A Look Toward the Heavens and the Almagest Ptolemy’S Theorem

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The Heavens

Garner

Ancient Greek Astronomy

Claudius Ptolemy A Look Toward the Heavens and the Almagest Ptolemy’s Theorem

The History of Mathematics, Part 10 Homework

Chuck Garner, Ph.D.

Department of Mathematics Rockdale Magnet School for Science and Technology

February 24, 2021

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Outline Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ancient Greek Astronomy Ptolemy’s Theorem

Homework

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Outline Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ancient Greek Astronomy Ptolemy’s Theorem

Homework

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Aristotle Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Aristotle Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Aristarchus Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

Aristarchus of Samos 310 BC-230 BC

6 / 31 I First to give a heliocentric model of the solar system

I Archimedes uses Aristarchus’ theory in The Sand-Reckoner

I Idea was not popular among the Greeks

I Used the equivalent of sin α α tan α < < sin β β tan β

in his works (0 < β < α < π/2)

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Aristarchus Garner

Ancient Greek I Reasoned mathematically that Mercury and Venus Astronomy revolved around the sun Claudius Ptolemy and the Almagest

So the sun is much bigger than earth... Ptolemy’s Theorem

So the earth revolves about the sun as well Homework

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Aristarchus Garner

Ancient Greek I Reasoned mathematically that Mercury and Venus Astronomy revolved around the sun Claudius Ptolemy and the Almagest

So the sun is much bigger than earth... Ptolemy’s Theorem

So the earth revolves about the sun as well Homework

I First to give a heliocentric model of the solar system

I Archimedes uses Aristarchus’ theory in The Sand-Reckoner

I Idea was not popular among the Greeks

I Used the equivalent of sin α α tan α < < sin β β tan β

in his works (0 < β < α < π/2)

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Hipparchus Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

Hipparchus of Rhodes 190 BC-120 BC “Never deceive a friend.”

8 / 31 I Catalogued 850 stars

I Advocated use of latitude and longitude

I May have been the ﬁrst to introduce dividing a circle into 360 parts

I Constructed a table of chords

The Heavens

Hipparchus Garner

Ancient Greek Astronomy

I Careful and precise astronomical observer Claudius Ptolemy and the Almagest I Determined the mean lunar month to within one Ptolemy’s Theorem

second Homework

I Accurate calculation of the inclination of the ecliptic

I Discovered procession of equinoxes

I Computed lunar parallax

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Hipparchus Garner

Ancient Greek Astronomy

I Careful and precise astronomical observer Claudius Ptolemy and the Almagest I Determined the mean lunar month to within one Ptolemy’s Theorem

second Homework

I Accurate calculation of the inclination of the ecliptic

I Discovered procession of equinoxes

I Computed lunar parallax

I Catalogued 850 stars

I Advocated use of latitude and longitude

I May have been the ﬁrst to introduce dividing a circle into 360 parts

I Constructed a table of chords

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Outline Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ancient Greek Astronomy Ptolemy’s Theorem

Homework

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Ptolemy Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

Claudius Ptolemy 90 AD-170 AD “When I trace at my pleasure the windings to and fro of the heavenly bodies, I no longer touch the earth with my feet: I stand in the presence of Zeus himself and take my ﬁll of ambrosia, food of the gods.”

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Ptolemy Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest I Improved Hipparchus’ table of chords Ptolemy’s Theorem

I Wrote the deﬁnitive Greek work on astronomy in 13 Homework books

I Mathematiki Syntaxis; trans. Mathematical Collection

I Became known as Megisti Syntaxis; trans. Greatest Collection

I Islamic mathematicians traslated it to al-Magisti

I Latin translation of Arabic became Almagest

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Retrogade Motion Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Epicycles Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Ptolemy Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem I Was not concerned with why only in a geometric Homework model of motion

I Ptolemy calculated where planets and stars would be; to do this, he needed a way to measure chords in circles...

I Almagest surpassed all other astronomical works

I Did for astronomy what Elements did for geometry

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The Table of Chords Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest The table gives the lengths of chords of all central angles Ptolemy’s Theorem 1 ◦ ◦ Homework of a circle from 2 to 180 in half-degree intervals I Radius of the circle is divided into 60 equal parts

I Chord lengths expressed in base-60 in terms of a radius-part

I Let’s use “crd α” to mean the length of a chord of angle α

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The Table of Chords Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

crd α α

17 / 31 I Let’s relate chords to something more modern...

The Heavens

The Table of Chords Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest I So Ptolemy has Ptolemy’s Theorem

Homework crd 36◦ = 37p 40 5500

which means the chord of angle 36◦ is equal to

I 37 parts of the 60-part radius, plus

I 4/60 of one of these radius parts, plus

I 55/60 of one of the 60 parts of the radius part

I Note similarity to degrees, minutes, seconds

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The Table of Chords Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest I So Ptolemy has Ptolemy’s Theorem

Homework crd 36◦ = 37p 40 5500

which means the chord of angle 36◦ is equal to

I 37 parts of the 60-part radius, plus

I 4/60 of one of these radius parts, plus

I 55/60 of one of the 60 parts of the radius part

I Note similarity to degrees, minutes, seconds

I Let’s relate chords to something more modern...

18 / 31 So the Table of Chords gives sines in quarter-degree intervals from 0◦ to 90◦.

The Heavens

The Table of Chords Garner

Ancient Greek Astronomy M Claudius Ptolemy A B and the Almagest α α Ptolemy’s Theorem Homework O

AM 2 · AM AB crd (2α) sin α = = = = OA 2 · OA 2 · 60 120

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The Table of Chords Garner

Ancient Greek Astronomy M Claudius Ptolemy A B and the Almagest α α Ptolemy’s Theorem Homework O

AM 2 · AM AB crd (2α) sin α = = = = OA 2 · OA 2 · 60 120

So the Table of Chords gives sines in quarter-degree intervals from 0◦ to 90◦.

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Outline Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ancient Greek Astronomy Ptolemy’s Theorem

Homework

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Ptolemy’s Theorem Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem This geometrical gem is not found in the Elements Homework Theorem In a cyclic quadrilateral the product of the diagonals is equal to the sum of the products of the two pairs of opposite sides.

21 / 31 From similar triangles 4ABC and 4DEC, we have AB/AC = DE/DC, whence AB · DC = DE · AC. Therefore

AD · BC + AB · DC = BE · AC + DE · AC = AC(BE + DE) = AC · BD.

The Heavens

Proof of Ptolemy’s Theorem Garner

Ancient Greek AD/AC = BE/BC, whence Astronomy Claudius Ptolemy AD · BC = BE · AC. and the Almagest Ptolemy’s Theorem

Homework

Let E be on BD such that ∠BCE = ∠ACD. Then 4BCE ∼ 4ACD, so that

22 / 31 Therefore

AD · BC + AB · DC = BE · AC + DE · AC = AC(BE + DE) = AC · BD.

The Heavens

Proof of Ptolemy’s Theorem Garner

Ancient Greek AD/AC = BE/BC, whence Astronomy Claudius Ptolemy AD · BC = BE · AC. and the Almagest From similar triangles Ptolemy’s Theorem 4ABC and 4DEC, we have Homework AB/AC = DE/DC, whence AB · DC = DE · AC.

Let E be on BD such that ∠BCE = ∠ACD. Then 4BCE ∼ 4ACD, so that

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Proof of Ptolemy’s Theorem Garner

AD · BC + AB · DC BE AC DE AC Let E be on BD such that = · + · = AC(BE + DE) ∠BCE = ∠ACD. Then 4BCE ∼ 4ACD, so that = AC · BD.

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Ptolemy’s Corollaries Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Corollary (1) Homework If a and b are the chords of two arcs of a circle of unit radius, then a√ b√ s = 4 − b2 + 4 − a2 2 2 is the chord of the sum of the two arcs.

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Ptolemy’s Corollaries Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Corollary (2) Homework If a ≥ b are the chords of two arcs of a circle of unit radius, then a√ b√ d = 4 − b2 − 4 − a2 2 2 is the chord of the diﬀerence of the two arcs.

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Ptolemy’s Corollaries Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem Corollary (3) Homework If t is the chord of a minor arc of a circle of unit radius, then q √ h = 2 − 4 − t2 is the chord of half the arc.

25 / 31 Using Corollary 3, we may successively ﬁnd chords of 12◦, ◦ ◦ 3 ◦ 3 ◦ 6 , 3 , 2 , and 4 , obtaining

3 ◦ 3 ◦ crd 2 = 1.5708 and crd 4 = 0.7854.

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Calculating the Chord of 1 Degree Garner

I From “Incommensurables” an isosceles triangle with Ancient Greek equal sides 1 and vertex angle 36◦ has base equal to Astronomy Claudius Ptolemy the golden ratio 0.6180 and the Almagest Ptolemy’s Theorem I Hence, in a circle of unit radius, Homework crd 36◦ = 60 × 0.6180 = 37.082.

◦ I In a circle of unit radius, crd 60 = 1, so by Corollary 2: crd 24◦ = crd (60◦ − 36◦) = 22.918.

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Calculating the Chord of 1 Degree Garner

◦ I In a circle of unit radius, crd 60 = 1, so by Corollary 2: crd 24◦ = crd (60◦ − 36◦) = 22.918. Using Corollary 3, we may successively ﬁnd chords of 12◦, ◦ ◦ 3 ◦ 3 ◦ 6 , 3 , 2 , and 4 , obtaining

3 ◦ 3 ◦ crd 2 = 1.5708 and crd 4 = 0.7854.

26 / 31 Also, crd (3/2)◦ 3/2 3 < = , crd 1◦ 1 2 or 2 3◦ 2 crd 1◦ > crd = × 1.5708 = 1.0472. 3 2 3 Hence, crd 1◦ = 1.0472.

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Calculating the Chord of 1 Degree Garner crd α α ◦ ◦ By the relation < , for 0 < β < α < 90 , we Ancient Greek crd β β Astronomy have ◦ Claudius Ptolemy crd 1 1 4 and the Almagest < = , crd (3/4)◦ 3/4 3 Ptolemy’s Theorem or Homework 4 3◦ 4 crd 1◦ < crd = × 0.7854 = 1.0472. 3 4 3 Also, crd (3/2)◦ 3/2 3 < = , crd 1◦ 1 2 or 2 3◦ 2 crd 1◦ > crd = × 1.5708 = 1.0472. 3 2 3 Hence, crd 1◦ = 1.0472. 27 / 31 The Heavens

The Almagest Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

Ninth century copy, in Greek, from the Vatican Library – oldest known copy of the Almagest

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Ptolemy’s Table of Chords Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Outline Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ancient Greek Astronomy Ptolemy’s Theorem

Homework

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework

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Homework Garner

Ancient Greek Astronomy

Claudius Ptolemy and the Almagest

Ptolemy’s Theorem

Homework I A general survey of Greek mathematics; Math Through the Ages, pages 15–24

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