## Mathematics

### 1     Introduction

Mathematics probably ranks as the Greeks’ greatest achievement, in the eyes of many modern scientists. And amongst the general public it is fair to say that some Greek mathematicians are better known than any other figures from antiquity, with the possible exception of Alexander the Great. For example, Pythagoras, Euclid and Archimedes are household names two thousand years after they lived and wrote their mathematics. The Greek alphabet, or some of it anyway, is widely known today because modern mathematics uses it by preference for symbols. For example, a, b and q for angles, the amazing p,1 S as the summation sign, and the c-square test. The Greeks gave us ‘square’ and ‘cube’ numbers, e.g. 22 and 33. ‘Squaring the circle’ is modern English idiom for an impossible task, the original task, set by the Greeks, being the mathematical problem of constructing a square with the same area as a given circle;2 Aristophanes mentions this problem.3 Famously, Plato forbade the geometrically-challenged to enter his Akademy.4 He also found mathematically interesting numbers for his ideal Republic.5 Greek mathematical texts are rich in mathematical concepts, methods and results. Like Greek plays, they contain timeless insights and truths. Also like literary works they require analysis and interpretation.

The influence of Greek mathematics continues through the ages. For example, arithmetic, music, geometry, and astronomy – pure number, applied number, stationary magnitude, and magnitude in motion respectively – began life in Plato’s ideal Republic6 and constituted the quadrivium of sciences for the late Roman and Middle Ages.7 The dramatic discovery of a proof of Fermat’s last theorem in 1993-4 shows how even current mathematical activity originates in Greek mathematical activity.8

There are essentially three approaches to the study of Greek mathematics. The traditional approach, which might be called implicit translation, is to seek the mathematical facts discovered by the Greeks, and to present them in a way which is easily comprehensible to moderns. T B L Heath’s and B van der Waerden’s works are of this type.9 The second approach is to try to understand what the Greeks were saying in their own terms (with careful attention to the manuscript tradition), which might be called explicit transliteration. J Klein’s, W Knorr’s and D Fowler’s works are of this type. We will come back to these two approaches in a moment. The third approach is to treat the mathematical texts as any other surviving literary work, and engage in source criticism: to ask questions of the aims and motives of the author in writing a particular work, in a particular genre, and for a particular audience. W Knorr began to explore these questions explicitly in his later studies (e.g. Knorr 1986).

There is a debate between implicit translation and explicit transliteration that is akin to the debate between doing classics in translation and doing classics in the original language. Mathematics is a language, a highly symbolic one. In translation, some ideas are lost and others are implanted. If Greek mathematics looks familiar to us, it is at least partly because it has been translated from ancient into modern mathematical concepts and notation. The preponderance of studies based on implicit translation has made Greek mathematics more understandable and familiar to us than it should be.10 This implicit translation extends even to what we consider the most basic feature of mathematics: number. In any ordinary ancient text (i.e. not mathematical treatises) where a number appears, that number will often appear in translation in numerical form, e.g. 100, when it usually appears in the original in verbal form, one hundred.11 As de Ste Croix commented with respect to Heath’s assumption that the Greeks performed arithmetical operations in columns, ‘[Heath] was trying in effect to smuggle into Greek arithmetic (of course without the slightest intention to deceive) a partial substitute for the place-value which is missing from the numeral notation, involving the use of a blank space corresponding to our zero. The whole conception, however, is fundamentally wrong, both on factual grounds and because it is a based on a misconception of the functioning of the alphabetic numeral notation’.12

To take a less subtle example, in most modern translations of ancient mathematical texts algebraic notation is conspicuous in the notes – or worse, the text – whereas in the Greek text it is conspicuous by its absence.13 Even in mathematical texts, in the original there is no +, or -, or ´, or ÷, or =, or even √ .14 This last is despite the fact that the Greeks spent a lot of time and effort on the problem of incommensurable quantities (e.g.√2), which they discovered. They conceived and spoke about ‘roots’ in terms of the side of a square: given a certain area – pictured as a square – what is the length of one side? That length is what we call the square root. (And if the number cannot be pictured as a square, but forms a rectangle or a triangle for example, then it cannot have a square root.15) Those readers who are uncomfortable with mathematics might find the Greek way of talking and thinking about these things more comprehensible than the modern way.16

For addition, subtraction and so on, what we find in the Greek text (instead of +, - and so on) is either nothing to indicate the operations to be performed on the numbers, or a preposition such as ‘with’ or ‘from’. How the operations are described varies from author to author. There is no completely standard word, number or operation order, so that one sometimes has to work out from the mathematical context what is meant in any particular part of a calculation: in an example given by Thomas in the Loeb17 one could read 10/71ths or 10 + 1/71ths. It is not mere pedantry to insist on the difference between these ways of referring to mathematical operations and our own. For modern mathematicians, for whom exactness, accuracy and clarity are the highest virtues, the ambiguities of some Greek mathematics would be a nightmare.

On the other hand, in order to reach understanding of something unfamiliar, we have to manipulate that unfamiliar thing with familiar concepts until we can understand it and see the differences and similarities between the two.18 It is like the word polis, used by some ancient historians in preference to any English word or phrase in order to try to avoid importing conceptual baggage from our time and language into Greek times and language. Nevertheless, when we try to explain to students what polis means, we have to use English words and phrases. Transliteration is not translation, but to understand a transliterated word does require discussion in one’s own language and concepts. In my view there is a place and a need for both approaches in the history of mathematics – and science in general – in the same way that there is a place and a need for classics education in translation and in the original. There is much of mathematical interest here, as well as of historical interest.

Increasing specialisation is the boon and the bane of modern intellectual life: boon because it advances understanding of a topic; bane because that topic is microscopic in scale relative to intellectual life as a whole, which grows out of control. We need syntheses19 to prevent the world of learning disaggregating into a kaleidescopic image of shattered fragments. If the advances of any topic are to be communicated outside an increasingly tiny field of scholars working on it and capable of understanding it in its own terms,20 then they must be made comprehensible to people outside that specialist field; they must, in short, be translated, whether from an ancient language such as Greek or Akkadian, or from a technical concept such as multiplication,21 of which there are four different types in Babylonian mathematics.

Recent studies on ancient literacy have emphasised that the non-existence of gaps between words, of punctuation, of contents, of index, and even the practicalities of handling papyrus rolls rather than sheets of text, must have had an impact on the reading habits of the ancient Greeks.22 In like manner the use of letters for numbers23 and the lack of symbols for mathematical operations – on top of the usual literary habits – must have had an impact on numeracy.

I will first discuss numbers in everyday life, then how numbers were written, and then geometry. I am starting with low-brow mathematics, such as practical computation, because our evidence for this kind of maths is a lot earlier than that for high-brow mathematics, which does not really get going until the Hellenistic period. Practical mathematics is also tied intimately to the history of the period which created and used it.

### 2     Ready reckoning and logistike

Historians of Greek mathematics tend to concentrate on what might be called ‘high mathematics’: the geometrical wizardry and axiomatic certainties of e.g. Euclid. In this they are following a long established tradition. The study of mathematics has long been influenced by the quest for general concepts and results – since Euclid, in fact. Practical computation was looked down on by Plato24 and by some others henceforth, and philosophers held a more or less dismissive attitude toward the men who actually counted or calculated things. The distinction was recognised in ordinary Roman life. For example, the calculatores were explicitly barred from the tax exemptions granted to some teachers and healers in the Roman empire.25 This type of mathematics was the type which was most used in everyday life, however, and it may have had an influence on the development of ‘pure’ mathematics too.26 A few ancient historians with interests in financial matters have discussed the type of reckoning used for official (state) and private purposes.27

A certain level of numeracy was required not just by traders and money-lenders calculating yields.28 Citizens serving as state officials needed to be able to measure, count or compute figures which were relevant for their office, and other citizens serving as state auditors needed to be able to check those figures. The most primitive type of reckoning, namely keeping tallies, began with notched sticks, and continues with the chalk marks on slates and five-bar-gates on paper we still make today. Darius used such a tally device when he left with the Greek force guarding the bridge over the Bosphorus a leather thong with 60 knots in it, and told them to undo one knot each day (Hdt 4.98). If he was not back by the time they untied the last knot, they could leave. With a method such as this, one does not need to be able to count beyond 1; one simply needs to ‘accumulate’ or ‘tick off’ notches, marks, strokes, knots or any other kind of marking system on the principle of a one-to-one correspondence. In the developed democracy, treasurers of all sorts and officials such as the logistai, poletai, booty-sellers on campaign – even the juror concerned with his pay – all knew how to use more sophisticated methods of reckoning. In Aristophanes’ comedy about jurors, the main characters perform a quick calculation on stage (Wasps 656-664, Sommerstein trans, slightly modified):

ANTIKLEON: ‘First of all reckon up roughly, not with counters but just on your hands, the amount of tribute that comes in to us altogether from the allied states, and apart from that the taxes, one by one, and the many one-hundreths, court fees, mines, markets, harbours, rentals, confiscations. We get a total for all these of nearly two thousand talents. Out of this now put down pay for the jurors for a year, six thousand of them (never yet have more this land inhabited): we get, I think, a hundred and fifty talents.

PROKLEON: ‘You mean our pay doesn’t even come to one tenth of the revenue!

‘Roughly, on your fingers will do’ suggests that finger-counting is merely a second-best method available to the ordinary Athenian in the street, who can also handle very large numbers with relative ease. Unfortunately, whilst the answers to sums are often given in literary or epigraphic sources – as in the Aristophanes’ passage just quoted, or the Athenian Tribute Lists – the methods by which the sums were calculated are not.

Given the size of the numbers and the preponderance of 60s as well as 10s in Athenian financial units, the finger-counting method involved was perhaps that which uses the 3 bones of each finger (but not the thumb which does the pointing operation) on one hand, so that one can count to 12 using one hand, and then use each (whole) finger of the other hand to record the powers of 12, leading to a total of 60 when all fingers and thumb of the second hand are employed. The number 12 is a much more useful base for practical purposes than 10, for, as Galen 29 remarked, 12 has four divisors (2, 3, 4, 6), in contrast to base 10’s two (2, 5), and is not less convenient anatomically than the decimal base. Or perhaps Prokleon used a much more complicated system of finger-counting of the type explained by the Venerable Bede 30 in the C7 AD, which seems to have a long pedigree, at least for the smaller numbers (Bede stops at a million).

Besides magistrates, significant numeracy was needed by anyone involved with state dues. The obvious categories are traders, tax-farmers and litigants. Traders paid (normally) 1/50th as a tax on all imports and exports (the pentekoste, normally subject to implicit translation and called the ‘two-percent’ tax). Tax-farmers needed to estimate fairly accurately what they could collect before submitting their bids for e.g. the pornikon telos (the prostitute tax) or the metoikion (the metic tax) or risk financial ruin. Numeracy was needed by litigants to calculate fees, fines and rewards. Most people using the Athenian courts had to pay court fees. In private cases the fees were called prytaneia, and were set at a proportion of the value of the case.31 In public cases there were two sorts of fees, parastasis, about which we know no details, and parakatabole, which was a deposit of 1/5th or 1/10th of the value of the case (depending on the type of case), which was forfeited by the litigant who lost or the prosecutor who failed to secure 1/5th of the votes of the jury. Some penalties for the convicted and rewards for the plaintiff were also set as a portion or multiples of the total value of the case. An example of the first is the epobelia, a fine of 1/6th of the value of the case.32 An example of the second is the reward for the successful prosecutor of 1/3rd of the value of the property confiscated from the convicted defendant in an apographe lawsuit.

The Athenians of Perikles’ day were familiar with computations involving six. For example, there were 6 obols in a drachma, 60 minas in a talent, 6,000 drachmai in a talent , and 1/60th of imperial tribute was dedicated to Athena. This does not mean they worked on base 6, or base 60 (the sexagesimal system) in ordinary life. They worked in base 10, like we do. Just because sixes and sixties feature strongly in their system of units for money and financial thinking it does not follow that they used a base 6 or 60 counting system. It is easy to recognise what system is at work by looking at which number names are abbreviated in the acrophonic system: five, ten, one hundred, one thousand, ten thousand, and no others; or at the groups in the alphabetic system: the first nine letters represent 1-9, the next nine letters represent 10-90, and the third nine letters represent 100-900 (on which see further below §3). The fact that the monetary units used in Athenian financial inscriptions involve multiples and fractions of 6 (and 100) is incidental to the base system in operation and the way they counted and recorded quantities.33

The Athenian tribute lists, which record the monies given to Athena, specify the amounts given by the different allies down to obols: it is debated whether these apparently very precise figures were arrived at by computation of 1/60th of what was given, or merely reflect conversion into Attic currency of monies paid in foreign staters or darics, or even in plate, which could have been weighed and then recorded in Attic currency figures.34 Quantification in antiquity was a problem only for those interested in comparisons, like us. Across the Mediterranean there were numerous different systems of weights and measures in operation simultaneously. But most people lived and worked in one area with one system, and were probably as, if not less, interested in foreign weights and measures as most of us are. The variety presents a problem for the historian, however, especially those seeking to compare ancient with modern answers. For example, the various different lengths of the stade is a notorious crux in Eratosthenes’ computation of the circumference of the earth: take one particular length of stade, and his answer was very close to ours; take another, and his answer is a lot less accurate.35 The same problem could have troubled the ancient reader of any ancient text referring to a specific quantity of a specific measure, if accuracy was important, especially if a measure by a certain name was not in use in their area. ‘Handful’, on the other hand, would give a rough idea to anyone reading it, and would do for most purposes. Consequently such approximate measures, usually based on body parts, tend to dominate in texts with practical applications.

Alexander the Great was very concerned with logistical matters: he needed to be, to keep his army alive. Any number of men, moving through enemy territory, were able to carry not more than 7 days’ supplies (at most 5 if required to carry their own water too, even on half-rations).36 They depended for their survival on securing sufficient supplies from urban centres with stored food, or being able to live off the land, which was possible only at certain times of year. Alexander did not normally depend on supply lines, but relied on capturing adequate supplies as he moved forward. For this purpose he sent out reconnaissance parties, gathered intelligence from the natives, and sometimes split his army into two or more smaller units following different routes to reduce the logistical demand on the routes chosen. Troop numbers were fluctuating constantly as some died, some veterans were settled or sent home, and some reinforcements arrived. And these changing logistical demands had to be worked out at least several days in advance to be of any use. Someone amongst his advisors, if not he himself, was performing the required calculations during the entire expedition. Troops were also paid, and got into debt, records of which were kept with the army on the move, e.g. his troops were said to have been endebted to the tune of either 10,000 or 20,000 talents37 by the time they got back from India to Susa in 324.

One of the best set of accounts that de Ste Croix could find in all surviving Greco-Roman documentary evidence was two soldiers’ pay sheets,38 dating from 83-4 AD. In the late empire the military again seems to have been more precise and accurate than most other groups in life, or so Vegetius opines in his Epitome of Military Science 2.19: ‘The administration of the entire legion, including special services, military services, and money, is recorded daily in the Acts with one might say greater exactitude than records of military and civil taxation are noted down in official files’ (Milner trans.). The officers responsible were called the librarii (§2.7), and since soldiers might receive single pay, pay and a half, or double pay, some computations were required. We also hear, in §2.20, details about the keeping of accounts of soldiers’ savings depositied ‘with the standards’. Calculations of a different sort were needed for pitching camp: ‘the general does not go wrong when he knows what space can hold how many fighting men’ (§3.15, Milner trans.). The mensores paced out the square footage for the camp and the parts within it, every time they set camp (which was frequent). Further calculations were sometimes required during the actual fighting: for example, Vegetius gives us an empirical and a calculated method to find the height of a city wall (§4.30): fire an arrow with a thin thread at the top of the wall, then measure the length of the thread. Or measure the wall’s shadow, and a ten-foot rod’s shadow, and compute the wall’s height from the ratio. This is to find the solution to the perennial problem of ensuring that scaling ladders are neither too long nor too short for the job in hand.39

The mention of pebbles in some literary contexts, such as ‘In writing and reckoning with pebbles the Greeks move the hand from left to right, but the Egyptians from right to left’ (Herodotos 2.36.2), implies the use of abacus, but the nature of such an abacus in Greek society is very uncertain. It is debated whether the Salamis stone (and similar slabs found in Greece) is an abacus or a gaming table.40 All certain surviving ancient examples of abaci are Roman in date. Pebbles (psephoi) feature in C5 and C4 Greek society most strongly in political contexts, for voting in the assembly or the courts, giving rise to the term psephisma for ‘decree’, something decided by vote. In the C4 any old pebbles were replaced by official bronze disks, which nevertheless retained the name and thus extended the meaning of psephoi. In early Greek mathematics pebbles are most obvious in the context of figured numbers, where their use laid out in regular patterns explains the notion of square, triangular, rectangular or any other shape numbers.

Simple lines in the dust and any old tokens or markers, such as pebbles lying to hand in the same dust, can serve as an abacus or calculating tool.41 All that is needed is some straight lines to make rows or columns, and something to put in those columns. Labels for each row or column are handy but not necessary if the user is familiar with the practice of using an abacus and has a regular method of working (left to right, or right to left, top to bottom, or bottom to top; start with smallest unit and progress to largest, or vice versa.) The Salamis stone is very big (5’ x 2’5"), very heavy (solid marble), and must have been very expensive to make (apart from the cost of transporting the stone, it is carefully chiselled). It is dated to the C6 or C5 BC, and it was sited in a sanctuary.42  Thus it is probably a large version of a smaller scale object made and used by mortals who thought that their god would like one too, in the same way that they gave huge dress pins to goddesses. This does not help resolve whether it was an abacus or gaming board however.

Lang tried to show that some errors in surviving Greek arithmetic in literary and epigraphic sources could have arisen more easily through careless use of an abacus than through mistakes in written calculations. Personally, I do not believe we have enough evidence to decide the matter. Herodotos43, for example, may or may not have used an abacus of some sort to compute the number of medimnoi of grain consumed per day by Xerxes’ army (for which he failed correctly to divide 5,283,220 by 48). As for Athenian financial documents, the problems are legion. Most stones are incomplete; where they are sufficiently complete, rarely is there any totalling; when there are totals that we can check, errors are not uncommon.44 If the purpose of these records was to assert the honesty of the officials responsible for the monies concerned, and if they were ‘checked’ by the auditors whose function it was to check them, one wonders whether it is incompetence, disinterest, or fraud that is preserved in these errors. I think it unduly optimistic to assume that all errors are accidents of calculation or chisel, and that if we ‘correct’ here and there the sums will add up: some of them may not add up because they never did add up.

### 3     Numbers and arithmetike

Our number symbols 0, 1,..., 9 are called Arabic numerals, and they mean nothing but numbers to us. They also meant nothing but numbers to the Arabs, who got them from India, which was an area of many mutually foreign languages and alphabets. The Greeks did not have dedicated symbols for numbers. To understand the significance of this, consider letters: for us, as for the Greeks, alpha is just alpha. For an ancient Semite, however, it was ‘ox’ badly pronounced.45 So it is with numerals: in the society which first uses them, they have meaning beyond number. Abstracted from that context, they can take on new meaning. So Ptolemy’s use of ‘o’ as an abbreviation for ouden (nothing) in a particular context46 could be read as a symbol for zero in India.47

The earliest Greek formal method of representing cardinal numbers (one, two, three etc.) is the acrophonic system.48 Besides | for the unit, the first letter of the word for five different numbers is used as an abbreviation for that number. The five are:

P for pente, five (the right leg is drawn somewhat shorter than the left);

D for deka, ten;

H for hekaton, one hundred;

X for khilioi, one thousand; and (a relatively late arrival)

M for myrioi, ten thousand.

Compound numbers for 50, 500, 5,000 and 50,000 were made up of small versions of the ten power being drawn inside a large P. Numbers in between were additive, so that one thousand two hundred and thirty-four, for example, was written XHHDDD||||. This is how numbers appear on the many official stelai from C5 and C4 Athens.

This was replaced in due course and in mathematical contexts by the alphabetic system, which used a slightly modified Greek alphabet of 27 letters – the ordinary Greek alphabet has only 24 letters, so another 3 were adopted from the Phoenician alphabet.

FIGURE 1 to follow

The first nine letters represent the numbers 1 to 9. The next nine represent the tens from 10 to 90. The third nine letters represent the hundreds from 100 to 900. This is not a place-value system, and large numbers are sometimes written in ascending order, sometimes in descending. So 111, for example, could be written as ‘air’ or ‘ria’. Since ‘a’ means 1, ‘i’ means 10 and ‘r’ means 100 wherever they stand in the order, the order really doesn’t matter. To put it another way and illustrate the point about letters serving as numbers, 111 could be written as ‘air’ or ‘ria’. By transliterating the Greek letters into English ones I trust that ‘air’ will have been read as ‘air’ and not as ‘one hundred and eleven’, despite following immediately upon the statement that this is a way of writing the number 111, and being enclosed in apostrophes, as Greek letters serving as numbers were (usually) signalled by some distinguishing mark. For ancient Greek readers, Greek letters will always have been letters first and mathematical symbols second, whereas for us they are abstract symbols first and Greek letters second.

Moreover, to the Greeks, numbers themselves were less abstract than they are today, particularly the sort of small numbers used commonly in ordinary life. This is best illustrated in the existence of the dual in Greek grammar (when there are two somethings, so singular for one, dual for two, and plural for three or more), and in the fact that Greek (and Roman) numbers up to four are adjectives which are declined to agree with the subject in case and gender. Beyond four they are indeclinable,49 but the inflection is carried in the definite article which accompanies the number. This reveals how the thing and the quantity of the thing are connected in thought – they must agree. ‘One’ is even irregular, apparently coming from two different roots: heis or hen, when the one is male or neuter, but mia when it is female.

So far, we have considered some aspects of practical computation and the Greek number system that underpinned them. The number system concerns whole numbers and fractions. In practical computation for building and surveying however, quantities can be more complex. Some arise geometrically as ratios, and turn out to be irrational numbers.50 A rational number is one which can be expressed as p/q, where p and q are whole numbers and q≠0. An irrational number cannot be so expressed, e.g. √2. The irrationals and their approximation by rationals is an important topic for historians of Greek mathematics for many reasons. Greek geometry involves irrational ratios and measurements that can be manipulated easily by geometrical methods but not represented in their number system and therefore in calculations.51 These ratios arise naturally, commonly involving √2, √3, … or π. It is commonly held that there was a crisis in Greek mathematics caused by the discovery of irrationals, specifically of quantities inexpressible in the number system. This traditional view sees the crisis as causing a shift in the focus of mathematical activity from number (on which e.g. the Pythagoreans had concentrated) to geometry.

Recently this view has been challenged. Fowler (1990 p. 10) has argued that early Greek mathematics (up to the C2 BC) was ‘completely non-arithmetised’, but Knorr disagrees with him.52 The idea of a crisis in the foundations of Greek mathematics presumes a solid body of Greek arithmetic, the existence of which Fowler disputes. This debate about exactly how the archaic and classical Greeks thought of quantities has implications for and should be related to other disciplines, e.g. architecture and astronomy, which currently presume the traditional view. In architecture, for example, de Jong 1989 argues that the proportions used in temples built before c. 400 BC are arithmetical (e.g. 1:3), whilst from c. 400 BC geometrical proportions take over (especially ). Meanwhile, in astronomy, Berggren 1991 says that until the mid second century BC perfectly adequate arithmetic methods were used in Greek astronomy, whereafter they were succeeded by geometrical (spheric) methods. He suggests that Euclid’s Phainomena was the first attempt to provide a well-founded (geometric) method for making calculations which had previously been done by arithmetic methods. These calculations included the rising or setting times for certain stars or signs, or the length of daylight at any particular location on any particular day of the year. Berggren suggests that Euclid did this by adapting the pre-existing body of knowledge on solid geometry to fit the two-sphere model of Eudoxus (one sphere for earth, with the celestial sphere around it).

### 4     Geometry

Geometry is arguably the Greeks’ greatest scientific achievement. Unlike, say, Aristotle’s physics or Galen’s medicine, Euclid’s theorems are still true and his methods are still admired. For millenia his books have been studied and referenced, though they are no longer used as a school text-book.53 He entitled his principal work Elements, and it was intended to be a foundational work in the subject, a starting point. The same Greek word (stoikheia) also means the letters in the alphabet, and Euclid’s elements are to geometry what letters are to language: the building blocks or basic components.

One of the most outstanding features of Euclid’s work is its structure: the first book contains a number of definitions, postulates and common notions, and the following twelve books endeavour to introduce or assume no extraneous material as they progress, but only to construct from definitions and propositions already done. Thus, for any proposition one can trace back the reasoning for a particular result through earlier propositions until one comes back ultimately to the original postulates and common notions. This trace can be illustrated by drawing a proof tree, of which an example is given below in Figure 3, to illustrate the reasoning for Pythagoras’ Theorem. Of course Euclid was not infallible, and there are occasionally holes in the arguments, but these should not be allowed to detract from the overall aim and success of his method. Another outstanding feature is the thoroughness with which propositions are proved, as will become apparent in the example given below. Let us first review the Elements.

Book 1 builds from twenty-three definitions, five postulates, and nine common notions.54 The definitions explain the basic terms of geometry, what is meant by words such as ‘point’ or ‘line’. The common notions are axioms or self-evident truths; statements that any sensible person would take as true, although it is not possible to prove them. For example, Common Notion 1 is ‘Things which are equal to the same thing are also equal to one another’. The postulates are unproved assertions about geometry. The first three postulates are assertions that amount to the possibility of doing geometry.

Postulate 1 ‘[It is possible] to draw a straight line from any point to any point’.

Postulate 2 ‘[It is possible] to produce a finite straight line continuously in a straight line’.

Postulate 3 ‘[It is possible] to describe a circle with any centre and diameter’.

The fourth and fifth postulates are different: they are premises which the beginner must accept as given, and their validity, and their classification as postulates (rather than as e.g. axioms), have been subjects of contention among mathematicians since antiquity.

Postulate 4 ‘All right angles are equal to one another’.

Postulate 5 ‘If a straight line falling on two straight lines make the interior angles on the same sides less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles’.

Euclid thought that this last assertion was undemonstrable, and hence made it a postulate. The fifth (parallel) postulate is notorious. Attempts to prove it from the preceeding four started shortly after, and the problem attracted mathematicians for two millenia. This research led to many postulates which are equivalent with the fifth. In Gerolamo Saccheri’s Euclides ab omni naevo vindicatus of 1733 an equivalent postulate is rejected in the hope of deriving a contradiction, and thereby proving the fifth postulate by reductio ad absurdum. But the argument failed. Consequently Saccheri’s work produced the first theorems of what is called non-Euclidean geometry.55

After the definitions, common notions and postulates, Book 1 discusses triangles, parallels and parallelograms. Book 2 gives two more definitions, and then carries on to the transformation of a rectilinear area of any shape into a parallelogram of any shape. Book 3 gives eleven more definitions, to deal with the circle. Book 4 has another seven definitions and moves on to deal with triangles and regular polygons which are drawn inside and around circles. Book 5 gives eighteen more definitions and introduces the theory of ratios, most of which was worked out by Euclid’s famous predecessor Eudoxus. Book 6 adds four more definitions and applies the theory of ratios to plane geometry. Books 7, 8 and 9 start with twenty-two definitions at the beginning of Book 7, and then deal with arithmetic and the theory of numbers.56 Book 10, after four more definitions, deals with the irrationals (alogoi or arrhtoi, lit. ‘inexpressible’). Books 11, 12 and 13 turn to solid geometry, introduced with twenty-eight definitions. So the entire work of 13 books, covering this huge range of mathematical topics, rests on a mere five postulates and nine common notions. On those few assertions which constitute a minimal undemonstrated bedrock, the framework of ancient geometry was constructed, in belt and braces fashion, proof by proof. It is a tremendous intellectual achievement.

Pythagoras’ theorem appears as Proposition 47 in Book 1, and after giving the text I will sketch part of the proof tree, showing how each part of the argument builds upon prior material. Geometry is all about diagrams, and the text of the proposition is easily understood by consulting the diagram constantly. As in botany, a picture can speak a thousand words, and we can see that something is true, even if we do not understand the proof that it is so.

[FIGURE 2 to follow]

[Proposition]

‘Let ABC be a right-angled triangle having the angle BAC right; I say that the square on BC is equal to the squares on BA, AC.

[Proof]

For let there be described on BC the square BDEC, and on BA, AC the squares FB, GC,57 and through A let AH be drawn parallel to either BD or CE, and let AD, JC be joined. Then, since each of the angles BAC, BAF is right, it follows that with a straight line BA and at the point A on it, two straight lines AC, AF, not lying on the same side, make the adjacent angles equal to two right angles; therefore CA is in a straight line with AF.58 For the same reasons BA is also in a straight line with AG. And since the angle DBC is equal to the angle JBA, for each is right, let the angle ABC be added to each; the whole angle DBA is therefore equal to the whole angle JBC.59 And since DB is equal to BC, and JB to BA, the two [sides] DB, BA are equal to the two [sides] BC, JB respectively; and the angle DBA is equal to the angle JBC. The base AD is therefore equal to the base JC, and the triangle ABD is equal to the triangle JBC.60 Now the parallelogram BH is double the triangle ABD, for they have the same base BD and are in the same parallels BD, AH.61 And the square FB is double the triangle JBC, for they have the same base JB and are in the same parallels JB, FC. Therefore the parallelogram BH is equal to the square FB. Similarly, if AE, BK are joined, it can also be proved that the parallelogram CH is equal to the square GC. Therefore the whole square BDEC is equal to the two squares FB, GC.62 And the square BDEC is described on BC, while the squares FB, GC are described on BA, AC. Therefore the square on the side BC is equal to the squares on the sides BA, AC.

Therefore in right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle; which was to be proved.63

[FIGURE 3 to follow]

Because of the mathematical importance of the Elements, the text and its debt to earlier mathematicians have been discussed in some detail in several general histories of mathematics64 as well as in histories of Greek mathematics.65 Such histories are usually addressed to mathematicians.

The Greek discovery of geometrical magnitudes that cannot be expressed as rational numbers – such as the diagonal of a square with sides of size one unit – is remarkable. The fact is used twice by Aristotle as a (presumably) well-known example of argument by reductio ad absurdam (Prior Analytics 1.23.41a26 and 1.44.55.a37), and is clearly older than him. The irrationals present problems for theories of geometric magnitude such as Eudoxus’ theory of proportion which underpins Elements book 5 and the construction of irrational magnitudes in book 10. The C5th and C4th origins of the theory of incommensurable magnitudes in the work of Pythagoras, Theodorus, Theaetetus, Archytas and Eudoxus has been examined in depth by Knorr 1975.

The presumed arithmetical basis of Greek geometry has been challenged by Fowler in The mathematics of Plato’s Academy. For the ancient historian without sufficient maths to follow the technical arguments in detail here (and I am amongst them), we must consider the other arguments used in the debate. Fowler has a very strong such argument in the following: surviving MSS and papyri are all from the end of the C3 or later, and 99% of them (p. 219) are from the C9 AD or later. There are only a few isolated scraps and tatters, amounting to 1% of the total, for the period from the end of the C3 BC to the C9 AD. People of the C9 AD copying old texts available to them but lost to us probably engaged in implicit translation66 in order to make those ancient texts comprehensible in their own times (we are dealing with a gap of up to and over a thousand years between e.g. Plato or Euclid and these scribes). From this very sound argument Fowler deduces that whatever Greek mathematics was in Plato’s or Euclid’s times, our evidence is filtered through the light of an arithmetic tradition that entered Greek mathematics only in the C2 BC, and developed continually, leaving ratio theories aside. This methodological argument can stand alone, and is relevant for ancient science in general, for the same circumstances prevail.67

The gaps in Euclid’s reasoning are of two kinds. First there are implicit assumptions, some of which qualify as axioms, such as the continuity of lines and circles: if two lines or circles meet, a missing postulate is required to say that they do so in a point (rather than in a gap between points). Secondly, some proofs implicitly depend on diagrams that do not depict all possible cases, and so some of the theorems, while true in general, have proofs for special cases only.

The achievement of Euclid’s Elements is its amazing organisation, and the rigour, depth and scale of its analysis. Modern mathematical standards originate in the Elements. The work has a superb order of results, exemplary proofs, a brilliant choice of axioms, and hundreds of theorems. And Euclid has consequently had a profound influence on the development of mathematics and its application, not least through non-Euclidean geometry.

#### Notes

1. For an enjoyable, as well as educational, book on this, see Blatner 1997.

2. Using only ruler and compass and a finite number of steps, to keep within Euclidian terms. This was proved to be impossible to do in 1882 (when Lindemann proved that π was a transcendental number).

3. Birds 1001-5, Meton speaking, production date 414 BC.

4. Famously but probably not accurately, for this story comes from an author who wrote 1400 years after Plato lived, Tzetzes, Khiliades 8.972-3.

5. E.g. 5040 citizens, a number with 59 divisors including all the numbers from 1 to 10, as he observes, Laws 737e, 738a.

6. 7.521c-31c, on which see Fowler 1990 chapter 4.

7. On which see Stahl 1971. The other three subjects (the trivium) were grammar (which covered language and literature), dialectic (logic), and rhetoric (expression).

8. Fermat’s famous last theorem was written in 1637 in the margin of his copy of DiophantosArithmetica, Book 2, next to Problem 8: ‘to divide a square number into two other square numbers’. This inspired Fermat to assert that ‘It is impossible to divide a cube into two other cubes, or a fourth power or, in general, any number which is a power greater than the second into two powers of the same denomination’. Or in modern symbolic terms, the equation xn + yn = zn has no integer solution when n is greater than 2. This is in contrast to Pythagorean triples, where x2 + y2 = z2, of which there are, as Euclid proved (in words not algebra), an infinite number of solutions. On Wiles’s proof of Fermat’s theorem, see Singh 1997.

9. E.g. Heath 1897, van der Waerden 1983. Anglin and Lambek 1995 continue this tradition with the dubious defence that ‘a presentation faithful to the original sources, while catering to the serious scholar, would bore most students to tears’ (p. vi). Even if true for students of mathematics – for whom the book was written – the opposite is more likely to be true for students of Greece and Rome.

10. See for example Heath’s chapter in Livingstone 1921. Unguru’s complaint about this sort of implicit translation, made 20 years ago (1979), is repeated and developed by Knorr 1991.

11. In fact this statement applies even to some mathematical texts, but these are rare cases; see Fowler 1990 p. 221. See Pliny NH 33.133 on the increase in the magnitude of numbers needed during Roman history (‘in the old days there was no number standing for more than one hundred thousand’). The same thing happened before and after: Homer calls ten thousand just that, deka khillioi, but later Greeks used myrioi (which hitherto meant countless) for ten thousand. The word million appeared in fourteenth century Italy. Now we have billion and trillion. Greek and Roman methods of measuring and numerating are well treated in Richardson 1985, which is especially useful for people reading ancient texts in the original, as it explains all the common forms of expression of mathematical ideas and technical usages, with the focus squarely on non-mathematical texts such as the Athenian orators, Cicero, and the Greek and Roman historians.

12. de Ste Croix 1956 p. 56, with full defence of this charge on pp. 56-9.

13. The same problem dogs Mesopotamian mathematics. For an excellent discussion of that, with frequent, relevant and important reference to Greek mathematics, see J Høyrup 1996. As Høyrup showed elsewhere, the operations performed in so-called Babylonian ‘algebra’ are not (and could not be) arithmetical operations with numbers. They are analytical operations on geometric figures.

14. All these symbols are post-antiquity. The symbol + is first used as an abbreviation for latin ‘et’ (and) in an MS dated 1417. The symbols + and - first appear in print in 1489, where they refer to surpluses and deficits in business problems, not as operations or positive/negative numbers. The symbol √ is first used in 1525. The equality sign first appears in print in 1557. The multiplication symbol ´ first appears in print in 1618, and the division symbol ÷ was first used in 1659. See Cajori 1928-9.

15. Figured numbers are well discussed in Heath 1921 pp. 76-84.

16. As Fowler points out (1990 p. 21), the Greeks think and talk about their geometrical figures literally, as if the shapes were in front of them, being manipulated by hand. We, by contrast, turn geometry into arithmetic and then turn arithmetic into algebra, and think and write about the subject abstractly. See also his comments p. 68.

17. Greek Mathematical Works vol 1 p.45. This example from Archimedes, and the text (to which Thomas does not refer) is Measurement of a circle, given on pp. 316-32, with the offending text on p. 332. Fowler p. 240f contrasts his own similar but more literal translation of this part of the text with Heath’s description of the same proposition (there reproduced for convenience), which well illustrates the ‘maths in translation’ versus ‘maths in the original’ debate.

18. Bunt, Jones and Bedient 1976 put the familiar and unfamiliar side by side for Greek, Egyptian and Babylonian mathematics; see esp. § 6.12 on ‘The difference between the Euclidean and the modern method of comparing areas’ and chapter 7, ‘Greek mathematics after Euclid: Euclidean versus modern methods’.

19. Hakfoort 1991 tries to explain both the absence of syntheses over the last generation and the sort of synthesis which might now be written in a ‘post-positivist philosophical vacuum’. Lindberg 1992 is a traditional type of synthesis and has been well received, though is better on medieval science than on ancient. Serres 1995 is a synthesis in the French style, which rejoices in the variety of philosophical and disciplinarian views held by its various contributors. It is well worth reading, but poorly referenced for the Greek chapters.

20. For example, Høyrup admits (1996 p. 22) to being the sole representative of a certain approach to ancient mathematics, namely, ‘recasting theories about the transmission of Babylonian mathematical knowledge and techniques to later cultures (with appurtenant transformation) and about the relation between practitioners’ mathematics, scribal mathematics and ‘scientific’ mathematics’. Most of the ‘recasting’ concerns the contextualisation of mathematics in the culture which produced it.

21. ‘Multiplied by’ is often expressed by epi plus dative in a mathematical context. This is the term used in expressions of interest rates (tokoi), for example. The Greeks were more likely to add than multiply, even for a sum as simple as 5 lots of 9: see Gow 1884 p. 51, where there is also a concise explanation of the ancient way of conceiving division, or Fowler 1990 pp. 14-16. There is an excellent explanation of technical terms in Greek mathematics (with special reference to Apollonios) in Heath 1896 pp. clvii-clxx.

22. There is a demonstration of Greek writing habits in Fowler 1990 p. 205. On literacy in general see e.g. Harris 1989 or Thomas 1992.

23. Thus letters could be mistakenly read as numbers and vice versa. In the Codex Constantinopolitanus, for example, some scribe took the word ‘lemma’ for a fraction (lh = 1/28th m = 1/40 th ma = 1/41 th) and included it in a computation. See Bruins 1964 vol. 3 p. 221 on fol. 77r. Apollonios of Perga (amongst others) even played with the double meaning of letters, by adding up the values of the letters in a poem to demonstrate his method of expresing large numbers (in the tradition of Archimedes’ Sand-reckoner); Heiberg 1922 p. 65. This dual meaning of letters offered an easy method of coding (and mystifying) information, expressing words as numbers.

24. See e.g. Philebus 56d-57d, where a distinction is drawn between the calculation and measurement employed in building and commerce and the calculation and geometry practiced by philosophers, the latter being described as ‘far superior’. He takes a different view in Laws 817e-820 (esp. 819c).

25. See e.g. Digest of Roman Law 27.1.15.5 (Modestinus on immunities). It is however interesting that enough of a case for their inclusion had been made to warrant the writing into to the law that they did not qualify! For discussion of the privileges in question, see Duncan-Jones 1990 pp.160-3.

26. As Fowler notes 1992 p. 134 n.4, commercial practice certainly effected mathematics in Renaissance Italy, and a treatise on decimal numbers which is now considered fundamental was dedicated to a lively assortment of trades and professions using calculations in 1585.

27. The best of which is still de Ste Croix 1956, on which see also Macve 1985. Also Tod 1950 and references there to his earlier papers on Greek numeral systems and notation.

28. This is how we should normally think of returns on loans, rather than as ‘interest’, which is expressed (and calculated) with reference to time, since Greek returns are not intrinsically tied to the passage of certain lengths of time. See Cohen 1992, esp. pp. 44-6

29. Affections and errors of the soul 2.5 (5.83-4 K). Ancient scientists were polymaths and their works do not fit neatly into modern disciplinary categories. One would not today expect discussion of this point in a text about the soul, written by a physician.

30. The Reckoning of Time 1, which concerns calculating and speaking with the fingers. Now available in English translation by F Wallis 1999.

31. If the case was worth less than 100 drachmai, there was no fee. For a case valued between 100 and 1000 drachmai, the fee was 3 dr. payable by each party; for over 1000 dr. the fee was 30 dr. per party. See Harrison 1971 p. 93 for details.

32. The name refers to the obol, because this penalty is calculated at one obol fine per drachma value.

33. We can also look at the small denominations: the khalkous, krithe and lepta were 1/48th, 1/72th, and 1/336th of a drachma respectively. None of these fractions are obvious ones to choose in a sexagesimal system (and there is no coin representing 1/60th), but all are divisible by 6.

34. See Vickers 1992. Greek coins had bullion value, they were not tokens, so precious metal objects could serve as large denominations.

35. Stades about which we have reliable information and which were in use in Eratosthenes’ time vary from about 7.5 stades to a Roman mile to 10 stades to a Roman mile. On the scale of the circumference of the earth, the difference is very significant.

36. ‘The ratio between the army’s consumption rate and its carrying capability remains constant no matter how many personnel or pack animals are used to carry supplies’, Engels 1978 p.21.

37. The lower figure is given by Curtius and Plutarch; the higher by Arrian and Justin.

38. de Ste Croix 1956 pp. 39-40, with accompanying Plate III and Figure IV.

39. A problem on which Polybios had expounded at some length, 9.19, having criticized Philip of Macedon for making mistakes in this area (5.97.5-6).

40. See Lang 1964, Pritchett 1965, Lang 1965.

41. Alan Turing showed in 1936 that any calculation can be made with just 2 tokens, symbolically represented by 1 and 0 using a Turing Machine, which is nothing more than a set of simple rules. Modern computers work with just these two symbols, translated for practical purposes into electrical ‘on’ and ‘off’.

42. This is true of other tables thought to have been either abaci or gaming tables, such as that found at the Amphiairion.

43. Herodotos 7.187. Work out the correct answer for yourself! Note that the Grene translation in the Chicago 1987 edition has an error here: the result that Herodotos gave should be 110,340 not 1,100,340. Numbers are as prone to error now as they were in Herodotos’ time, it seems, whether arising from innumeracy or copying/typographic slips. Even though he apparently got it wrong, it is interesting that Herodotos thought of performing this calculation.

44. Errors are very common in literary texts, including those originals which have survived on papyri and which errors cannot thus be attributed to poor transcription by uncomprehending copyists, as they may be with medieval MSS. Byzantine scholars, at least, not infrequently corrected errors in the original rather than introduced them.

45. ‘Aleph’. " was the symbol for ox: it is clearly the head and horns. In Greek hands, abstracted from ox, it fell over, and then came to be drawn upside down, as A. The original aleph symbol is revived in modern mathematical logic, where it is the standard notation meaning ‘for all’.

46. The contexts in which ‘o’ appears are fractions or positions in sexagesimal place-value notation, such as the ‘Table of straight lines in the circle’ (i.e. chords) in Almagest 1.11, where Ptolemy needs to indicate nothing in a particular place.

47. Flegg 1989 p. 106f and 110.

48. This system of numeration is used almost exclusively in inscriptions, and almost exclusively for cardinal numbers; ordinals (first, second, third etc.) are written out in words. The system is otherwise known as the Attic or Herodianic system, after the grammarian Herodianus who first explained it. Its use is not confined to Attika but is found in other areas too, down to about the C2 BC, and it later continues in sporadic use, much as we still use Roman numerals sometimes. See Ifrah 1985 p. 230 fig. 14-23 for the many variations in the signs used in acrophonic systems across Greece (not just the Attic version given in text here).

49. Some peoples of the world have had singular, dual, trial (3 somethings), quadrual (four somethings), and then plural for anythings greater than four. [Aristotle] Problems 15.3 claims that the Thracians ‘alone among men’ count in fours ‘because their memory, like that of children, cannot extend further and they do not use a large number of anything’. Without counting, our ability to perceive quantities at a glance works up to four, but then quickly deteriorates; this is perhaps why in Greek acrophonic and Roman numerals the symbol for 1 may be repeated up to four times but then there is a change to a (one) new symbol for 5, so that one is not faced with trying to read (rather than count) IIIII. See Ifrah p. 6-7 and fig.9-7 over pp.137-141 illustrating the same phenomenon in the numerical notations used by many early societies.

50. I use the terms rational and irrational number with their current meanings, which are not those of Euclid. For discussion of the terms see Gow 1923 p. 78-9 and Heath Euclid vol 3 pp. 11-12.

51. Put simply in modern terminolgy, the Greeks did not construct what we call the real numbers (as with squaring the circle this problem was not satisfactorily resolved until the nineteenth century AD).

52. See e.g. Knorr 1991 (following a paper by Fowler), esp. § 3.

53. Until the latter half of the C19, Euclid’s Elements were unchallenged as a textbook for schools in England; see Richards 1988 chapter 4.

54. In this I am following the manuscripts. All MSS have 9 common notions, but 4 of them are mathematically redundant, and consequently since antiquity their genuineness has been questioned. Most modern texts ignore the redundant ones and say that there are five common notions. But it is precisely their redundancy that argues in favour of the genuineness of these four notions, in my view.

55. The development of non-Euclidean geometry belongs to the C19 (e.g. publications by Lobachevskii 1829 and Boylai 1831). For a short review see Gray 1987.

56. For example, Book 9 Prop. 20 is ‘there are more prime numbers than any number’ i.e. the number of prime numbers is infinite.

57. See Prop. 46.

58. See Prop. 14.

59. See Common Notion 2.

60. See Prop. 4.

61. See Prop. 41.

62. See Common Notion 2.

63. Thomas trans. in the Loeb Greek Mathematical Works 1 pp. 179-185, but I have used English letters where Thomas uses Greek.

64. E.g. by Kline 1972; Boyer and Merzback 1989.

65. E.g. by Heath, Tannery, van der Waerden, Knorr and Fowler.

66. This is not Fowler’s term; he talks of numerical material being ‘modernised and uniformised in what might then have been considered unimportant ways’, 1992 p. 134, and goes on to give, in an illustrated Annex, wonderful examples of similar processes at work in modern editions.

67. For example, though written in the C2 AD, Galen's On the use of the pulse survives in two Greek manuscripts of the C14-16, one of which is a copy of the other, and one Arabic MS of the C? [if you know when Istefan ibn Basil lived, pleased let me know], which was a translation into Arabic of a Syriac translation of the Greek made in the C9-10. There was another Greek  MS in existence in the C16 which was used for the Aldine edition of the text, but that is now lost, Furley & Wilkinson 1984 pp.187-8.

To follow