Claudius Ptolemy: A Life in Five Chapters

Chapter 1 · A Life Without a Record

What we know, and how little it is

c. 100 – c. 170 · Alexandria

Almost every famous figure from the ancient world comes with anecdotes — Archimedes in his bath, Eratosthenes and his nickname, Euclid and the king. Ptolemy comes with none.

What can be said is short. His name is Greek and Roman together: Claudius suggests Roman citizenship, Ptolemaeus an Egyptian Greek family. He worked at Alexandria. His recorded astronomical observations run from about the year 127 to about 141, which fixes his working life to the middle of the second century. Everything else — his teachers, his household, his character, whether he was rich or poor — is unknown.

This is worth sitting with, because it changes what studying him means. There is no life to interpret. There is a body of work, and the only questions that can be asked are questions about the work: what did he take from others, what did he add, how careful was he, and where did he go wrong.

He worked in a city that had been the centre of Greek learning for four hundred years and was by then part of the Roman empire. The Library's greatest days were behind it, but the tradition was intact: he had access to Hipparchus, to Euclid, to Menelaus, and to Babylonian eclipse records going back centuries. His achievement is in large part an achievement of synthesis, and he never pretends otherwise. He names Hipparchus constantly.

Why this matters

A reminder that for most of the ancient world the person is lost and only the work survives — and that the work is what can honestly be judged.

A man known entirely through what he wrote. What would you ask him?

Ask Ptolemy

• “What do you take from Hipparchus, and what is your own?”
• “What was Alexandria like by your time?”
• “How did you get hold of Babylonian eclipse records?”
• “Does it trouble you that nothing of your life survives?”
• “What made you decide to write one enormous book instead of many small ones?”

Chapter 2 · The Table of Chords

A length in a stated circle, not a bare ratio

c. 150 · Alexandria

To predict where a planet will be, you must be able to turn angles into lengths. Ptolemy needed that constantly, so Book I of the Almagest builds the tool: a table giving, for every arc in a circle at half-degree steps, the length of the straight line joining the arc's two ends. That line is called the chord.

The detail that matters most is the one modern readers skip. His circle has a diameter of one hundred and twenty parts. A chord in his table is therefore a LENGTH in a particular circle, not a bare ratio between two sides. Ask him for the chord of sixty degrees and he gives you sixty parts — of that circle. In a circle twice as wide, everything doubles.

He built the table from the cases that can be constructed exactly with straight edge and compasses: the sides of the inscribed hexagon, square, pentagon and decagon. From these he reaches everything else using a theorem about a four-sided figure inscribed in a circle, which yields the chord of a sum and the chord of a difference, together with a rule for halving an arc.

One entry defeats construction entirely: the chord of a single degree. It cannot be built by these methods. So he traps it between two chords he can construct, squeezes the two bounds together until they agree to the accuracy he needs, and takes the value. He says exactly what he has done, which is why we can still judge it.

The half-chord — what later became the sine, and what schools use now — is an Indian innovation of several centuries after him. Ptolemy never has it.

Why this matters

Every trigonometric table descends from this one, and the thing it lacks — a pure ratio independent of size — is exactly what school trigonometry is built on.

A table of lengths that served astronomy for fourteen centuries. What would you ask him?

Ask Ptolemy

• “What is a chord, and why does it depend on the circle?”
• “Why choose a diameter of one hundred and twenty parts?”
• “How did you get from four constructible chords to a whole table?”
• “Why could you not construct the chord of one degree?”
• “What would you have gained from the half-chord, if you had known it?”

Chapter 3 · The System

Wrong at the centre and right about the appearances

c. 150 · Alexandria

The Almagest is thirteen books, and its purpose is prediction. Given a date, where will the moon be? When will the next eclipse fall? Where is Mars tonight?

The Earth sits motionless at the centre. Around it the sun, moon and planets move on circles — but not simple ones, because simple ones do not fit the observations. Ptolemy uses a circle carried on a circle, so that a planet can appear to slow, stop, and travel backwards against the stars for a while, which is what planets are actually seen to do. He offsets the centre of the main circle away from the Earth, and he introduces a further point about which the motion is uniform. Each of these devices exists because without it the model disagrees with what is observed.

That is the honest description of his system: it is false about the arrangement of the solar system and remarkably good at saying where things will appear. It stayed the standard for fourteen centuries because it worked, in the only sense that could be tested at the time.

There is a serious charge against him. From the tenth century onwards, astronomers noticed that many of his reported observations agree with his theory more precisely than his instruments should have allowed, and that some appear to be computed backwards from the theory rather than made. Modern scholars disagree sharply about how far this goes and how much of it is ordinary ancient practice rather than deception. It has not been settled, and it should not be smoothed over.

Why this matters

The clearest example anywhere of a model that is wrong about reality and genuinely predictive — and of how hard it then is to dislodge.

A system that worked for fourteen hundred years and was false. What would you ask him?

Ask Ptolemy

• “Why did you keep the Earth at the centre?”
• “Why does a planet need a circle carried on another circle?”
• “Is a model that predicts well but is false a good model?”
• “Did you adjust your observations to fit your theory?”
• “What would have persuaded you that the Earth moves?”

Chapter 4 · Flattening the Sphere

The Geography, the Analemma, and the shadow of a leaning line

c. 150s · Alexandria

Ptolemy's second great work is the Geography, and it does for the Earth what the Almagest does for the sky. Some eight thousand places are listed by position, and the book sets out how to project a round world onto a flat sheet — how to decide which distortions to accept, since some distortion is unavoidable.

His map has a famous flaw. He took the Earth's circumference from Marinus of Tyre rather than from Eratosthenes, and Marinus' figure is too small. The known world therefore occupies a larger share of the globe than it should, and the ocean between Europe and Asia looks crossable. Columbus, reading Ptolemy fourteen centuries later, was encouraged by exactly that error.

The Analemma is a shorter and stranger work, and mathematically it is the most modern-feeling thing he wrote. It solves problems about the sun's position on the celestial sphere by dropping perpendiculars onto three planes at right angles to one another — the meridian, the horizon, the plane through east and west — and then reading the answer off a flat diagram. The three-dimensional problem is not solved in three dimensions. It is projected down until it becomes a set of flat figures with right angles in them, and those are solved with the tools he already has.

That is exactly the strategy a student uses on a cuboid or a pyramid today: find the flat figure hiding inside the solid, draw it separately, and work on it there.

Why this matters

The method of the Analemma — reduce the solid to a flat figure and solve that — is still how three-dimensional geometry is done at school.

A round world drawn flat, and a sphere reduced to three plane diagrams. What would you ask him?

Ask Ptolemy

• “How do you draw a round world on a flat sheet?”
• “Why did you use Marinus' size for the Earth instead of Eratosthenes'?”
• “What is the analemma actually doing?”
• “Why solve a problem on the sphere by dropping it onto planes?”
• “How did you get positions for eight thousand places?”

Chapter 5 · Fourteen Centuries

Greek to Arabic to Latin, and then out

c. 170 – 1543 · Alexandria · Baghdad · Toledo · Europe

Ptolemy died around 170; the date is a guess. His book did not.

The Syntaxis was translated into Arabic in ninth-century Baghdad, and it is from the Arabic superlative — al-majisti, the greatest — that the name Almagest comes. Islamic astronomers used it, checked it, corrected its parameters and criticised its physical assumptions for six hundred years; the tables of al-Battani and the Maragha school's models are direct engagements with it. It reached Latin Europe in the twelfth century through translations made in Spain, and became the astronomy curriculum of the medieval university.

When Copernicus published in 1543, he was not overturning a vague tradition. He was answering Ptolemy, in Ptolemy's own terms, with a book laid out on the same plan. Kepler's ellipses finally broke the circles, and Newton explained why the planets move as they do at all.

What survived the wreck is the mathematics. The table of chords was superseded by tables of half-chords, but the method of building a table — start from what can be constructed exactly, use a sum-and-difference rule to reach the rest, and squeeze the entries that resist — is his. His geography shaped European mapmaking for a thousand years, mistakes included. And his central standard, that a model is judged by whether it reproduces what is observed, is the one that eventually destroyed his own system, which is the fairest thing that can be said about it.

Why this matters

The standard Ptolemy insisted on — agreement with observation — is what eventually overturned him, which is how science is supposed to work.

A book that outlived its cosmology by centuries. What would you ask him?

Ask Ptolemy

• “How does it strike you that your book was named by its translators?”
• “What did the astronomers of Baghdad correct in your work?”
• “Copernicus answered you point by point. What would you say back?”
• “Which part of your work do you think deserved to last?”
• “Your own standard was agreement with observation. Did it defeat you fairly?”

What Ptolemy changed

The Almagest was the standard work on astronomy for roughly fourteen centuries, in Greek, Arabic and Latin, and its table of chords is the ancestor of every trigonometric table since — including the method of building one from constructible cases and a sum-and-difference rule. The Geography's coordinates shaped European mapmaking for a thousand years. The Analemma's strategy of solving a problem on a sphere by projecting it onto planes is still how three-dimensional geometry is taught.

A debate that continues

Since the tenth century, astronomers have argued that many observations reported in the Almagest agree with his theory more closely than his instruments allowed, and may have been computed rather than made. Modern historians disagree about how serious this is. His Earth-centred cosmology was wrong, and his adoption of Marinus' undersized Earth misled European geography — and eventually Columbus — for centuries.

Keep exploring — ask Ptolemy

• “Can a model be valuable if it is false about what it describes?”
• “Where is the line between tidying data and manufacturing it?”
• “Why did it take fourteen centuries to replace a system everyone could test?”

Related lives

• Eratosthenes — Who Measured the World From a Shadow
• Regiomontanus — Who Made Triangles A Subject
• Ibn al-Haytham — Who Tested What the Ancients Wrote

Related themes

Trigonometry · Astronomy before Copernicus · Maps and projection