6 Indian and Islamic Mathematics
6.1 India, the Hindu–Arabic Numerals & Zero
The Indian/South Asian subcontinent is bordered to the north by
the Himalayan mountains and to the east by dense jungle. Its pri-
mary historical frontier comprised the fertile Indus valley to the
west, now the central corridor of Pakistan, where recorded civi-
lization dates to at least 2500 BC. During the first millennium BC,
Hinduism developed as an amalgamation of previous practices
and beliefs; Buddhism and Jainism began to spread in the later
part of this period, particularly in the Ganges valley further east.
Alexander the Great’s conquests reached the Indus in 326 BC,
bringing Greek, Babylonian and Egyptian knowledge in his wake.
The Greek overlords he left behind were rapidly overthrown and
the subcontinent became largely unified under the Mauryan Empire for the next 150 years. After this
came 1000 years of shifting control with several invasions from the west by the Persians. Islam con-
quered the Indus around AD 1000, with most of India becoming part of the Islamic Mughal Empire
by the 1500s; after the Mughal decline and fragmentation, the British became dominant in 1857.
The modern political situation reflects this complicated history. India gained independence from
Britain in 1947 after World War II and was shortly thereafter partitioned according to religion: the
greater Indus valley and the lower Ganges/Brahmaputra comprise the modern Islamic states of Pak-
istan and Bangladesh, with the majority of the landmass becoming the nominally secular but majority
Hindu country of India. The upper Indus valley (Kashmir) remains contested and has been the site
of several military conflicts between India, Pakistan and China.
Ancient India’s contributions to world knowledge and development are significant; it is estimated
that India accounted for 25–30% of the world’s economy during the 1
st
millenium AD! It was more-
over a technological and cultural crossroads between East (China) and West (Greece, Persia, Rome,
etc.); while some trade and knowledge passed north of the Himalayas directly between China and
the Middle East/Europe, far more percolated slowly through India, being improved upon and given
back in turn.
Brahmi Numerals & Numerical Naming Our primary focus is on possibly the most important
practical mathematical development in history: the decimal positional system of enumeration, com-
plete with fully-functional zero. The Brahmi numerals, one of the earliest antecedents of modern
numerals, first appeared around the 3
rd
century BC.
1 2 3 4 5 6 7 8 9 10
The example dates from around 100 BC and was used in Mumbai/Bombay. Additional symbols
denoted multiples of 10, 100, 1000, 10000, etc. As with Chinese characters, the system was partly
positional (800 would be written by prefixing the symbol for 100 by that for 8) and there was no
symbol or placeholder for zero.
54
Symbols are only part of the story. The modern approach to naming numbers and constructing large
numbers can also be linked to the same period. The table below gives old Sanskrit names.
1 2 3 4 5 6 7 8 9
eka dvi tri catur pancha sat sapta asta nava
10 20 30 40 50 60 70 80 90
dasa vimsati trimsati catvarimsat panchasat sasti saptati asiti navati
100 1000 10000 100000 1000000 10
7
10
8
10
9
10
10
sata sahasra ayuta niyuta prayuta arbuda nyarbuda samudra madhya
Many European languages have Sanskrit roots; it should be no surprise that several ancient Sanskrit
numbers are similar (e.g., dva in Russian, quatre in French). The construction of larger numbers should
also seem familiar: for example tri sahasra sat sata panchasat nava is precisely how we read 3659.
Such familiarity has its limits, for old Sanskrit verbiage doesn’t map perfectly onto modern English.
For instance, old Sanskrit had distinct words for powers of 10 up to (at least!) 10
62
, and employed a
version of pre-subtraction: e.g., ekanna-niyuta meant ‘one less than 100000,’ or 99999.
Gwalior Numerals During the first few centuries AD, a fully positional decimal place system came
into being. The earliest evidence comes from a manuscript found in Bakhsh
¯
al
¯
ı (Pakistan) in 1881,
which has been carbon-dated to the 3
rd
or 4
th
century. The manuscript contains the earliest known
version of the modern symbol for zero, a circular dot. It is conjectured that the decimal place sys-
tem was inspired by the Chinese counting-board method, though convincing proof has yet to be
uncovered. Regardless of attribution, Chinese mathematicians were copying the method by the 8
th
century.
The examples below are better understood than the Bakhsh
¯
al
¯
ı manuscript and come from Gwalior
(northern India) around 876.
0 1 2 3 4 5 6 7 8 9 10
The similarity with modern numerals is clear; 0, 1, 2, 3, 4, 7, 9, 10 are very familiar. Zero has evolved
from the Bakhsh
¯
al
¯
ı dot to a hollow circle. The symbols for 2 and 3 are conjectured to have devel-
oped in an attempt to write earlier versions (e.g. the Brahmi numerals) cursively; try writing three
horizontal strokes quickly. . .
The system is fully positional. Below are the numbers 270 and 30984:
Sanskrit is written left-to-right, with the leftmost digits representing the largest powers of 10. Note
how zero is used as a placeholder to clarify position so that, e.g., 27, 207, and 270 are clearly distin-
guishable.
55
Zero On the right is a table of modern Sanskrit names and numerals; the digits and names are
certainly similar to their Gwalior counterparts.
The Sanskrit shuuny´a means void or emptiness. It is related
to svi (hollow), which in turn derives from an ancient word
meaning to grow. This reflects a major idea within religions
of the area, with the void being the source of all things, of
creation and creativity. Contemplation of the void (the doc-
trine of Shunyata) is recommended before composing mu-
sic, creating art, etc. This contrasts with the Abrahamic re-
ligions where the void is something to be feared; an early
conception of hell was the eternal absence of God.
The Gwalior numerals travelled westwards, with Europe eventually inheriting the system via Islam;
as such they are today known the Hindu–Arabic numerals. Here is a short version of the etymological
journey of zero into European languages.
• Shunya was transliterated to sifr in Arabic where the double-meaning persisted: al-sifr was the
number zero, while safira meant it was empty.
• The term came to Europe in the 12
th
-13
th
centuries courtesy of Fibonacci where it became cifra.
This was blended with zephyrum (west wind/zephyr) providing an alternate spelling.
• Cifra ultimately became the words cipher (English), chiffre (French) and ziffer (German), meaning
a figure, digit, or code.
• Zephyrum became zefiro in Italian and zero in Venetian.
Zero and the Hindu–Arabic numerals also travelled eastwards, with Qin Jiushao introducing the
zero symbol into China in the 13
th
century.
Our modern understanding of zero is a fusion of several concepts:
Numerical positioning For instance, to distinguish 101 from 11.
Absence of a quantity 101 contains no 10’s.
Symbol First a dot (bindu), then a circle (chidra/randhra meaning hole). The relationship between
shunya and a symbol was established by AD 2-300, as this quote from AD 400 (Vasavadatta)
illustrates
The stars shone forth, like zero dots [shunya-bindu] scattered as if on a blue rug. The
Creator reckoned the total with a bit of the moon for chalk.
Mathematical operations By the time of Brahmagupta (7
th
C.), a mathematical text might contain a sec-
tion called shunya-gania, with computations involving zero, including addition, multiplication,
subtraction, effects on ±-signs, division and the relationship with ∞ (ananta). In the 12
th
C.,
Bhaskaracharya stated:
If you were to divide by zero you would get a number that was “as infinite as the god
Vishnu.”
56
Other ancient cultures had one or more of these aspects of zero, but the Indians were the first to put
them all together.
• The Egyptian hieroglyph nfr (beautiful/complete) indicated zero remainder in calculations as
early as 1700 BC and was also used as a reference point/level in buildings.
• Very late in Babylonian times, a placeholder symbol was used to separate powers of 60. It was
not used as a number.
• With the Chinese counting board, an empty space served as a placeholder.
• Various Mesoamerican cultures, such as the Maya, had a zero symbol that was used as a place-
holder, particularly when writing dates.
‘Real’ Indian Mathematics
Indian mathematicians made great progress on several fronts, not merely the decimal place system.
Much ancient work was influenced by religion. For in-
stance, the pre-Hindu sulbasutras contained instructions
for laying out altars using ruler-and-compass construc-
tions. These could be quite complex, as the construction
of the base of the Mahavedi (great altar) shows: The cen-
ter line is divided left-to-right in the ratio
1 : 7 : 12 : 11 : 5
and the altar contains five distinct Pythagorean triples!
30 pada 24 pada
36 pada
Of particular importance to our continuing narrative is Indian work on trigonometry. Here are some
highlights:
• The early 5
th
C. text Pait¯amahasiddh¯anta is assumed to be an extension of Hipparchus’ work,
since it contains a table of chords based on a circle of radius 57,18; rather than Ptolemy’s 60.
• Indian mathematicians instituted the use of half-chords, in line with our modern understanding
of sine. Indeed the word sine is the result of a long sequence of (mis)translations and transliter-
ations via Arabic and Latin from the Sanskrit jy¯a-ardha (chord-half).
30
The Indians also began to
distinguish ‘base sine’ and ‘perpendicular sine’ (cosine).
• Created tables of sines/half-chords from 0° to 90° in steps of 3
3
4
°, using linear interpolation to
approximate values in between. By 650, Bhramagupta had much better approximations, using
quadratic polynomials to interpolate. By 1530, Indian mathematicians had discovered cubic
and higher approximations (essentially Taylor polynomials 130 years before Newton) for even
greater accuracy of sine, cosine and arctangent.
Navigation was one of the drivers of this development. While Mediterranean sailors rarely strayed
long out of sight of land, the Indians sailed the ocean and required accurate measurements to find
their latitude.
30
This is also the root of the word sinus meaning bay or gulf (e.g., in your nose).
57
Exercises 6.1. 1. The Mahavedi (pg. 57) contains five Pythagorean triples; find them.
2. To simplify square root expressions, Bhaskara used the formula
q
a +
√
b =
r
1
2
a +
p
a
2
−b
+
r
1
2
a −
p
a
2
−b
Prove Bhaskara’s formula and use it to simplify
p
2 +
√
3.
3. Here is an Indian method for ‘finding’ a circle whose area is equal to a given square.
In a square ABCD, let M be the intersection of the diago-
nals. Draw a circle with M as the center and MA the radius;
let ME be the radius of the circle perpendicular to the side
AD and cutting AD at G. Let GN =
1
3
GE. Then MN is the
radius of the desired circle.
Show that if AB = s and MN = r, then
r
s
=
2 +
√
2
6
Show that this implies a value for π equal to 3.088311755.
A
B
C
D
M
E
G
N
4. Solve the following problem of Mah
¯
av
¯
ıra.
Of a collection of mango fruits, the king took 1/6; the queen took 1/5 of the remain-
der, and the three chief princes took 1/4, 1/3, 1/2 of what remained at each step. The
youngest child took the remaining three mangoes. O you, who are clever in work-
ing miscellaneous problems on fractions, give out the measure of that collection of
mangoes.
58
6.2 Islamic Mathematics I: Algebra
Muhammad ibn Abdullah was born in Mecca (modern Saudi Arabia) in 570. Around 610 he began
preaching Islam (submission to the will of God)—the third of the major Abrahamic religions, (chrono-
logically) following Judaism and Christianity. After several years of exile, he returned with an army,
conquering Mecca a few years before his death in 632.
Through military conquest, Muhammad’s successors expanded the caliphate (empire) at a truly re-
markable speed. At the time of his death, the Arabian peninsula was Islamic. By 660 Islam had
reached Libya and most of Persia, and by 750 extended from Iberia & Morocco to Afganistan & Pak-
istan. Serious schisms eventually arose
31
and several successor empires emerged, the longest-lasting
of which was the Ottoman Empire (c. 1300–1922). Even though centralized political control ended
long ago, Islam remains dominant in the region pictured below (with the notable exceptions of Spain
and Portugal) and over a greater region of Africa and south-east Asia (e.g., Indonesia).
As with the Romans, early Muslims permitted conquered peoples—including Jews and Christians
(people of the book)—to maintain their culture, provided they acknowledged their overlords and paid
taxes. Those who converted to Islam were welcomed as full citizens, though deconversion (apostasy)
was not tolerated. Many of the great Islamic thinkers were born on the periphery and travelled to
the great centers of learning, particularly Baghdad during the Islamic golden age (8
th
–13
th
centuries).
Knowledge was also absorbed from Alexandria and western India (Pakistan). In the mid-700s paper-
making came from China, greatly facilitating the dissemination and consolidation of knowledge.
Schools (madrassas) reflected a strong cultural and religious focus on learning.
The Islamic golden age overlapped the European dark ages (c. 500–1200) following the fall of Rome,
during which European philosophical development stagnated. By 1200, the crusades
32
were well
underway and Islam had come to be seen as the enemy of Christian Europe. The infusion of knowl-
edge that came to Europe from Islam around this time helped spur the European renaissance & later
scientific revolution. Among European scholars almost to the present day, it was fashionable to credit
Islam merely with the preservation of ancient ‘European’ knowledge; a claim both fanciful and chau-
vinistic, but plainly stemming from medieval animosity.
31
In particular between the Sunni and Shia branches of the faith. Much of the modern-day tension between Saudi Arabia
and Iran stems from this rupture.
32
A series of religious–military campaigns 1096–1291 with the goal of wresting control of the Holy Land, particularly
Jerusalem, from Islam.
59
Algebra & Algorithms
Proof and axiomatics were learned from Greek texts such as the Elements. Like the Greeks, Islamic
scholars gave primacy to geometry and proved algebraic relations in a geometric manner.
33
Practical
and accurate calculation was more important than to the Greeks, and great advances were made in
this area. This included completing the development of the Indian decimal place system (hence the
dual credit Hindu–Arabic numerals).
The second most obvious legacy of Islamic mathematics is encountered daily in every mathematics
classroom. Algebra
34
comes from the Arabic al-˘gabr, meaning restoring. It originally referred to mov-
ing a deficient (negative) quantity from one side of an equation to another. A second term al-muqabala
(comparing/balancing) meant to subtract the same positive quantity from both sides of an equation.
Al-˘gabr: x
2
+ 7x = 4 −2x
2
=⇒ 3x
2
+ 7x = 4
Al-muq¯abala: x
2
+ 7x = 4 + 5x =⇒ x
2
+ 2x = 4
Islamic scholars did not use symbols or equations in a modern sense; statements were instead written
out in sentences.
Muhammad ibn M ¯us¯a al-Khw¯arizm¯ı (780–850) Born near the Aral Sea in modern Uzbekistan, al-
Khw
¯
arizm
¯
ı eventually became chief librarian at the great school of learning, the House of Wisdom, in
Baghdad. His Compendious book on the calculation by restoring and balancing
35
(820) is a synthesis of
Babylonian methods and Euclidean axiomatics; an algorithm demonstrated a solution, followed by
a geometric proof. After being translated into Latin in the 1100s it became a standard textbook of Eu-
ropean mathematics, displacing Euclid in places due to its greater emphasis on practical calculation.
The word algorithm reflects its importance: the Latin dixit algorismi literally means al-Kw¯arizm¯ı says.
Example. Here is al-Khw
¯
arizm
¯
ı’s approach to the equation x
2
+ 4x = 60, or, more properly:
What must be the square which, when increased by four of its roots, amounts to sixty?
The algorithm may be applied to any equation of the form x
2
+ ax = b
where a, b > 0: here a is the number of ‘roots,’ and b the total ‘amount.’
• Halve the number of roots
2 =
1
2
a
• Multiply by itself
4 =
1
4
a
2
• Add to the total amount
64 =
1
4
a
2
+ b
• Take the root of this
8 =
q
1
4
a
2
+ b
• Subtract half the number of roots
6 =
q
1
4
a
2
+ b −
a
2
x
2
x
2
Al-Kw
¯
arizm
¯
ı essentially constructs the quadratic formula =
−a+
√
a
2
+4b
2
, while the pictorial justifica-
tion is Euclid’s (Elements, Thm II. 4). The geometry should be obvious: the original square (x
2
) has
been increased by four of its roots; the algorithm is simply ‘completing the square’ (x + 2)
2
= 64.
33
Like Book II of the Elements. Such Greek texts were venerated by Islamic scholars; recognizing the depth of Ptolemy’s
work on astronomy and trigonometry, they bestowed the name by which it is now known, the Almagest (Great Work).
34
Many words beginning al- are of Arabic origin (alkali, albatross, etc.), as are others that have been latinized (elixir).
35
Al-kit¯ab al-mukhtasar f¯ı his¯ab al-˘gabr wa’l-muq¯abala.
60
Other algorithms were supplied to solve every type of quadratic.
It is hard to notice from our example, but the crucial development from a math-history point of view
is the abstraction, in a modern sense the algebra; al-Khw
¯
arizm
¯
ı’s approach applies equally to numbers
as it does to geometric objects, a very different approach to the geometry-focused Greeks.
As an example of the power of this idea, consider how Ab
¯
u K
¯
amil (Egypt 850–930) generalized Eu-
clid’s Book II geometric-algebra arguments to permit substitution, provided the resulting equation
was quadratic.
If y =
1 + x
3 + x
and y
2
+ y = 1 then x =
√
5
Ab
¯
u K
¯
amil essentially substitutes y =
1+x
3+x
into the quadratic (with solution y =
√
5−1
2
). While al-
Khw
¯
arizm
¯
ı’s methods were geometrically justified, when combined in this fashion the entire pro-
cess no-longer admits a straightforward geometric interpretation. This method of substitution was
an early step towards establishing the modern primacy of algebra and number over geometry and
length.
Over the following centuries, this algebraic approach was further improved. In particular, Omar
Khayyam (1048–1131) produced ground-breaking work on cubic equations, astronomy, the binomial
theorem, and irrational numbers.
Exercises 6.2. 1. Solve the equations
1
2
x
2
+ 5x = 28 and 2x
2
+ 10x = 48 using al-Khw
¯
arizm
¯
ı’s
methods (first multiply or divide by 2).
2. Al-Khw
¯
arizm
¯
ı gives the following algorithm for solving the equation bx + c = x
2
.
• Halve the number of roots.
• Multiply this by itself.
• Add this square to the number.
• Extract the square root.
• Add this to half the roots.
Translate this into a formula. Give a geometric argument
for the validity of the approach using the picture: HC
has length b where G is the midpoint; rectangle ABRH
has area c; KHGT and AMLG are squares; and the large
square ABDC has side-length x.
AB
C
D
G
H
M
L
R
N
K
T
3. Solve the following problems by Ab
¯
u K
¯
amil (use modern algebra!).
(a) Suppose 10 is divided into two parts and the product of one part by itself equals the
product of the other part by the square root of 10. Find the parts.
(b) Suppose 10 is divided into two parts, each of which is divided by the other, and the sum
of the quotients equals the square-root of 5. Find the parts.
61
6.3 Islamic Mathematics II: Spherical Trigonometry and the Qibla
Late 8
th
century Indian work on trigonometry, linking back to Hipparchus, was known in Baghdad,
as was the work of Ptolemy. Islamic scholars were interested in trigonometry for reasons beyond
mere astronomy. A primary requirement in Islam is to face the Ka’aba in the Great Mosque at Mecca
when at prayer: this is the qibla (direction in Arabic). A mosque is typically built so that one wall
faces Mecca for convenience; if not possible, an arrow indicating the qibla might be placed in an
alcove. In Muhammad’s time (when Muslims faced Jerusalem not Mecca), determining the qibla
was relatively easy, though as Islam spread the curvature of the earth made determination more
difficult. The religious impetus behind this problem motivated Islamic mathematics for centuries,
and the methods developed (with minor modifications) are still used today, though in modern times
the mathematics is very much hidden behind GPS technology!
Terminology and Trigonometric Tables Scholars worked with the Indian half-chord (sine), and with
circles of various radii. Al-Batt
¯
an
¯
ı (c. 858–929) introduced an early version of cosine as the complemen-
tary half-chord for angles less than 90°, and an analogue of the modern function versine:
36
versin θ = 1 −cos θ
Al-B
¯
ır
¯
un
¯
ı (973–1048) defined versions of tangent, cotangent, secant and cosecant by projecting (e.g.,
a sundial) onto either a horizontal or a vertical plane. In the second picture below, the gnomon is the
vertical stick of length 1. With this definition, al-B
¯
ır
¯
un
¯
ı moves towards the modern consideration of
trigonometry in terms of triangles rather than circles.
sin α
1
cos α
versin α
α
α
crd 2α
1
tan α = cot β
sec α = csc β
β
α
Trigonometric tables with improved accuracy over Ptolemy were created for all these ‘functions.’
Ab
¯
u al-Waf
¯
a (940–998) and his descendants computed sine & tangent values for every minute of arc
accurate to five sexagesimal places (one part in 777 million!) via repeated applications of the half-angle
formula and interpolating using the downwards concavity of the sine function (draw a picture!):
sin(α + β) −sin α < sin α −sin(α − β) whenever 0° < α − β < α + β < 90°
36
Versed sine refers to the measurement of a length in a reversed direction (perpendicular) to that of sine.
62
Calculating the Qibla In what follows we observe several conventions:
• A single letter A refers to a point or to the angle measure in a triangle with vertex A.
•
⌢
AB means the great-circle arc joining points A, B or its arc-length. A spherical triangle △ABC
comprises three points on a sphere joined by great-circle arcs.
• AB means the straight line joining A, B with length
|
AB
|
.
• All results are modernized and applied to a unit sphere (center O). The arc-length along a
great-circle therefore equals the central angle subtended by that arc in radians:
⌢
AB= ∡AOB.
3D movable versions of all pictures are online—click them!
Ptolemy and the Indians had already done some relevant work, though
Ptolemy’s approach relies heavily on Menelaus’ Theorem (c. AD 100).
Theorem (Menelaus). For the pictured configuration of spherical triangles
on a sphere of radius 1,
sin
⌢
CE
sin
⌢
AE
=
sin
⌢
CF
sin
⌢
DF
·
sin
⌢
BD
sin
⌢
AB
Applying Menelaus is difficult since one typically needs to create many new
spherical triangles. Al-Waf
¯
a simplified matters with an alternative result.
Theorem (Al-Waf¯a). If △ABC and △ADE are spherical triangles with right angles at B, D and a
common acute angle at A, then
sin
⌢
BC
sin
⌢
AC
=
sin
⌢
DE
sin
⌢
AE
In fact these ratios equal sin α where α is the acute angle, though al-Waf
¯
a didn’t say this.
Proof. Let O be the center of the sphere. Project C orthogonally to the plane containing O, A, B to
produce K, then project K to OA to get L.
Consider the right-angled planar triangle CKL. Since α
is the angle between two planes, we have α = ∡CLK.
Moreover
|
CK
|
= sin ∡COK = sin ∡COB = sin
⌢
BC
|
CL
|
= sin ∡COL = sin ∡COA = sin
⌢
AC
The usual sine formula for plane triangles says
sin α =
|
CK
|
|
CL
|
=
sin
⌢
BC
sin
⌢
AC
The same ratio is obtained for △ADE.
63
By dropping a perpendicular in a spherical triangle, Al-Waf
¯
a’s result quickly yields the spherical sine
rule. For the pictured triangle, drop the perpendicular to H ∈
⌢
AB from C. Al-Waf
¯
a says
sin B =
sin h
sin a
and sin A =
sin h
sin b
By equating the sin h terms and permuting symmetrically, we’ve
proved:
Corollary (Sine rule). If a, b, c are the side-lengths of a spherical
triangle with angles A, B, C, then
sin a
sin A
=
sin b
sin B
=
sin c
sin C
Al-Waf
¯
a’s proof was a little more complicated. He extended
⌢
AB
and
⌢
BC to quarter-circles resulting in a spherical triangle with
right-angles at D and E. Since
⌢
DE is an arc with central angle
B, we have
⌢
DE= B. Since
⌢
BD= 90°, Al-Waf
¯
a’s theorem implies
sin h
sin a
=
sin B
sin 90°
=⇒ sin h = sin a sin B
Mirroring this by extending
⌢
AB past B and equating the sin h
terms yields the result.
Using this approach, al-Waf
¯
a could solve spherical triangles and
thus compute the qibla. As with his sine rule argument, his method
required several auxiliary triangles and is difficult to follow.
Al-B
¯
ır
¯
un
¯
ı further simplified matters by developing what is essen-
tially the cosine rule. We apply his method to find the qibla from a
location L (remember: our sphere (Earth!) has radius 1).
Let M be Mecca and N the north pole. The qibla is β, the initial
bearing from L to M. Our (known) initial data are the latitudes
and longitudes of L, M, specifically:
• α is the difference in the longitudes.
• b, c are the colatitudes
37
of M, L respectively.
The cosine rule follows from Ptolemy’s Theorem (pg. 44). Extend
⌢
NL to Q with the same latitude as
M. Similarly let P ∈
⌢
NM have the same latitude as L. By symmetry, L, P, Q, M are coplanar, whence
the quadrilateral □LPQM lies on the intersection of a plane and a sphere: a circle! Measured as
straight lines (chords) and using symmetry (
|
PQ
|
=
|
LM
|
and
|
LQ
|
=
|
PM
|
), Ptolemy says
|
LM
||
PQ
|
=
|
LQ
||
PM
|
+
|
LP
||
QM
|
=⇒
|
LM
|
2
=
|
LQ
|
2
+
|
LP
||
QM
|
37
Colatitude (equals 90° minus latitude) is measured southwards from the north pole. Since our model sphere has radius
1, the arc-lengths b, c equal the colatitudes in radians.
64
The great-circle arc-lengths on the sphere may be found from straight-line distances via the usual
chord relations: e.g.,
|
LM
|
= crd
⌢
LM= 2 sin
⌢
LM
2
Ptolemy’s theorem now becomes a relation between arc-lengths
sin
2
⌢
LM
2
= sin
2
b − c
2
+ sin
⌢
LP
2
sin
⌢
QM
2
By bisecting α we obtain two pairs of right-triangles; al-Waf
¯
a’s the-
orem tells us that
sin
α
2
=
sin
⌢
LP
2
sin c
=
sin
⌢
QM
2
sin b
=⇒ sin
2
⌢
LM
2
= sin
2
b − c
2
+ sin
2
α
2
sin c sin b (∗)
To complete the proof we apply the multiple-angle formulæ
sin
2
x
2
=
1
2
(1 −cos x) cos(b −c) = cos b cos c + sin b sin c
Corollary (Cosine rule). In a spherical triangle with sides a, b, c
and angle α opposite a, we have
cos a = cos b cos c + sin b sin c cos α
For our triangle of interest a =
⌢
LM. Given points L, M (and thus b, c, α), one uses the cosine rule to
compute a and then the sine rule to find the qibla β (whew!):
sin b
sin β
=
sin a
sin α
=⇒ sin β =
sin α sin b
sin a
Example. For fun, here is some real-world data. Mecca and London have, respectively, co-ordinates
21°25’ N, 39°49’ E and 51°30’ N, 8’ W. This corresponds to
α = 39°57
′
, b = 68°35
′
, c = 38°30
′
By al-B
¯
ır
¯
un
¯
ı’s cosine rule,
cos a = cos 68°35
′
cos 38°30
′
+ sin 68°35
′
sin 38°30
′
cos 39°57
′
=⇒ a = 43.110°
Since Earth’s circumference is 24900 miles, the distance London → Mecca is
43.110×24900
360
= 2981 miles.
Al-Waf
¯
a’s sine rule computes the qibla
β = 180° −sin
−1
sin α sin b
sin a
= 118°59
′
where we subtracted from 180° since the relevant angle is plainly obtuse. Check it yourself at the
Great Circle Mapper (the website uses airports for slightly different initial data).
65
Spherical Trigonometry Cheat Sheet
Let △ABC be a spherical triangle with side-lengths a, b, c on a sphere of radius 1.
Basic trigonometry. If △ABC is right-angled at C
sin A =
sin a
sin c
cos A =
tan b
tan c
tan A =
tan a
sin b
Al-Waf
¯
a essentially proved the first; the others follow from trig identities (cos
2
A = 1 −sin
2
A . . .)
Sine rule (Al-Waf
¯
a)
sin A
sin a
=
sin B
sin b
=
sin C
sin c
Cosine rule (Al-B
¯
ır
¯
un
¯
ı)
cos c = cos a cos b + sin a sin b cos C
The spherical Pythagorean Theorem is the special case cos c = cos a cos b (C = 90°).
If the sphere has radius r, simply divide all lengths by r before applying the results; e.g.,
sin A =
sin(a/r)
sin(c/r)
As r → ∞, we have sin
a
r
≈
a
r
and cos
a
r
≈ 1 −
a
2
2r
2
, which recover the flat (Euclidean geometry)
versions of these statements.
Examples. 1. On a sphere of radius 1, an equilateral triangle has side length
π
3
. Splitting it in half
creates two right-triangles with adjacent
π
6
and hypotenuse
π
3
. The angles in the triangle are
therefore
α = cos
−1
tan
π
6
tan
π
3
= cos
−1
1
3
≈ 70.53°
The angle sum in the triangle is 3α ≈ 211.59°!
2. An airfield is at C and two planes are at A and B. The bearings and distances to the aircraft are
45°, 2000 miles, and 90°, 4000 miles respectively. Find the distance between the aircraft.
This is just the cosine rule! We have a spherical triangle with sides 2000 and 4000 with angle
45° between them. If r = 4000 miles is Earth’s radius, then
cos
c
r
= cos
2000
r
cos
4000
r
+ sin
2000
r
sin
4000
r
cos 45°
= cos
1
2
cos 1 +
1
√
2
sin
1
2
sin 1
=⇒ c = 2833 miles
This is a little closer than the value (2947 miles) one would obtain from assuming a flat Earth!
Modern navigators typically use a slightly different, though equivalent, approach to minimize
the error inherent in estimating cosine for small values: look up the haversine formula if you’re
interested.
66
Exercises 6.3. 1. A right-isosceles triangle on the surface of a unit sphere has equal legs of length
π
4
. Find the length of the hypotenuse and the sum of the angles in the triangle.
2. Explain the observation on page 62 that
0° < α − β < α + β < 90° =⇒ sin(α + β) −sin α < sin α −sin(α − β)
is the downwards concavity of the sine function.
3. Suppose we have a spherical triangle (sphere radius 1) as on page 65 with data
c = 30°, b = 60°, α = 60°
(a) Use the cosine rule to find a.
(b) Compute the remaining angles in the triangle. What do you observe about the sum of the
angles α + β + γ?
4. Determine the qibla for Rome (latitude 41°53’ N, longitude 12°30’ E).
Repeat for the UCI campus (33°39’ N, 117°51’ W).
5. Al-B
¯
ır
¯
un
¯
ı devised a method for determining the radius r of the
earth by sighting the horizon from the top of a mountain of known
height h. He would measure α, the angle of depression from the
horizontal to which one sights the apparent horizon. Show that
r =
h cos α
1 − cos α
In a particular case, al-B
¯
ır
¯
un
¯
ı measures α = 34
′
from a mountain of
height 652; 3, 18 cubits. Assuming that a cubit equals 18
′′
, convert
your answer to miles and compare with the modern value. Discuss
the efficacy of this method.
r
r
h
α
6. On a sphere of radius r, Pythagoras’ Theorem may be stated
cos
c
r
= cos
a
r
cos
b
r
(∗)
where c is the hypotenuse and a, b the other side-lengths. Use the
Maclaurin series cos x =
∞
∑
n=0
(−1)
n
x
2n
(2n)!
to expand (∗) to degree 4.
Suppose a, b are constant so that c is a function of r. Prove that
lim
r→∞
c
2
= a
2
+ b
2
. Why does this make sense?
7. Construct a triangle on the surface of a sphere of radius r by taking
two lines of longitude making an angle θ from the north pole to the
equator. Prove that the area of the triangle is
A = r
2
θ
What does Pythagoras (∗) say for this triangle?
(Hint: What fraction of the sphere is covered by the triangle?)
67
