4 Ancient Astronomy & Trigonometry
Astronomy is one of the most important drivers of mathematical development; early trigonometry
(literally triangle-measure) was largely developed in order to facilitate astronomical computations, for
which there are many practical benefits.
Calendars The phases of the moon (whence month), the seasons, and the solar year are paramount.
Without accurate calendars, food production, gathering and hunting are more difficult: When
will the rains come? When should we plant/harvest? When will the buffalo return?
Navigation The simplest navigational observation in the northern hemisphere is that the stars appear
to orbit Polaris (the pole star), thus providing a fixed reference point/direction in the night sky.
As humans travelled further, accurate computations became increasingly important.
Religion and Astrology In modern times, we distinguish astronomy (the science) from astrology
(how the heavens influence our lives), but for most of human history the two were inseparable.
In our light-polluted urban world, it is hard to imagine the significance the night sky held for our
ancestors, even 100–200 years ago. Almost all religions imbue the heavens with meaning, which
historically provided a huge religious/astrological imperative for mathematical and technological
development. Here are just a few examples of the relationship between astronomy, astrology and
culture.
• The concept of heaven as the domain of the gods, whether explicitly in the sky or simply atop a
high mountain (e.g., Olympus in Greek mythology, Moses ascending Mt. Sinai, etc.).
• Many ancient structures were constructed in alignment with heavenly objects:
– Ancient Egyptians viewed the region around Polaris as their heaven; pyramids included
shafts emanating from the burial chamber so that the deceased could ‘ascend to the stars.’
– Several Mayan temples and observatories appear to be oriented to the solstices (see be-
low). Such alignments can be found elsewhere in the Americas and throughout the world.
– Venus and Sirius, respectively the brightest planet and star in the night sky, were also
important objects of alignment.
• The modern (western) zodiac comes from Babylon, dating to before 1000 BC. A tablet dated
to 686 BC describes 60–70 constellations and stars with aspects that are familiar to modern as-
trologers, including Taurus, Leo, Scorpio and Capricorn. During the same time-period Chinese
and Indian astronomers developed different systems of constellations.
16
• Calendars mark religious festivals, practices and even the age of the world.
– The traditional Hebrew calendar dates the beginning of the world to 3760 BC.
– The Mayan long count calendar dates the creation of the world to 3114 BC.
– The modern Gregorian calendar arose to facilitate an accurate determination of Easter.
• The star in the east is associated to the birth of Jesus in Christianity.
• Muslims orient themselves towards Mecca when at prayer; we’ll see later how this direction
(the qibla) may be computed, but the required data is astronomical.
16
Chinese astronomy has 28 constellations (or mansions). As a point of comparison, Taurus corresponds roughly to the
Chinese ‘White Tiger of the West’ (Baihu, and similar terms in various East-Asian languages).
36
4.1 Astronomical Terminology and Early Measurement
Seasonal variation exists because the earth’s axis is tilted approximately 23.5° with respect to the eclip-
tic (sun-earth orbital plane). Summer, in a given hemisphere, is when the earth’s axis is tilted towards
the sun, resulting in more sunlight and longer days. Astronomically, the seasons are determined by
four dates:
Summer/Winter Solstice (c. 21
st
June/December) The north pole is maximally tilted towards/away
from the sun. Solstice comes from Latin (Roman) meaning ‘sun stationary.’ The location of
the rising/setting sun and its maximal (noon) elevation changes throughout the year, with
extremes on the solstices: the summer solstice being when the setting sun is most northerly
and (north of the tropics) the noon sun is highest in the sky. Indeed the tropics (of Can-
cer/Capricorn) are the lines of latitude where the sun is directly overhead at noon on one
solstice.
Vernal/Autumnal Equinox (c. 21
st
March/September) Earth’s axis is perpendicular to the sun-earth
orbital radius. Equinox means equal night: day and night both last approximately 12 hours
everywhere since Earth’s axis passes through the day-night boundary.
The picture shows the orientation of the ecliptic, the earth’s axis and the day-night boundary.
Measurements can now be conducted relative to this set-up.
Fixed stars These form the background with respect to which everything else is measured. The ecliptic
is the sun’s apparent path over the year set against the fixed stars. Planets (wandering stars) are
also seen to move relative to this background.
Celestial longitude Measured from zero to 360° around the ecliptic with 0° at the vernal equinox. One
degree corresponds approximately to the sun’s apparent daily motion. The ecliptic is divided
into twelve equal segments: Aries is 0–30° (March to April 21
st
); Taurus is 30–60°, etc.
Celestial latitude Measured in degrees north or south of the ecliptic; the sun has latitude zero.
This formulation was largely co-opted by the Greeks from Babylon. The Greeks kept the Babylonian
base-60 degrees-minutes-seconds system, which, with minor modifications, persists to this day.
17
17
In modern times latitude and longitude (declination/right-ascension) are measured with respect to the earth’s equa-
torial plane rather than the ecliptic. Such equatorial co-ordinates are first known to have been introduced by Hipparchus of
Nicaea (below). Right-ascension is measured in hours-minutes-seconds rather than degrees, with 24 hours = 360°, though
modern scientific practice is to use decimals rather than the sexagesimal minutes and seconds.
37
The Circumference of the Earth
Eratosthenes of Cyrene (c. 200 BC, pg. 34) performed one of the earliest accurate estimations of the
circumference of the earth. His idea was to measure the sun’s rays at noon in two different places.
• Syene (modern-day Aswan, Egypt) is approximately
5,000 stadia south of Alexandria.
• When the sun is directly overhead at Syene, it is in-
clined 7°12
′
=
1
50
·360° at Alexandria.
• Earth’s circumference is therefore approximately 50 ·
5, 000 = 250, 000 stadia.
rays
sun’s
A
S
Eratosthenes’ original calculation is lost, though it was a little more complicated than the above.
From other (shorter) distances, historians have inferred that Eratosthenes’ stadion was ≈ 172 yards,
making his approximation for the circumference of the earth ≈ 24, 500 miles, astonishingly accu-
rate in comparison to the modern value of ≈ 25, 000 miles. Later mathematicians provided other
estimates based on other locations, but the basic method was the same.
Modelling the Heavens
Early Greek analysis reflects several assumptions.
• Spheres and circles are perfect, matching the ‘perfect design’ of the universe. The earth is a
sphere and the fixed stars (constellations) lie on a larger ‘celestial sphere.’ Models relied on
spheres and circles rotating at constant rates.
• The earth is stationary, so the celestial sphere rotates around it once per day.
• The planets lie on concentric spherical shells also centered on the earth.
When such assumptions are tested by observation, two major contradictions are immediate:
Variable brightness The apparent brightness of heavenly bodies, particularly planets, is not constant.
Retrograde motion Planets mostly follow the east-west motion of the heavens, though are sometimes
seen to slow down and reverse course.
If planets are moving at constant speed around circles centered on the earth, then how can these
phenomena be explained? The attempt to produce accurate models while ‘saving the phenomena’ of
spherical/circular motion led to the development of new mathematics.
One of the earliest known models is due to Eudoxus of Knidos (c. 370 BC, pg. 21). Eudoxus developed
a concentric-sphere model where planets and the sun are attached to separate spheres, each of which
has its poles attached to the sphere outside it; the outermost sphere is that of the fixed stars. The
motion generated by such a model
18
is highly complex. Eudoxus’ approach is capable of producing
retrograde motion, but not the variable brightness of stars and planets.
18
The link is to a very nice flash animation of Eudoxus’ model that would have been far beyond Eudoxus’ ability to
visualize and measure.
38
Epicycles & Eccentric Orbits Apollonius of Perga (2
nd
/3
rd
C. BC) is most
famous for his study of conic sections, but is relevant in this section for
developing two models for solar/planetary motion.
In his eccenter model, a planetary/solar orbit is a circle (the deferent) whose
center is not the earth. This straightforwardly addresses the problem of
variable brightness since the planet is not a fixed distance from the earth.
The obvious criticism is why? What philosophical justification could there
be for the eccenter? Eudoxus’ model may have been complex and essen-
tially impossible to compute with, but was more in line with the assump-
tions of spherical/circular motion.
Eccenter
Earth
Planet/Sun
Apollonius’ second approach used epicycles: small circles attached
to a larger circle—you’ll be familiar with epicycles if you’ve played
with the toy Spirograph. An observer at the center sees the apparent
brightness change, and potentially observes retrograde motion. In
modern language, the motion is parametrized by the vector-valued
function
x(t) = R
cos ωt
sin ωt
+ r
cos ψt
sin ψt
where R, r, ω, ψ are the radii and frequencies (rad/s) of the circles.
Combining these models allowed Apollonius to describe very complex motion. Calculation was
difficult however, requiring finding lengths of chords of various circles from a given angle, and vice
versa. It is from this requirement that some of the earliest notions of trigonometry arise.
One might ask why the Greeks didn’t make the ‘obvious’ fix and place the sun at the center of the
cosmos. In fact Aristarchus of Samos (c. 310–230 BC) did precisely this, suggesting that the fixed stars
were really just other suns at exceptional distance! However, the great thinkers of the time (Plato,
Aristotle, etc.) had a very strong objection to Aristarchus’ approach: parallax.
distant stars
nearer star
Earth and Sun
θ
If the earth moves around the sun and the fixed stars are really independent objects, then the posi-
tion of a nearer star should appear to change throughout the year. The angle θ in the picture is the
parallax of the nearer star. Unfortunately for Aristarchus, the Greeks were incapable of observing
any parallax.
19
It took 2000 years before the work of Copernicus and Kepler in the 15-1600s forced
astronomers to take heliocentric models seriously (Helios is the Greek sun-god).
19
The astronomical unit of one parsec is the distance of a star exhibiting one arc-second (
1
3600
°) of parallax, roughly 3.3
light-years or 3 ×10
13
km, an unimaginable distance to anyone before the scientific revolution. The nearest star to the sun
is Proxima Centauri at 4.2 light years = 0.77 parsecs: is it any wonder the Greeks rejected the hypothesis?!
39
Hipparchus of Nicaea/Rhodes (c. 190–120 BC)
Born in Nicaea (northern Turkey) but doing much of his work on the Mediterranean island of Rhodes,
Hipparchus was one of the pre-eminent Greek astronomers. He made use of Babylonian eclipse data
to fit Apollonius’ eccenter and epicycle models to the observed motion of the moon. As part of this
work, he needed to be able to accurately compute chords of circles; his resulting chord tables are
acknowledged as the earliest tables of trigonometric values.
In an imitation of Hipparchus’ approach, we define a function
crd which returns the length of the chord in a given circle sub-
tended by a given angle. Translated to modern language,
crd α = 2r sin
α
2
α
r
crd α
crd(180
◦
−α)
Hipparchus chose a circle with circumference 360° (in fact he used 60 ·360 = 21600 arc-minutes), the
result being that r =
21600
2π
≈ 57, 18 (written base 60!). Note that this is sixty times the number of
degrees per radian.
20
His chord table was constructed starting with two obvious values:
crd 60° = r = 57, 18 crd 90° =
√
2r = 81, 2
Since (Thales) the large triangle is right-angled, the Pythagorean theorem was used to obtain chords
for angles 180° −α. In modern language
crd(180° −α) =
q
(2r)
2
−(crd α)
2
= 2r
q
1 −sin
2
( α/2) = 2r cos
α
2
Pythagoras was again used to halve and double angles in an approach analogous to Archimedes’
quadrature of the circle (pg. 33). We rewrite the argument in this language.
In the picture, we double the angle α; plainly M is the mid-
point of AD and
|
DB
|
= crd(180° −2α). Since ∠BDA = 90°,
it follows that BD is parallel to OM and so
|
OM
|
=
1
2
|
BD
|
=
1
2
crd(180° −2α)
Now apply Pythagoras to △CMD:
α
α
A
B
C
D
O
M
(
crd α
)
2
=
1
2
crd 2α
2
+
r −
1
2
crd(180° −2α)
2
(
|
CD
|
2
=
|
DM
|
2
+
|
CM
|
2
)
=
1
4
(crd 2α)
2
+ r
2
−r crd(180° −2α) +
1
4
crd(180° −2α)
2
=
1
4
(crd 2α)
2
+ r
2
−r crd(180° −2α) +
1
4
(4r
2
−(crd 2α)
2
)
= 2r
2
−r crd(180° −2α)
In modern notation this is one of the double-angle trigonometric identities!
4r
2
sin
2
α
2
= 2r
2
−2r
2
cos α ⇐⇒ cos α = 1 −2 sin
2
α
2
20
One radian is the angle subtended by an arc equal in length to the radius of a circle. Hipparchus essentially does this
in reverse; the circumference is fixed so that degree now measures both subtended angle and circumferential distance.
40
Example To calculate crd 30°, we start with crd 60° = r. Then
crd 120° =
p
4r
2
−r
2
=
√
3 r
=⇒ crd 30° =
q
2r
2
−r crd(180° −60°) =
q
2r
2
−
√
3r
2
=
q
2 −
√
3 r
In modern language this yields an exact value for sin 15°:
crd 30° = 2r sin 15° =⇒ sin 15° =
1
2
q
2 −
√
3
Continuing this process, we obtain crd 150° =
p
2 +
√
3 r, whence
crd 15°)
2
= 2r
2
−r crd 150° = (2 −
q
2 +
√
3) r =⇒ crd 15° =
r
2 −
q
2 +
√
3 r
Again translating: sin 7.5° =
1
2
q
2 −
p
2 +
√
3.
By applying this approach, Hipparchus computed the chord of each of the angles 7.5°, 15°, . . . , 172.5°,
in steps of 7.5°. Of course everything was an estimate since he had to rely on approximations for
square-roots. All Hipparchus’ original work is now lost. We primarily know of his work by reference.
In particular, the above method of chords is probably due to Hipparchus, although we see it first in
the work of Ptolemy, as we’ll consider next.
Exercises 4.1. 1. Calculate crd 150°, crd 165°, and crd 172
1
2
° using the method of Hipparchus.
(Leave your answers as a multiple of r = crd 60°)
2. Sirius, the brightest star in the sky, is 2.64 parsecs (8.6 light-years) from the sun. Use modern
trigonometry to find its parallax.
3. The tropic of cancer is the line of latitude (approximately) 23.5° north of the equator marking
the locations where the sun is directly overhead at noon on the summer solstice.
21
At the arctic
circle on the winter solstice, the sun is precisely on the horizon.
(a) Explain why the latitude of the arctic circle is 66.5° north.
(b) Find the angle the sun makes above the horizon at the arctic circle at noon on the summer
solstice.
4. Consider the epicycle model where the position vector of a planet is given by
x(t) = R
cos ωt
sin ωt
+ r
cos ψt
sin ψt
(a) Suppose R = 4 and r = 1, ω = 1 and ψ = 2, so that the epicycle rotates twice every orbit.
Sketch a picture of the full orbit.
(b) Suppose that ω, ψ are positive constants. Prove that an observer will see retrograde motion
if and only if rψ > Rω.
(Hint: differentiate x
′
( t) and think about its direction)
21
Syene (pg. 38) is almost exactly on the Tropic of Cancer.
41
4.2 Ptolemy’s Almagest
Born in Egypt and living much of his life in Alexandria, Claudius Ptolemy (c. AD 100–170) was a
Greek/Egyptian/Roman
22
astronomer and mathematician. Around AD 150, he produced the Math-
ematica Syntaxis, better known as the Almagest; the latter term is derived from the Arabic al-mageisti
(great work), reflecting its importance to later Islamic learning.
The Almagest is essentially a textbook on geocentric cosmology. It shows how to compute the motions
of the moon, sun and planets, describing lunar parallax, eclipses, the constellations, and elementary
spherical trigonometry, probably courtesy of Menelaus (c. AD 100). It contains our best evidence as
to the accomplishments of Hipparchus and describes his calculations. The text formed the basis of
Western/Islamic astronomical theory through the 1600s.
Ptolemy’s Calculations Ptolemy used several innovations to compute more chords and at a far
greater accuracy than Hipparchus.
Initial Data Ptolemy took r = 60 so that crd 60° = 60. He also had more initial data:
crd 90° = 60
√
2, crd 36° = 30(
√
5 −1), crd 72° = 30
q
10 −2
√
5
Halving/Doubling Angles Ptolemy used what was probably Hipparchus’ method:
crd
2
α = 2r
2
−r crd(180° −2α) = 60
120 −crd(180° − 2α)
crd(180° −α) =
q
(2r)
2
−crd
2
α =
q
120
2
−crd
2
α
with square-roots approximated to the desired accuracy. For example,
crd 30° =
q
60( 120 −crd 120°) =
r
60
120 −60
√
3
= 60
q
2 −
√
3 ≈ 31; 3, 30
Multiple-Angle Formula Ptolemy computed crd 12° = crd(72° − 60°) , then halved this for angles of
6°, 3°, 1.5°, and 0.75°. Chords for all integer multiples of 1.5° were computed using addition
formula.
Interpolation The observation α < β =⇒
crd β
crd α
<
β
α
allowed Ptolemy to compute chords for every
half-degree to the incredible accuracy of two sexagesimal places. For approximating between
half-degrees, his table indicated how much should be added for each arc-minute (
1
60
°). For
example, the second line of Ptolemy’s table reads
1° 1; 2, 50 ; 1, 2, 50
The first two columns state that crd 1° = 1; 2, 50 to two sexagesimal places.
23
The third entry
says, for example, that
crd 1°5
′
≈ 1; 2, 50 + 5(; 1, 2, 50) = 1; 8, 4, 10 ≈ 1; 8; 4
To obtain these arc-minute approximations, it is believed Ptolemy computed half-angle chords
to an accuracy of five sexagesimal places (1 part in over 750 million!). The construction of the
chord-table was truly a gargantuan task, one for which Ptolemy almost certainly had assistance.
22
Ptolemy (Ptolemaeus) is a Greek name, while Claudius is Roman, reflecting the changing cultural situation in Egypt.
23
This is 1 +
2
60
+
50
60
2
= 1.0472222 . . . = 120 sin
1.00003625...
2
°, an already phenomenal level of accuracy.
42
How did Ptolemy know the values of crd 36° and crd 72°? Everything necessary is in the Elements.
Theorem. 1. (Thm XIII. 9) In a circle, the sides of a regular inscribed hexagon and decagon are in
the golden ratio (this ratio is 60 : crd 36° in Ptolemy).
2. (Thm XIII. 10) In a circle, the square on an inscribed pentagon equals the sum of the squares on
an inscribed hexagon and decagon.
Purely Euclidean proofs are too difficult for us, so here is a way to see things in modern notation.
1. Let AB = x be the side of a regular decagon in-
scribed in a unit circle with center O.
△OAB is isosceles with angles 36°, 72°, 72°.
Let C lie on OB such that AC = x.
Count angles to see that △OAB and △ABC are
similar, that ∠OAC = 36° and so OC = x.
Similarity now tells us that
x =
1 − x
x
=⇒ x =
√
5 −1
2
In a circle of radius 60, this gives the exact value
crd 36° = 60x = 30(
√
5 −1)
1
36
◦
O
A
B
C
D
x
y
2. Now let AD = y be the side of a regular pentagon inscribed in the same circle. Applying
Pythagoras, we see that
y
2
2
+
1 − x
2
2
= x
2
Since x
2
= 1 − x, this multiplies out to give Euclid’s result
y
2
= 1
2
+ x
2
from which we obtain the exact value
crd 72° = 60y = 30
q
10 −2
√
5
While these values were geometrically precise, Ptolemy used sexagesimal approximations to square-
roots to obtain
crd 36° = 37; 4, 55 crd 72° = 70; 32, 3
While these are the values stated in his tables, he must have used a far higher degree of accuracy in
order to obtain similarly accurate values for other chords.
43
Angle-addition Computation of crd(α ± β) was facilitated by versions of the multiple-angle for-
mulæ of modern trigonometry.
Theorem (Ptolemy’s Theorem). Suppose a quadrilateral is inscribed in a circle. Then the product
of the diagonals equals the sum of the products of the opposite sides.
24
Proof. Choose E on AC such that ∠ABE
∼
=
∠DBC. Then ∠ABD
∼
=
∠EBC. Since ∠BAE
∼
=
∠BDC are
inscribed angles of the same arc BC, we obtain two pairs of similar triangles:
A
B
C
D
E
A
B
C
D
E
△ABE ∼ △DBC △ABD ∼ △EBC
The proof follows immediately: since
|
AE
|
|
CD
|
=
|
AB
|
|
BD
|
and
|
CE
|
|
AD
|
=
|
BC
|
|
BD
|
, we have
|
AC
||
BD
|
= (
|
AE
|
+
|
CE
|
)
|
BD
|
=
|
AB
||
CD
|
+
|
AD
||
BC
|
Corollary. If α > β, then
120 crd(α − β) = crd α crd(180° − β) −crd β crd(180° − α)
In modern language, divide out by 120
2
to obtain
sin
α − β
2
= sin
α
2
cos
β
2
−sin
β
2
cos
α
2
A
B
C
D
O
β
α
Proof. If
|
AD
|
= 120 is a diameter of the pictured circle, then Ptolemy’s Theorem says
crd α crd(180° − β) = crd β crd(180° − α) + 120 crd(α − β)
Similar expressions for crd(α + β) and crd(180° −( α ± β)) were also obtained, essentially recovering
all versions of the multiple-angle formulæ for sin(α ± β) and cos(α ± β).
24
It is generally considered that this result predates Ptolemy, though there is some debate as to whether it belongs in
the Elements. Book VI traditionally contains 33 propositions, however some editions append four corollaries, of which
Ptolemy’s Theorem is the last (Thm VI. D).
44
Examples 1. Here is how Ptolemy might have calculated crd 42°. Let α = 72° and β = 30°, then
120 crd 42° = crd 72° crd 150° −crd 30° crd 108°
Since crd 72° = 30
p
10 −2
√
5 is known, and
crd 108° = crd(180° − 72°) =
q
120
2
−crd
2
72° = 30
q
6 + 2
√
5
we see that
crd 42° =
1
120
30
q
10 −2
√
5 ·60
q
2 +
√
3 −60
q
2 −
√
3 ·30
q
6 + 2
√
5
= 15
q
10 −2
√
5 ·
q
2 +
√
3 −
q
2 −
√
3 ·
q
6 + 2
√
5
≈ 43; 0, 15 ≈ 43.0042
Note all the square-roots which had to be approximated!
2. The Almagest also contained many practical examples. Here is one such.
A stick of length 1 is placed in the ground. The angle of ele-
vation of the sun is 72°. What is the length of its shadow?
Ptolemy says to draw a picture. The lower isosceles triangle
has base angles 72°, and the length of the shadow is ℓ. The
ratio of the chords is then computed:
1 : ℓ = crd 144° : crd 36°
=⇒ ℓ =
crd 36°
crd 144°
=
30(
√
5 −1)
30
p
10 + 2
√
5
≈ 0.32491
This is precisely cot 72°, though Ptolemy had no such notion.
36
◦
1
ℓ
Exercises 4.2. 1. What are the exact values of sin 36° and sin 18°?
2. (a) Restate the interpolation formula α < β =⇒
crd β
crd α
<
β
α
in terms of the sine function. What
facts about
sin x
x
does this reflect?
(b) Find crd 57
′
(arc-minutes!) to two sexagesimal places.
3. Find the exact value of crd 54°
4. Prove the following using Ptolemy’s Theorem. What is this in modern language?
120 crd
180° −(α + β)
= crd(180° − α) crd(180° − β) −crd α crd β
5. Use Ptolemy’s Theorem to establish a version of the double-angle formula: sin α = 2 sin
α
2
cos
α
2
.
(Hint: draw a symmetric quadrilateral one of whose diagonals is a diameter)
6. Calculate the length of a noon shadow of a pole of length 60 using Ptolemy’s methods:
(a) On the vernal equinox at latitude 40°.
(b) At latitude 36° north on both the summer and winter solstices.
(Hint: recall Exercise 4.1.3)
45
