The Oikoumene of Ptolemy

I was reading recently about the Geographia of Ptolemy (written around 150 AD). This classic book applied Greek mathematical skills to mapping and map projection – and if there was one thing the Greeks were good at, it was mathematics. According to Neugebauer, Ptolemy believed the Oikoumene, the inhabited portion of the world, to range from Thule (63° North) to 16°25′ South, and 90 degrees East and West of Syene in Egypt.

The map above illustrates this Oikoumene, with a modern population overlay in red (data from SEDAC). Ptolemy was not too far wrong – today this region holds 80.6% of the world’s population, and the percentage would have been greater in antiquity.

Also shown on the map are some of the many cities listed in the Geographia. Open circles show Ptolemy’s coordinates (from here, adjusted to a Syene meridian), and filled circles show true positions. Ptolemy had reasonably good latitude values (an average error of 1.2° for the sample shown on the map), but much worse longitude values (an average error of 6.8°). The longitude error is mostly systemic – Ptolemy’s estimate of 18,000 miles or 29,000 km for the circumference of the earth was only 72% of the true value (several centuries earlier, Eratosthenes had come up with a much better estimate). If Ptolemy’s longitudes are adjusted for this, the average error is only 1.5°.

However, Ptolemy’s book deserves considerable respect – it is not surprising that it was used for more than a thousand years.

A Medieval Calendar

The beautiful image above (click to zoom) represents the month of September in the Très Riches Heures du Duc de Berry, a book of hours from the 1400s. In the background of the main picture is the Château de Saumur, with its height exaggerated (almost doubled). For comparison, below is a modern photograph of the château (by Kamel15) stretched vertically ×2:

The foreground of the main picture shows the grape harvest. At the top is a complex calendar. On the inner track, around the chariot of the sun, in red and black numerals, are the days of the month. On the outer track, in red and blue numerals, is a zodiacal calendar, showing the last days of Virgo and the beginning of Libra. Adjacent to the inner track are blue letters which relate to the 19-year Metonic cycle. Combining those letters with an appropriate table will show the phases of the moon for a given year.

The manuscript uses the Hindu-Arabic numerals first introduced to Europe by Fibonacci in his Liber Abaci of 1202. They are not quite the same as the ones we use today:

It is interesting to compare those digits with the ones in this German manuscript of 1459 by Hans Talhoffer (although Talhoffer actually mixes two different styles of 5). Then again, the letters of the alphabet have also changed since that time.


Spineless by Juli Berwald: a book review


Spineless by Juli Berwald (2017)

I recently read Spineless: The Science of Jellyfish and the Art of Growing a Backbone by Juli Berwald (not to be confused with Spineless by Susan Middleton). Part memoir and part science writing, this book is very well-written, and moderately full of information about the oft-ignored jellyfish clan (some readers will find the combination of autobiography and science not to their liking). The book cover appears to be mostly derived from this Haeckel print:

I myself was fascinated by the jellyfish in the Monterey Bay Aquarium, but I did not fall in love with jellyfish the way that Juli Berwald did. This book does not quite do the job either, perhaps because of the distractions of the (not all that compelling) autobiographical material. It also seemed difficult to reconcile the author’s concerns about global warming with her high-carbon lifestyle.


Sea nettles at the Monterey Bay Aquarium (photo by “Omegacentrix”)

Still, this book is well worth a read. Other reviews of the book are linked from the author’s Wikipedia article.

* * *
Spineless by Juli Berwald: 3 stars


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.


Looking back: 2009


Washington, DC in June 2009

In 2009, I had the privilege of visiting the United States twice (in June and November).

This was the year that saw the launch of the Lunar Reconnaissance Orbiter (which imaged, among other things, the Apollo 11 landing site), the Kepler space telescope (designed to look for exoplanets), the Herschel space observatory (an infrared telescope studying star formation), the Planck spaceprobe (which studied the cosmic microwave background), and the Wide-field Infrared Survey Explorer (an infrared telescope looking for minor planets and star clusters).


Apollo 11 landing site, imaged by the LRO (with photographs from 1969 inset)

More metaphorically, Bitcoin and the programming language Go were also launched. US Airways Flight 1549, on the other hand, was skillfully landed in a river. In archaeology, hoards were discovered in Staffordshire (gold and silver metalwork) and Shrewsbury (Roman coins). Australian Hospital Ship Centaur, torpedoed in 1943, was discovered off the Queensland coast.

Books of 2009 included Wolf Hall by Hilary Mantel (set in 1500–1535; a TV series of 2015), The Windup Girl by Paolo Bacigalupi (dystopian science fiction; Nebula Award winner), and The Maze Runner by James Dashner (young adult dystopian sci-fi; a film of 2014). Books that I later reviewed include The Lassa Ward by Ross Donaldson and God’s Philosophers by James Hannam.

Movies of 2009 included Avatar (rather disappointing), 2012 (a little silly), Angels & Demons (a travesty), Up (Pixar/Disney), Coraline (designed to give children nightmares), District 9 (designed to give adults nightmares), Julie & Julia (a film about cooking), The Imaginarium of Doctor Parnassus (a film about mirrors), and Sherlock Holmes (a lot of fun). On the whole, a good year for films.

In this series: 1978, 1980, 1982, 1984, 1989, 1991, 1994, 2000, 2004, 2006, 2009.


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 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.