Six Easy Leactures on Ancient Mathematical Astronomy - D. Duke [Univ of FL Lecture Mtls] WW

7/18/2019 Six Easy Leactures on Ancient Mathematical Astronomy - D. Duke [Univ of FL Lecture Mtls] WW

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Course website: www.scs.fsu.edu/~dduke/lectures

Dennis Duke

Six Easy Lectures on Ancient Mathematical Astronomy

Department of Physics and School of Computational Science

Florida State University

http://www.scs.fsu.edu/~dduke/lectures






Lecture 1

• Where, When and Who

• Almagest Books 1 and 2• the celestial sphere

• numbers and angles (sexagesimal base-60)

• obliquity and latitude and the related instruments

• plane geometry and trigonometry, the chord tables

• spherical trigonometry, circles on the celestial sphere



the Where: Ptolemy’s World A.D. 150



The world as our story begins. The East



and the West



the Greek’s near their peak



Alexander the Great’s empire



Strabo’s Geography (1st-2nd

century B.C.)



Ptolemy’s World Map (1st century A.D.)



Most of what we have from antiquity was preserved andtransmitted to us by the Islamic societies of the 8

th – 13

th

centuries A.D.



Who and When: Ancient Astronomers

Homer/Hesiod -750 Aratus -270

Meton/Euctomen -430 Timocharis -260Eudoxus -380 Aristarchus -240

Aristotle -340 Archimedes -220

Heraclides -330 Eratosthenes -210

Callippus -330 Apollonius -200

Autolycus/Euclid -330 Hipparchus -130Aristyllus -300 Posidonius -100

Berosus -300 Geminus -50

Theon of Smyrna 120

Ptolemy Almagest 150Theon of Alexandria 350



Relevant Famous People

Plato -375 philosopher

Alexander the Great -330 conquered Babylon

Strabo 10 Geography Pliny 70 Natural History

Plutarch 100 Concerning Nature

The Face in the Moon Marinus of Tyre 120 geography (Ptolemy’s source)



Later Famous Astronomers(and Ptolemy influenced every one of them)

Hipparchus -130

Ptolemy Almagest 150

Aryabhata (India) 500

al-Sufi (Islam) 950

al-Tusi/Urdi/Shatir 1250Ulugh Beg 1420

Copernicus 1540

Tycho Brahe 1570

Kepler 1620

Newton 1680



Almagest , Book I begins:

and a bit later:





similarly (see the excerpts on the supplementary reading page):Theon of Smyrna (about A.D. 120)

Strabo Geography (about A.D. 5)

Geminus (about 50 B.C.)

Hipparchus (about 130 B.C.)Autolycus (about 300 B.C.), and Euclid’s Phenomena is similar

Eudoxus (about 320 B.C.)

Aristotle (about 350 B.C.)

Hesiod (about 750 B.C.)

Homer (about 780 B.C.)

It is fair to say that Ptolemy makes the best effort to give fairly cogent

arguments, usually astronomical, to support all of these assumptions.



For example:





Ptolemy is probably summarizing the winning arguments in an old debate, going

back as far as Aristarchus in about 240 B.C.:





Celestial Sphere





The oblique circle (the ecliptic, the path of the Sun, Moon and planets)









Using a gnomon

α = geographical latitude

β = twice the obliquity of the ecliptic



Gnomon’s are also the basis of sundials:



How were these angles measured other than using a gnomon?Ptolemy describes two instruments:



Expressing Numbers

Even today we measure angles in degrees,

minutes, seconds, and we also measure time

in hours, minutes, seconds.

In both cases there are 60 minutes per

degree or hour, and 60 seconds per minute.

Apparently this began in Babylon, no later

than early first millenium B.C. and probablya lot earlier, since we have many 1000’s of

surviving clay tablets covered with such

numbers.



Ptolemy also used this base-60 sexagesimal number format, at least for the

fractional part of the number. Thus he expressed the number 1 14 300

365 −+ as

15 12 14 60 12365 365 ( )60 3600 60 3600 3600

14 4836560 3600

365;14,48

+ − = + + −

= + +

=



The integer part of the number was given in decimal.

With a good set of multiplication and division tables, whicheveryone had, manual arithmetic was no harder for them than it is

for us.



Ptolemy used mostly plane geometry and trigonometry, with alittle spherical trig when he needed it, which was not often.

For plane trig he had only one construct – the chord – rather than

our sine, cosine, tangent, etc, and this was enough.



He also had good tables of the chord function, and was quitecapable of interpolation, just as we (used to) do it.



Ptolemy says that he will present a ‘simple and efficient’ way to compute the chords, but he doesn’tactually say the table was computed that way, or even that he computed it. In fact, there is good

reason to think that it was not computed using his methods, or that he was the person who computed

it. Unfortunately, however, we have no evidence about who did compute it.



As we will see in Lectures 2 and 3, it is likely that Hipparchus also had a good command of

trigonometry, both plane and spherical, but he also probably had a simpler trig table. Most peopleassume he also used the chord construct, but there is no evidence for this, and there is some reason to

think he used instead the sine. Angle(degrees) Chord

0 0

7 ½ 450

15 897

22 ½ 134130 1780

37 ½ 2210

45 2631

52 ½ 3041

60 3438

67 ½ 3820

75 4186

82 ½ 453390 4862

97 ½ 5169

105 5455

112 ½ 5717

120 5954

127 ½ 6166

135 6352

142 ½ 6511150 6641

157 ½ 6743

165 6817

172 ½ 6861

180 6875

/360 60 21,6003438

2 2 R

π π

′⋅= =

216006875 D

π

=



There is also no reason to think that Hipparchus invented trignonometry and tables, either chord or

sine. In fact, a work of Archimedes shows the explicit computation of about 2/3’s of the entries inHipparchus’ (supposed) table, and computing the other entries would be straightforward.

Archimedes gets 781 1

4 2

66 153sin1

2017 4673

< <

(equivalent to 78

0.03272 sin1 0.03274< <

)

which leads to10 1

371 7

3 π < <

circumscribed inscribed circumscribedinscribed

Base



Angle a c a c Base 3438

Base

3438

3 6/8 153 2339 3/8 780 11926 225 225

7 4/8 153 1172 1/8 780 5975 7/8 449 44911 2/8 169 866 2/8 70 358 7/8 671 670

15 153 591 1/8 780 3013 6/8 890 890

18 6/8 571 1776 2/8 2911 9056 1/8 1105 1105

22 4/8 169 441 5/8 70 182 7/8 1315 1316

26 2/8 744 1682 3/83793 6/8 8577 3/8 1520 1520

30 153 306 780 1560 1719 171933 6/8 408 734 3/8 169 304 2/8 1910 1909

37 4/8 571 937 7/8 2911 4781 7/8 2093 2093

41 2/8 1162 1/81762 3/85924 6/8 8985 6/8 2267 2267

45 169 239 70 99 2431 2431

• columns 2–5 come from Archimedes, while columns 6–7 are just

3,438ac

×

• notice that Archimedes is working entirely in sine and cosine, never chord• there is no doubt that Hipparchus was familiar with Archimedes’ work on this

• about all we can conclude is that Archimedes, Hipparchus, or someone in between

might have computed the first trig table this way



We can, in fact, go even farther back into the very early history of trigonometry by considering

Aristarchus’ On Sizes and Distances, and we shall see that a plausible case can be made that his paper could easily have been the inspiration for Archimedes’ paper. The problem Aristarchus posed

was to find the ratio of the distance of the Earth to the Moon to the distance of the Earth to the Sun

[as we will see in Lecture 4]. He solved this problem by assuming that when that the Moon is at

quadrature, meaning it appears half-illuminated from Earth and so the angle Sun-Moon-Earth is 90°,

the Sun-Moon elongation is 87°, and so the Earth-Moon elongation as seen from the Sun would be

3°. Thus his problem is solved if he can estimate the ratio of opposite side to hypotenuse for a right

triangle with an angle of 3°, or simply what we call sin 3°.

Aristarchus proceeded to solve this problem is a way that is very similar to, but not as systematic as,

the method used by Archimedes. By considering circumscribed (Fig. 2 below) and inscribed

triangles (Fig 3 below) and assuming a bound on2

Aristarchus effectively establishes bounds onsin 3° as1 120 18

sin3< <

and, although he does not mention it, this also establishes bounds on π as

13

3 3π < <

clearly not as good as Aristarchus got just a few years later.







Actually, the sine (not chord!) table that we suppose was used by Hipparchus

shows up clearly in Indian astronomical texts of the 5th and 6th centuries A.D.For example, Aryabhata writes in The Aryabhatiya (ca. A.D. 500) verse I.10:

10. The sines reckoned in minutes of arc are 225, 224,

222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154,143, 131, 119, 106, 93, 79, 65, 51, 37, 22,7.

and later he explains how to compute these in verse II.12:

12. By what number the second sine is less than the first sine, and by the quotient obtained by dividing the sum of

the preceding sines by the first sine, by the sum of these

quantities the following sines are less than the first sine.

These are clearly not sines but rather the differences of adjacent terms in thetable of sines. The base is 3,438, just as Hipparchus used.



Many similar examples (to be seen in coming weeks) lead to what I call the

Neugebauer – Pingree – van der Waerden Hypothesis:

The texts of ancient Indian astronomy give us a sort of wormhole through

space-time back into an otherwise inaccessible era of Greco-Roman

developments in astronomy.



Thus the essentially universally accepted view that the astronomy we find in

the Indian texts is pre-Ptolemaic. Summarizing the prevailing opinion, Neugebauer wrote in 1956:

“Ptolemy’s modification of the lunar theory is of importance for the problem

of transmission of Greek astronomy to India. The essentially Greek origin ofthe Surya-Siddhanta and related works cannot be doubted – terminology, use

of units and computational methods, epicyclic models as well as local

tradition – all indicate Greek origin. But it was realized at an early date in the

investigation of Hindu astronomy that the Indian theories show no influence

of the Ptolemaic refinements of the lunar theory. This is confirmed by theplanetary theory, which also lacks a characteristic Ptolemaic

construction, namely, the “ punctum aequans,” to use a medieval

terminology”.

This fundamental idea will be explored much further in coming lectures.



Ptolemy’s obliquity and latitude of Alexandria

Ptolemy uses 02 47 ;42 ,3ε = ′ ′′ but in reality he should have gotten about 47°;21´.

Now 21´ is a fairly large error for this kind of measurement, about 2/3rd the size

of the Moon. What is not surprising is that Ptolemy made such an error, but that

he got exactly the same values used by Eratosthenes and Hipparchus, who

should have gotten about 47°;27´.

This kind of thing occurs frequently throughout the Almagest .



For the geographical latitude, Ptolemy writes:

and later in Almagest 5.12:

Actually, the latitude of Alexandria is between 31°;13´and 31°;19´,depending on

exactly where Ptolemy worked (probably closer to the more northern limit).

Ptolemy’s value 30°;58´follows exactly from an equinoctial shadow ratio of 5/3,and was probably also a value he inherited from some old tradition.



Spherical trigonometry solves problems related to circles on a sphere.

A particular problem is to compute the angles between the ecliptic, the equator,

and the horizon. Another is to compute the time required for a given segment ofthe ecliptic to rise or set above or below the horizon. Another is to compute the

length of the longest (or shortest) day at any given geographic latitude.



from Almagest Book 2.6, for the parallel of the Tropic of Cancer:



and some parallels further north:



so Ptolemy is systematically computing what the shadow lengths will be at a

sequence of geographical longitudes from the equator to the arctic circle.



This had been going

on for centuries. Inabout 200 B.C.

Eratosthenes had

managed to determine

the circumference of

the Earth.

Strabo, writing about

A.D. 5, gives and

interesting account ofthe work of both

Eratosthenes and

Hipparchus in this

area (see the

supplementaryreading).



Eratosthenes is said to have measured the angle as 7 1/5 degrees, and took the

distance from Syene to Alexandria as 5,000 stades, giving

15

360 5,000 50 5,000 250,0007 Earth

C = × = × = stades

which he rounded to 252,000 stades to make it divisible by 60 (and also 360).





Ancient Astronomy

Lecture 2Course website: www.scs.fsu.edu/~dduke/lectures





Lecture 2

• Almagest Book 3

•

the length of the year• the length of the seasons

• the geometric models

• the length of the day

•

the background• lost episodes in solar history



•

Book 3 of the Almagest is about the Sun.

• The Sun is first in Ptolemy’s logical structure,

followed by the Moon, then the fixed stars, and finally

the wandering stars (planets).

•

Ptolemy says he is following Hipparchus’ theory of the

Sun (a claim confirmed by Theon of Smyrna).

•

Probably nothing in Book 3 is original with Ptolemy,

apart from the four equinox and solstice ‘observations’

in Almagest 3.1 and 3.7.



What are the major questions to be answered?

1. What is a ‘year’?

2.

Is the length of the year constant?

3. What is the length of the year?

4. What are the lengths of the seasons?

5.

How does the speed of the Sun vary throughout the year?

6. What kind of geometrical model would account for the phenomena

(observations)?

7. How does the length of the day vary?



1.

What is a ‘year’?

There are several choices:

(a) return to the same star on the ecliptic (sidereal year).

(b) return to the same declination δ (e.g. the same place on the equator).

(c) return to the same speed (anomalistic).

(d)

return to the same latitude (distance from the ecliptic).

Ptolemy, and probably Hipparchus before him, chose option (b), usually called

the tropical year, since you could define it as the time it takes for the Sun to

return to a tropic circle, i.e. a solstice (summer or winter). Ptolemy actually

measures relative to the vernal (spring) equinox.









2.


3.

What is the length of the year?

4.

What are the lengths of the seasons?

5.



(observations)?




2–3. Is the length of the year constant, and how long?

Ptolemy says that Hipparchus measured the number of days between

successive equinoxes, first autumn:

h i



then spring:

so Hipparchus finds that with a few exceptions the year is 365 ¼ days.Further, the exceptions could easily be observation uncertainties, so Ptolemy

finds no reason to doubt that the year length is constant.

G tti ‘ i ’ l th



Getting a ‘precise’ year length.

Ptolemy says that Hipparchus found a year length of 1 14 300

365+ − days (probably

from the interval between the summer solstices in 280 B.C. and 135 B.C.)

Ptolemy then says

Pt l i th d t f th t i f 132 d 139



Ptolemy gives the dates for the autumn equinoxes of 132 and 139

and the spring equinox and summer solstice of 140, all “mostcarefully observed”, and compares them to the fall equinox of

-146, the spring equinox of -145, and the summer solstice of -431:

(1)

-146/9/27 midnight to 132/9/25 2 pm

(2) -146/9/27 midnight to 139/9/26 7 am

(3) -145/3/24 6 am to 140/3/22 1 pm

(4) -431/6/27 6 am to 140/6/25 2 am

in each case you count the number of intervening days, divide by

the number of years, and the year length is 1 14 300

365+ − days.

The correct value is about 1 14 133

− , about 6 minutes shorter. 365+






2.




5.



(observations)?


4 5 the length of the seasons and the varying speed of the



4–5. the length of the seasons and the varying speed of the

Sun

Early History of Season Lengths

based on a 365 day calendar (except for Hipparchus)

Summer Autumn Winter Spring

Democritus (460 B.C.) 91 91 91 92

Euctomen (432 B.C.) 90 90 92 93

Eudoxus (380 B.C.) 91 92 91 91Callippus (340 B.C.) 92 89 90 94

Geminus (200 B.C.) 92 89 89 95

Hipparchus (130 B.C.) 92½ 88⅛ 90⅛ 94½

accurate (134 B.C.) 92⅓ 88⅔ 90¼ 94

other than Hipparchus, it is not at all certain that any of these were based on

observation of equinoxes or solstices.

Ptolemy says that Hipparchus assumed season lengths 94½ days for Spring and



Ptolemy says that Hipparchus assumed season lengths 94½ days for Spring and

92½ days for Summer, but he does not say how Hipparchus got these values.

This tells us that the Sun

does not appear to move

around the ecliptic at a

uniform speed.






2.




5.



(observations)?


6 What kind of geometrical model would account for the phenomena




(observations)?

www.scs.fsu.edu/~dduke/models

from the season lengths we know angles TZN and PZK, and using

http://www.scs.fsu.edu/~dduke/models




from the season lengths we know angles TZN and PZK , and using

simple geometry gives EZ = e = 2;30 and angle TEZ = 65;30°.

VE

SS WS

AE

How accurate is the model? Not bad in Hipparchus’ time….



How accurate is the model? Not bad in Hipparchus time….

-0.4

0 50 100 150 200 250 300 350

day of year

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

e r r o r ( d

e g r e e s )

average

e = 2;30e = 2;10

e = 2;30

e = 2;10

average error

135 B.C.

much less so in Ptolemy’s time (large average error).



y ( g g )

-0.5

0

0.5

1

1.5

2

e r r o r

( d e g r e e s )

0 50 100 150 200 250 300 350

day of year

0 50 100 150 200 250 300 350

day of year

average

e = 2;30

e = 2;10

average

e = 2;30

e = 2;10

e = 2;30

e = 2;10

average error

135 B.C.

A.D. 137

-0.5

0

0.5

1

1.5

2

e r r o r

( d e g r e e s )

average

e = 2;30

e = 2;10

average

e = 2;30

e = 2;10

e = 2;30

e = 2;10

average error

135 B.C.

A.D. 137

Some key terms:



y

mean motion refers to the average speed of some celestial body. They

knew that the speed could vary around the orbit, but they knew the

average speed was the distance around many orbits divided by the time

for many orbits. Mean motion is regular .

mean position refers to where the body would be if it always traveled

with mean speed . In reality the true position of the body would usually

be ahead of or behind the mean position.

motion in anomaly is the regular motion that actually causes the true

motion to differ from the mean motion, so the true motion appears to be

irregular. Thus irregular motions result from a compounding of regular

motions (mean and anomaly).

the equation is an angle that is the difference between the true position



q g p

and the mean position of a body. Thus

true = mean + equation

mean positionangle

true position

angle





2.




5.



(observations)?





the length of the day is determined mostly by how fast the Earth rotates about its

axis (or the celestial sphere, to the ancients).

However, because

(b)

the Sun is moving on the oblique ecliptic, and

(c) the speed of the Sun varies on the ecliptic,

the actual length of time between successive noon’s varies slightly.

From day to day the variation is very small, but it does accumulate so that a dayin February can be about 15 minutes shorter than average, while a day in

November can be about 15 minutes longer than average, etc.

Ptolemy understood this very well, but does not tell us how he learned it.



Main features of Hipparchus’ solar model as reported by



Ptolemy:

• there is only one variation: the speed around the ecliptic.

• the eccentric and epicycle versions give equivalent explanations.

•

apogee is the direction of slowest motion on the ecliptic, perigee is thedirection of fastest motion.

• the direction of the apogee is always 65½° from the vernal equinox and the

eccentricity is always 2;30 (compared to 60). Both are determined from the

season lengths for Spring and Summer (94½ and 92½ days).

•

Ptolemy insists the model must predict that the time from slowest to mean(average) speed is greater than the time from mean to fastest speed, “for we

find that this accords with the phenomena [observations]”.

• the Sun is always on the ecliptic, never north or south (no latitude).

The Background



A closer look at Ptolemy’s ‘most carefully observed’ equinox and solstice dates.

(1) -146/9/27 midnight to 132/9/25 2 pm

(2)

-146/9/27 midnight to 139/9/26 7 am(3)

-145/3/24 6 am to 140/3/22 1 pm

(4) -431/6/27 6 am to 140/6/25 2 am

suppose we compute the expected date of each event by multiplying the number

of intervening years by the assumed days per year, 1 14 300

365+ − . We get

correct (1) 132/9/25 1:46 pm (9/24 4 am)

(2) 139/9/26 7:12 am (9/24 9 pm)

(3)

140/3/22 1:12 pm (3/21 4 pm)

(4)

140/6/25 2:19 pm (6/23 1 am)

Was it just too hard for Ptolemy to get right?



-161/9/27 6 pm (9/27 2 am)

-158/9/27 6 am (9/26 8 pm)

-157/9/27 noon (9/27 2 am)

-146/9/27 midnight (9/26 6 pm)

-145/3/24 6 am (3/24 3 pm)-145/9/27 6 am (9/26 11pm)

-144/3/23 noon (3/23 9 pm)

-143/3/23 6 pm (3/24 2 am)

-142/3/24 midnight (3/24 8 am)

-142/9/26 6 pm (9/26 5 pm)

-141/3/24 6 am (3/24 2 pm)

-140/3/23 noon (3/23 8 pm)

-134/3/24 midnight (3/24 7 am)

-133/3/24 6 am (3/24 1 pm)

-132/3/23 noon (3/23 6 pm)-131/3/23 6 pm (3/24 midnight

-130/3/24 midnight (3/24 6 am)

-129/3/24 6 am (3/24 noon)

-128/3/23 noon (3/23 6 pm)

-127/3/23 6 pm (3/23 11pm)

so apparently Hipparchus was generally accurate to the nearest ¼ day.

Clearly Ptolemy computed those four dates. Why did he do that? There is noobjective evidence to help us, so we can only speculate. What we can say with

some certainty is that this was the rule, not the exception, for Ptolemy.

Ptolemy tells us that his solar model is the same as Hipparchus’ but gives us no



other background information. In Book 12 he does mention that Apollonius ofPerge (ca. 200 B.C.) had proved a rather complicated theorem involving the

epicycle model, so it seems likely that epicycles and eccentrics, and their

equivalence, had been studied for several centuries.

Also, as we shall see for the Moon and the planets, those models require bothmoving apogee directions and (effectively) oscillating eccentricities. One might

think that for uniformity and unity Ptolemy would make the solar model more

like the models for the Moon and the planets, but he does not.

What Ptolemy does not tell us is that there was a lot of other activity developing



solar models both near his time and going back centuries.

Here are some examples:

(1) There are clear indications in the Almagest that Hipparchus himself used

solar models different from the standard one attributed to him by Theon andPtolemy. For example, the two pairs of eclipse longitude differences that

Hipparchus uses to find the unusual lunar eccentricities in Almagest 4.11 may

also be used to deduce the underlying solar models, and the resulting parameters

are equally unusual: e=7;48 and A=76;25° for Trio A, and e=3;11 and A=46;09

° for Trio B. Although attempts have been made to understand the underlying

models, the analyses are neither conclusive nor satisfying. The solar parameters

are so bizarre that we might be tempted to speculate that Hipparchus is

somehow trying to use a lunar theory to learn something about the time variation

of solar theory (the trios date to about –380 and –200), and so it is perhapsinteresting that in both trio analyses the eclipses all occur near equinoxes and

solstices [more about this case in Lecture 3].

(2) Almagest 5.3 and 5.5 give three timed solar longitudes due to Hipparchus,



and these imply a solar model with parameters e = 2;16 and A = 69;05°,although it might be that the underlying model is actually based on season

lengths of 94¼ days and 92½ days, for which the parameters are instead e = 2;19

and A = 67;08°. Either way, the value of e is significantly improved over the

‘standard’ Hipparchan value 2;30.

(3) Theon of Smyrna mentions, quite matter-of-factly, a solar model with

periods of 365¼ days in longitude, 365½ days in anomaly, and 18

365 days in

latitude. He also mentions that the Sun strays from the ecliptic by ±½°. Solar

latitude was mentioned as early as Eudoxus, and must have had some level ofuse, since not only Theon but also Pliny mentions it, and Hipparchus felt

compelled to deny its existence. Ptolemy never mentions solar latitude.

(4) P. Oxy LXI.4163 is a fragment of a papyrus table from Oxyrhynchus



that gives a template for daily longitudes of the Sun to degrees andminutes starting from the day of summer solstice. All indications are that

it is not based on the usual Hipparchan parameters.

(5) P. Oxy LIX.4162 is similar to P. Oxy LXI.4163 but appears to count

days starting when the Sun is at perigee and puts the cardinal points at 8° of the signs. In this case the indications are strong that the underlying

theory is kinematical, but even if it is, it seems not likely to be based on

the usual Hipparchan parameters.

(6) P. Oxy. LXI.4148 is a table of dates of summer solstices over a series

of years. The dates are in error by about five days in the years covered in

the fragment and are based on a year of length 365;15,22,46 days. There

are indications that the dates might have begun from a known

Hipparchan summer solstice measurement of –127 June 26 at sunrise.

For more information on the astronomical papyri of Oxyrhynchus see

http://www.chass.utoronto.ca/~ajones/oxy/

We know from the Almagest that Hipparchus knew the times of

i d l i f b ¼ d





equinoxes and summer solstices to an average accuracy of about ¼ day.Since no ancient source explains how these times were determined, we

need to consider just how an ancient astronomer would measure the time

of an equinox or a solstice to that level of accuracy.

By definition,

•

an equinox occurs at the moment the Sun touches the equator, so its

declinationδ

= 0°

.

•

and a solstice occurs when the Sun touches either tropic circle

(Cancer to the north, Capricorn to the south), so its declination23;43δ ε = =

It is clear from practical considerations that no one could have reliably

d ti l i l t d th t h th S ’ d li ti



and routinely simply noted the moment when the Sun’s declination wasat a given value: 0° for an equinox or ±23;43° for a solstice.

• On the one hand, about half of the events will occur at night, when

the Sun is not visible.

•

On the other hand, even if the event happens in daylight, it is not

always the case that the Sun will be unobscured by clouds and in a

position in the sky favorable for measuring the declinationaccurately.

In addition, for the solstices it is impossible to achieve ¼ day accuracy

with naked eye observations of any kind within a day or so of the event

since the declination of the Sun is changing extremely slowly near a

solstice.

It is most likely, then, that equinoxes and solstices were determined by

b i l ltit d f i f d b f d ft th t



observing noon solar altitudes for a series of days before and after the events.

When the Sun is crossing the meridian at noon, it is relatively easy to measure

its altitude, and then knowing the geographical latitude, to compute the

declination. From the declination, it is easy to compute the Sun’s position on the

ecliptic (the longitude), and we know that Hipparchus knew how to do it.

But it is only at noon that such an easy determination is possible. It is then fairly

straightforward to estimate the time that the Sun’s declination reaches some

specific targeted value: 0° for an equinox, and maximum or minimum for a

solstice.

That series of daily altitude measurements were used to determine the time of

cardinal events can hardly be doubted, even though no surviving ancient source

has documented such an episode. Especially for the solstices, it is essentially the

only viable option for achieving ¼ day accuracy.

In fact, however, you don’t really need equinoxes or solstices. Any trio of timed

longit des o ld be adeq ate Ptolem pro ides t o s ch anal ses for the



longitudes would be adequate. Ptolemy provides two such analyses for theMoon and one each for Mars, Jupiter and Saturn. Hipparchus, and perhaps his

predecessors and certainly his successors, knew the method, so it seems

inconceivable that it was not used multiple times to also determine solar model

parameters.

C

M3

M1

M2

O

B

R

e

P

Finally, Ptolemy insists the model must predict that the time from slowest to

mean (average) speed is greater than the time from mean to fastest speed “for



mean (average) speed is greater than the time from mean to fastest speed, forwe find that this accords with the phenomena [observations]”.

First, the time differences Ptolemy refers to are undetectable using naked eye

observations, so this is something he is pushing not from empirical

observations, but from some unstated theoretical (or philosophical) bias.

Second, there is a perfectly good model for the solar motion that violates

Ptolemy’s rule: the concentric equant. Using the concentric equant one finds

that the time from least speed to mean speed is equal to the time from mean

speed to greatest speed.

In the concentric equant model the Earth is at the center E of the deferent, but

the center of uniform motion Z of the Sun S is displaced some distance e from



the center of uniform motion Z of the Sun S is displaced some distance e fromthe center. Even though the Sun is now always at the same distance R = ES from

the Earth, the model still produces an apparent speed variation in the motion of

the Sun such that in one direction (the direction EZ) the Sun seems to be moving

slowest, and in the opposite direction it seems to be moving fastest.

The concentric equant model for the Sun is repeatedly attested in Indian texts,

all of which are generally supposed to be of Greco-Roman origin, and the

accurate value e = 2;10 is routinely used.

www.scs.fsu.edu/~dduke/models

For the equation of time, remember that it has two causes:

(a)

the Sun is moving on the oblique ecliptic and(b) h d f h i h li i





(a) the Sun is moving on the oblique ecliptic, and(b) the speed of the Sun varies on the ecliptic,

so the actual length of time between successive noon’s varies slightly.

It turns out that Geminus, writing in about 50 B.C., mentions the equation of

time but for him only (a) is involved.

In the ancient Indian texts, on the other hand, the equation of time is attributed

only to (b). These texts are supposed to originate from Greco-Roman sources

from the time period between about 100 B.C. and A.D. 100, or post-Hipparchus

and pre-Ptolemy.

Early History of Season Lengths





Summer Autumn Winter Spring

Democritus (460 B.C.) 91 91 91 92

Euctomen (432 B.C.) 90 90 92 93

Eudoxus (380 B.C.) 91 92 91 91Callippus (340 B.C.) 92 89 90 94

Geminus (200 B.C.) 92 89 89 95

Hipparchus (130 B.C.) 92½ 88⅛ 90⅛ 94½

accurate (134 B.C.) 92⅓

88⅔

90¼ 94

other than Hipparchus, it is not at all certain that any of these were based on

observation of equinoxes or solstices.



A i t A t



Ancient Astronomy

Lecture 3February 14, 2007


Lecture 3





Lecture 3

• Almagest Books 4 – 6

• the Moon

•

the problem of parallax

• the length of the various months

• the first geometric model

•

the second geometric model• sizes and distances of the Sun and Moon

• the background

the Moon and the Sun are both about the same size as viewed from Earth: they

both subtend about ½° in the sky.



y

the distance to the Moon is not

negligible compared to the size of

the Earth.

best observations are times of lunar eclipses: at that time we can compute the

position of the Sun, and we then know that the Moon is exactly 180° away.



p , y y



by the way, a solar eclipse is similar but a bit more complicated.

http://sunearth.gsfc.nasa.gov/eclipse/SEanimate/SE2001/SE2017Aug21T.GIF



drawn to scale:



What is a ‘month’?

There are several:



There are several:

(a) return to the same star on the ecliptic (sidereal). 27d 07h 43m 12s

(b)

return to the same declination δ (tropical). 27d 07h 43m 05s

(c) return to the same speed (anomalistic). 27d 13h 18m 33s

(d) return to the same latitude (draconitic). 27d 05h 05m 36s

(e)

return to the same angle from the Sun (synodic). 29d 12h 44m 03s

The synodic month – from one new moon or full moon to the next – is the one

we use in daily conversation.

Sidereal Month (return to same longitude or fixed star)

Tropical Month (return to the same equinox or solstice)



Anomalistic Month (return to same speed, e.g. fastest or slowest)



Draconitic Month (return to the nodes)



Synodic Month (return to the Sun)





to get an eclipse we must have the Sun-Earth-lunar nodes lines up, and the

Moon fairly near a node (with about≤

7°

). On average we get about two eclipses per year, somewhere.



Period Relations

Periodic (Saros)d m a d t



6585⅓d = 223m = 239a = 242d = 241t + 10⅔° (about 18y)

Exeligmos (3x Saros)

19,756d = 669m = 717a = 726d = 723t + 32° (about 54y)

Hipparchus (Babylonian)

126,007d 1h = 4267m = 4573a = 726d = 4612t – 7½° (about 345y)

and 5458m = 5923d

Note that 126007d 1h / 4267m = 29d 12h 44m 02s (compared to 29d 12h 44m 03s)

All of these come from centuries of eclipse records in Babylon, starting around

750 B.C. if not earlier (remember that Alexander the Great conquered Babylonin 323 B.C.)

Ptolemy and Hipparchus found that regarding just new moon and full moon,

when the Sun and Moon are in a line with the Earth, a simple model would

work.

http://www scs fsu edu/ dduke/models htm

http://www.scs.fsu.edu/~dduke/models.htm





However, in the more general case the simple model fails and Ptolemy uses a

more complicated model. http://www.scs.fsu.edu/~dduke/models.htm







Sizes and Distances of the Sun and Moon

Ptolemy gives an analysis which is extremely delicate to compute.

http://en.wikipedia.org/wiki/Image:HipparchusConstruction.png



Ptolemy takes

θ = 0;31,20°

φ = 2 3/5 θ = 1;21,20°

L = 64;10

to get S = 1210 but for example φ / θ = 2 2/5 makes S < 0

The Background

Ptolemy’s usual fudging

http://en.wikipedia.org/wiki/Image:HipparchusConstruction.png



Luni-Solar calendars

Babylonian models

Ptolemy’s fudging

for the simple model he produces two trios of lunar eclipses:



-720 Mar 19/20 7:30 pm 133 May 6/7 11:15 pm

-719 Mar 8/9 11:10 pm 134 Oct 20/21 11:00 pm-719 Sep 1/2 8:30 pm 136 Mar 5/6 4:00 am

Analysis of both trios gives virtually identical results, and

changing any of the times by even a few minutes substantiallychanges the results.

Later, in Almagest 4.11, he gives two more trios and once again

gets the very same answers. Such coincidences are very unlikely.

for the complicated model

(a)

Ptolemy wants to know the maximum angle the true Moon candiffer from the average Moon. In the case of the simple model this

i 5° P l d b i hi h h l



is 5°. Ptolemy produces two observations which he analyzes to get

a maximum angle of 7;40° (in both cases). But he neglected

parallax, and if he had included it he would have gotten 7;31° and7;49° for the two cases.

(b) Ptolemy needs to know the size of his new central epicycle, so he

produces two observation that both give him 10;19. In both cases

he miscomputes but still manages to get the same answer.

(c)

Ptolemy’s complicated model makes the apparent size of the Moon

vary by almost a factor two. In reality it varies by about 15%

(maximum to minimum).

for the sizes and distances Ptolemy has to very carefully analyze

eclipses from 523 B.C. and 621 B.C. (why so ancient?). In the end hefinds

19S



19S L

However, in about 240 B.C. Aristarchus, in a completely different kindof analysis, also found / 19S L . He assumed only that the angle Moon-

Earth-Sun was 87° at half-moon.

In between Ptolemy and Aristarchus, Hipparchus used slightly

different parameters to get a much different answer:



Hipparchus takes

θ = 0;16,37°

φ = 2 1/2 θ = 0;41,33° L = 67;20

to get S = 490, or he

assumed S = 490 andcomputed L = 67;20

Ptolemy takes

θ = 0;15,40°

φ = 2 3/5 θ = 0;40,40° L = 64;10

to get S = 1210

the correct answers are about L = 60 and S = 23,000

Luni-Solar Calendars

The fact that the month is just a bit longer than 29½ days caused a lot of bother

in establishing a workable calendar that keeps months properly aligned with the



in establishing a workable calendar that keeps months properly aligned with the

year and its seasons.

Early try: Meton and Euctomen (about 430 B.C.): the Metonic calendar

19 years = 235 months = 6940 days

= 12 years of 12 months plus 7 years of 13 months

There are 125 full (30-day) months and 110 hollow (29-day) months

Resulting year is 365 5/19 days

Resulting month is 29 + 1/2 + 3/94 days

365 5/9 is longer than 365 ¼ by 1/76 day. Hence Callippus (about 330 B.C.)

suggested a new calendar with four successive 19-year Metonic cycles but

leaving out 1 day from one of the cycles:



76 years = 940 months = 27,759 days

= 4 x 235 months = 4 x 6,940 days – 1 day

Resulting year is 365 1/4 days

Resulting month is 29 + 1/2 + 29/940 days

The fraction 29/940 is about 1/32.4 whereas a slightly more accurate value is1/33, and this was known to Geminus and hence would have been widely

known.

There may have been even older and simpler calendars. Geminus describes ones

with 8 years = 99 months and 16 years = 198 months and 160 years = 1979

months. In all of these either the month or year length is not good enough.

The Antikythera Mechanism







the pin-and-slot mechanism to simulate Hipparchus’ model





Babylonian Astronomy

During the late 1800’s some 50,000 or so clay tablets were sent to the British

Museum from Babylon and Uruk.



About 250 of the tablets related to astronomy were studied by two Jesuit priests,

Fathers Epping and Strassmaier in the late 1800’s and followed by Father

Kugler in the early 1900’s.





The work of the three Fathers revealed a previously unsuspected history of very

involved mathematical astronomy developed in Babylon starting about 450 B.C.

Before their work science in Babylon was generally associated with ideas like

magic mysticism and astrology These people were often referred to as the



magic, mysticism, and astrology. These people were often referred to as the

Chaldeans.

Whereas the Greek models were designed to give the position of the Sun or

Moon at any moment in time, the Babylonians were interested in predicting the

times and position of sequences of quasi-periodic events – new moon, full

moon, etc.

The Babylonians used a purely lunar calendar. The “lunar month” begins on the

evening when the lunar crescent is first visible shortly after sunset.



Such a definition has a number of intrinsic difficulties, and Babylonian lunar

theory was developed to deal with these complications.

How many days are in a “lunar month”? Each such month is either 29 or 30

days, but we need to know which in advance.



days, but we need to know which in advance.

This clearly involves both the varying speed of the Moon and the varying speedof the Sun. Remember that the Moon covers about 13° per day and the Sun

about 1° per day, but these are averages. So we must account for the departure

from average throughout each month.

There are seasonal changes due to the angle between the ecliptic and the horizonand also changes due to the varying latitude of the Moon.

setting



westernhorizon ☼

☼Spring

Fall

setting



westernhorizon

☼Spring

latitude

≤5°

setting

latitude

≤5°



western

horizon

☼

Spring

≤5

The ‘astronomical diaries’ were kept for many centuries and are night-by-night

accounts of where the various celestial objects were to be found:



The result was a list of eclipses covering about six centuries, which Hipparchus

apparently had access to.

In addition, the Babylonians kept extensive records of several centuries of

observations of the times between rising/setting of the Moon and the Sun.



The lunar theories

Each tablet is a set of columns of

numbers





Almost all of these changes vary fairly smoothly somewhat like sine and cosine.

The Babylonian astronomers invented schemes for approximating this kind of

variation.



Nothing survives to tell us how these schemes were created. What do survive are

a small number of ‘procedure texts’ which give the rules the scribes need to

compute each column.





Ancient Astronomy



Lecture 4Course website: www.scs.fsu.edu/~dduke/lectures

Lecture 4

• Almagest Books 7–8

th t





• the stars

• precession• the constellations

• rising and setting and the calendar

• the background

Ptolemy is now ready to discuss the stars. Recall that

• first, he measures the Sun w.r.t. the equinoxes and

solstices



• then he measures the Moon w.r.t. the Sun

• now he will measure the stars w.r.t. the Moon

•

next he will measure the planets w.r.t. the stars

But first he must deal with a small complication:

the stars move!

The goal is to measure the position of stars on the celestial

sphere. Let’s see what is involved.



Throughout each night the stars rise in the east and set in the west.



since the latitude of Alexandria is about 31° (similar to Tallahassee) the

celestial equator is about 59° above the southern horizon.



the declination coordinate is the distance of each star from the celestial

equator. It is easiest to measure when the star crosses the southern

meridian.



stars rotate along circles of constant declination parallel to the celestial

equator



from north pole to south pole run parallel lines of constant right

ascension, always perpendicular to the lines of declination.



so one convenient set of coordinates is (right ascension, declination).





Ptolemy says he used an armillary sphere to measure the position of a star.



The problem is: these coordinate systems do not stay fixed w.r.t. each other.



The real reason is that the Earth is like a spinning top, hence the name precession.



Ptolemy tells us that Hipparchus was quite careful about this, e.g. he wondered

if just the stars near the ecliptic were involved, or all the stars? To test this he

left a list of star alignments good in his time, and invited future observers tocheck them. Ptolemy did, and added some new ones:



To determine the speed of precession several kinds of observations

were used. First, eclipses when the Moon was near a star:



The changes in

declination for

stars near an

equinox:



Lunar occultations of stars a few hundred years apart in time:



Ptolemy recognizes 48 constellations: 21 north of the zodiac, 12 in

the zodiac, and 15 south of the zodiac. He gives the ecliptic

longitude and latitude of 1,028 stars (including 3 duplicates shared

by two constellations).

Ursa Minor (Little Dipper) longitude latitude mag

Description d m d m V name



Description d m d m V name

1 The star on the end of the tai. UMi 11 60 10 + 66 0 3 1Alp UMi2 The one next to it on the tail UMi1 2 62 30 + 70 0 4 23Del UMi

3The one next to that, before the place where the tail joins[the body] UMi 13 70 10 + 74 20 4 22Eps UMi

4The southernmost of the stars in the advance side of therectangle UMi1 4 89 40 + 75 40 4 16Zet UMi

5 The northernmost of [those in] the same side UMi15 93 40 + 77 40 4 21Eta UMi

6 The southern star in the rear.side UMi 16 107 30 + 72 50 2 7Bet UMi

7 The northern one in the same side UMi 17 116 10 + 74 50 2 13Gam UMi

8The star lying on a straight line with the stars in the rear side[of the rectangle] and south of them UMi1 8 i 103 0 + 71 10 4 5 UMi



For most people in antiquity the real interest in the stars resulted from their

relation to the annual calendar. Ptolemy and many people before him published

something like this (e.g. 14 hrs is the longest day at latitude 31°):

Epiphi (the 11th month).

1. Summer solstice. 13 ½ hours: the middle one of the belt of Orion rises.14 hours: the one on the trailing shoulder of Orion rises.



2. 15 ½ hours: the bright one of Perseus rises in the evening.5. 14 hours: the one common to Eridanus and the foot of Orion rises. 15hours: the one on the leading shoulder of Orion rises.6. 13 ½ hours: the one on the head of the leading twin [of Gemini] rises.14 hours: the middle one of the belt of Orion rises, and the last one ofEridanus rises, and the one on

the head of the leading twin [of Gemini] rises.7. 14 ½ hours: the bright one of Corona Borealis sets in the morning.8. 15 hours: the one on the head of the leading twin [of Gemini] rises. 15½ hours: the one common to Pegasus and Andromeda rises in theevening.

9. 15 ½ hours: the one on the head of the leading twin [of Gemini] rises.

latitude 31°

14 hrs

longest day



latitude 41°

15½ hrs

longest day





The Background

Constellations were known to the Babylonians. Many but not allare related to the Greek versions.

The constellations were well-organized in Greece no later than 380



B.C. and probably considerably earlier. We know this from thefamous poem of Aratus written about 270 B.C. that was derived

from two works by Eudoxus.







In about 130 B.C. Hipparchus wrote a Commentary to Aratus and

Eudoxus that, for the most part,

(a) severely criticized most earlier astronomers for not being

accurate enough, and



(b) gave Hipparchus’ own version of the rising and setting of

constellations that established a new level of accuracy and

precision.

For example, Hipparchus wrote:

1.1.5 Since I observe that even on the most important points Aratus conflicts both with phenomena and with the things that really happen, but that in practically all details not only

other commentators but even Attalus agree with him, I thought it good — for the sake of your

learning and the common benefit of others — to make an accounting of the things that seem to

me to be erroneous. I undertook to do this not because I chose to enhance my image by

refuting others. That is hollow and altogether mean; indeed, I think, on the contrary, that we

i i d ll h i ki h l i k f h



must give gratitude to all who engage in taking upon themselves rigorous tasks for thecommon benefit. However, I undertook this so that neither you nor others who seek wisdom

might stray from scientific knowledge concerning phenomena in the universe. Many have

suffered this; and it is easy to understand why. For the charm of poems acquires for their

statements a certain reliability, and almost everybody who has commented upon this poet

submits to his statements.1.1.8 Eudoxus treated the same material concerning phenomena as did Aratus, but with greater

understanding. Naturally, then, the poetry is also regarded as trustworthy because so many

great mathematical astronomers concur. And yet, it is not appropriate that one assail Aratus,

even if he happens to err in certain points. For he wrote the Phaenomena closely following

Eudoxus’ material, but without observing for himself and without promising to report the

opinion of mathematical astronomers in matters concerning the heavens; this is where Aratusmakes mistakes in his Phaenomena.

Hipparchus’ own version of the rising of the Crab:

3.3.1b When the Crab is rising, together with it rises the zodiac from 23° of the Twins

until 18° of the Crab. On the meridian is the portion from 5° of the Fishes until 1m° of

the Ram. And the first star to rise is the one in the tip of the northern Claw; the last is that

in the tip of the southern Claw.

Of others on the meridian, the first is the bright star in Andromeda’s head; the last is the

l di t f th th i th R ’ h d d th b i ht i d t l i t d



leading star of the three in the Ram’s head, and the bright, unassigned star lying towardthe south along the middle of the Sea-monster’s body, and the southern of the following

stars in the quadrilateral of the Sea-monster, and Andromeda’s left foot which is a little

short of the meridian.

The Crab rises in 1⅔ hours.

In fact this is pretty accurate in 130 B.C.:





It turns out that using all the data that Hipparchus gives we can

(a) conclude that he had an extensive catalog of star coordinates, and

(b) figure out the errors on many of his star positions.

The correlation of the Commentary and Almagest errors should be small

if the catalogs are independent, but large if the catalogs share a common

heritage



heritage.

It is clear that Ptolemy copied his star coordinates from Hipparchus.

-1 5

-1 5 -1 0 -5 0 5 1 0

C o m m en ta r y E rr o rs (d eg re es )

-1 0

-5

0

5

1 0

A l m a g e s t E r r o r s ( d e g r e e s )

99 6

99 6

9 95

99 5

805

80 5

91 8

9 18

9 92

892

892

918

9 18

What Ptolemy almost certainly did was take Hipparchus’

coordinates, probably in equatorial right ascension and declination,

convert them to ecliptical longitude and latitude, and then add 2⅔° to the longitudes to account for 265 years of precession at 1° per

100 years.



This in spite of his explicit claim:



Hipparchus does have some strange things, though:

1.4.1 Eudoxus is in ignorance concerning the North Pole, when he saysthis:

There is a certain star which remains ever in the same spot; and this

star is the Pole of the world.

Upon the pole lies not even one star; rather it is an empty place beside



Upon the pole lies not even one star; rather it is an empty place besidewhich lie three stars. With these the point on the Pole forms nearly a

square, according to Pytheas of Massilia. It is difficult to know what Eudoxus or Hipparchus is referring to,though:





There were probably a number of other star catalogs around. Star

globes seem to have been popular, and two very nice originals

have survived: the Farnese Atlas and the Mainz Globe.











Ancient AstronomyLectures 5-6





Lectures 5-6

• Almagest Books 9–13

• geocentric vs. heliocentric point of view

•

the wandering stars, or planets• the two anomalies





g , p• the two anomalies

• the eccentric plus epicycle and its problems

• the equant

•

latitude• distances

• the background

In reality the Earth and all the other planets revolve around the Sun.

Nevertheless, we can imagine a reference frame in which the Earth isat rest, and ask “what would a correct theory look like in that reference

frame?”

Answer: it would look very much like the theory created by the Greek



Answer: it would look very much like the theory created by the Greek

astronomers.

And note: modern astronomers compute first the planets orbiting the

Sun, and then have to figure out the position of the planet relative to

the Earth.



geocentric = ego-centric = more"natural"

Problems for heliocentric theory:

• Earth in motion??? can't feel it

•

no parallax seen in stars

Kepler’s Three Laws of planetary motion:

1. orbits are ellipses, Sun at focus 2. equal area in equal time

http://abyss.uoregon.edu/~js/glossary/parallax.html

http://abyss.uoregon.edu/~js/glossary/parallax.html



3.

2

3 1Pa = for each planet



Instead of the Earth circling the Sun, we would have the Sun

circling the Earth.



For the outer planets, the radius of the epicycle is always

parallel to the direction of the Sun from the earth.



http://www.csit.fsu.edu/~dduke/juphelio.html

All of the planets have, from time to time, a retrograde

motion, i.e. the slow motion from west to east stops,

then reverses into an easy to west motion, then stopsagain and resumes a west to east motion.

http://www.astronomynotes.com/nakedeye/animations/retrograde-anim.htm







In reality this happens because planets closer to the Sun

move faster than planets farther from the Sun.

In a geocentric theory, this happens because of the

counter-clockwise motion on the epicycle.

http://www.astro.utoronto.ca/~zhu/ast210/both.htmlhttp://www.scs.fsu.edu/~dduke/models.htm

http://www.astro.utoronto.ca/~zhu/ast210/both.html



http://www.astro.utoronto.ca/~zhu/ast210/both.html



p

And note the close relation to the Sun. In a heliocentric

picture it is clear that in retrograde the Sun–Earth–planet arein a line. In a geocentric picture it is not required, but the

Greeks knew that had to assume it to be true.

Counting the number of retrograde episodes and planetary

orbits over many years gives the period relations:

planet years orbits retrogrades

Saturn 59 2 57

Jupiter 71 6 65 Mars 79 42 37





Venus 8 13 5

Mercury 46 191 145

Note that the solar year is somehow involved for every

planet! Such relations are completely ad hoc in a geocentric

view but exactly as expected in a heliocentric view.

In the Almagest Ptolemy says little about the distances to the

planets:



For all the models Ptolemy assumes a deferent circle of radius R =

60 and an epicycle of radius r < 60. Comparing the heliocentric

distances and the Almagest geocentric distances gives

modern Almagest

planet a r r

Mercury 0.3871 23;14 22;30Venus 0 7233 43;24 43;10



Venus 0.7233 43;24 43;10

Mars 1.5237 39;22 39;30

Jupiter 5.2028 11;32 11;30

Saturn 9.5388 6;17 6;30

As far as we know, nobody after Aristarchus (ca. 230 B.C.) and

before Copernicus (A.D. 1540) was willing to make the leap to theheliocentric picture.

Like the Sun and Moon, the speed of the planets also varies

smoothly as they circle the zodiac, so the planetary orbits each

have an apogee and a perigee.







A new idea, the equant, solves the problem.



The equant is very similar to Kepler’s ellipse, and accounts

very well for Kepler’s 1st and 2

nd Laws.

www.scs.fsu.edu/~dduke/kepler.html

http://people.scs.fsu.edu/~dduke/kepler3.html

Combining the periods and distances gives Kepler’s 3rd Law:

a a3

Period P2

P2/a

3

Mercury 0.38 0.05 0.24 0.06 1.10

Venus 0.72 0.37 0.62 0.38 1.02Earth 1.00 1.00 1.00 1.00 1.00M 1 52 3 50 1 88 3 54 1 01

http://www.scs.fsu.edu/~dduke/kepler.html

http://www.scs.fsu.edu/~dduke/kepler.html





Mars 1.52 3.50 1.88 3.54 1.01Jupiter 5.22 142.02 11.83 140.03 0.99

Saturn 9.23 786.53 29.50 870.25 1.11

So the Almagest models are indeed very much like the real

planetary orbits when viewed from Earth.

For some reason Ptolemy

makes the model forMercury more complicated.



http://people.scs.fsu.edu/~dduke/mercury.html

Like the Moon, the planet orbits are tilted relative to the Sun’s

orbit.

outer inner





Outer planet http://people.scs.fsu.edu/~dduke/latitude.html

Inner planet http://people.scs.fsu.edu/~dduke/latitude2.html

Note that these make good sense in a heliocentric view.

In the Planetary Hypotheses Ptolemy writes:

http://people.scs.fsu.edu/~dduke/latitude.html

http://people.scs.fsu.edu/~dduke/latitude2.html

http://people.scs.fsu.edu/~dduke/latitude2.html

http://people.scs.fsu.edu/~dduke/latitude.html



http://people.scs.fsu.edu/~dduke/ptolemy.html

Using his “nesting” assumption Ptolemy gets:





Early Greek Planetary Theories

The Keskintos Inscription (found on Rhodes about 1890) and probably carved about 100 B.C.









The texts of ancient Indian astronomy give us a sort of

wormhole through space-time back into an otherwise

inaccessible era of Greco-Roman developments in astronomy.



Indian Planetary Theories

Conventional wisdom:

“The orbits of the planets are concentric with the center of the earth. The single

inequalities recognized in the cases of the two luminaries are explained by manda-

epicycle (corresponding functionally to the Ptolemaic eccentricity of the Sun and lunar

epicycle, respectively), the two inequalities recognized in the case of the five star-

planets by a manda-epicycle (corresponding to the Ptolemaic eccentricity) and a

sighra-epicycle (corresponding to the Ptolemaic epicycle). The further refinementsof the Ptolemaic models are unknown to the Indian astronomers.”



P

E E

P

The Indian theories have even longer period relations:

planet years orbits retrogrades

Saturn 4,320,000 146,564 4,173,436

Jupiter 4,320,000 364,224 3,955,776

Mars 4,320,000 2,296,824 2,023,176Venus 4,320,000 4,320,000 2,702,388



Mercury 4,320,000 4,320,000 13,617,020

In fact, the numbers the Indians text quote for Venus and Mercury are the

number of heliocentric revolutions for each planet in 4,320,000 years:

Venus: 7,022,388 = 4,320,000 + 2,702,388

Mercury 17,937,020 = 4,320,000 + 13,617,020

eccentric (manda) sin ( ) sinq eα α = −

sin

tan ( )1 cos

r p

r

γ γ

γ =

+

epicycles (sighra)

(1)

Aα λ λ = −

1

21 ( )qν λ α = +

(2)

1S γ λ ν = −

1

22 1 ( ) pν ν γ = +

(3)

2 Aα ν λ = − 3 ( )qν λ α = +

Aryabhata’s text says:

half the mandaphala obtained fromthe apsis is minus and plus to the

mean planet. Half from the

sigraphala is minus and plus to the



(4) 3S

γ λ ν = − 3 ( ) pλ ν γ = +

sigraphala is minus and plus to the

manda planets. From the apsis they

become sphutamadhya [true-mean]. From the sigraphala they

become sphuta [true].

-1

0

1

2

Almagest Sunrise

Jupiter



-2

500 502 504 506 508 510 512

Most of the difference is due to poor orbit parameters in the

Sunrise model.

What happens if we use identical orbit elements in both models?

-5

0

5

10

15

Equant Eccentric SunriseMars

-1.5

-0.5

Equant Eccentric SunriseJupiter



-15

-10

500 502 504 506 508 510 512-2.5

500 502 504 506 508 510 512

Therefore, it is clear that the Almagest equant and the Indian

models share the same mathematical basis.

Arabic astronomers were very unhappy with the equant since it

violates Aristotle’s principle of uniform motion in a circle. By

about A.D. 1250 they had developed several alternatives that are asgood as the equant and use only uniform motion.

http://people.scs.fsu.edu/~dduke/arabmars.html

The same issues bothered Copernicus (ca. 1520-1540) and he

used the same models although we do not know how he





used the same models, although we do not know how he

learned about them.

Documents

Six Easy Leactures on Ancient Mathematical Astronomy - D. Duke [Univ of FL Lecture Mtls] WW