# 0 and 1 in Greek mathematics

Following up on an earlier post about zero in Greek mathematics and this timeline of zero, I want to say something more about the role of 0 (zero) and 1 (one) in ancient Greek thought. Unfortunately, some of the discussion on Greek mathematics out there is a bit like this:

## 0 and 1 as quantities

The ancient Greeks could obviously count, and they had bankers, so they understood credits and debts, and the idea of your bank account being empty. However, they had not reached the brilliant insight of Brahmagupta, around 628 AD, that you could multiply a debt (−) and a debt (−) to get a credit (+).

The ancient Greeks had three words for “one” (εἷς = heis, μία = mia, ἑν = hen), depending on gender. So, in the opening line of Plato’s Timaeus, Socrates counts: “One, two, three; but where, my dear Timaeus, is the fourth of those who were yesterday my guests … ? (εἷς, δύο, τρεῖς: ὁ δὲ δὴ τέταρτος ἡμῖν, ὦ φίλε Τίμαιε, ποῦ τῶν χθὲς μὲν δαιτυμόνων … ; )

The Greeks had two words for “nothing” or “zero” (μηδέν = mēden and οὐδέν = ouden). So, in the Christian New Testament, in John 21:11, some fisherman count fish and get 153, but in Luke 5:5, Simon Peter says “Master, we toiled all night and took nothing (οὐδὲν)!

## 0 and 1 in calculations

In ordinary (non-positional) Greek numerals, the Greeks used α = 1, ι = 10, and ρ = 100. There was no special symbol for zero. Greek mathematicians, such as Archimedes, wrote numbers out in words when stating a theorem.

Greek astronomers, who performed more complex calculations, used the Babylonian base-60 system. Sexagesimal “digits” from 1 to 59 were written in ordinary Greek numerals, with variations of ō for zero. The overbar was necessary to distinguish ō from the letter ο, which denoted the number 70 (since an overbar was a standard way of indicating abbreviations, it is likely that the symbol ō was an abbreviation for οὐδὲν).

Initially (around 100 AD) the overbar was quite fancy, and it became shorter and simpler over time, eventually disappearing altogether. Here it is in a French edition of Ptolemy’s Almagest of c. 150 AD:

In Greek-influenced Latin astronomical calculations, such as those used by Christians to calculate the date of Easter, “NULLA” or “N” was used for zero as a value. Such calculations date from the third century AD. Here (from Gallica) is part of a beautiful late example from around 700 AD (the calendar of St. Willibrord):

Outside of astronomy, zero does not seem to get mentioned much, although Aristotle, in his Physics (Book 4, Part 8) points out, as if it is a well-known fact, that “there is no ratio of zero (nothing) to a number (οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν),” i.e. that you cannot divide by zero. Here Aristotle may have been ahead of Brahmagupta, who thought that 0/0 = 0.

## 0 and 1 as formal numbers?

We now turn to the formal theory of numbers, in the Elements of Euclid and other works. This is mathematics in a surprisingly modern style, with formal proofs and (more or less) formal definitions. In book VII of the Elements (Definitions 1 & 2), Euclid defines the technical terms μονάς = monas (unit) and ἀριθμὸς = arithmos (number):

1. A monas (unit) is that by virtue of which each of the things that exist is called one (μονάς ἐστιν, καθ᾽ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται).
2. An arithmos (number) is a multitude composed of units (ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος).

So 1 is the monas (unit), and the technical definition of arithmos excludes 0 and 1, just as today the technical definition of natural number is taken by some mathematicians to exclude 0. However, in informal Greek language, 1 was still a number, and Greek mathematicians were not at all consistent about excluding 1. It remained a number for the purpose of doing arithmetic. Around 100 AD, for example, Nicomachus of Gerasa (in his Introduction to Arithmetic, Book 1, VIII, 9–12) discusses the powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512 = α, β, δ, η, ιϛ, λβ, ξδ, ρκη, σνϛ, φιβ) and notes that “it is the property of all these terms when they are added together successively to be equal to the next in the series, lacking a monas (συμβέβηκε δὲ πᾱ́σαις ταῖς ἐκθέσεσι συντεθειμέναις σωρηδὸν ἴσαις εἶναι τῷ μετ’ αὐτὰς παρὰ μονάδα).” In the same work (Book 1, XIX, 9), he provides a multiplication table for the numbers 1 through 10:

The issue here is that Euclid was aware of the fundamental theorem of arithmetic, i.e. that every positive integer can be decomposed into a bag (multiset) of prime factors, in no particular order, e.g. 60 = 2×2×3×5 = 2×2×5×3 = 2×5×2×3 = 5×2×2×3 = 5×2×3×2 = 2×5×3×2 = 2×3×5×2 = 2×3×2×5 = 3×2×2×5 = 3×2×5×2 = 3×5×2×2 = 5×3×2×2.

Euclid proves most of this theorem in propositions 30, 31 and 32 of his Book VII and proposition 14 of his Book IX. The number 0 is obviously excluded from consideration here, and the number 1 is special because it represents the empty bag (even today we recognise that 1 is a special case, because it is not a prime number, and it is not composed of prime factors either – although, as late as a century ago, there were mathematicians who called 1 prime, which causes all kinds of problems):

• If two numbers (arithmoi) by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers (ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, τὸν δὲ γενόμενον ἐξ αὐτῶν μετρῇ τις πρῶτος ἀριθμός, καὶ ἕνα τῶν ἐξ ἀρχῆς μετρήσει) – i.e. if a prime p divides ab, then it divides a or b or both
• Any composite number is measured by some prime number (ἅπας σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – i.e. it has a prime factor
• Any number (arithmos) either is prime or is measured by some prime number (ἅπας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – this would not be true for 1
• If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it (ἐὰν ἐλάχιστος ἀριθμὸς ὑπὸ πρώτων ἀριθμῶν μετρῆται, ὑπ᾽ οὐδενὸς ἄλλου πρώτου ἀριθμοῦ μετρηθήσεται παρὲξ τῶν ἐξ ἀρχῆς μετρούντων) – this is a partial expression of the uniqueness of prime factorisation

The special property of 1, the monas or unit, was sometimes expressed (e.g. by Nicomachus of Gerasa) by saying that it is the “beginning of arithmoi … but not itself an arithmos.” As we have already seen, nobody was consistent about this, and there was, of course, no problem in doing arithmetic with 1. Everybody agreed that 1 + 2 + 3 + 4 = 10. In modern mathematics, we would avoid problems by saying that natural numbers are produced using the successor function S, and distinguish that function from the number S(0) = 1.

The words monas and arithmos occur in other Greek writers, not always in the Euclidean technical sense. For example, in a discussion of causes and properties in the Phaedo (105c), Plato tells us that “if you ask what causes an arithmos to be odd, I shall not say oddness, but the monas (οὐδ᾽ ᾧ ἂν ἀριθμῷ τί ἐγγένηται περιττὸς ἔσται, οὐκ ἐρῶ ᾧ ἂν περιττότης, ἀλλ᾽ ᾧ ἂν μονάς).” Aristotle, in his Metaphysics, spends some time on the philosophical question of what the monas really is.

In general, the ancient Greeks seem to have had quite a sophisticated understanding of 0 and 1, though hampered by poor vocabulary and a lack of good symbols. Outside of applied mathematics and astronomy, they mostly worked with what we would call the multiplicative group of the positive rational numbers. What they were missing was any awareness of negative numbers as mathematical (not just financial) concepts. That had to wait until Brahmagupta, and when it came, 0 suddenly became a whole lot more interesting, because it eventually became possible to define more advanced mathematical concepts like fields.

# Some Oldest Manuscripts

The chart below (click to zoom) shows the dates of ten significant written works:

Each work is indicated by a vertical line, which runs from the date of writing to the date of the oldest surviving complete copy that I am aware of (marked by a dark circle). Open circles show some of the older partial or fragmentary manuscripts (these act as important checks on the reliability of later copies).

Two threshold periods (marked with arrow) are worth remarking on. First, Gutenberg’s printing press – after its invention, we still have at least one first edition for many important works. Second, the invention of Carolingian minuscule – many older works were re-copied into the new, legible script after that time. They were then widely distributed to monasteries around Europe, so that survival from that period has been fairly good. In the Byzantine Empire, Greek minuscule had a similar effect.

The Bible is a special case (I have highlighted one particular gospel on the chart). It was copied so widely (and so early) that many ancient manuscripts survive.

# Zero in Greek mathematics

I recently read The Nothing That Is: A Natural History of Zero by Robert M. Kaplan. Zero is an important concept in mathematics. But where did it come from?

## The Babylonian zero

From around 2000 BC, the Babylonians used a positional number system with base 60. Initially a space was used to represent zero. Vertical wedges mean 1, and chevrons mean 10:

This number (which we can write as 2 ; 0 ; 13) means 2 × 3600 + 0 × 60 + 13 = 7213. Four thousand years later, we still use the same system when dealing with angles or with time: 2 hours, no minutes, and 13 seconds is 7213 seconds.

Later, the Babylonians introduced a variety of explicit symbols for zero. By 400 BC, a pair of angled wedges was used:

The Babylonians were happy to move the decimal point (actually, “sexagesimal point”) forwards and backwards to facilitate calculation. The number ½, for example, was treated the same as 30 (which is half of 60). In much the same way, 20th century users of the slide rule treated 50, 5, and 0.5 as the same number. What is 0.5 ÷ 20? The calculation is done as 5 ÷ 2 = 2.5. Only at the end do you think about where the decimal point should go (0.025).

## Greek mathematics in words

Kaplan says about zero that “the Greeks had no word for it.” Is that true?

Much of Greek mathematics was done in words. For example, the famous Proposition 3 in the Measurement of a Circle (Κύκλου μέτρησις) by Archimedes reads:

Παντὸς κύκλου ἡ περίμετρος τῆς διαμέτρου τριπλασίων ἐστί, καὶ ἔτι ὑπερέχει ἐλάσσονι μὲν ἤ ἑβδόμῳ μέρει τῆς διαμέτρου, μείζονι δὲ ἢ δέκα ἑβδομηκοστομόνοις.

Phonetically, that is:

Pantos kuklou hē perimetros tēs diametrou triplasiōn esti, kai eti huperechei elassoni men ē hebdomō merei tēs diametrou, meizoni de ē deka hebdomēkostomonois.

Or, in English:

The perimeter of every circle is triple the diameter plus an amount less than one seventh of the diameter and greater than ten seventy-firsts.

In modern notation, we would express that far more briefly as 10/71 < π − 3 < 1/7 or 3.141 < π < 3.143.

The Greek words for zero were the two words for “nothing” – μηδέν (mēden) and οὐδέν (ouden). Around 100 AD, Nicomachus of Gerasa (Gerasa is now the city of Jerash, Jordan), wrote in his Introduction to Arithmetic (Book 2, VI, 3) that:

οὐδέν οὐδενί συντεθὲν … οὐδέν ποιεῖ (ouden oudeni suntethen … ouden poiei)

That is, zero (nothing) can be added:

nothing and nothing, added together, … make nothing

However, we cannot divide by zero. Aristotle, in Book 4, Lectio 12 of his Physics tells us that:

οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν (oude to mēden pros arithmon)

That is, 1/0, 2/0, and so forth make no sense:

there is no ratio of zero (nothing) to a number

If we view arithmetic primarily as a game of multiplying, dividing, taking ratios, and finding prime factors, then poor old zero really does have to sit on the sidelines (in modern terms, zero is not part of a multiplicative group).

## Greek calculation

For business calculations, surveying, numerical tables, and most other mathematical calculations (e.g. the proof of Archimedes’ Proposition 3), the Greeks used a non-positional decimal system, based on 24 letters and 3 obsolete letters. In its later form, this was as follows:

Units Tens Hundreds
α = 1 ι = 10 ρ = 100
β = 2 κ = 20 σ = 200
γ = 3 λ = 30 τ = 300
δ = 4 μ = 40 υ = 400
ε = 5 ν = 50 φ = 500
ϛ (stigma) = 6 ξ = 60 χ = 600
ζ = 7 ο = 70 ψ = 700
η = 8 π = 80 ω = 800
θ = 9 ϙ (koppa) = 90 ϡ (sampi) = 900

For users of R:

``````to.greek.digits <- function (v) { # v is a vector of numbers
if (any(v < 1 | v > 999)) stop("Can only do Greek digits for 1..999")
else {
s <- intToUtf8(c(0x3b1:0x3b5,0x3db,0x3b6:0x3c0,0x3d9,0x3c1,0x3c3:0x3c9,0x3e1))
greek <- strsplit(s, "", fixed=TRUE)[[1]]
d <- function(i, power=1) { if (i == 0) "" else greek[i + (power - 1) * 9] }
f <- function(x) { paste0(d(x %/% 100, 3), d((x %/% 10) %% 10, 2), d(x %% 10)) }
sapply(v, f)
}
}``````

For example, the “number of the beast” (666) as written in Byzantine manuscripts of the Bible is χξϛ (older manuscripts spell the number out in words: ἑξακόσιοι ἑξήκοντα ἕξ = hexakosioi hexēkonta hex).

This Greek system of numerals did not include zero – but then again, it was used in situations where zero was not needed.

## Greek geometry

Most of Greek mathematics was geometric in nature, rather than based on calculation. For example, the famous Pythagorean Theorem tells us that the areas of two squares add up to give the area of a third.

In geometry, zero was represented as a line of zero length (i.e. a point) or as a rectangle of zero area (i.e. a line). This is implicit in Euclid’s first two definitions (σημεῖόν ἐστιν, οὗ μέρος οὐθέν = a point is that which has no part; γραμμὴ δὲ μῆκος ἀπλατές = a line is breadthless length).

In the Pythagorean Theorem, lines are multiplied by themselves to give areas, and the sum of the two smaller areas gives the third (image: Ntozis)

## Graeco-Babylonian mathematics

In astronomy, the Greeks continued to use the Babylonian sexagesimal system (much as we do today, with our “degrees, minutes, and seconds”). Numbers were written using the alphabetic system described above, and at the time of Ptolemy, zero was written like this (appearing in numerous papyri from 100 AD onwards, with occasional variations):

For example, 7213 seconds would be β ō ιγ = 2 0 13 (for another example, see the image below). The circle here may be an abbreviation for οὐδέν = nothing (just as early Christian Easter calculations used N for Nulla to mean zero). The overbar is necessary to distinguish ō from ο = 70 (it also resembles the overbars used in sacred abbreviations).

This use of a circle to mean zero was passed on to the Arabs and to India, which means that our modern symbol 0 is, in fact, Graeco-Babylonian in origin (the contribution of Indian mathematicians such as Brahmagupta was not the introduction of zero, but the theory of negative numbers). I had not realised this before; from now on I will say ouden every time I read “zero.”

Part of a table from a French edition of Ptolemy’s Almagest of c. 150 AD. For the angles x = ½°, 1°, and 1½°, the table shows 120 sin(x/2). The (sexagesimal) values, in the columns headed ΕΥΘΕΙΩΝ, are ō λα κε = 0 31 25 = 0.5236, α β ν = 1 2 50 = 1.0472, and α λδ ιε = 1 34 15 = 1.5708. The columns on the right are an aid to interpolation. Notice that zero occurs six times.

# Eight Greek inscriptions

I love ancient inscriptions. They provide a connection to people of the past, they provide an insight into how people thought, and they demonstrate how the experience of writing has changed over the past five thousand years or so. Here are eight Greek inscriptions and documents that interest me – some historical, some religious, and one mathematical.

Six of the eight inscriptions

## 1. The inscription that is no longer there, 480 BC

Our first inscription was inscribed at the site of the Battle of Thermopylae (480 BC), where Leonidas and his 300 Spartans (plus several thousand allies) died trying to hold off a vastly superior Persian army. The inscription no longer exists (though there is a modern copy at the site), but the wording has been preserved by Herodotus (Histories 7.228.2):

Ω ΞΕΙΝ ΑΓΓΕΛΛΕΙΝ
ΛΑΚΕΔΑΙΜΟΝΙΟΙΣ ΟΤΙ ΤΗΔΕ
ΚΕΙΜΕΘΑ ΤΟΙΣ ΚΕΙΝΩΝ
ΡΗΜΑΣΙ ΠΕΙΘΟΜΕΝΟΙ.

Ō ksein’, angellein
Lakedaimoniois hoti tēide
keimetha, tois keinōn
rhēmasi peithomenoi.

I’ve always thought that there was a degree of sarcasm in this laconic epigram – after all, the Spartans had declared war on the Persians (rather informally, by throwing the Persian ambassadors down a well), but then stayed home, leaving Leonidas and his personal honour guard (plus the allies) to do the actual fighting. My (rather free) personal translation would therefore be:

Go tell the Spartans,
Stranger passing by,
We listened to their words,
And here we lie.

The battle of Thermopylae, 480 BC (illustration: John Steeple Davis)

## 2. The Rosetta Stone, 196 BC

The rich history of the Rosetta Stone has always fascinated me (and I made a point of seeing the Stone when I visited the British Museum). The Stone records a decree of 196 BC from Ptolemy V, inscribed using three forms of writing – Egyptian hieroglyphs, Egyptian demotic script, and a Greek translation. The Stone was therefore a valuable input to the eventual decoding of Egyptian hieroglyphs. Romance practically drips off the Stone.

The Rosetta Stone in the British Museum (photo: Hans Hillewaert)

## 3. The Theodotus inscription, before 70 AD

The Theodotus inscription in Jerusalem was located in a 1st century synagogue near the Temple (this dating is generally accepted). It reads as follows (with [square brackets] denoting missing letters):

ΘΕΟΔΟΤΟΣ ΟΥΕΤΤΕΝΟΥ ΙΕΡΕΥΣ ΚΑΙ
ΑΡΧΙΣΥΝΑΓΩΓΟΣ ΥΙΟΣ ΑΡΧΙΣΥΝ[ΑΓΩ]
Γ[Ο]Υ ΥΙΟΝΟΣ ΑΡΧΙΣΥΝ[Α]ΓΩΓΟΥ ΩΚΟ-
ΔΟΜΗΣΕ ΤΗΝ ΣΥΝΑΓΩΓ[Η]Ν ΕΙΣ ΑΝ[ΑΓ]ΝΩ-
Σ[Ι]Ν ΝΟΜΟΥ ΚΑΙ ΕΙΣ [Δ]ΙΔΑΧΗΝ ΕΝΤΟΛΩΝ ΚΑΙ
ΤΟΝ ΞΕΝΩΝΑ ΚΑ[Ι ΤΑ] ΔΩΜΑΤΑ ΚΑΙ ΤΑ ΧΡΗ-
Σ[Τ]ΗΡΙΑ ΤΩΝ ΥΔΑΤΩΝ ΕΙΣ ΚΑΤΑΛΥΜΑ ΤΟΙ-
Σ [Χ]ΡΗZΟΥΣΙΝ ΑΠΟ ΤΗΣ ΞΕ[Ν]ΗΣ ΗΝ ΕΘΕΜΕ-
Λ[ΙΩ]ΣΑΝ ΟΙ ΠΑΤΕΡΕΣ [Α]ΥΤΟΥ ΚΑΙ ΟΙ ΠΡΕ-
Σ[Β]ΥΤΕΡΟΙ ΚΑΙ ΣΙΜΩΝ[Ι]ΔΗΣ.

In translation:

Theodotus, son of Vettenus [or, of the gens Vettia], priest and
archisynagogue [leader of the synagogue], son of an archisynagogue,
grandson of an archisynagogue, built
the synagogue for the reading of
the Law and for teaching the commandments;
also the hostel, and the rooms, and the water
fittings, for lodging
needy strangers. Its foundation was laid
by his fathers, and the
elders, and Simonides.

The inscription is interesting in a number of ways. Along with other similar inscriptions, it demonstrates the existence of Greek-language synagogues in 1st Palestine. The title ἀρχισυνάγωγος (archisynagōgos) also occurs in the New Testament (nine times, starting at Mark 5:22), so is clearly a title of the time-period. Some scholars have suggested that Theodotos was a freed slave, who had made his fortune and returned from Italy to the land of his fathers (in which case there is a very slight possibility that the synagogue with the inscription might have been the “synagogue of the Freedmen” mentioned in Acts 6:9).

The Theodotus inscription in the Israel Museum, Jerusalem (photo: Oren Rozen)

## 4. The Delphi inscription, 52 AD

The Temple of Apollo at Delphi (photo: Luarvick)

The Delphi inscription is a letter of around 52 AD from the Roman emperor Claudius. It was inscribed on stone at the Temple of Apollo at Delphi (above), although it now exists only as nine fragments. Of particular interest is this line (see also the photograph below):

[IOU]ΝΙΟΣ ΓΑΛΛΙΩΝ Ο Φ[ΙΛΟΣ] ΜΟΥ ΚΑ[Ι ΑΝΘΥ]ΠΑΤΟΣ …

[Jou]nios Galliōn ‘o ph[ilos] mou ka[i anthu]patos …

This is a reference to Lucius Junius Gallio Annaeanus, who was briefly proconsul (anthupatos) of the Roman senatorial province of Achaea (southern Greece) at the time:

Junius Gallio, my friend and proconsul …

This same anthupatos Gallio appears in the New Testament (Acts 18:12–17: “Γαλλίωνος δὲ ἀνθυπάτου ὄντος τῆς Ἀχαΐας …”), and therefore provides a way of dating the events described there.

One of the fragments of the Delphi inscription, highlighting the name ΓΑΛΛΙΩΝ = Gallio (photo: Gérard)

## 5. Papyrus Oxyrhynchus 29, c. 100 AD

I have written before about Papyrus Oxyrhynchus 29. It contains the statement of Proposition 5 of Book 2 of Euclid’s Elements, with an accompanying diagram (plus just a few letters of the last line of the preceding proposition). In modern Greek capitals, it reads:

ΕΑΝ ΕΥΘΕΙΑ ΓΡΑΜΜΗ
ΤΜΗΘΗ ΕΙΣ ΙΣΑ ΚΑΙ ΑΝ-
ΙΣΑ ΤΟ ΥΠΟ ΤΩΝ ΑΝΙ-
ΣΩΝ ΤΗΣ ΟΛΗΣ ΤΜΗΜ[ΑΤ]ΩΝ ΠΕΡΙΕΧΟΜΕΝΟΝ
ΟΡΘΟΓΩΝΙΟΝ ΜΕΤΑ Τ[Ο]Υ ΑΠΟ ΤΗΣ ΜΕΤΟΞΥ
ΤΩΝ ΤΟΜΩΝ ΤΕΤ[ΡΑ]ΓΩΝΟΥ ΙΣΟΝ ΕΣΤΙΝ
ΤΩ ΑΠΟ ΤΗΣ ΗΜΙΣΕΙ-
ΑΣ ΤΕΤΡΑΓΩΝΟΥ

However, the actual document (image below) uses “Ϲ” for the modern “Σ,” and “ω” for the modern “Ω”:

ΕΑΝ ΕΥΘΕΙΑ ΓΡΑΜΜΗ
ΤΜΗΘΗ ΕΙϹ ΙϹΑ ΚΑΙ ΑΝ-
ΙϹΑ ΤΟ ΥΠΟ ΤωΝ ΑΝΙ-
ϹωΝ ΤΗϹ ΟΛΗϹ ΤΜΗΜ[ΑΤ]ωΝ ΠΕΡΙΕΧΟΜΕΝΟΝ
ΟΡΘΟΓωΝΙΟΝ ΜΕΤΑ Τ[Ο]Υ ΑΠΟ ΤΗϹ ΜΕΤΟΞΥ
ΤωΝ ΤΟΜωΝ ΤΕΤ[ΡΑ]ΓωΝΟΥ ΙϹΟΝ ΕϹΤΙΝ
Τω ΑΠΟ ΤΗϹ ΗΜΙϹΕΙ-
ΑϹ ΤΕΤΡΑΓωΝΟΥ

This manuscript is important because, being from 75–125 AD, it dates to only four centuries after the original was written in 300 BC – most manuscripts of Euclid are twelve centuries or more after (in fact, it pre-dates the alterations made to the work by Theon of Alexandria in the 4th century AD). The manuscript also contains one of the oldest extant Greek mathematical diagrams. The text is identical to the accepted Greek text, except for two spelling variations and one one grammatical error (τετραγώνου for τετραγώνῳ on the last line, perhaps as the result of the mental influence of the preceding word in the genitive):

ἐὰν εὐθεῖα γραμμὴ
τμηθῇ εἰς ἴσα καὶ ἄνισα,
τὸ ὑπὸ τῶν ἀνίσων τῆς ὅλης τμημάτων περιεχόμενον ὀρθογώνιον
μετὰ τοῦ ἀπὸ τῆς μεταξὺ τῶν τομῶν τετραγώνου
ἴσον ἐστὶ τῷ ἀπὸ τῆς ἡμισείας τετραγώνῳ.

It is really just a geometric way of expressing the equality (x + y)2 = x2 + 2xy + y2, but in English it reads as follows:

If a straight line
be cut into equal and unequal [segments] (x + y + x and y),
the rectangle contained by the unequal segments of the whole (i.e. (x + y + x)y = 2xy + y2)
together with the square on the straight line between the points of section (+ x2)
is equal to the square on the half (= (x + y)2).

The proof of the proposition is missing, however, and there are no labels on the diagram. I suspect that the manuscript was a teaching tool of some kind (either an aide-mémoire or an exam question). Alternatively, it may have been part of an illustrated index to the Elements.

Papyrus Oxyrhynchus 29 (photo: Bill Casselman)

## 6. Rylands Library Papyrus P52, c. 140 AD

Papyrus P52 is a small fragment written in a similar style to Papyrus Oxyrhynchus 29, but is dated a few decades later (to around 140 AD). In modern Greek capitals, it reads:

ΟΙ ΙΟΥΔΑΙ[ΟΙ]· ΗΜΕ[ΙΝ ΟΥΚ ΕΞΕΣΤΙΝ ΑΠΟΚΤΕΙΝΑΙ]
ΟΥΔΕΝΑ. ΙΝΑ Ο Λ[ΟΓΟΣ ΤΟΥ ΙΗΣΟΥ ΠΛΗΡΩΘΗ ΟΝ ΕΙ]
ΠΕΝ ΣΗΜΑΙΝΩ[Ν ΠΟΙΩ ΘΑΝΑΤΩ ΗΜΕΛΛΕΝ ΑΠΟ]
ΘΝΗΣΚΕΙΝ. ΙΣ[ΗΛΘΕΝ ΟΥΝ ΠΑΛΙΝ ΕΙΣ ΤΟ ΠΡΑΙΤΩ]
ΡΙΟΝ Ο Π[ΙΛΑΤΟΣ ΚΑΙ ΕΦΩΝΗΣΕΝ ΤΟΝ ΙΗΣΟΥΝ]
ΚΑΙ ΕΙΠ[ΕΝ ΑΥΤΩ· ΣΥ ΕΙ O ΒΑΣΙΛΕΥΣ ΤΩΝ ΙΟΥ]
[Δ]ΑΙΩN;

The reverse side also has writing:

[ΒΑΣΙΛΕΥΣ ΕΙΜΙ. ΕΓΩ ΕΙΣ TO]ΥΤΟ Γ[Ε]ΓΕΝΝΗΜΑΙ
[ΚΑΙ (ΕΙΣ ΤΟΥΤΟ) ΕΛΗΛΥΘΑ ΕΙΣ ΤΟΝ ΚΟ]ΣΜΟΝ, ΙΝΑ ΜΑΡΤΥ-
[ΡΗΣΩ ΤΗ ΑΛΗΘΕΙΑ· ΠΑΣ Ο ΩΝ] ΕΚ ΤΗΣ ΑΛΗΘΕI-
[ΑΣ ΑΚΟΥΕΙ ΜΟΥ ΤΗΣ ΦΩΝΗΣ]. ΛΕΓΕΙ ΑΥΤΩ
[Ο ΠΙΛΑΤΟΣ· ΤΙ ΕΣΤΙΝ ΑΛΗΘΕΙΑ; Κ]ΑΙ ΤΟΥΤΟ
[ΕΙΠΩΝ ΠΑΛΙΝ ΕΞΗΛΘΕΝ ΠΡΟΣ] ΤΟΥΣ Ι[ΟΥ]
[ΔΑΙΟΥΣ ΚΑΙ ΛΕΓΕΙ ΑΥΤΟΙΣ· ΕΓΩ ΟΥΔ]ΕΜΙ[ΑΝ]
[ΕΥΡΙΣΚΩ ΕΝ ΑΥΤΩ ΑΙΤΙΑΝ].

Some clever detective work has identified the fragment as being from a manuscript of the New Testament gospel of John (John 18:31b–33 and 18:37b–38), permitting the reconstruction of the missing letters. The fragment is from the top inner corner of a book page (books with bound two-sided pages were a relatively new technology at the time, with many people still using scrolls). The fragment dates from less than a century after the gospel of John was written (and possibly just a few decades), thus helping in dating that work. There is no indication of any textual difference from later manuscripts – even the text on the missing parts of the front page seems of the right amount. The only exception is in the second line of the reverse side – there’s not quite enough room for the expected wording, and it seems likely that the duplicated words ΕΙΣ ΤΟΥΤΟ were not present.

… the Jews, “It is not lawful for us to put anyone to death.” This was to fulfil the word that Jesus had spoken to show by what kind of death he was going to die. So Pilate entered the Praetorium again and called Jesus and said to him, “Are you the King of the Jews?” …
… I am a king. For this purpose I was born and for this purpose I have come into the world – to bear witness to the truth. Everyone who is of the truth listens to my voice.” Pilate said to him, “What is truth?” After he had said this, he went back outside to the Jews and told them, “I find no guilt in him.”

Papyrus P52 (front and back) in the John Rylands Library

## 7. The Akeptous inscription in the Megiddo church, c. 250 AD

The Akeptous inscription is one of a number of inscriptions found in the mosaic floor of a 3rd century church which was discovered in 2005 while digging inside the Megiddo Prison in Israel (the date is just slightly later than the Dura-Europos church in Syria). The Akeptous inscription reads:

ΠΡΟϹΗΝΙΚΕΝ
ΑΚΕΠΤΟΥϹ,
Η ΦΙΛΟΘΕΟϹ,
ΤΗΝ ΤΡΑΠΕ-
ZΑΝ {Θω} {ΙΥ} {Χω}
ΜΝΗΜΟϹΥΝΟΝ

Phonetically:

Prosēniken Akeptous, ‘ē philotheos, tēn trapezan Th(e)ō Ι(ēso)u Ch(rist)ō mnēmosunon.

In English translation:

A gift of Akeptous, she who loves God, this table is for God Jesus Christ, a memorial.

Brief as it is, the inscription has several interesting features. First, Jesus Christ is being explicitly referred to as God, which tells us something about Christian beliefs of the time. Second, the inscription uses nomina sacra – divine names (“God,” “Jesus,” and “Christ”) are abbreviated with first and last letter, plus an overbar (this is denoted by curly brackets in the Greek text above). Third, the inscription records the gift of a prominent (presumably wealthy) female church member (the feminine definite article shows that Akeptous was female). And fourth, the reference to the construction of a table suggests that there were architectural features in the church to support the celebration of Communion, which tells us something about liturgy.

The Akeptous inscription in the Megiddo church

## 8. The Codex Sinaiticus, c. 340 AD

Our final inscription is a portion of the Codex Sinaiticus, a 4thcentury manuscript of the Christian Bible, containing the earliest complete copy of the New Testament. This Bible is a century later than the Megiddo church, and two centuries after Papyrus P52. Unlike Papyrus P52, it is written on vellum made from animal skins, and is written in beautiful calligraphic script. I have selected the passage John 1:1–3a:

ΕΝ ΑΡΧΗ ΗΝ Ο ΛΟΓΟϹ,
ΚΑΙ Ο ΛΟΓΟϹ ΗΝ
ΠΡΟϹ ΤΟΝ {ΘΝ}, ΚΑΙ
{ΘϹ} ΗΝ Ο ΛΟΓΟϹ. ΟΥ-
ΤΟϹ ΗΝ ΕΝ ΑΡΧΗ
ΠΡΟϹ ΤΟΝ {ΘΝ}. ΠΑ[Ν]-
ΤΑ ΔΙ ΑΥΤΟΥ ΕΓΕΝΕ-
ΤΟ, ΚΑΙ ΧΩΡΙϹ ΑΥΤΟΥ
ΕΓΕΝΕΤΟ ΟΥΔΕΝ

In English:

In the beginning was the Logos, and the Logos was with God, and the Logos was God. He was in the beginning with God. All things through him were made, and apart from him was not one thing made …

In the Greek, nomina sacra for “God” can be seen, together with a number of corrections (including, on the last line, an expansion of the contraction ΟΥΔΕΝ = “nothing” to ΟΥΔΕ ΕΝ = “not one thing”). Spaces between words had still not been invented, nor had punctuation or lowercase letters, which means that it is almost impossible to make sense of the text unless it is read aloud (or at least subvocalised). Fortunately, things have changed in the last seventeen centuries!

John 1:1–3a in the Codex Sinaiticus

# A History of Science in 12 Books

Here are twelve influential books covering the history of science and mathematics. All of them have changed the world in some way:

1: Euclid’s Elements (c. 300 BC). Possibly the most influential mathematics book ever written, and used as a textbook for more than 2,000 years.

2: De rerum natura by Lucretius (c. 50 BC). An Epicurean, atomistic view of the universe, expressed as a lengthy poem.

3: The Vienna Dioscurides (c. 510 AD). Based on earlier Greek works, this illustrated guide to botany continued to have an influence for centuries after it was written.

4: De humani corporis fabrica by Andreas Vesalius (1543). The first modern anatomy book.

5: Galileo’s Dialogue Concerning the Two Chief World Systems (1632). The brilliant sales pitch for the idea that the Earth goes around the Sun.

6: Audubon’s The Birds of America (1827–1838). A classic work of ornithology.

7: Darwin’s On the Origin of Species (1859). The book which started the evolutionary ball rolling.

8: Beilstein’s Handbook of Organic Chemistry (1881). Still (revised, in digital form) the definitive reference work in organic chemistry.

9: Relativity: The Special and the General Theory by Albert Einstein (1916). An explanation of relativity by the man himself.

10: Éléments de mathématique by “Nicolas Bourbaki” (1935 onwards). A reworking of mathematics which gave us words like “injective.”

11: Algorithms + Data Structures = Programs by Niklaus Wirth (1976). One of the early influential books on structured programming.

12: Introduction to VLSI Systems by Carver Mead and Lynn Conway (1980). The book which revolutionised silicon chip design.

That’s four books of biology, four of other science, two of mathematics, and two of modern IT. I welcome any suggestions for other books I should have included.

The great geometer Euclid (according to Proclus) once told King Ptolemy I of Egypt that there was “no royal road to geometry.” Even for the rich and powerful, there is no easy way of absorbing difficult mathematical concepts – just as there is no easy way of becoming a concert-level pianist.

The legendary XKCD gives us a modern take on Euclid’s remark:

# Geometry 1900 years ago

Papyrus Oxyrhynchus 29 (not to be confused with New Testament Papyrus 29) is a papyrus from the Oxyrhynchus collection, containing the statement of Proposition 5 of Book 2 of Euclid’s Elements, with an accompanying diagram. In modern notation, the proposition is ab + (ab)2/4 = (a+b)2/4. Euclid states the proposition as follows (the first paragraph is on the papyrus):

If a straight line be cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.

PROOF: For let a straight line AB be cut into equal segments at C and into unequal segments at D; I say that the rectangle contained by AD, DB together with the square on CD is equal to the square on CB.

For let the square CEFB be described on CB, and let BE be joined; through D let DG be drawn parallel to either CE or BF, through H again let KM be drawn parallel to either AB or EF, and again through A let AK be drawn parallel to either CL or BM.

Then, since the complement CH is equal to the complement HF, let DM be added to each; therefore the whole CM is equal to the whole DF.

But CM is equal to AL, since AC is also equal to CB; therefore AL is also equal to DF. Let CH be added to each; therefore the whole AH is equal to the gnomon NOP.

But AH is the rectangle AD, DB, for DH is equal to DB, therefore the gnomon NOP is also equal to the rectangle AD, DB.

Let LG, which is equal to the square on CD, be added to each; therefore the gnomon NOP and LG are equal to the rectangle contained by AD, DB and the square on CD.

But the gnomon NOP and LG are the whole square CEFB, which is described on CB; therefore the rectangle contained by AD, DB together with the square on CD is equal to the square on CB. Therefore etc. Q. E. D.

The papyrus is in Greek capitals; in modern letters it reads like this:

Modern scholars date the fragment to AD 75–125. It is not of great quality, with poor handwriting, spelling errors (μετοξὺ for μεταξὺ, and τετραγώνου for τετραγώνῳ on the last line), and missing labels on the diagram (making it of limited use, and perhaps explaining why it was found in an ancient trash pile). However, unlike the New Testament with its hundreds of manuscripts, there is not much of Euclid before AD 900, which makes this fragment historically very significant. It contains one of the oldest extant Greek mathematical diagrams.

See more here.