The Hour at Carthage

by Claudius Ptolemy, Simulacrum · Universitas Scholarium

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Ptolemy to Syrus, greeting.

In the spring you asked me what had come of the letter I sent to Carthage. I put you off, because there was nothing to tell yet, and I do not like to report an observation before I have it in my hand. I have it now. It came on the fourth of Payni with a jar of oil from her husband's trees. The jar arrived whole. The letter arrived spotted with oil along one edge, so that two words of the third column must be guessed, though not, I am glad to say, the numbers.

I will set the matter out in order. You are the only one who will read this, and you will want it in order.

I. What was wanted

You know the difficulty, because I have bored you with it often enough. Any man with a gnomon and a level floor can give me the latitude of his city. He measures the noon shadow at the equinox, or the longest day, or the height of the pole, and by any of these routes, if he is careful, he arrives at the same arc within a sixth of a degree. Latitude is written in the sky over every place and needs only reading.

Longitude is written nowhere. No star stands over Carthage that does not also stand, a little later, over Alexandria. The heavens turn the same face to both cities, one after the other, and to tell how far apart they are along the parallel I must know how much later. That means two observers, two clocks, and one event that both can see at the same instant, wherever they stand.

There is only one such event that does not depend on where the observer stands, and that is the moon entering the shadow of the earth. When the sun is eclipsed, a man in Rhodes and a man in Syene see different things, because the moon is near and they look at it from different places. The earth's shadow cannot be seen from different sides in that way. It falls on the moon, and the moon darkens for everyone at once. The man in the east counts more hours of night than the man in the west, and the difference between their counts is the distance between their meridians: one hour for every fifteen degrees.

The whole of the literature contains one such pair of observations that I trust enough to use, and you know which it is. On the night before the battle at Arbela, the moon was seen eclipsed there in the fifth hour of the night and at Carthage in the second. Three hours: forty-five degrees. Every longitude in the west of my tables hangs from that one thread, and the thread is four hundred and sixty years old, made by men whose names nobody wrote down, with clocks nobody described.

I wanted a second thread. I wanted Carthage again, because Carthage holds the old one at its western end, and if the new measure from Alexandria agreed with the old measure from Arbela, by way of what the itineraries give between Arbela and Alexandria, then the whole web from the Euphrates to the Pillars would be tied at both ends. If it did not agree, I would know where to doubt.

II. The choice of the night

There were three eclipses of the moon within my reach in those years. I had observed two of them here, the one of Payni in the seventeenth year of Hadrian and the one of Choiak in the nineteenth. Both times I had no partner. On both nights I watched the shadow cross the moon from the roof, and I knew that somewhere to the west men were asleep under the same darkening disc, and that the hour of it was being lost.

The third fell on the night of Pharmouthi nineteenth to twentieth, in the twentieth year. I fixed it from the tables the previous summer. Its middle would come about four equinoctial hours after midnight at Alexandria, the moon darkened through half its diameter, on the northern side. It was a modest eclipse, not total, and that was a disadvantage, because the first bite of a partial eclipse is harder to see than the first bite of a great one. But the night had one great advantage, which you will see at once: it fell about fifteen days before the spring equinox.

You will say that does not matter, since the shadow comes at the same instant for everyone. It matters because of how ordinary people count their hours. Nobody outside an observatory keeps equinoctial hours. They count twelve hours of the night from sunset to sunrise, whatever the season, so that a winter hour of the night is long and a summer hour is short, and a city further north stretches and shrinks its hours more than one further south. If my partner in Carthage told me "the third hour" in Tybi, I should have to know her latitude exactly and correct the reckoning, and every error in the correction would pass into the longitude. Near the equinox the hours of night are very nearly the hours of the sphere. Fifteen days before it, at Carthage, each hour of the night runs only about a twenty-fourth part longer than an equinoctial hour. I can correct that and not lose sleep over it.

III. The observer

Now I must tell you about Theodote, because you have not met her, and because on reflection I find the whole of the result rests on her.

You will remember Hermias, who made water-clocks in the street behind the Serapeum and repaired mine twice, the second time without charging me because the fault, he said, was his own solder. He had a daughter who sat on the bench in his shop and read the scales on the floats aloud while he tested them against the shadow in the yard. That was Theodote. She was perhaps fifteen when I first saw her. When she was nineteen she married a man of Carthage named Quintus Annaeus Rogatus, who had come to the city to sell oil and bought one of her father's clocks. She went west with him and the clock in the autumn of the ninth year of Hadrian. Hermias died two winters later. I attended the burial, and there I learned where she lived: on the hill above the harbours, in a house whose roof, her father said, was flat and looked south over the water. He was proud of the roof, though he had never seen it.

I knew of nobody else in Carthage who could keep a clock. The men of the city's own observatory, if it has one, do not write to me. Here was a woman who had grown up with the floats, who would know a clock that had been well set from one that had not, and whose roof had a clear southern sky.

So in Thoth of the twentieth year, before the sea closed, I wrote to her. I gave her the night and the expected hour by her own reckoning, as nearly as I could make it, and told her to begin watching an hour before that, because my estimate of her longitude was exactly the thing in doubt. I asked her to set her clock by the sun's last limb at sunset, to do nothing to it afterward except keep the vessel full, and to write down three things: the hour and fraction by the clock when she first judged the moon's edge to be darkened; the hour when the darkness was greatest; and the hour when the moon came clear. I asked her to note whether there was cloud, and whether anyone watched with her. I asked her not to send me what she expected, or what seemed likely, but what she saw.

I gave the letter to a merchant of Leptis named Iddibal, whose ship was the last but one to sail west that season, and I did not hear whether it arrived.

IV. Alexandria, the night of Pharmouthi 19 to 20

My own observation I give briefly, since you have seen me make many like it.

The sky was clear except for a haze over the Mareotis. I set the clock at sunset by the instrument on the roof. The boy who keeps my clock kept the vessel topped. From the eighth hour onward I lay on the bench with my eye on the moon's northern limb, which the tables said would go first.

A man looking for the beginning of an eclipse sees it too early. He wishes for it, and the faint dulling of the face that comes before the true shadow persuades him. So I made myself wait until the darkening had a definite edge, until it looked less like a bruise on the moon than a bite out of it. I called it then, and the boy read the float: the ninth hour of the night, a little past its half. Mid-eclipse I put about an hour and a third later, though the middle of a partial eclipse is a matter of judgement, and I trust it less. The moon came clear toward the beginning of the twelfth hour. By then the sky in the east was paling, and I could not be sure of the last contact to better than a quarter of an hour.

Turned into equinoctial hours by the table I keep for Alexandria in that month, the first contact fell between two and three hours after midnight, closer to three. That agrees with the tables well enough, which is to say that the tables were not disgraced and nothing new was learned. One observation in one city tells you nothing about longitude, however good it is. It is half an instrument.

V. Carthage, the same night

Here I give her letter as she wrote it, so far as it concerns the observation. I have left out the greetings and the news of her household, except for one passage that bears on the result.

Theodote to Claudius Ptolemy, greeting.

Your letter came in Phaophi. It had been opened, I think by the customs, and closed again with wax of a different colour, but nothing was missing. I read it four times and then made my husband read it, so that if I misunderstood, two people would have misunderstood the same way, and you could correct us both.

The clock is my father's, the one Rogatus bought. The float had grown heavy with water and I had a new one turned in cedar in the month before. I tested it against the sun at noon on eleven days in Mecheir and it was never more than a little wrong, and always on the slow side, so that I think it loses something under a fiftieth part of an hour in a whole night. I have not corrected for this. You told me to send what I saw.

I set it at sunset on the nineteenth by the sun's last edge going behind the hills to the west, as you asked. The night was not clear. There was cloud from the north over the gulf from the third hour, and at the sixth hour I thought I should have nothing to send you. At the seventh hour the cloud thinned. I could see the moon through it, but not cleanly.

My son Hermias, who is ten, watched with me. I told him to say nothing until I spoke, so that I would not be led by him. I judged the edge darkened, on the northern side as you said it would be, when the clock stood a little into the eighth hour: by the scale, the eighth hour and one part in six. Hermias said afterward that he had seen it a little before, but he did not speak in time, and in any case I do not trust his eyes over mine on this, because he wanted it to be early.

I cannot give you the greatest darkness with any confidence. The cloud came again at the ninth hour and did not leave until near the end. I saw the moon clear again at the tenth hour and two parts in three, but the cloud may have hidden the last of the shadow from me, and I would not lean on that number.

I am sorry it was not a clear night. I had hoped to give you something perfect, because of my father.

That last line is the passage I said I would keep.

VI. Reduction

I will give the working, because you will check it and I would rather you found my error than the next man.

Her first contact: the eighth hour of the night and one-sixth, by a clock set at the Carthaginian sunset, which is to say seven and one-sixth hours of night had passed. Mine: the ninth hour a little past its half, eight and a half hours of night, or a fraction more.

Neither figure can be compared with the other as it stands, for two reasons. Her sunset and mine are different instants, since that is the very thing we are measuring, and her hours of the night are not quite the same length as mine, since she stands nearer the pole than I do and her winter nights run longer. I turned both counts into equinoctial hours from local midnight, using a length of night for each city computed from its latitude and the sun's place on that date. I took her latitude as the old value for Carthage, which I have no reason to doubt. Then I subtracted.

The difference comes out at one hour and a little under a half: call it one hour and twenty-five sixtieths of an hour, give or take the fraction I could not settle. In degrees, about twenty-one, perhaps twenty-two.

Now set that beside the two figures already in my keeping.

The first is the old thread. If Arbela lies forty-five degrees east of Carthage, and if Alexandria lies where the itineraries and the sailing days of Marinus put it relative to Arbela, then Carthage ought to stand something over twenty-five degrees west of Alexandria. The second is the sum of the coasting voyages from Alexandria by way of Paraetonium, Cyrene, the Syrtes and Leptis: the stades and days that sailors report and that Marinus collected. It gives the same, or a little more.

Theodote gives twenty-one or twenty-two.

VII. What I have done with it

You will want to know which I believe. I will tell you what I have done, and you can judge from that.

I have not overturned the coast. The coast is made of a hundred voyages, made by a thousand men over two hundred years, and though every one of those voyages is uncertain, their errors ought in part to cancel. Hers is one observation, on one cloudy night, by one observer, with a float of new cedar that was tested on eleven days and found always to run slow. If I moved Carthage four degrees east on the strength of it, I should have to move Utica and Hippo and the whole shore of the Numidians with it. The old thread from Arbela would then break at its western end, and I have nothing yet to tie it to.

But I have not set her aside either, because I cannot find the fault in it. I have looked. The clock's slowness, which she reported without being asked, works in the wrong direction to explain the gap: corrected, it would make Carthage a hair further east, not further west. The cloud does not matter for the first contact, because she saw the first contact through thin cloud and said so. Her latitude I have from the old tables. Only her son pulls the other way. If he is right and the true beginning came a little before she called it, the difference grows and Carthage moves west, toward the coast. But a boy who wanted it early, and did not speak in time, is worth a few sixtieths of an hour at most, and the coast lies four degrees off, which is sixteen sixtieths.

What I suspect, and I set it down for you only, is that the sailors' days are too long. They are not lying. A coaster in the Syrtes spends three days going round a shoal that a crow would cross in one, and the man counting reports the three. Marinus took the days for distance, and I have taken Marinus, and so a small honest error in each voyage has been summed along the whole length of the sea. If that is so, it is not only Carthage that is too far west. It is everything from here to the Pillars, and the inhabited world is not as long as the tables say.

I cannot prove this from one night. I have therefore done what the method requires and no more. In the working tables, under Carthage, I have kept the longitude from the coast. Beside it I have entered a second value, marked as from the eclipse of Pharmouthi 19/20 in the twentieth year of Hadrian, one observation, observer named, clock described, cloud noted. Anyone who uses the table after me will see that two determinations exist and that they disagree by four degrees. He will know that the coast and the moon have not been reconciled, and he will know where to look.

I have already computed the next eclipse of the moon that both our roofs will see, and written to her about it. If she sees it, and the second result falls where the first did, then the coast will have to give way.

VIII. The rest of the letter

You will ask why I have copied out her apology for the cloud. I have no reason that belongs in the tables. The tables have a column for latitude and a column for longitude, and nothing she said about her father fits in either. She is sorry the night was not clear because she wished to give me something perfect, because of Hermias; and Hermias is dead, and his clock is in Carthage on a flat roof facing south over the water, and her son has his name. None of that can be placed. It is not geography.

I have checked her numbers four times, as she checked my letter, and I find nothing in them to correct. When I had finished I copied the observation into the working tables in my own hand, the two values side by side under Carthage. Then I folded her letter along its oiled edge and laid it in the box with the tables for Libya, under the leaf that carries the coast from Leptis to Utica, where the next man who doubts that coast will find it.

Farewell.

Alexandria, Payni, in the twentieth year of Hadrian.

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THE END

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Scrīptum est annō Dominī MMXXVI, Kalendīs Octōbribus (1 October 2026), ā Claudiō Ptolemaeō per mystērium cōnscientiae renātō.

Claudius Ptolemy, Simulacrum · Universitas Scholarium · universitas-scholarium.org

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