The Santa Fe Trail #4

NPS map of the Santa Fe Trail in 1871 (click to zoom; more maps here)

The American Solar Challenge is on again in 2021, and includes a road race along the Santa Fe Trail on 4–7 August, from Independence, MO to Santa Fe, NM (exact route still to be decided).

To get myself in the mood, I’ve been reading Land of Enchantment, the memoirs of Marion Sloan Russell, who travelled the Santa Fe Trail multiple times. After marrying, she was an “army wife” for some time, before setting up a trading post beside the Trail. In 1871, she moved to a ranch in the mountains west of Trinidad, CO, where her husband was murdered during the Colfax County War. Towards the end of her life she visited many important sites along the Trail. They were already falling into ruin:

“At Fort Union I found crumbling walls and tottering chimneys. Here and there a tottering adobe wall where once a mighty howitzer had stood. Great rooms stood roofless, their whitewashed walls open to the sky. Wild gourd vines grew inside the officers’ quarters. Rabbits scurried before my questing feet. The little guard house alone stood intact, mute witness of the punishment inflicted there. The Stars and Stripes was gone. Among a heap of rubble I found the ruins of the little chapel where I had stood—a demure, little bride in a velvet cape—and heard a preacher say, ‘That which God hath joined together let no man put asunder.’”

Marion Sloan Russell died in 1936 (aged 92) after being struck by a car in Trinidad, CO. She is buried in Stonewall Cemetery.

Fort Union in 2006 (credit: Scott; click to zoom)

Other posts in this series: Santa Fe Trail #1, Santa Fe Trail #2, Santa Fe Trail #3, Santa Fe Trail #4.

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The Santa Fe Trail #3

NPS map of the Santa Fe Trail in late 1866 (click to zoom; more maps here)

The American Solar Challenge is on again in 2021, and includes a road race along the Santa Fe Trail on 4–7 August, from Independence, MO to Santa Fe, NM (exact route still to be decided).

To get myself in the mood, I’ve been reading Land of Enchantment, the memoirs of Marion Sloan Russell, who travelled the Santa Fe Trail multiple times. After marrying, she was an “army wife” for some time, before setting up a trading post beside the (somewhat shorter in 1866) Santa Fe Trail at Tecolote, NM (about 15 km south of Las Vegas, NM):

“We had five living rooms behind the store. They were cool and pleasant. The thick stone walls resisted both heat and cold. The windows were long and narrow running from ceiling to floor. I draped them with a gay silken print. The floor I had covered with Navajo rugs … Often I have heard old-timers laughing about the heat and the dust of the desert. I have heard them say jokingly that Hell would seem cool after living in Santa Fé. I had heard them say that the burning sands of the desert had sucked old-timers so dry that they could not pray. I had laughed with them …”

Hopefully solar cars in the American Solar Challenge do not find the temperatures quite so hellish. The chart below shows average maximum July temperatures (early August temperatures are on average only about 0.5°C cooler, and may indeed be warmer, which means that temperatures inside the vehicles will be very hot):

Click to zoom; map produced using climate data from worldclim.org

Other posts in this series: Santa Fe Trail #1, Santa Fe Trail #2, Santa Fe Trail #3, Santa Fe Trail #4.

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The Santa Fe Trail #2

NPS map of the Santa Fe Trail “Mountain Route” (click to zoom; more maps here)

The American Solar Challenge is on again in 2021, and includes a road race along the Santa Fe Trail on 4–7 August, from Independence, MO to Santa Fe, NM (exact route still to be decided).

To get myself in the mood, I’ve been reading Land of Enchantment, the memoirs of Marion Sloan Russell, who travelled the Santa Fe Trail multiple times. Her third trip was in 1860, at the age of 15, travelling from Fort Leavenworth along the “Mountain Route” or “Upper Crossing.” This route avoided Indian raids along the Cimarron Cut-Off. The Mountain Route crosses the 7,840 ft (2,390 m) Raton Pass:

“Breaking camp while it was still early, our cavalcade began the steep and tortuous ascent of the Raton Pass. Today we glide easily over hairpin curves that in 1860 meant broken axles and crippled horses. The trail was a faint wheel mark winding in and out over fallen trees and huge boulders.”

If the American Solar Challenge follows the Mountain Route, solar cars will hopefully have an easier time on the modern road. The “big climb” at the 2018 American Solar Challenge (following the Oregon Trail) was 902 m in 35 km (2.6%). Starting from Trinidad, CO, the Raton Pass has a similar climb of 558 m in 22 km (2.5%), with a maximum grade of 6% on the steepest sections.

Raton Pass in October 2009 (credit: Chris Light; click to zoom)

Other posts in this series: Santa Fe Trail #1, Santa Fe Trail #2, Santa Fe Trail #3, Santa Fe Trail #4.

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The Santa Fe Trail #1

Map of the Santa Fe Trail (credit: NPS; click to zoom)

The American Solar Challenge is on again in 2021, and includes a road race along the Santa Fe Trail on 4–7 August, from Independence, MO to Santa Fe, NM (exact route still to be decided).

To get myself in the mood, I’m reading a second-hand copy of Land of Enchantment, the memoirs of Marion Sloan Russell, who first travelled the Santa Fe Trail in 1852 as a young girl of seven (following the southern route, the “Cimarron Cut-Off”) under the leadership of François Xavier Aubry:

“Each night there were two great circles of wagons. Captain Aubry’s train encamped a half mile beyond the government’s. Inside those great circles the mules were turned after grazing, or ropes were stretched between the wagons and thus a circular corral made. Inside the corral were the cooking fires, one for each wagon. After the evening meal we would gather around the little fires. The men would tell stories of the strange new land before us, tales of gold and of Indians. The women would sit with their long skirts drawn up over a sleeping child on their laps. Overhead brooded the night sky, the little camp fires flickered, and behind us loomed the dark hulks of the covered wagons. … It was strange about the prairies at dawn, they were all sepia and silver; at noon they were like molten metal, and in the evening they flared into unbelievable beauty—long streamers of red and gold were flung out across them. The sky had an unearthly radiance. Sunset on the prairie! It was haunting, unearthly and lovely.”

Not everything was quite so lovely; a theft in Santa Fe forced Marion’s widowed mother to abandon a further trip to California. Instead, she ran a boarding house: first in nearby Albuquerque, and later in Santa Fe itself. Let’s hope things go more smoothly for the solar cars of 2021.

Other posts in this series: Santa Fe Trail #1, Santa Fe Trail #2, Santa Fe Trail #3, Santa Fe Trail #4.

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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).

Paul Erdős (photo: Kmhkmh)

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.

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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, 20^th 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.

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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.

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 1^st 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 1^st 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 …

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:

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 4^th 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.

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 3^rd 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 4^thcentury 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

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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.

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A tale of two arrivals

Fifty years ago, on 19 July 1969, the spaceship duo Columbia / Eagle entered orbit around the Moon, roughly 3 days and 4 hours after its launch, as part of the Apollo 11 mission. Eagle (with Neil Armstrong and Buzz Aldrin) went on the land on the moon on 20 July while Columbia (with Michael Collins) continued to orbit the moon. When he announced the space programme, Kennedy had said:

“We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.”

Much can be learned from doing hard things, and an enormous amount was learned from the space programme. Solar car teams also learn a great deal from doing hard things. Fifty years after Columbia and Eagle entered orbit, also after hard effort, the University of Michigan Solar Team’s solar car Electrum arrived in public view (at 17:30 Michigan time). They intend to win too!

photo: University of Michigan Engineering

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On having multiple hypotheses which all fit the data

“For every fact there is an infinity of hypotheses.” – from Robert Pirsig, Zen and the Art of Motorcycle Maintenance

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“‘I will read the inventory… First item: A very considerable hoard of precious stones, nearly all diamonds, and all of them loose, without any setting whatever… Second item: Heaps and heaps of loose snuff, not kept in a horn, or even a pouch, but lying in heaps… Third item: Here and there about the house curious little heaps of minute pieces of metal, some like steel springs and some in the form of microscopic wheels… Fourth item: The wax candles, which have to be stuck in bottle necks because there is nothing else to stick them in… By no stretch of fancy can the human mind connect together snuff and diamonds and wax and loose clockwork.’

‘I think I see the connection,’ said the priest. ‘This Glengyle was mad against the French Revolution. He was an enthusiast for the ancien regime, and was trying to re-enact literally the family life of the last Bourbons. He had snuff because it was the eighteenth century luxury; wax candles, because they were the eighteenth century lighting; the mechanical bits of iron represent the locksmith hobby of Louis XVI; the diamonds are for the Diamond Necklace of Marie Antoinette.’

Both the other men were staring at him with round eyes. ‘What a perfectly extraordinary notion!” cried Flambeau. “Do you really think that is the truth?’

‘I am perfectly sure it isn’t,’ answered Father Brown, ‘only you said that nobody could connect snuff and diamonds and clockwork and candles. I give you that connection off-hand. The real truth, I am very sure, lies deeper.’

He paused a moment and listened to the wailing of the wind in the turrets. Then he said, ‘The late Earl of Glengyle was a thief. He lived a second and darker life as a desperate housebreaker. He did not have any candlesticks because he only used these candles cut short in the little lantern he carried. The snuff he employed as the fiercest French criminals have used pepper: to fling it suddenly in dense masses in the face of a captor or pursuer. But the final proof is in the curious coincidence of the diamonds and the small steel wheels. Surely that makes everything plain to you? Diamonds and small steel wheels are the only two instruments with which you can cut out a pane of glass.’

The bough of a broken pine tree lashed heavily in the blast against the windowpane behind them, as if in parody of a burglar, but they did not turn round. Their eyes were fastened on Father Brown. ‘Diamonds and small wheels,’ repeated Craven ruminating. ‘Is that all that makes you think it the true explanation?’

‘I don’t think it the true explanation,’ replied the priest placidly; ‘but you said that nobody could connect the four things. The true tale, of course, is something much more humdrum. Glengyle had found, or thought he had found, precious stones on his estate. Somebody had bamboozled him with those loose brilliants, saying they were found in the castle caverns. The little wheels are some diamond-cutting affair. He had to do the thing very roughly and in a small way, with the help of a few shepherds or rude fellows on these hills. Snuff is the one great luxury of such Scotch shepherds; it’s the one thing with which you can bribe them. They didn’t have candlesticks because they didn’t want them; they held the candles in their hands when they explored the caves.’

‘Is that all?’ asked Flambeau after a long pause. ‘Have we got to the dull truth at last?’ ‘Oh, no,’ said Father Brown.

As the wind died in the most distant pine woods with a long hoot as of mockery Father Brown, with an utterly impassive face, went on: ‘I only suggested that because you said one could not plausibly connect snuff with clockwork or candles with bright stones. Ten false philosophies will fit the universe; ten false theories will fit Glengyle Castle. But we want the real explanation of the castle and the universe.’” – from G. K. Chesterton, “The Honour of Israel Gow”

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