Lisa asked Oxford University Press for a copy of Aristotle's Fragmenta Selecta in return for reviewing a book. It arrived all busted up.
Is this someone's idea of a joke? A fragmented Fragmenta?
December 7, 2024
In 1543, Nicolaus Copernicus published a book arguing that the Earth revolves around the Sun: De revolutionibus orbium coelestium.
This is sometimes painted as a sudden triumph of rationality over the foolish yet long-standing belief that the Sun and all the planets revolve around the Earth. As usual, this triumphalist narrative is oversimplified. In the history of science, everything is always more complicated than you think.
First, Aristarchus had come up with a heliocentric theory way back around 250 BC. While Copernicus probably didn't know all the details, he did know that Aristarchus said the Earth moves. Copernicus mentioned this in an early unpublished version of De revolutionibus.
Copernicus also had some precursors in the Middle Ages, though it's not clear whether he was influenced by them.
In the 1300's, the philosopher Jean Buridan argued that the Earth might not be at the center of the Universe, and that it might be rotating. He claimed — correctly in the first case, and only a bit incorrectly in the second — that there's no real way to tell. But he pointed out that it makes more sense to have the Earth rotating than have the Sun, Moon, planets and stars all revolving around it, because
it is easier to move a small thing than a large one.In 1377 Nicole Oresme continued this line of thought, making the same points in great detail, only to conclude by saying
Yet everyone holds, and I think myself, that the heavens do move and not the Earth, for "God created the orb of the Earth, which will not be moved" [Psalms 104:5], notwithstanding the arguments to the contrary.Everyone seems to take this last-minute reversal of views at face value, but I have trouble believing he really meant it. Maybe he wanted to play it safe with the Church. I think I detect a wry sense of humor, too.
Sometime between 410 and 420 AD, Martianus Capella came out with a book saying Mercury and Venus orbit the Sun, while the other planets orbit the Earth!
This picture is from a much later book by the German astronomer Valentin Naboth, in 1573. But it illustrates Capella's theory — and as we'll see, his theory was rather well-known in western Europe starting in the 800s.
First of all, take a minute to think about how reasonable this theory is. Mercury and Venus are the two planets closer to the Sun than we are. So, unlike the other planets, we can never possibly see them more than 90° away from the Sun. In fact Venus never gets more than 48° from the Sun, and Mercury stays even closer. So it looks like these planets are orbiting the Sun, not the Earth!
But who was this guy, and why did he matter?
Martianus Capella was a jurist and writer who lived in the city of Madauros, which is now in Algeria, but in his day was in Numidia, one of six African provinces of the Roman Empire. He's famous for a book with the wacky title De nuptiis Philologiae et Mercurii, which means On the Marriage of Philology and Mercury. It was an allegorical story, in prose and verse, describing the courtship and wedding of Mercury (who stood for "intelligent or profitable pursuit") and the maiden Philologia (who stood for "the love of letters and study"). Among the wedding gifts are seven maids who will be Philology's servants. They are the seven liberal arts:
In seven chapters, the seven maids explain these subjects. What matters for us is the chapter on astronomy, which explains the structure of the Solar System.
This book De nuptiis Philologiae et Mercurii became very important after the decline and fall of the Roman Empire, mainly as a guide to the liberal arts. In fact, if you went to a college that claimed to offer a liberal arts education, you were indirectly affected by this book!
Here is a painting by Botticelli from about 1485, called A Young Man Being Introduced to the Seven Liberal Arts:
I'm no expert on this, but it seems as the Roman Empire declined there was a gradual dumbing down of scholarship, with original and profound works by folks like Aristotle, Euclid, and Ptolemy eventually being lost in western Europe — though preserved in more civilized parts of the world, like Baghdad and the Byzantine Empire. In the west, eventually all that was left were easy-to-read popularizations by people like Pliny the Elder, Boethius, Macrobius, Cassiodorus... and Martianus Capella!
By the end of the 800s, many copies of Capella's book De nuptiis Philologiae et Mercurii were available. Let's see how that happened!
To set the stage: Charlemagne became King of the Franks in 768 AD. Being a forward-looking fellow, he brought in Alcuin, headmaster of the cathedral school in York and "the most learned man anywhere to be found", to help organize education in his kingdom.
Alcuin set up schools for boys and girls, systematized the curriculum, raised the standards of scholarship, and encouraged the study of liberal arts. Yes: the liberal arts as described by Martianus Capella! For Alcuin this was all in the service of Christianity. But scholars, being scholars, took advantage of this opportunity to start copying the ancient books that were available, writing commentaries on them, and the like.
In 800, Charlemagne became emperor of what's now called the Carolingian Empire. When Charlemagne died in 814 a war broke out, but it ended in 847. Though divided into three parts, the empire flourished until about 877, when it began sinking due to internal struggles, attacks from Vikings in the north, etc.
The heyday of culture in the Carolingian Empire, roughly 768–877, is sometimes called the Carolingian Renaissance because of the flourishing of culture and learning brought about by Alcuin and his successors. To get a sense of this: between 550 and 750 AD, only 265 books have been preserved from Western Europe. From the Carolingian Renaissance we have over 7000.
However, there was still a huge deficit of the classical texts we now consider most important. As far as I can tell, the works of Aristotle, Eratosthenes, Euclid, Ptolemy and Archimedes were completely missing in the Carolingian Empire. I seem to recall that from Plato only the Timaeus was available at this time. But Martianus Capella's De nuptiis Philologiae et Mercurii was very popular. Hundreds of copies were made, and many survive even to this day! Thus, his theory of the Solar System, where Mercury and Venus orbited the Sun but other planets orbited the Earth, must have had an out-sized impact on cosmology at this time.
Here is part of a page from one of the first known copies of De nuptiis Philologiae et Mercurii:
It's called VLF 48, and it's now at the university library in Leiden. Most scholars say it dates to 850 AD, though Mariken Teeuwen has a paper claiming it goes back to 830 AD.
You'll notice that in addition to the main text, there's a lot of commentary in smaller letters! This may have been added later. Nobody knows who wrote it, or even whether it was a single person. It's called the Anonymous Commentary. This commentary was copied into many of the later versions of the book, so it's important.
Alas, now I need to throw a wet blanket on that, and show how poorly Martianus Capella's cosmology was understood at this time!
The Anonymous Commentary actually describes three variants of Capella's theory of the orbits of Mercury and Venus. One of them is good, one seems bad, and one seems very bad. Yet subsequent commentators in the Carolingian Empire didn't seem to recognize this fact and discard the bad ones.
These three variants were drawn as diagrams in the margin of VLF 48, but Robert Eastwood has nicely put them side by side here:
The one at right, which the commentary attributes to the "Platonists", shows the orbit of Mercury around the Sun surrounded by the larger orbit of Venus. This is good.
The one in the middle, which the commentary attributes to Martianus Capella himself, shows the orbits of Mercury and Venus crossing each other. This seems bad.
The one at left, which the commentary attributes to Pliny, shows orbits for Mercury and Venus that are cut off when they meet the orbit of the Sun, not complete circles. This seems very bad — so bad that I can't help but hope there's some reasonable interpretation that I'm missing. (Maybe just that these planets get hidden when they go behind the Sun?)
Robert Eastwood attributes the two bad models to a purely textual approach to astronomy, where commentators tried to interpret texts and compare them to other texts, without doing observations. I'm still puzzled.
That's 30 years after Copernicus came out with his book... but the clarification probably happened earlier. And Copernicus did mention Martianus Capella's work. In fact, he used it to argue for a heliocentric theory! In Chapter 10 of De Revolutionibus he wrote:
In my judgement, therefore, we should not in the least disregard what was familiar to Martianus Capella, the author of an encyclopedia, and to certain other Latin writers. For according to them, Venus and Mercury revolve around the sun as their center. This is the reason, in their opinion, why these planets diverge no farther from the sun than is permitted by the curvature of their revolutions. For they do not encircle the earth, like the other planets, but "have opposite circles". Then what else do these authors mean but that the center of their spheres is near the sun? Thus Mercury's sphere will surely be enclosed within Venus', which by common consent is more than twice as big, and inside that wide region it will occupy a space adequate for itself. If anyone seizes this opportunity to link Saturn, Jupiter, and Mars also to that center, provided he understands their spheres to be so large that together with Venus and Mercury the earth too is enclosed inside and encircled, he will not be mistaken, as is shown by the regular pattern of their motions.For [these outer planets] are always closest to the earth, as is well known, about the time of their evening rising, that is, when they are in opposition to the sun, with the earth between them and the sun. On the other hand, they are at their farthest from the earth at the time of their evening setting, when they become invisible in the vicinity of the sun, namely, when we have the sun between them and the earth. These facts are enough to show that their center belongs more to then sun, and is identical with the center around which Venus and Mercury likewise execute their revolutions.
This diversity of theories is fascinating... even though everyone holds, and I think myself, that the Earth revolves around the Sun.
Above is a picture of the "Hypothesis Tychonica", from a book written in 1643.
You King Gelon are aware the 'universe' is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while its radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.The last sentence, which Archimedes went on to criticize, seems to be a way of saying that the fixed stars are at an infinite distance from us.
For Aristarchus' influence on Copernicus, see:
And if we should admit that the motion of the Sun and Moon could be demonstrated even if the Earth is fixed, then with respect to the other wandering bodies there is less agreement. It is credible that for these and similar causes (and not because of the reasons that Aristotle mentions and rejects), Philolaus believed in the mobility of the Earth and some even say that Aristarchus of Samos was of that opinion. But since such things could not be comprehended except by a keen intellect and continuing diligence, Plato does not conceal the fact that there were very few philosophers in that time who mastered the study of celestial motions.For Buridan on the location and possible motion of the Earth, see:
For the full text of Copernicus' book, translated into English, go here.
December 21, 2024
A farmer with a wolf, a goat, and a cabbage must cross a river by boat. The boat can carry only the farmer and a single item. If left unattended together, the wolf would eat the goat, or the goat would eat the cabbage. How can they cross the river without anything being eaten?
You probably know this puzzle. There are two efficient solutions, related by a symmetry that switches the wolf and the cabbage.
But what you might not know is that this puzzle goes back to a book written around 800 AD, sometimes attributed to Charlemagne's advisor Alcuin! Charlemagne brought this monk from York to help set up the educational system of his empire. But Alcuin had a great fondness for logic. Nobody is sure if he wrote this book — but it's fascinating nonetheless.
It has either 53 or 56 logic puzzles, depending on the version. It's called Propositiones ad Acuendos Juvenes, or Problems to Sharpen the Young. If the wolf, goat and cabbage problem is too easy for you, you might like this one:
Three men, each with a sister, must cross a boat which can carry only two people, so that a woman whose brother is not present is never left in the company of another man.There are also trick puzzles, like this:
A man has 300 pigs. He ordered all of them slaughtered in 3 days, but with an odd number killed each day. What number were to be killed each day?Wikipedia says this was given to punish unruly students — presumably students who didn't know that the sum of three odd numbers is always odd.
It's fascinating to think that while some Franks were fighting Saxons and Lombards, students in more peaceful parts of the empire were solving these puzzles!
The book also has some of the first recorded packing problems, like this:
There is a triangular city which has one side of 100 feet, another side of 100 feet, and a third of 90 feet. Inside of this, I want to build rectangular houses in such a way that each house is 20 feet in length, 10 feet in width. How many houses can I fit in the city?This is hard! There's a nice paper about all the packing problems in this book:
However, Alcuin — or whoever wrote the book — didn't really solve the problem. They just used an approximate Egyptian formula for the area of a triangle in terms of its side lengths, and divided that by the area of the houses! This is consistent with the generally crappy knowledge of math in Charlemagne's empire.
There's an even harder packing problem in this book, which again isn't solved correctly.
For more on this book of puzzles, check out this:
He then goes on to discuss the problem of the wolf, the goat and the cabbage — though in this paper, presumably more historically accurate, it's a wolf, a goat and a bag of oats. Then he returns to the big question:A medieval mathematical manuscript
The medieval cleric Alcuin (AD 735–804) is credited in surviving manuscripts with being the originator of a collection of fifty-three mathematical and logical puzzles, the Propositiones ad acuendos iuvenes (‘Problems for Sharpening the Young’). There is no direct evidence to connect the collection with Alcuin either as compiler or creator, but even modern commentators continue to associate his name with the puzzles. There are at least fourteen extant or partial copies of the Propositiones, which date from the ninth to the fifteenth century, suggesting that the collection was in popular use at least from the time of Alcuin onwards. Michael Gorman confidently listed the Propositiones among ninety spurious prose works of Alcuin for two reasons: firstly, because Alcuin always attached a dedicatory letter to his works, and there is none here, and secondly, because the work falls, Gorman thinks, among those documents which Alcuin would have had neither the time nor the energy to write in the period AD 782–800. Alcuin himself admitted to having only ‘stolen hours’ at night in which to write a life of Willibrord. Despite Gorman’s view that the work is pseudo-Alcuin, it is reasonable to ask if there is internal evidence in the puzzles which would support or oppose the assigning of the collection to Alcuin himself, committed as he was to the promotion of educational subjects, including mathematics, in the course of the Carolingian ‘renaissance’.The majority of the problem types in the Propositiones are known from earlier Chinese, Indian, Egyptian, Byzantine, Greek, and Roman sources, whilst others appear in works by Boethius, Metrodorus, and Isidore of Seville. Among the puzzles, however, there are a small number of important types that are not as yet known from earlier sources. These include the so-called ‘river-crossing problems’, ‘strange family problems’, a transportation (or ‘desert-crossing’) problem, and a problem that relies on the summation of an arithmetical series. The case for the mathematical creativity of the author, if there was only one author, rests on these. One puzzle has outstripped all the others in having popular appeal down to the present day. This puzzle, passing presumably from later medieval reproductions of the Propositiones into oral tradition, concerns a farmer who needs to ferry three items across a river in a boat.
He then goes on to discuss Alcuin's fascination with numerology, acrostics, and making up nicknames for his friends. It's too bad we'll never know for sure what, if anything, Alcuin had to do with this book of puzzles!Did Alcuin compose the Propositiones?
In a letter to Charlemagne in AD 799, three years after Alcuin had moved from the Palace School to an abbacy at Tours, Alcuin wrote that he was sending, along with some examples of grammar and correct expression, quidquid figuras arithmeticas laetitiae causa (‘certain arithmetical curiosities for [your] pleasure’). He added that he would write these on an empty sheet that Charlemagne sent to him and suggested that ‘our friend and helper Beselel will [...] be able to look up the problems in an arithmetic book’. Beselel was Alcuin’s nickname for the technically skilled Einhard, later Charlemagne’s biographer. It would be convenient if the reference in Alcuin’s letter were to the Propositiones, but for many reasons it is almost certainly not. For one thing, fifty-three propositions and their solutions would not fit on the blank side of a folio, given that they occupy ten folio sides in most of the manuscripts in which they are currently found. Secondly, since Alcuin’s letter to Charlemagne dealt primarily with the importance of exactness of language, a grammarian and self-conscious Latinist like Alcuin would not describe the Propositiones as figurae arithmeticae, or refer to an ‘arithmetic’ book in a case where the solutions required both geometry and logic methods. Thirdly, the idea of Einhard looking up the problems in an arithmetic book is an odd one, given that the Propositiones is usually found with answers, even if in most cases it does not show how these answers were arrived at.Apart from the reasons given by Gorman for Alcuin not having been the author of the Propositiones — the effort and time involved and the absence of a dedication — there are deeper, internal reasons for concluding that Alcuin was not the author. The river-crossing puzzles, along with the ‘transportation problem’, and a puzzle about the number of birds on a100-step ladder took a considerable amount of time and mathematical sophistication to compose. Accompanying them are two types of puzzles that appear repeatedly in the collection and that demand far lesssophistication: area problems, often fanciful (such as problem 29: ‘how many rectangular houses will fit into a town which has a circular wall?’); and division problems where a ‘lord of the manor’ wishes to divide grain among his household (problems 32–5, and others). Repetition of problems suggests that the Propositiones was intended for use as a practice book, but this supposed pedagogical purpose stumbles (even if it does not fall) on two counts. Firstly, the solutions to most of the mensuration problems, like the one involving a circular town, rely on late Roman methods for approximating the areas of common figures (circles, triangles, etc.) and these methods can be quite wrong. Secondly, the lack of worked solutions in the Propositiones deprived the student of the opportunity to learn the method required to solve another question of the same type. Solution methods might have been lost in transcription, but their almost total absence makes this unlikely. Mathematical methods (algebra in particular) that post-dated the Carolingian era would have been necessary to provide elegant and instructive solutions to many of the problems, and one cannot escape the suspicion that trial-and-error was the method used by whoever supplied answers (without worked solutions) to the Propositiones: a guess was made, refined, and then a new guess made. Such an approach is difficult to systemise, and even more difficult to describe in writing for the benefit of students. We are left with a mixture of ‘complex’ mathematical problems, such as those cited earlier, and simpler questions whose answers are mostly not justified.
This lack of uniformity suggests that it was a compiler rather than a composer who produced the first edition of the Propositiones. If Alcuin was involved, he was no more than a medium through which the fifty-three puzzles were assembled. No one person was the author of the Propositiones problems, because no mathematically sophisticated person would have authored the weaker ones, and nobody who was not mathematically sophisticated could have authored the others. Furthermore, with the more sophisticated problems, there is a noticeable absence of the kind of refinement and repetition that is common in modern textbooks.