger than either Babylonian or Egyptian. The oldest account (Aristotle, ~350 BC) says Thales of Miletus (~580 BC) discovered certain proofs in geometry. Pythagoras (d. ~500 BC) almost certainly did significant new mathematics. This seems to have been the beginning. The golden age of Athens, around 480-400 BC, which is what we normally think of when we think of the glory of ancient Greece, hardly rates a footnote in the history of mathematics. Mathematics was being done somewhere but not there. Pythagoras, who was born to the east, on the island Samos, near Asia Minor, may have travelled to both Egypt and Babylon (as a captive, in one story). He eventually migrated west, bypassing Athens, and founded a kind of utopian society in Southern Italy, what is sometimes called Magna Graecia. There continued to be people called Pythagoreans, followers of Pythagoras in some sense, for centuries afterward. (According to tradition, some of the Pythagorean leaders were women. The whole tradition is fascinating, but frustratingly obscure.) Some Pythagoreans were mathematicians whose names and accomplishments we know of by hearsay, but the written documents we have are from Athens, a mathematical backwater. Plato (~380 BC) had great respect for mathematics, and even travelled west to learn from the Pythagoreans; the main result of this was not new mathematics but rather a kind of mystical reverence for mathematics as the key to the universe, transmitted through the writings of Plato to the Renaissance. Our main source for this period is Aristotle, Plato's successor at the Academy in Athens, but who, unlike Plato, disliked mathematics. He has little to say about the Pythagoreans, and what little he does say makes them seem faintly ridiculous. Aristotle's attempt to write encyclopaedic books on all the sciences is conspicuous, in this way of looking at it, in leaving out, almost entirely, the one science which proved to be of permanent value, Greek mathematics.
Alexandria
The conquests of Alexander of Macedon (a pupil of Aristotle!) united, however briefly, Babylon, Egypt, and Greece. Conventional history says that Alexander's empire fell apart at his death (323 BC), but if you look at mathematics, the empire didn't fall apart at all. It was the beginning of a golden age, called the Hellenistic Age. A new city in Egypt, Alexandria, founded by the young conqueror, was endowed with a great library and support for scholars, like a modern research institute. Almost immediately, mathematical work of brilliance was produced at Alexandria, and continued to be produced, not only there, but all over the Greek Mediterranean. Euclid's Elements were written there around 300 BC, within one generation of the founding of Alexandria. The work of Archimedes followed within another generation or two. (Some of the works of Archimedes are in the form of letters to his colleagues at Alexandria from his very comfortable position at Syracuse in Sicily.) Now I personally wonder where these scholars came from, who got Alexandria off to such a fast start. Clearly there was a vigorous mathematical establishment ready to take advantage of Alexander's support, but where did they come from? Not Athens. Since most of the historical record represents Athens, we don't really know what else was happening. Who was Euclid, for example, and where did he come from? We don't know. Were these the descendants of Pythagoreans, and did they arrive from Magna Graecia? The existence of Archimedes makes it clear that there was mathematics in Sicily. The historical trail has gone cold, however.
Rome
The great Hellenistic mathematics was done early, soon after the founding of Alexandria. The last of the really great works to come down to us is by Apollonius, written around 200 BC. The astronomy of Hipparchus, which drew upon Babylonian sources as well as contemporary measurements at Alexandria, was somewhat later, around 100 BC, but it does not survive. We can guess that it drew upon Babylonian arithmetic as well, because Ptolemy, who updated Hipparchus' work around 100 AD, uses Babylonian arithmetic. Ptolemy's astronomy, very impressive in its way, was the standard until the time of Galileo, over 1500 years later.
Why did Hellenistic mathematics, which seemed to be so vigorous, die out so soon? It is not as if these Greeks were pursuing dead-end questions. Quite the contrary. When their works were translated and studied again in the Renaissance, they stimulated development which has been going on continually ever since, at what seems an ever quickening pace. And it is not as if such work became impossible to do. Alexandria and its library continued to exist, and to turn out rather mediocre and derivative books, for centuries. The honest answer is that no one knows.
I can't help noticing, though, that the last great works in any location seemed to coincide with the arrival of the Romans. Archimedes was actually killed by a Roman soldier at the siege of Syracuse in 212 BC, an episode which was part of the larger Roman campaign against Carthage. There was no mathematics in Sicily after that. The last great works in the eastern Mediterranean coincide pretty nearly with the Roman arrival there (remember Antony and Cleopatra) around 40 BC. I can't help picturing the Roman expansion eastward like a cloud of poison gas, snuffing out intellectual life. The institutions remained, formally, but they were somehow no longer alive.
There is a famous account of Archimedes at the siege of Syracuse written by Plutarch, a Romanized Greek. You can hear a lot of Roman attitudes in it, and it is a very famous story. It occurs in an excerpt from his Life of Marcellus, a Roman general. (This is a clue already: Plutarch isn't writing the life of Archimedes, but Marcellus!)
An excerpt from Plutarch's Life of Marcellus:
[Marcellus is about to attack Syracuse.] "...he had erected an engine of artillery on a huge platform supported by eight galleys fastened together, and with this sailed up to the city wall, confidently relying on the extent and splendour of his equipment and of his own great fame. But all of this proved to be of no account in the eyes of Archimedes and in comparison with the engines of Archimedes. To these he had by no means devoted himself as work worthy of his serious effort, but most of them were mere accessories of a geometry practiced for amusement, since in bygone days Hiero the king had eagerly desired and at last persuaded him to turn his art somewhat from abstract notions to material things, and by applying his philosophy somehow to the needs which make themselves felt, to render it more evident to the common mind.
[Plutarch is eager to tell us how impractical and abstract Greek mathematics was, and how far from real problems.]
"For the art of mechanics, now so celebrated and admired, was first originated by Eudoxus and Archytas, who embellished geometry with its subtleties, and gave to problems incapable of proof by word and diagram, a support derived from mechanical illustrations that were patent to the senses. For instance, in solving the problem of finding two mean proportional lines, a necessary requisite for many geometrical figures, both mathematicians had recourse to mechanical arrangements, adapting to their purposes certain intermediate portions of curved lines and sections. But Plato was incensed at this, and inveighed against them as corrupters and destroyers of the pure excellence of geometry, which thus turned her back upon the incorporeal things of abstract thought and descended to the things of sense, making use, moreover, of objects which required much mean and manual labour. For this reason mechanics was made entirely distinct from geometry, and being for a long time ignored by philosophers, came to be regarded as one of the military arts.
"And yet even Archimedes, who was a kinsman and friend of King Hiero, wrote to him that with any given force it was possible to move any given weight; and emboldened, as we are told, by the strength of his demonstration, he declared that, if there were another world, and he could go to it, he could move this. [Do you recognize a reference to our starting point here?] Hiero was astonished, and begged him to put his proposition into execution, and show him some great weight moved by a slight force. Archimedes therefore fixed upon a three-masted merchantman of the royal fleet, which had been dragged ashore by the great labours of many men, and after putting on board many passengers and the customary freight, he seated himself at a distance from her, and without any great effort, but quietly setting in motion with his hand a system of compound pulleys, drew her towards him smoothly and evenly, as though she were gliding through the water. Amazed at this, then, and comprehending the power of his art, the king persuaded Archimedes to prepare for him offensive and defensive engines to be used in every kind of siege warfare. These he had never used himself, because he spent the greater part of his life in freedom from war and amid the festal rites of peace; but at the present time his apparatus stood the Syracusans in good stead, and, with the apparatus, its fabricator.
"When, therefore, the Romans assaulted them by sea and land, the Syracusans were stricken dumb with terror; they thought that nothing could withstand so furious an onset by such forces. But Archimedes began to ply his engines, and shot against the land forces of the assailants all sorts of missiles and immense masses of stones, which came down with incredible din and speed; nothing whatever could ward off their weight, but they knocked down in heaps those who stood in their way, and threw their ranks into confusion. At the same time huge beams were suddenly projected over the ships from the walls, which sank some of them with great weights plunging down from on high; others were seized at the prow by iron claws, or beaks like the beaks of cranes, drawn straight up into the air, and then plunged stern foremost into the depths, or were turned round and round by means of enginery within the city, and dashed upon the steep cliffs that jutted out beneath the wall of the city, with great destruction of the fighting men on board, who perished in the wrecks. Frequently, too, a ship would be lifted out of the water into mid-air, whirled hither and thither as it hung there, a dreadful spectacle, until its crew had been thrown out and hurled in all directions, when it would fall empty upon the walls, or slip away from the clutch that had held it. As for the engine which Marcellus was bringing up on the bridge of ships, and which was called 'sambuca' from some resemblance it had to the musical instrument of the same name, while it was still some distance off in its approach to the wall, a stone of ten talents' weight was discharged at it, then a second and a third; some of these, falling upon it with great din and surge of wave, crushed the foundation of the engine, shattered its frame-work, and dislodged it from the platform, so that Marcellus in perplexity ordered his ships to sail back as fast as they could, and his land forces to retire.
[Marcellus tries several more assaults, all driven back.]
"... However Marcellus made his escape, and jesting with his own artificers and engineers, "Let us stop," said he, "fighting against this geometrical Briareus [a Titan from mythology], who uses our ships like cups to ladle water from the sea, and has whipped and driven off in disgrace our sambuca, and with the many missiles which he shoots against us all at once, outdoes the hundred-handed monsters of mythology." For in reality all the rest of the Syracusans were but a body for the designs of Archimedes, and his the one soul moving and managing everything; for all other weapons lay idle, and his alone were then employed by the city both in offence and defence. At last the Romans became so fearful that, whenever they saw a bit of rope or a stick of timber projecting a little over the wall, "There it is," they cried, "Archimedes is training some engine upon us," and turned their backs and fled. Seeing this, Marcellus desisted from all fighting and assault, and thenceforth depended on a long siege.
[Again Plutarch insists on the impracticality of pure mathematics:]
"And yet Archimedes possessed such a lofty spirit, so profound a soul, and such a wealth of scientific theory, that although his inventions had won for him a name and fame for superhuman sagacity, he would not consent to leave behind him any treatise on this subject, but regarding the work of an engineer and every art that ministers to the needs of life as ignoble and vulgar, he devoted his earnest efforts only to those studies the subtlety and charm of which are not affected by the claims of necessity. These studies, he thought, are not to be compared with any others; in them the subject matter vie with the demonstration, the former supplying grandeur and beauty, the latter precision and surpassing power. For it is not possible to find in geometry more profound and difficult questions treated in simpler and purer terms. Some attribute this success to his natural endowments; others think it due to excessive labour that everything he did seemed to have been performed without labour and with ease. For no one could by his own efforts discover the proof, and yet as soon as he learns it from him, he thinks he might have discovered it himself; so smooth and rapid is the path by which he leads one to the desired conclusion. And therfore we may not disbelieve the stories told about him, how, under the lasting charm of some familiar and domestic Siren, he forgot even his food and neglected the care of his person; and how, when he was dragged by main force , as he often was, to the place for bathing and anointing his body, he would trace geometrical figures in the ashes, and draw lines with his finger in the oil by which his body was anointed, being possessed by a great delight, and in very truth a captive of the Muses.
[Marcellus takes the city by stealth. The death of Archimedes:]
"But what most of all afflicted Marcellus was the death of Archimedes. For it chanced that he was by himself, working out some problem with the aid of a diagram, and having fixed his thoughts and his eyes as well upon the matter of his study, he was not aware of the incursion of the Romans or of the capture of the city. Suddenly a soldier came upon him and ordered him to go with him to Marcellus. This Archimedes refused to do until he had worked out his problem and established his demonstration, whereupon the soldier flew into a passion, drew his sword, and dispatched him. Others, however, say that the Roman came upon him with drawn sword threatening to kill him at once, and that Archimedes, when he saw him, earnestly besought him to wait a little while, that he might not leave the result that he was seeking incomplete and without demonstration; but the soldier paid no heed to him and made an end of him. There is also a third story, that as Archimedes was carrying to Marcellus some of his mathematical instruments, such as sun-dials and spheres and quadrants, by means of which he made the magnitude of the sun appreciable to the eye, some soldiers fell in with him and thinking that he was carrying gold in the box, slew him. However it is generally agreed that Marcellus was afflicted at his death, and turned away from his slayer as from a polluted person and sought out the kindred of Archimedes and paid them honour. "
Roman Mathematics
This section can be short. There isn't any Roman mathematics. Their number system, still familiar to us, is even clumsier than the Egyptian, which it resembles. I am not sure how the Romans represented fractions -- I do not recall any mention of it, if they represented fractions at all, but I would guess they used the Egyptian system. The remarkable thing is that the Romans incorporated the Greek regions into their empire, but never learned any Greek mathematics. When Europeans began to recover Greek mathematics, first in Arabic translation and then in the original, they eagerly translated it into Latin so that it would be available to them. The question that comes to my mind is, why wasn't it in Latin already?? A Roman who wanted to learn mathematics would first learn Greek, and only then learn mathematics. This system produced no Roman mathematicians at all.
What Romans did get of the mathematical traditions of the eastern part of their empire was trivia, like personal horoscopes, which became popular in this period. There is also a kind of mystical fascination with the abstract science of the Greeks, and a naive credulity, which one can hear in the Plutarch section, and which reaches a bizarre high point in the didactic poem De Rerum Natura of Lucretius, ("On the nature of things"). Lucretius makes great claims for his science, of the nature of religious claims, but what ignorance! He ridicules the notion that the earth is round, although Alexandrian geometers had long before accurately determined not only its shape but its size. He is mystified how the sun gets around to the east again each morning, and suggests that it may not be the same sun which went down in the west, but a new one, etc. It is truly embarrassing.
The religious ferment of the Roman empire (see the Faustina exhibit at our Art Museum!) led eventually to the adoption of Christianity as a state religion under Constantine, but there is one last chapter to mention. Constantine was followed by Julian the Apostate (not a Christian, as you might have guessed), who tried to introduce a new religion whose principal figure was -- Pythagoras! There must have been a Pythagorean movement, because two biographies had been written shortly before, by Porphry and by Iamblichus. Both survive, and are available in modern translations. Needless to say, this was a LONG time after the historical Pythagoras, about 800 years, and it is not clear how much light they shed on him. They scarcely mention mathematics at all. On the other hand, they tell stories, which, while ludicrous in themselves, clearly point to some early knowledge which is being remembered now in garbled form. In fact, that is what all Roman science seems to be. They repeated what they did not understand very well from the Greeks. The more it was repeated, the more garbled and fragmented it became.
The Romans were great builders, there is no denying it. Apparently you didn't need much mathematics to build, as we also noted in the case of the Egyptians. [The mathematicians of the Renaissance happily cited the Romans, however, as they pressed their services on princes, dukes, kings, and sovereigns of all kinds. Surely, in the Roman example, you could see how necessary mathematics was to the well run state!]
Let us leave the sorry degenerate period that Rome ushered in, and go back to the happy days of early Alexandria. Euclid is famous for geometry, but much of the Elements is concerned with number theory, represented in geometrical form. We have briefly mentioned the prime numbers, those numbers which can't be represented as products of smaller numbers. The first primes are 2, 3, 5, 7, 11, 13, 17, ... etc. The numbers which are missing from this short, partial list have 2 or 3 as factors, so are not prime. As we go higher there are more and more ways that a number might fail to be prime, in the sense that there are more and more lower primes which might divide it. The primes become rarer and rarer. One naturally wonders if perhaps there are only a finite number of them, and beyond a certain point all numbers are divisible by something. Euclid proved that this does not happen: there are an infinite number of primes (Book IX, Proposition 20). His proof is simple and compelling: suppose there were only a finite number of primes, which we could call {p1, p2, ..., pN}. Now form the number (p1*p2*...*pN)+1, i.e, 1 more than the product of all of them. This is an integer, so it has a factorization into primes. But none of {p1,p2,...,pN} goes into it evenly! They all leave remainder 1. Thus the prime factors of this number are new, different primes, and the assumption that we had listed them all is contradicted. Since no finite list contains them all, there are an infinite number of them.
This argument is famous and beautiful, and it suggest the following easy little game. Pick a finite set of primes, find their product, add one, and factor. Verify that the prime factors are all new primes -- just for fun. It illustrates the main idea in the proof. I will do an example: I choose the set {3,7}, and form 3*7+1=22. Factoring, I get prime factors {2,11}, which are clearly new. Your assignment: do this twice, with interesting choices for the sets -- make your choices distinctive enough that no one else will have a result identical to yours!
1. The Works of Archimedes, ed. T.L. Heath, Cambridge University Press, 1897, pp. 189-194.
2. O. Neugebauer, The Exact Sciences in Antiquity, Princeton University Press, 1952, p. 79.
3. Plutarch's Lives, The Loeb Classical Library, G.P. Putnam's Sons, 1917, pp. 471-487.