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 B.C.

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 (on whom see also chapter 6 n. 22).

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 Diophantos’ Arithmetica, 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 e.g. Knorr 1986.

¹¹ 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.

¹² 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 khilioi, 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.

¹³ de Ste Croix, 1956, p. 56, with full defence of this charge on pp. 56-9.

¹⁴ 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.

¹⁵ All these symbols are post-antiquity. The symbol + is first used as an abbreviation for Latin ‘et’ (and) in a 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.

¹⁶ Figured numbers are well discussed in Heath 1921, pp. 76-84.

¹⁷ 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.

¹⁸ 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. 240 f. 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.

¹⁹ 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’.

²⁰ 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 disciplinary views held by its various contributors. It is well worth reading, but poorly referenced for the Greek chapters.

²¹ 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 contextualization of mathematics in the culture which produced it.

²² ‘Multiplied by’ is often expressed by έπí plus dative in a mathematical context. This is the term used in expressions of interest rates (τόκοί), 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.

²³ 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.

²⁴ 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 (Дг; = l/28^th μ = l/40^th μα = l/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 expressing 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.

²⁵ 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 practised by philosophers, the latter being described as ‘far superior’. He takes a different view in Laws 817e- 820 (esp. 819c).

²⁶ 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 the law that they did not qualify! For discussion of the privileges in question, see Duncan-Jones 1990 pp. 160-3.

²⁷ As Fowler notes, 1992 p. 134 n. 4, commercial practice certainly affected 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.

²⁸ 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.

²⁹ 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.

³⁰ Wasps 656-64, Sommerstein trans., slightly modified.

³¹ Affections and Errors of the Soul 2.5 (5.83-4K). As I pointed out in chapter 1, 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.

³² De temporum ratione 1, which concerns calculating and speaking with the fingers.

³³ The prostitutes’ tax, paid by prostitutes.

³⁴ The metics’ tax, paid by resident aliens.

³⁵ If the case was worth less than 100 drachmai, there was no fee. For a case valued between 100 and 1000 drachai, 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.

³⁶ The name refers to the obol, because this penalty is calculated at one obol fine per drachma value.

³⁷ Contra Nixon and Price 1992.

³⁸ 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.

³⁹ See Vickers 1992. Greek coins had bullion value, they were not tokens, so precious metal objects could serve as large denominations.

⁴⁰ Stades about which we have reliable information and which were in use in Eratosthenes’ time vary from about 7.5 to 10 stades to a Roman mile. On the scale of the circumference of the earth, the difference is very significant.

⁴¹ ‘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.

⁴² See Engels op. cit., esp. chapter 1.

⁴³ The lower figure is given by Curtius and Plutarch; the higher by Arrian and Justin.

⁴⁴ de Ste Croix 1956 pp. 39-40, with accompanying Plate III and Figure IV.

⁴⁵ A problem on which Polubios had expounded at some length, 9.19, having criticized Philip of Macedon for making mistakes in this area (5.97.5-6).

⁴⁶ ‘In writing and reckoning with pebbles the Greeks move the hand from left to right, but the Egyptians from right to left’.

⁴⁷ See Lang 1964, Pritchett 1965, Lang 1965. Pullan 1970 (which is well illustrated with archaeological evidence and diagrams) follows Lang but is not always reliable.

⁴⁸ 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.

⁴⁹ This is true of other tables thought to have been either abaci or gaming tables, such as that found at the Amphiareion.

⁵⁰ 7.187. Note that the Grene translation in the Chicago 1987 edition has an error here: the result 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.

⁵¹ Errors are very common in literary texts, including those originals which have survived on papyri and in 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.

⁵² ‘Aleph’. ˏ 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’.

⁵³ 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.

⁵⁴ Flegg 1989 pp. 106 f. and 110.

⁵⁵ 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 B.C., 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 the text here).

⁵⁶ 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 pp. 6-7 and fig. 9-7 over pp. 137-41 illustrating the same phenomenon in the numerical notations used by many early societies.

⁵⁷ 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, 1884, pp. 78-9 and Heath Euclid vol. 3 pp. 11-12.

⁵⁸ Put simply in modern terminology, 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 A.D.).

⁵⁹ Fowler 1990, p. 10.

⁶⁰ See e.g. Knorr 1991 (following a paper by Fowler), esp. §3.

⁶¹ Until the latter half of the C19, Euclid’s Elements were unchallenged as a textbook for schools in England; see Richards 1988 chapter 4.

⁶² 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.

⁶³ The development of non-Euclidean geometry belongs to the C19 (e.g. publications by Lobachevskii 1829 and Boyali 1831). For a short review see Gray 1987.

⁶⁴ For example, Book 9 Prop. 20 ‘there are more prime numbers than any number’ i.e. the number of prime numbers is infinite.

⁶⁵ See Prop. 46.

⁶⁶ See Prop. 14.

⁶⁷ See Common Notion 2.

⁶⁸ See Prop. 4.

⁶⁹ See Prop. 41.

⁷⁰ See Common Notion 2.

⁷¹ Thomas trans, in the Loeb Greek Mathematical Works 1, pp. 179-85, but I have used English letters where Thomas uses Greek.

⁷² E.g. by Kline 1972; Boyer and Merzback 1989.

⁷³ E.g. by Heath, Tannery, van der Waerden, Knorr, and Fowler.

⁷⁴ 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.

⁷⁵ For example, although written (probably) in the mid-C4 A.D., Pelagonius’ Ars veterinaria survives in one MS, a late C15 copy of a C7 or C8 MS.