Complexity and Randomness revisited

I have posted before (post 1 and post 2) about order, complexity, and randomness. The image above shows the spectrum from organised order to random disorder, with structured complexity somewhere in between. The three textual examples below illustrate the same idea.

Regular Complex Random
AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA … It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, … ShrfT e6IJ5 eRU5s nNcat qnI8N m-cm5 seZ6v 5GeYc w2jpg Vp5Lx V4fR7 hhoc- 81ZHi 5qntn ErQ2- uv3UE MnFpy rLD0Y DI3GW p23UF FQwl1 BgP36 RK6Gb 6lpzR nV03H W5X3z 2f1u8 OgpXy tY-6H HkwEU s0xLN 9W8H …

These three examples, and many intermediate cases, can be distinguished by the amount of information they contain. The leading way of measuring that information is with Kolmogorov complexity. The Kolmogorov complexity of a block of text is the length of the shortest program producing that text. Kolmogorov complexity is difficult to calculate in practice, but an approximation is the size of the compressed file produced by good compression software, such as 7-Zip. The chart below shows the number of bytes (a byte is 8 bits) for a compressed version of A Tale of Two Cities, a block of the letter ‘A’ repeated to the same length, and a block of random characters of the same length:

The random characters are chosen to have 64 possible options, which need 6 bits to describe, so a compression to about 75% of the original size is as expected. The novel by Dickens compresses to 31% of its original size.

But does this chart show information? Grassberger notes that Kolmogorov complexity is essentially just a measure of randomness. On this definition, random-number generators would be the best source of new information – but that’s not what most people mean by “information.”

An improvement is to introduce an equivalence relation “means the same.” We write X ≈ Y if X and Y have the same meaning. In particular, versions of A Tale of Two Cities with different capitalisation have the same meaning. Likewise, all meaningless random sequences have the same meaning. The complexity of a block of text is then the length of the shortest program producing something with the same meaning as that text (i.e. the complexity of X is the length of the shortest program producing some Y with X ≈ Y).

In particular, the complexity of a specific block of random text is the length of the shortest program producing random text (my R program for random text is 263 bytes), and we can approximate the complexity of A Tale of Two Cities by compressing an uppercase version of the novel. This definition of complexity starts to look a lot more like what we normally mean by “information.” The novel contains a large amount of information, while random sequences or “AAAAA…” contain almost none:

Those who hold that information satisfies the rule ex nihilo nihil fit can thus be reassured that random-number generators cannot create new information out of nothing. However, if we combine random-number generators with a selection procedure which filters out anything that “means the same” as meaningless sequences, we can indeed create new information, as genetic algorithms and genetic programming have demonstrated – although Stuart Kauffman and others believe that the evolution of biological complexity also requires additional principles, like self-organisation.

Something is going on with the primes…

The chart below illustrates the Erdős–Kac theorem. This relates to the number of distinct prime factors of large numbers (integer sequence A001221 in the On-Line Encyclopedia):

Number No. of prime factors No. of distinct prime factors
1 0 0
2 (prime) 1 1
3 (prime) 1 1
4 = 2×2 2 1
5 (prime) 1 1
6 = 2×3 2 2
7 (prime) 1 1
8 = 2×2×2 3 1
9 = 3×3 2 1
10 = 2×5 2 2
11 (prime) 1 1
12 = 2×2×3 3 2
13 (prime) 1 1
14 = 2×7 2 2
15 = 3×5 2 2
16 = 2×2×2×2 4 1

The Erdős–Kac theorem says that, for large numbers n, the number of distinct prime factors of numbers near n approaches a normal distribution with mean and variance log(log(n)), where the logarithms are to the base e. That seems to be saying that prime numbers are (in some sense) randomly distributed, which is very odd indeed.

In the chart, the observed mean of 3.32 is close to log(log(109)) = 3.03, although the observed variance of 1.36 is smaller. The sample in the chart includes 17 numbers with 8 distinct factors, including 1,000,081,530 = 2×3×3×5×7×19×29×43×67 (9 factors, 8 of which are distinct).

The Erdős–Kac theorem led to an episode where, following the death of Paul Erdős in 1996, Carl Pomerance spoke about the theorem at a conference session in honour of Erdős in 1997. Quoting Albert Einstein (“God does not play dice with the universe”), Pomerance went on to say that he would like to think that Erdős and [Mark] Kac replied “Maybe so, but something is going on with the primes.” The quote is now widely misattributed to Erdős himself.

Complexity in medicine: some thoughts

I have been thinking recently about medicine and complexity, as a result of several conversations over many years. In particular, the Cynefin framework developed by Dave Snowden (see diagram below) seems a useful lens to use (this thought is not original to me – see among others, the articles “The Cynefin framework: applying an understanding of complexity to medicine” by Ben Gray and “Cynefin as reference framework to facilitate insight and decision-making in complex contexts of biomedical research” by Gerd Kemperman). I will also refer to two case studies from the book Five Patients by Michael Crichton, which is still quite relevant, in spite of being written in 1969.

The Cynefin framework developed by Dave Snowden. The central dark area is that of Disorder/Confusion, where it is not clear which of the four quadrants apply (image: Dave Snowden).

The Cynefin framework divides problems into four quadrants: Obvious, Complicated, Complex, and Chaotic. In addition, the domain of Disorder/Confusion reflects problems where there is no clarity about which of the other domains apply. In medicine, this reflects cases where multiple factors are at work – potentially, multiple chronic conditions as well as one or more acute ones. These conditions can exist in all four quadrants. Ben Gray gives the example of a child with a broken arm linked to both a vitamin deficiency and an abusive home environment. Several quite different interventions may be required.

The Obvious Quadrant

The quadrant of the Obvious applies to conditions with clear cause and effect, where there is a single right answer. According to Dave Snowden, the appropriate response is to sense what is going on, categorise the situation as one on a standard list, and then to respond in the way that people have been trained to do. This response may be trivial (a band-aid, say), or it may involve enormous professional skill. In medicine, much of nursing falls in this quadrant, as does much of surgery.

Michael Crichton’s Five Patients discuses the case of Peter Luchesi, a man admitted to Massachusetts General Hospital during 1969 with a crushed arm and nearly severed hand, as the result of an industrial accident:

Three inches above the left wrist the forearm had been mashed. Bones stuck out at all angles; reddish areas of muscle with silver fascial coats were exposed in many places. The entire arm about the injury was badly swollen, but the hand was still normal size, although it looked shrunken and atrophic in comparison. The color of the hand was deep blue-gray.

Carefully, Appel picked up the hand, which flopped loosely at the wrist. He checked pulses and found none below the elbow. He touched the fingers of the hand with a pin and asked if Luchesi could feel it; results were confusing, but there appeared to be some loss of sensation. He asked if the patient could move any of his fingers; he could not.

Meanwhile, the orthopedic resident, Dr. Robert Hussey, arrived and examined the hand. He concluded that both bones in the forearm, the radius and ulna, were broken and suggested the hand be elevated; he proceeded to do this.

Outside the door to the room, one of the admitting men stopped Appel. ‘Are you going to take it, or try to keep it?’

‘Hell, we’re going to keep it,’ Appel said. ‘That’s a good hand.’

Once the surgeons had sensed the problem and categorised it as an arm reconstruction, a team of three surgeons, two nurses, and an anaesthetist (all highly trained in their respective fields) then spent more than 6 hours in the operating theatre, repairing bone, tendons, and blood vessels. Certainly not trivial, but a case of professionals doing what they were trained to do.

The Complicated Quadrant

Public Domain image

The Complicated quadrant is the realm of diagnosis. Information is collected – in medicine, that generally means patient history, blood tests, scans, etc. – and is then subjected to analysis. This identifies the nature of the problem (in an ideal world, at least), which in turn indicates the appropriate response.

Diagnosis by physicians typically searches for the cause of an illness, while diagnosis by nurses typically focuses on severity. This reflects differences in the responses that physicians and nurses have been trained to provide (the triage officer in a modern hospital is typically a nurse).

Decades of work have gone into automating the diagnosis process – initially using statistical analysis, later using expert systems, and most recently using machine learning. At present, the tool of choice is still the human brain.

In general, modern medicine excels when it operates in the Obvious and Complicated quadrants.

The Complex Quadrant

The Complex quadrant is the realm of interactions. It is inherently very difficult to deal with, and cause and effect are difficult to disentangle. The paradigm of information collection and analysis fails, because each probe of the system changes it in some way. The best approach is a sequence of experiments, following each probe with a response that seems reasonable, and hoping to find an underlying pattern or a treatment that works. Michael Crichton provides this example:

Until his admission, John O’Connor, a fifty-year-old railroad dispatcher from Charlestown, was in perfect health. He had never been sick a day in his life.

On the morning of his admission, he awoke early, complaining of vague abdominal pain. He vomited once, bringing up clear material, and had some diarrhea. He went to see his family doctor, who said that he had no fever and his white cell count was normal. He told Mr. O’Connor that it was probably gastroenteritis, and advised him to rest and take paregoric to settle his stomach.

In the afternoon, Mr. O’Connor began to feel warm. He then had two shaking chills. His wife suggested he call his doctor once again, but when Mr. O’Connor went to the phone, he collapsed. At 5 p.m. his wife brought him to the MGH emergency ward, where he was noted to have a temperature of 108 °F [42 °C] and a white count of 37,000 (normal count: 5,000–10,000).

The patient was wildly delirious; it required ten people to hold him down as he thrashed about. He spoke only nonsense words and groans, and did not respond to his name. …

One difficulty here was that John O’Connor could not speak, and so could not provide information about where he felt pain. He appeared to suffer from septicaemia (blood poisoning) due to a bacterial infection in his gall bladder, urinary tract, GI tract, pericardium, lungs, or some other organ. Antibiotics were given almost immediately, to save his life. These eliminated the bacteria from his blood, but did not tackle the root infection. They also made it difficult to identify the bacteria involved, or to locate the root infection, thus hampering any kind of targeted response. In the end (after 30 days in hospital!) John O’Connor was cured, but the hospital never did locate the original root infection.

Similar problems occur with infants (Michael Crichton notes that “Classically, the fever of unknown origin is a pediatric problem, and classically it is a problem for the same reasons it was a problem with Mr. O’Connor—the patient cannot tell you how he feels or what hurts”). As Kemperman notes, medical treatment of the elderly often also falls in the Complex domain, with multiple interacting chronic conditions, and multiple interacting drug treatments. Medical treatment of mental illness is also Complex, as the brain adapts to one treatment regimen, and the doctor must experiment to find another that stabilises the patient.

Similarly Complex is the day-to-day maintenance of wellness (see the Food and Wellness section below) which often falls outside of mainstream medicine.

The Chaotic Quadrant

The Chaotic quadrant is even more difficult than the Complex one. Things are changing so rapidly that information collection and experimentation are impossible. The only possible response is a dance of acting and reacting, attempting to stabilise the situation enough that it moves from Chaotic to Complex. Emergency medicine generally falls in this quadrant – immediate responses are necessary to stop the patient dying. In the airline industry, the ultimate (and extremely rare) nightmare of total engine failure shortly after takeoff (as in US Airways Flight 1549) sits here too – each second of delay sees gravity take its toll.

Success in the Chaotic domain requires considerable experience. In cases where the problem is a rare one, this experience must be created synthetically using simulation-based training.

Food and Wellness

Michael Crichton notes that “The hospital is oriented toward curative treatment of established disease at an advanced or critical stage. Increasingly, the hospital population tends to consist of patients with more and more acute illnesses, until even cancer must accept a somewhat secondary position.” There is, however, a need for managing the Complex space of minor variations from wellness, using low-impact forms of treatment, such as variations in diet. Some sections of this field are reasonably well understood, including:

Traditional culture often addresses this space as well. For example, Chinese culture classifies foods as Yin (cooling) or Yang (heaty) – although there is little formal evidence on the validity of this classification.

There remain many unknowns, however, and responses to food are highly individual anyway. There may be a place here for electronic apps that record daily food intake, medicine doses, activities, etc., along with a subjective wellness rating. Time series analysis may be able to find patterns in such data – for example, I might have an increased chance of a migraine two days after eating fish. Once identified, such patterns suggest obvious changes in one’s diet or daily schedule. Other techniques for managing this Complex healthcare space are also urgently needed.

The three men and their sisters

The medieval Propositiones ad Acuendos Juvenes (“Problems to Sharpen the Young”) is attributed to Alcuin of York (735–804), a leading figure in the “Carolingian Renaissance.” He is the middle person in the image above.

Along with the more famous problem of the wolf, the goat, and the cabbage, Propositiones ad Acuendos Juvenes contains the problem of the three men and their sisters. Three men, each accompanied by a sister, wish to cross a river in a boat that holds only two people. To protect each woman’s honour, no woman can be left with another man unless her brother is also present (and if that seems strange, remember that Alcuin was writing more than 1,200 years ago). In Latin, the problem is:

“Tres fratres erant qui singulas sorores habebant, et fluvium transire debebant (erat enim unicuique illorum concupiscientia in sorore proximi sui), qui venientes ad fluvium non invenerunt nisi parvam naviculam, in qua non potuerunt amplius nisi duo ex illis transire. Dicat, qui potest, qualiter fluvium transierunt, ne una quidem earum ex ipsis maculata sit?”

The diagram below (click to zoom) shows the state graph for this problem. The solution is left (per tradition) as an exercise for the reader (but to see Alcuin’s solution, highlight the white text below the diagram).

Miss A and Mr A cross
Mr A returns (leaving Miss A on the far side)
Miss B and Miss C cross
Miss A returns (leaving Misses B and C on the far side)
Mr B and Mr C cross
Mr B and Miss B return (leaving Miss C and Mr C on the far side)
Mr A and Mr B cross
Miss C returns (leaving 3 men on the far side)
Miss A and Miss C cross
Mr B returns (leaving the A’s and C’s on the far side)
Mr B and Miss B cross

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 Babylonian zero was never used at the end of a number. 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):


Phonetically, that reads:

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):


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):


Phonetically, that reads:

[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:


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


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:


The reverse side also has writing:


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.

In English, the passage reads:

… 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:

ZΑΝ {Θω} {ΙΥ} {Χω}


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:


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

And returning him safely to the earth

In 1961, John F. Kennedy told Congress: “I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the earth.

The Moon landing on 20 July 1969 achieved the first part of that goal. The second part was yet to come (in 1970, that would prove to be the hard part).

But on 21 July 1969, at 17:54 UTC, the spacecraft Eagle lifted its metaphorical wings and took off from the Moon (well, the upper ascent stage took off, as shown in the photograph below). There followed a rendezvous with Columbia, a flight back to Earth, and an eventual splashdown on 24 July. Mission accomplished.