The 1950 International Congress of Mathematicians was held in Cambridge, Massachusetts, USA from 30 August to 6 September. Rome was one of the invited plenary speakers. He delivered his address at 9 a.m. on Friday 1 September in the Fogg Large Room.

Before giving a cut-down version of Rome's address, we give a little information to put the lecture into context. The commentary on the Almagest mixes three Greek sources and the Greek text was first published by Joachim Camerarius in 1538. Rome produced a critical addition published in three volumes 1931, 1936, 1943. In Rome's lecture he refers to the Almagest, a work by Ptolemy which contains details of observations and of the mathematical methods used in the astronomical work. It was written around 150 AD. Ptolemy also wrote the Handy tables, which contains expanded tables from the Almagest instructions to aid calculating using them.

#### The Calculation of an Eclipse of the Sun According to Theon of Alexandria

In English-speaking countries, Theon of Alexandria, and still more his daughter, Hypatia, are well known through a novel by Charles Kingsley. Whether or not there are still many people who actually have read it, I don't know. Novels grow old quickly, and that particular one is nearly a centenarian, having been published in 1853. It is also rather difficult to understand for anyone who is not acquainted at the same time with ancient Greek literature and history, the history of Greek science, and ancient church history, as well as the religious history of England in the Victorian Era, Wiseman, Pusey, Newman, and the other Tractarians. But I often remarked in England that the name Hypatia appealed to many people there, and I should not be surprised if it were the same here.

Now, I happen to be engaged in the editing of the Commentary of Theon of Alexandria on the Almagest of Ptolemy. In the course of that work, I found that Hypatia had made a revision of the third book of her father's big work, which was hitherto unknown, and which is now the only text of Hypatia actually published. By the way, those 150 pages show that Kingsley, in his chapter entitled Nephelococcugia (an allusion to "The Birds" of Aristophanes), was perhaps not altogether well inspired when he imagined Hypatia speaking like Proclus and making mystical speculations on mathematics. The only text we can read of hers is rather elementary, but quite sound, mathematics and astronomy. It is without all the mystical nonsense some people of the second Sophistic period were dreaming à propos of still more elementary mathematics.

My editorial work has just reached the Sixth Book of the Commentary on the Almagest, which explains the theory of eclipses of the sun and moon. As an example of the calculation of an eclipse of the sun, Theon has taken the eclipse of June 16, 364 A.D. We shall rapidly examine it. The Greek text of it is not yet completely "established" as we classical philologists say. Before going to the printer, it will undergo a few minor changes. But on the whole, I think it is already sufficiently secure to enable me to speak about it.

Theon says that he has observed the eclipse of the sun of June 16, 364, and he gives the times he has recorded. On the other hand, at several places in his Sixth Book, he calculates the different elements of the same eclipse, to show how it must be done, and also to prove that the results of the Handy tables are tolerably concordant with the results of the Almagest. He explains also the trigonometrical proofs of the Almagest, and occasionally adds some proofs of his own. I shall show you one of these trigonometrical computations, selecting a very short one, for the greatest defect of ancient trigonometry is that it is very cumbrous to handle, and generally frightfully long.

Both in the Almagest and in the Handy tables, the time of the eclipse is determined by seeking first the time of mean conjunction, then the time of true conjunction, and finally, taking the sun's and the moon's parallax into account, the time of apparent conjunction for the given place, which is here Alexandria. Having thus found the elements of the eclipse, he proceeds to compute the circumstances, the magnitude, and the time table, of which he gives two successive approximations. Finally, there is a calculation of the "prosneusis", an azimuthal direction that enables one to find the place on the sun's disc where the first and last contacts can be observed.

Having thus taken a bird's eye view of the calculation, we can now examine some details.

The date is given according to two systems: the "vague" year of 365 days, and the "Alexandrian" one of 365 days with a leap year every fourth year: the Julian calendar is based on Alexandrian astronomy. In the corresponding explanations, Theon gives a model of the calculations needed to pass from one system to the other. Those models enable us to fix with absolute certainty, if there were any doubt about it, when the leap years were started at Alexandria. The names of the months might seem more puzzling. Every month has 30 days, and after 12 months there are 5 or 6 complementary days. Thoth, Payni, and so on, are the original Egyptian names, which were used by the Alexandrian Greeks. Alexandria was a bilingual town. By the way, you know that the French revolution at the end of the 18th century tried to introduce a new calendar. In fact, it was simply the Alexandrian one, with new names, starting with the autumn equinox, because the "Alexandrian" year started about that time, August 29 or 30. How the French came to that idea is simple to find when one knows that the decimal system of weights and measures was in great part organised by Delambre, the man who made the triangulation of the arc of meridian Dunkirk-Paris-Barcelona, which was originally the base of the decimal metre. Now, that same man was also an historian of astronomy; his history of Greek astronomy is not yet completely superseded, after 150 years. So it is easy to guess where he found the original version of Floreal and Germinal and Messidor.

Theon says he has observed the eclipse. For the first contact he says he is quite sure of the time, and for the last, he says it is approximate. Now, Delambre finds that this result is too beautiful to be true. But Delambre has a knack of casting a doubt on the reality of all sorts of observations made by Greek astronomers. Strangely enough, people do not read Delambre any more (and in this, I think they are wrong), and still, all those hypercritical doubts are lingering about in general histories of astronomy.

First, note that Theon does not give exact figures for the minutes: he observed the first contact on the 24th of Thoth, year 1112 of the era of Nabonassar, $2 \large\frac{1}{2}\normalsize \large\frac{1}{3}\normalsize$ seasonal hours P.M. The last contact, approximately $4 \large\frac{1}{2}\normalsize$ seasonal hours P.M.

Then if we compare the calculated time tables given by Theon, we see clearly what he means: He finds the first contact according to the Almagest, at $3 \large\frac{1}{3}\normalsize$ equinoctial hours P.M. which he reduces to $2 \large\frac{1}{2}\normalsize \large\frac{1}{3}\normalsize$ seasonal hours P.M., but according to the Handy tables, he finds $3 \large\frac{1}{4}\normalsize$ equinoctial hours, and he reduces it again to $2 \large\frac{1}{2}\normalsize \large\frac{1}{3}\normalsize$ seasonal ones. The maximum phase, according to the Almagest, at $4 \large\frac{1}{3}\normalsize$ equinoctial hours, is taken to be equivalent to $3 \large\frac{1}{2}\normalsize \large\frac{1}{6}\normalsize$ seasonal ones; according to the Handy tables it is at $4 \large\frac{1}{4}\normalsize$ equinoctial hours or $3 \large\frac{1}{2}\normalsize \large\frac{1}{6}\normalsize$ seasonal hours. The last contact, according to the Almagest, is at $5 \large\frac{1}{4}\normalsize$ equinoctial hours, or $4 \large\frac{1}{2}\normalsize$ seasonal; according to the Handy tables it is at $5 \large\frac{1}{6}\normalsize$ equinoctial or $4 \large\frac{1}{2}\normalsize$ seasonal. Maybe some of those figures will be corrected when my text is completely "established", but it is quite clear that the objection of Delambre is not justified. In fact, I am under the impression that whenever Theon or Ptolemy uses Egyptian fractions instead of sexagesimal ones, he intends only to give a rougher approximation.

Can we get an idea of their approximations? Ptolemy (Almagest, ed. Heiberg, 1st vol., p. 505, 24) speaking of eclipses, considers $\large\frac{1}{16}\normalsize$ of an equinoctial hour as negligible. Theon considers as small an error as one that can be reckoned to 10 minutes.

It is interesting to compute our eclipse with modern tables. The concordance is fairly good. With the tables of P V Neugebauer, which are Schram's tables corrected to bring them into accord with Schoch's, I found for Alexandria: first contact, 15h 12m true time; maximum phase 16h 25m; last contact 17h 20m; magnitude 5.5. (Theon finds with the Almagest, 4; 39, 18 digits, and with the Handy tables 4; 58 digits.) This compares fairly well with the above results in equinoctial hours. If we use Oppolzer, Kanon der Finsternisse, we get the first contact at 15h 28m true time at Alexandria; maximum phase, 16h 20.88m; last contact, 17h 19m; magnitude 4.3. Ginzel, Spezieller Kanon, speaks of our eclipse, p. 213. I am afraid he has not been able to read the "Egyptian" fractions in the Old Basel edition of 1538, and his figures are all wrong.

But, writing this, I see already some of the readers preparing an objection: can we use a modern table to control the accuracy, to a few minutes, of an observation by Theon? I think we may not. At any rate, we may not always. The astronomers who make tables of eclipses choose the coefficients of their formulae to fit observations that they deem interesting. Oppolzer took a minimum of them, 7 from antiquity; 3 from the middle ages, with the result that his Kanon der Finsternisse does not fit exactly the observations from 1650 to our time. Ginzel gave the maximum weight to the eclipses of +71 and +1385. For the rest, he considered only 3 eclipses of antiquity (all after Plutarch, 2nd century A.D.) and all the medieval ones; but he, too, overlooked all the modern observations.

Schoch tried to fit in the modern observations, too. P V Neugebauer has shown, for instance, how Schoch's tables give an excellent description of the sun eclipse of April 17, +1912: For the point chosen, Senlis in France, the curve of centrality calculated with Schoch's tables is not farther from what most probably happened, according to observation, than the curves given by the current almanacs, which are all slightly different from one another. But in addition to this, Schoch took into account all the sun eclipses of antiquity he could know, and all the eclipses of the Middle Ages, For Greek antiquity, he depends on Fotheringham's A solution of ancient eclipses of the sun. Now Fotheringham has examined also the eclipse of June 16, 364. You see that if we try to control Theon by means of Schoch's tables, we are arguing in a circle.

I would make some remarks about that article of Fotheringham.

He calls his number 8, The eclipse of Hipparchus. When he wrote his article, he had to rely upon a partial edition of a few pages of Pappus, by Hultsch. Now the full text of Pappus' Fifth Book is printed. But there is much hesitation about the date of that eclipse, and I dare not say anything about it.

Fotheringham's number 11 is just our eclipse of June 16, 364. He took all the particulars about it from the only edition available, the Basel edition of 1538. There, the observed times of Theon are recorded distinctly as being equinoctial hours. The consequence of this was that the time supposed to be recorded by Theon could not be brought into line with the other eclipses recorded, and with the theory. Finally, Fotheringham admitted that "The time of Theon was 37, 32, and 38 minutes slow at the three observed phases of the eclipse."

First, I showed you that Theon did not give the time to the minute. Then, letting alone the question of night determination of time, which is of course irrelevant here, I am afraid it is impossible to admit that Theon had such wretched sundials at his disposal. But there is another thing: although my text of the Sixth Book is not yet definitively established, I am quite sure that the original text of Theon did not speak of equinoctial hours; in the best manuscript, the Medicaeus 28, 18, the word equinoctial is absent. Besides, Theon says that the calculated hours conform to the observed ones. But the conformity exists only with the calculated seasonal hours, and of course they cannot be mistaken for equinoctial ones, as the whole calculation is given. If the hours recorded by Theon are seasonal, they fit fairly well with the hour angles called "set A" by Fotheringham, which are obtained by combining Newcomb's tables with Brown's movements of the moon and Fotheringham's obliquity of ecliptic.

In consequence, if tables of syzygy were to be calculated again on the principles of Schoch, there would be some change in two or three of the equations.

One might ask how they proceeded to observe an eclipse of the sun. Up to now, I don't think a text has ever been discovered describing the method of observing it, when the sun is not quite close to the horizon. Probably the ancient astronomers found that detail too trivial to mention. Many systems have been suggested, but I venture to add that there is a possibility which all historians always seem to overlook or to discard. When making glass, the trouble is not in getting it coloured, but in getting it white. If we now consult the archaeologists, we shall see that the glassmakers were able to produce glass (and transparent glass, too) in practically all shades, up to black. Alexandria was especially renowned for its blue glass, from dark to light blue. On the other hand, if we look at the Optics of Ptolemy, we shall find that he could cut and polish glass to plane or cylindrical forms. The blue Alexandrian glass was exported as a half-finished product in the form of cubes, which were used by glassmakers very far away, up to the Rhine. I think it was not at all difficult for an astronomer at Alexandria to find a tolerably plane screen that would do nicely to observe the sun. And it is not extravagant to suppose that they had noticed it was possible to look at the sun through that screen. Think also of the famous emerald of Nero.

This suggestion that they might have resorted to glass screens is of course a mere hypothesis, but I think that hypothesis is not worse than the others generally advanced in that connection.

Ginzel points out that screens seem to have been used only since the 17th century. This does not prove that the ancient Greeks did not know them. On the other hand, it would be a grave exaggeration to say that, as they had coloured glass and could not possibly not see that it was an excellent screen, they surely used it. When you have in hand all the elements of the solution of a problem, you are not yet saved if you don't realize that you have found the solution. Of this I shall give you a clear example when I show you a trigonometrical solution.

Next we may perhaps find some interesting things, if we look at the calculations of mean conjunction according to the Almagest and the Handy tables.

Ptolemy says distinctly in the Almagest that the mean conjunction might be imputed simply by using the tables of the sun's and moon's mean motion. In fact, in the Handy tables, there is no table of mean syzygies. As the moon epicycle's centre at the time of mean syzygy must be on the apogee of the eccentric, we find the time of mean conjunction the only thing we have to do, when we have the moon's position for the beginning of the month we are inquiring about, is to seek in the tables for the epicycle's centre quantities that will complete the 360 degrees of a circle exactly. The days and hours corresponding to those arcs, are the sought days and hours of the mean conjunction, and the only thing we need to do is to fill in the other columns for the same days and hours to get all the required information.

In the Small commentary on the Handy tables, we find an alternative way which is interesting, because it is exactly the method that is followed to determine the date of Easter in the church calendar. Even the word "epact" is used by Theon. I say Theon, although I cannot yet guarantee that this place is not interpolated. But it seems prima facie to be authentic. The Ptolemaic astronomy has surely been used by the experts who tried to solve the vexed question of the Easter date. The council of Nicea took place in 325, just about the time when Pappus wrote his Commentary. I think the date of the equinox was fixed on the 21st of March, because the tables of the sun in the Almagest point to that date for 325 A.D. Tannery supposed that the council relied upon an observation of the equinox, but as the council took place somewhere in May, it was too late to make such an observation. It was much easier and more in accord with the habits of the period, too, to have it computed from the Almagest or from the Handy tables. So the system of epacts might also derive from the Ptolemaic School.

In the Almagest, Ptolemy has found it useful to make special tables of syzygies, one for new moons and another for full moons. He starts from the 1st of Thoth of year 1 in the era of Nabonassar. From the elongation at this moment, and the table of moon's mean elongation, he can find the time of the first new moon of that year. From there he starts his cycles of 25 years.

Now those cycles of 25 years have been quite recently studied by Professors Neugebauer and van der Waerden. Egyptian texts published by Professor Neugebauer use just the same lunar cycles of 25 years, and suppose that 25 "Egyptian" years are exactly as long as 309 synodic months. Ptolemy knows a more accurate equivalent, viz, 309 months minus 0; 2, 47, 5 days.

Did Ptolemy know Egyptian texts of that type? In more than one place in the Almagest, he assumes a distinctly polemical tone, to say that people thinking they can make tables valid for all times to come cannot be considered as seeking the truth. It is tempting to suppose that the Egyptian texts in question are just based upon those "eternal tables."

On the other hand, the Commentary of Theon on the Sixth Book makes another hypothesis which can be interesting, too. The table is supposed to have 45 lines. That number of 45 lines plays a great role in the Almagest. A whole lot of tables are expressedly mentioned as having 45 lines. Such a mention gave a very quick way of controlling whether the copy was accurately done. There are many places in the Almagest and its commentators where the reader is given practical hints on how to detect errors in his manuscript. That was of course the great danger to avoid when using tables. 45 lines is nearly the highest number of lines found on Greek literary papyri. Publishers were consequently bound to use the largest usual size of papyrus rolls when they copied the Almagest. Now, with 45 lines, and periods of 18 years, as in the other tables of the Almagest, we come only to 810 of the era of Nabonassar (63 A.D.). Hipparchus worked about 625 Nabonassar. But the time of Ptolemy is about 885 Nabonassar. Ptolemy has obviously kept the disposition of Hipparchus' tables for the mean motions of sun, moon, and planets. When using those tables after 810 Nabonassar, it is quite simple to put down first the data for 810 and then add a sufficient number of 18 year periods to reach the given dates. But that does not work with the table of syzygy. Accordingly, he took cycles of 25 years, which brought him up to 354 A.D., quite enough for him, but not for Theon. But Ptolemy was not a believer in "eternal tables." His successors, disregarding his warnings, because they were no more able to do original research work, simply put appropriate prolongations in their manuscripts. There are even manuscripts where one can see by different hands the successive prolongations of the Byzantine period, many centuries after the period of validity of the tables.

In the Handy tables, 25 year cycles are used everywhere, which, further, start from the first year of Philippus, not of Nabonassar.

The table of syzygies is also an exception in the Almagest for a second reason: there the cycles are tabulated under the ordinal number of the year, and not as everywhere else under the number of years elapsed since the beginning of the era. This last way of tabulating is the cause of many distractions. I have had several opportunities of experimenting with it. In the Handy tables, the cycle arguments are given as in the table of syzygies.

In fact, you see that Ptolemy has probably kept in the Almagest the disposition of Hipparchus' tables. He says expressly how he corrected them. But in the table of syzygies, which probably did not exist in Hipparchus' works, he tried another disposition, which he adopted completely in the Handy tables. That table of syzygies marks thus a transition between his two great astronomical works.

We might now have a look at a trigonometrical calculation. As I told you, I shall take the shortest possible case, because ancient trigonometry is always very simple (all problems of spherical trigonometry are solved by application of one theorem, always the same; and all problems of plane trigonometry are solved as I shall show you); but it is nearly always very long. That is the reason why I am sure that Ptolemy had a whole regiment of calculators, perhaps slaves, at his disposal. Otherwise, he could not possibly have made all the tables that he published under his name: I don't think it was possible for a calculator working very rapidly and quite used to the Greek methods, to find one single datum of the fourth column in the Handy table of parallaxes in less than two hours. The practical way of doing it is explained by Theon. I had to use it, with the help of a machine, to find a missing figure. It took me twice as much time as that. The existence of those calculators can even be detected, I think, in the table of trigonometrical ratios.

We shall take as an example the table of eclipses in the Almagest. There is a table for the maximum distance of the moon, and another for the minimum distance; and finally, a table of corrections for intermediate positions. That table of corrections is made on the assumption that the digits of the eclipse and the length of the moon's path during the eclipse vary proportionally to the distance of the moon from the earth. The same table is used also for parallaxes. Accordingly, the calculation I shall show you is not to be found in the Sixth Book of the Almagest, but in the fifth (ed. Heiberg, p. 433), where Ptolemy explains how he has computed the seventh column of his table of parallaxes.

The solution is based on the following principle: if we have inscribed triangle $ABC$, and if we have made a table of chords of all the arcs of a circle, taking the radius as unity, it is clear that $CB = crd 2 CAB \times DB$.

You see at once that all the triangles can be solved, with a little patience. You see also that the solution is general and applies to any triangle. Well, the Greeks never saw it. They applied it only to rectangular triangles, and for the others always took the way around of cutting them into two rectangular triangles. This is the example which I promised to show you of the fact that when all the elements of a solution are known, there may still remain the need to realize that the solution is found.

[Rome's example of the detailed calculation is omitted.]

It happens often that tables of the collection contained in the Handy tables have been obtained by interpolation. These interpolations are simply proportional, but whenever there are remainders to distribute, that distribution is not made at random. Ptolemy had instinctively, if not explicitly, the sensation that it was better to reproduce more or less in the interpolated terms the variation of the calculated terms. Theon, in his Great commentary on the Handy tables, calls the attention of his readers to that point. Another curiosity of the Handy tables is that, in order to save labour, Ptolemy sometimes uses a table that was not originally meant for the problem he is solving, but which gives a numerical answer to a sufficient degree of approximation. We met a curious example of that process in the calculation of the moon's speed according to the Handy tables: take in the ... number corresponding to the datum called "moon's centre," divide by 10 and add 0 degrees 30': you have the moon's speed with an approximation of to within 1 minute.

We might finish our very incomplete exploration of the Sixth Book of Theon by having a look at the calculation of prosneusis. As it is always difficult to see the first and last contacts of an eclipse, it may be useful to know beforehand in which part of the sun's disc it will happen. Our modern astronomical almanacs always mention it. The astronomers of the Ptolemaic school used to calculate the point where a great circle passing through the sun and moon's centres should cut the horizon. If they put one of the circles of a spherical astrolabe at the same time on the sun's centre, and on the azimuth so defined, that circle was also of the point of the sun's disc at which to look. That was, I think, the use of the prosneusis computation. I don't think it had any astrological significance. At any rate, I could not find, either in the Tetrabiblos, the great bible of astrology, nor in the catalogue of Cumont, any text showing an astrological significance such as would, for instance, point out to what country the omen would apply. If there is no astrological significance in the prosneusis, I don't see any possible use for the prosneusis except the one I mentioned.

Delambre pretends it was not used for that purpose, because he thinks that ancient astronomers did not care to observe the first contact accurately. I have spoken already of this hypercritical opinion.

It would have been possible to compute the prosneusis for each case very accurately. But, as Greek trigonometry was very cumbrous to handle, that computation would have been very long. Accuracy on that point was not required. Ptolemy has, consequently, given in the Almagest and in the Handy tables approximate methods that are said by Theon to give concordant results. In fact, if we try them both on the eclipse of June 16, 364, we get with the Almagest a prosneusis of 16°51' from W in the northern direction, and with the Handy tables, a prosneusis of 21°41' from W in the northern direction. If the circle of the astrolabe were put exactly on the centre of the sun, and on the points of the horizon indicated by those calculations, this shifting of 4° on the horizon would of course have an influence on the location of the point where the first contact was to be looked for. In our particular case, the circle of the astrolabe would indicate two points situated at 4° from one another on the circumference of the sun. But as the diameter of the sun is seen under an angle of $\large\frac{1}{2}\normalsize$°, that arc of 4° on the circumference of the sun would have been seen under an angle of less than 0°1', which could not be appreciated with naked eyes; in both cases, the circle would point exactly to the same spot of the sun's disc. Thus it can be explained why Theon calls both results equivalent.

This conclusion is at variance with the conclusion of the article that I published in 1948 on the prosneusis. And I am not sure that I shall not change my mind again before I publish the fourth volume of the commentaries on the Almagest, where those notes will be printed. In fact, some of the Greek texts we have been exploring now together are not quite ready for publication, and some of them are not even ready at all. But I thought you would prefer an exploration of this kind, even with its hesitations and mistakes, to looking at
things which are better settled but already published everywhere.

I express my thanks to Professor P S Jones of University of Michigan, who has kindly revised my text, and Professor Cohen, Harvard University, who assisted me in the correction of proofs. At the same time, as this address was delivered within a few yards of the Widener Library, I take the occasion to thank Isis and its founder, Professor G Sarton, for the help which is constantly received from that quarter by anyone wishing to study the history of science.

UNIVERSITY OF LOUVAIN,

Last Updated July 2020