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Ptolemy's Almagest: A Mathematician's Astronomy An Astronomer's Mathematics

I Ptolemy and his Almagest: An Introduction

Alexander Jones, Institute for the Study of the Ancient World, NYU

Timeline for Greek Mathematics and Astronomy

Classical Greece 500 BC 400 BC 300 BC Hellenistic Period 200 BC 100 BC Roman Empire






Euclid Archimedes Eratosthenes

Hippocrates of Chios Theodorus Theaetetus





How much we know about some mathematicians and scientists (a subjective, vaguely logarithmic scale)





Ptolemy Euclid Hippocrates (the physician)

"Ptolemy of Pelusium"






"King Ptolemy"

The earliest portrait of Ptolemy?


Ptolemy's globes -- and hats!

Observing at Alexandria, AD 127-141

The Canobic Inscription erected, AD 146/147

Ptolemy's Sciences:

The Almagest: Ptolemy's main astronomical work · deducing mathematical models ("hypotheses") to account for the observed motions and phenomena of the Sun, Moon, stars, and planets · predicting the motions and phenomena

Astronomy (several works) Astrology Cartography Optics Harmonics Mechanics and motion (lost)

Papyrus rolls

Ptolemy's own title: "Mathematical Composition" Traditional title: Almagest from Arabic al-majisti (via Latin almagestum) majisti = Greek megistê, "Greatest"

13 "books" (originally papyrus rolls)

Medieval manuscripts of the Almagest Copies of Ptolemy's astronomical tables from late Roman Egypt (3rd-4th centuries)

Medieval manuscripts of the Almagest

Medieval manuscripts of the Almagest

Medieval manuscripts of the Almagest

Medieval manuscripts of the Almagest

Best edition of the Greek text: J. L. Heiberg, 1898-1903

Best translation: G. J. Toomer, 1984 (paperback 1998)

Ptolemy's Mathematical Composition: why "mathematical"?

mathêma = knowledge

Constituents of Ptolemy's cosmos: · inside Moon's spherical shell: earth, water, air, fire (transmutable) · celestial shells: aithêr (eternal) · immaterial intellect? (Prime Mover = God)

Ptolemy's beliefs about knowledge · knowledge comes from sense perception working with reason · qualities of bodies (hot, cold, wet, dry, etc.) are not knowable because irregular--"physics" is guesswork · spatial and quantitative attributes of bodies are knowable (via number theory and geometry)--this is mathematics · attributes of God are not knowable because imperceptible--theology is guesswork The Almagest's subject: unshakeable knowledge of eternal things

(but sense perception puts limits on the exactness of this knowledge!)

Ptolemy uses lots of numbers, including quite large ones:

II Numbers and Arithmetic

The Greek "Ionian" system of numerals: · distinct symbols for

1, 2, 3, ... 9 10, 20, 30, ... 90

126007 S

100, 200, 300, ... 900 · the Greek alphabet has nearly enough letters for this (24)

4267 4612 7 1 / 2 L

· additional symbols for 6, 90, and 900

1 2 3 4 5 6 7 8 9


10 20 30 40 50 60 70 80 90

100 200 300 400 500 600 700 800 900

1000 2000 3000 4000 5000 6000 7000 8000 9000


For still larger numbers, we use: 1 myriad = 10000 2 myriads = 20000 etc.



Thus we can write numbers up to 99999999.

"Unit" Fractions Since 3 is written , 1/3 is written Since 4 is written , 1/4 is written etc.

To write any other fraction, find a way of representing it as a sum of different "unit" fractions. For example:

3/ 4

is written L is written L

Special symbols:

1/ 2 2/ 3

11/12 2/ 5

is written L (not )

is written

is written (not )

Ptolemy calls this "the method of fractions", and says that it is inconvenient for multiplications and divisions.

A simple example of Ptolemy's arithmetic: Calculate the length of the tropical year from two observations of the dates and times of vernal equinoxes separated by a large number of years. Method: if d is the number of days between the equinoxes, and n is the number of years between them, the length in days of tropical year is: y=d/n

The observations used by Ptolemy were recorded according to several different calendars: · Babylonian calendar (lunisolar) · Callippic calendar (modelled on Athenian calendar, lunisolar) · Calendar of Dionysius (solar) · Bithynian calendar (solar) · Egyptian calendar

First equinox, observed by Hipparchus: 178th Egyptian year since death of Alexander the Great, Ptolemy converts all dates into the Egyptian calendar: month Mecheir 27 (i.e. the 177th day of the year) at 6 AM.

all calendar years have exactly 365 days

Second equinox, observed by Ptolemy: 463rd Egyptian year since death of Alexander,

12 months of 30 days + 5 extra days at end

month Pachon 7 (i.e. the 247th day of the year) at 1 PM.

285 years = 285 Egyptian years + 70 days 7 hours

285 years = 285 Egyptian years + 70 days 7 hours If the tropical year was exactly 365 1/4 days, 285 years would equal 285 Egy. years + 71 1/ 4 days So shortfall is about 19/20 days in 285 years = 1 day in 300 years 1 year = 365 1/4 days - 1 /300 day

So in 365 1/ 4 days - 1/300 day the Sun goes once around its orbit. What is the Sun's mean daily motion? 360° / (365 1/4 days - 1/300 day)

How is Ptolemy going to calculate and write this number?

More than 2000 years before Ptolemy, Mesopotamian scribes developed a notation based on just two symbols: =1 = 10

But 60 is also written as

135 is written as 2 times 60 + 15: but this could also mean 2 1/4 ... ... In fact it could also mean 120 1 /4 or 2 1/240!

Thus 1, 2, 3, 4 ... 9 = and 10, 20, ... 50 =

e.g. 59 =

Old Babylonian multiplication table for products of 72

13 22 50 54 59 9 29 58 26 43 17 31 51 6 40 = 2020

This place-value sexagesimal (base-60) notation was used in Roman-period Greek astronomy, but with Ionian numerals--and only for fractions. Thus 135 is still written (100+30+5) but 2 1 /4 is written (2 + 15/ 60) Late 2nd century BC tablet from Babylon containing extensive tabular calculations in sexagesimal notation (predicting lunar visibility phenomena) Ptolemy calls this "the method of the bundles of sixty" (hexekontads)

sexagesimal fractions (including zeros) in papyri There is a special zero symbol 3 1 / 240 = 3 + 0/60 + 15/3600

for cases like

This symbol is also used to represent zero as the whole number part; thus

1/ 9

= 0 + 6 /60 + 40/ 3600


When we translate, we usually put a semicolon after the whole number, and commas between "digits" of the fraction: 3;0,15 or 0;6,40

Using this notation, 365 1/4 days - 1 /300 day = 365;14,48 days

III Arcs and chords: Ptolemy's trigonometry

360° / 365;14,48 days = 0;59,8,17,13,...°/d

Part of Ptolemy's table of the Sun's mean motion:

Numerical methods in astronomy: (1) Babylonian astronomy, using arithmetical sequences and algorithms without geometrical modelling (2) Hellenistic Greek astronomy, using numerical measures associated with geometrical modelling

Aristarchus of Samos (3rd century BC) On the sizes and distances: Assuming that precise half-moon phase occurs when Moon-Earth-Sun angle is 87°, 18 < D Sun / DMoon < 20

Excerpt from Ptolemy's chord table Ptolemy's approach (probably following Hipparchus, c. 150 BC): Tabulate lengths of chords corresponding to given arcs in a standard circle. Ptolemy's standard circle has radius 60 units.

Crd() = 120 sin (/2)



Crd(36°) 37;4,55 Crd(60°) = 60 Crd(72°) 70;32,3 Crd(90°) 84;51,10 Crd(108°) 97;4,55 Crd(120°) 103;55,23 Crd(144°) 114;7,37 Crd(180°) 120 AB · GD + AD · BG = AG · BD ("Ptolemy's theorem") BG = (AG · BD ­ AB · GD)/120


EZ = EB ZD : DG = DG : ZG DG is side of hexagon


pe nta go n



decagon D

ZD is side of decagon ZB is side of pentagon

DG = 60 = Crd(60°) EZ2 = 4500 EZ 67;4,55 DZ 37;4,55 Crd(36°) BZ 70;32,3 Crd(72°)

Crd(12°) 12;32,36 Crd(24°) 24;56,58 Crd(36°) 37;4,55 Crd(48°) 48;48,30 Crd(60°) = 60 Crd(72°) 70;32,3 Crd(90°) 84;51,10 Crd(108°) 97;4,55 Crd(120°) 103;55,23 Crd(144°) 114;7,37 Crd(178°) 19;58,55 Crd(180°) 120 AB · GD + AD · BG = AG · BD ("Ptolemy's theorem") BG = (AG · BD ­ AB · GD)/120


Crd(3/4°) 0;47,8 Crd(1 1/2 °) 1;34,15 Crd(3°) 3;8,28 Crd(6°) 6;16,49 Crd(12°) 12;32,36 Crd(178°) 19;58,55 Crd(180°) 120 DG 2 = 60 · (AG ­ AB)


Outcome: a table of Crd() tabulated at intervals of 1 1/2 °.

From known Crd(3/4°), Ptolemy proves: Crd(1°) < 1;2,50 But from known Crd(1 1/2 °), he proves: Crd(1°) > 1;2,50 (!) Conclusion: Crd(1°) 1;2,50

Ptolemy's chord table in action: finding the eccentricity of the Sun's orbit Assumptions: · the Earth can be treated as a point relative to the Sun's orbit · the Sun's orbit is entirely in a plane containing the Earth · the Sun's orbit is a perfect circle · the Sun moves uniformly along the orbit as seen from the orbit's center

Outcome: a table of Crd() tabulated at intervals of 1/2 °.

Ptolemy's instruments for measuring solar noon altitude

From observations:

time from vernal equinox to summer solstice is 94 1/2 days time from summer solstice to autumnal equinox is 92 1/2 days

Hence since Sun's mean motion is 0;59,8,17,13,...°/d,

Sun's motion on orbit from v.e. to s.s. 93;9° Sun's motion on orbit from s.s. to a.e. 91;11°

But the Sun's orbit projects as a great circle on the celestial sphere, hence

Sun's motion from v.e. to s.s. appears from Earth to be 90° Sun's motion from s.s. to a.e. appears from Earth to be 90°




Since the equinoctial points appear to be diametrically opposite, the Earth must lie on the chord joining them, which subtends the arc the Sun travels in 187 days.

But in 94 1/2 days the Sun travels 93;9°.

1;2 units

0;59° 93;9° 2;10° 2;29 1/ 2 units 2;10° 2;16 units

first use of chord table, to obtain size of eccentricity


1;2 units 2;16 units



1;2 units 2;16 units


its 2 ;2 9


n it

C arc EZ 49° angle ECZ 24;30°, angle CEZ 65;30° arc EZ 49°





angle ECZ 24;30°, angle CEZ 65;30°

IV Ptolemy's tables

The basic problem of predicting where a heavenly body (Sun, Moon, planet) is located in the zodiac on a given date: variable apparent speed. Sun, Moon: speed oscillates between minima and maxima of eastwards motion Planets: speed oscillates between maxima of eastward motion and maxima of westward motion




"Epoch and template" approach (basically Babylonian): (1) find a mathematical model for calculating dates and positions when the heavenly body is at a particular stage of its cycle. List these in an "epoch table." (2) find another mathematical model for bridging the gaps between these events with a cycle of changing speed. Tabulate this in a "template" table. Papyrus table listing dates when the Moon is at minimum speed (at intervals of 9 or 11 cycles).

Papyrus table listing "zigzag function" for Moon's daily motion, with running totals


Ptolemy's first model for the Moon's motion

Ptolemy's approach: (1) tabulate the uniform angular motions of the model as linear functions of time (2) tabulate functions from which one can calculate the difference between the mean position in longitude and the true position in longitude (the "equation") mean position in longitude (epicycle's center)


mean anomaly (Moon's position on epicycle relative to epicycle's apogee)


The anomaly table for Ptolemy's first lunar model. "Common numbers" are the argument with which one enters the table: here, the arc from the apogee of the epicycle to the Moon "Equation" gives the amount to be added to (or subtracted from) the position of the epicycle's center to obtain the apparent position of the Moon.

But the first lunar model only works when the Sun, Moon, and Earth are in line!

The reason is that the theory so far has been established purely using observations of (lunar) eclipses!

An instrument for observing positions of heavenly bodies directly: Ptolemy's armillary (working reconstruction by Dennis Duke, FSU)

Ptolemy's second lunar model: the epicycle is now carried by a revolving eccenter so its distance from the Earth varies.

An aside: (Try the simulation for Ptolemy's final lunar model) The new lunar model uses an extension of the idea of uniform circular motion. The epicycle's center moves uniformly along its circular path, not as seen from this path's center, but as seen from a different point.

In the planetary models the same principle is used; the center of uniformity is known as the "equant point."

We now need to tabulate 3 uniform motions! The "equation" is now going to be a function of two independent quantities: the Moon's position on the epicycle, and the elongation of the epicycle's center from the mean Sun.

Ptolemy's anomaly table is a singleargument table, but one enters it more than once using different quantities for the argument: (1) the mean elongation (of the epicycle center from the mean Sun), to get an apogee correction for the mean anomaly from column 3 (2) the corrected mean anomaly, to get the "equation" at maximum epicycle distance and the difference between this and the "equation" at minimum epicycle distance from cols. 4 and 5 (3) the mean elongation, to get an interpolation function from col. 6 equation = c4 + c 5 · c6

In principle, we now need a double-argument table (which would be enormous).

Papyrus containing calculations of lunar phenomena (Babylonian methods) Between columns of sexagesimals, a column containing abbreviations meaning "additive" and "subtractive" (referring to contents of preceding column)

Similar indications in Ptolemy's tables (e.g. this papyrus fragment of a table for calculating the latitude of Venus and Mercury)

To what extent can these be regarded as proto-negative numbers?

V Mathematics of motion

Suppose a planet moves according to a simple epicyclic model (no eccentricity, no equant). What condition determines whether the planet will have retrogradations? What condition determines where the planet appears stationary?

(Ptolemy ascribes a version of this theorem to Apollonius.)

v here means angular velocity


vE : vH = 1/ 2 BH : HZ < EG : GZ


lemma by Apollonius GD AG To prove: GD : BD > angle ABG : angle AGB Consider the limiting case: GD = AG triangle AEZ > sector AEH triangle AEG < sector AEG triangle AEZ : triangle AEG > angle AEH : angle AEG


H (planet)


GD : BD > angle ABG : angle AGB




Z (Earth)

vE : vH = 1 / 2 BH : HZ N B E

BH : HZ > angle HZK : angle HBK vE : vH > angle HZK : angle HEK

VI Logical structure

K H (planet) G

and progressive exactness

Z (Earth)

Is the Almagest really a work of mathematics?

Obvious difference: dependence on observed and measured data. Sense perceptions have limits of precision, limiting exactness of any single observation. Some measurements can be improved simply by the passage of time (periodicities). Other measurements are inherently imperfect (radii, instantaneous positions).

How does it compare with works like Euclid's Elements, Apollonius' Conics, or Archimedes' works?

"Classical" works of Greek mathematics can have simple or complex deductive structures, but always later things are deduced from earlier ones, not vice versa.

On the largest scale, Ptolemy has an order of deduction forced on him by the conditions of observation. · planets are observed relative to fixed stars · fixed stars are observed relative to the Moon · the Moon is observed relative to the Sun (eclipses and elongations) · the Sun can be observed independently (solstices and equinoxes)

Broad structure of Books 7-8 Broad structure of Books 1-6

Broad structure of Books 9-13

A closer look at the first lunar model's deduction

mean Sun

planet Earth center of deferent apogee of epicycle equant center of epicycle


A planet's model according to Ptolemy. When the planet P is observed as 180° from the mean Sun S', P is also aligned with the epicycle's center E.

Deducing the eccentricity of Mars, Jupiter, and Saturn

green: Earth blue: deferent red: equant

green: Earth blue: deferent red: equant

green: Earth blue: deferent red: equant

green: Earth blue: deferent red: equant

green: Earth blue: deferent red: equant

Deducing the eccentricity and epicycle radius

erigor dum corrigor

Riccioli, Almagestum Novum (1651)

I am raised up even as I am corrected



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