1 Ancient Egypt

Summary

• Recorded civilization in Nile valley from c. 3150 BC.

• Conquered 322 BC by Alexander the Great (Greece/Macedonia).

• Became a Roman province in 30 BC under Cleopatra.

• Discovery of the Rosetta Stone (AD 1799) containing Greek, hieroglyphic and demotic (post

hieratic, cursive) Egyptian script, allowed translation of ancient Egyptian writing.

• Primary mathematical sources: Rhind/Ahmes (A’h-mose) papyrus c. 1650 BC and the Moscow

papyrus c. 1700 BC.

Part of the Rhind papyrus is shown below. It contained two tables: unit

fraction representations of all

for n < 100 (ish) and expressions for

, . . . ,

in terms

of unit fractions. Also included were around 100 worked problems. Scribes would learn the

method by copying previous problems and changing the numbers.

The Moscow papyrus is shorter and more focused on geometric problems.

• Few other primary sources. Egyptians wrote on papyrus (plant-based form of paper) which

decomposes. Other Egyptian mathematics was likely absorbed uncredited by the Greeks.

• Practical/non-theoretical: worked problems on sums, linear equations, construction and land-

measurement (tax-collection). No clear distinction between exact and approximate solutions.

Rhind was a Scottish egyptologist. Ahmes was the name of the scribe who wrote/copied the papyrus. The Moscow

papyrus is named because it was sold to the Moscow Museum of Fine Art; its author is unknown.

Notation & Egyptian Fractions

The ancient Egyptians had two distinct systems for enumeration: hieroglyphic (dating at least to

5000 BC) and hieratic (c. 2000 BC). These changed over time, so we give only one version.

Hierogylphic enumeration Essentially decimal symbols for

numerals/powers of 10.

• Could be written in any direction: top-to-bottom, right-to-

left, etc., or just lumped together: e.g.

2349 =

• Slow to write, numbers take up a lot of space.

Numeral Hieroglyph

10 (heel bone)

100 (snare)

1000 (lotus ﬂower)

10000 (ﬁnger)

100000 (ﬁsh)

1000000 (person)

Hieratic enumeration We will largely ignore this

since it is written cursively.

• Different symbols for 1–9, 10–90, 100–900

etc., mapped onto hieratic alphabet.

• System copied later by the Greeks with their

own alphabet.

• Pros: less space, easier to write with ink,

each number requires fewer symbols.

• Cons: More symbols, slower calculations.

For instance, 23 would be written (approximately!) |||∧.

The Eygptians had no numeral for zero, though the hieroglyph nfr (beautiful/perfect) was used to

denote, for instance, the base ﬂoor of a building or to indicate balanced books in accounting.

Fractions Ancient Egyptians worked almost entirely with reciprocals of integers (

where n ∈ N).

In modern times, any fraction represented as a sum of reciprocals is called an Egyptian fraction; their

theory is still actively researched. For instance

is a representation of

as an Egyptian fraction.

In hieratic notation, a dot was placed above to denote the reciprocal: e.g.,

∧ is

In hieroglyphs, a reciprocal was represented by plac-

ing an oval over a numeral. We do this with a bar:

e.g. 2 =

. As with integers, combinations of frac-

tions could be written in any order/direction.

The only non-reciprocal fractions with special sym-

bols were

and

, and these only appeared late in

Egyptian civilization.

Fraction Hieroglyph Modern

103204

31 +

25 2 31

For instance an oval over one-and-a-half sticks for

, and an oval over three short-long-short sticks for

The Rhind papyrus contains a table, of which we reproduce part,

showing how to express

as Egyptian fractions for all odd integers

n < 100. The ﬁrst column denotes n, and the remaining columns the

Egyptian fraction representation. For instance, the ﬁrst row states

( 3 15)

5 3 15

7 4 28

9 6 18

11 6 66

13 8 52 104

15 10 30

There are several approaches to ﬁnding Egyptian fraction representations, and it can be proved that

any fraction may be written in such a form. If the denominator is odd, then a simple place to start is

with

where r =

m + n

Most lines in the Rhind table follow this formula, but not all. Note also that the formula often permits

multiple representations.

• The line for

has m = 1, n = 5 and r = 3; this is unique up to reordering.

•

may be represented

(

(m, n, r) = (3, 3, 3)

(m, n, r) = (1, 9, 5)

The ﬁrst of these is essentially useless and the second is not the expression from the table.

• In the table,

is written as a sum of three reciprocals instead of two.

The table reduced the need to divide and sped up the computation of harder fractions: for instance,

= 2 ·

= 2



104



Egyptian Calculations & Example Problems

Addition/Subtraction With hieroglyphs this is simple: Write out numbers one above another and

count up the symbols! Replace 10 of one by the next symbol. For subtraction, one might need to

convert a larger symbol to 10 of a smaller one. Essentially this is ‘carrying.’ A special symbol was

used to denote both addition and subtraction: its meaning changed depending on the direction the

text was read.

Multiplication This relied on a base-two algorithm. To compute ab:

1. Write 1, b

2. Repeatedly double each until the ﬁrst term is about to exceed a

3. Determine powers of 2 that sum to a

4. Sum the corresponding multiples of b

For example, to compute 13 ·15, we construct a table where the checked rows are summed.

1 15 ✓

2 30

4 60 ✓

8 120 ✓

=⇒ 13 · 15 = (1 + 4 + 8) ·15 = 15 + 60 + 120 = 195

Note how 13 = 1 + 4 + 8 = 2

+ 2

is essentially the binary representation. We stopped at the

fourth row since another doubling would have resulted in the ﬁrst term (16) exceeding 13. We could

instead have reversed the roles of the factors:

1 13 ✓

2 26 ✓

4 52 ✓

8 104 ✓

=⇒ 15 · 13 = (1 + 2 + 4 + 8) ·13 = 13 + 26 + 52 + 104 = 195

All you need is addition and the ability to multiply by 2!

Division This also relies on doubling/halving, though the answer is non-unique and might re-

quire some creativity. To ﬁnd

, think about solving the problem a = bx and apply a variant of the

multiplication algorithm to ﬁnd multiples of b summing to a. Here are some examples.

1. To compute 260 ÷13, we repeatedly double 13 until terms in the right column sum to 260.

1 13

2 26

4 52 ✓

8 104

16 208 ✓

Since 260 = 208 + 52 we conclude that 260 ÷13 = 16 + 4 = 20

2. To ﬁnd 5 ÷ 13 we start by dividing by 2 with the intent of making terms in the right column

sum to 5.

1 13

2 6 2

4 3 4 ✓

8 1 2 8 ✓

Since (3 4) + (1 2 8) = 4 2 4 8 is 8 (

) short of what we want, we continue the table in a different

way. First divide by 13, then continue halving until we obtain the desired 8 in the right column.

13 1

26 2

52 4

104 8 ✓

We conclude that 5 ÷13 = 4 8 104 =

104

. We could have proceeded differently to obtain

the same result as followed from the Rhind table:

5 = (3 4) + 1 + 2 + 4 =⇒ 5 ÷13 = 4 13 26 52

Practical application: Loaf-splitting A typical Egyptian problem might involve determining how

to split 5 loaves among 13 people. The previous calculation tells us that we could give each person

104

of a loaf. This might seem complicated but it has some advantages over the modern

approach where we’d ﬁrst cut each loaf into 13 equal pieces:

• The large chunks of bread are created by repeatedly cutting in half: this is easy to do accurately,

whereas cutting 13

parts is difﬁcult! The remaining 104

parts of a loaf would probably be

ignored as crumbs.

• The Egyptian approach only requires 34 cuts, as opposed to 60 in the modern style.

Linear equations Another common type of problem was how to solve linear equations. Solutions

were based on the method of false position. Essentially one guesses an approximate solution, then

modiﬁes it until it works. Here is problem 24 of the Rhind papyrus.

A heap plus a seventh of a heap is nineteen. What is the heap?

In modern algebra, representing ‘heap’ by x, we wish to solve x +

x = 19. Here is the Egyptian

method.

1. Guess intelligently: x = 14 is easy to divide by 7 and we obtain

x +

x = 16

2. Correct our guess: We want 19, not 16, so we multiply our guess

(14) by

= 1 8 16 to obtain the correct answer

x = 2 4 8 + 4 2 4 + 9 2 = 16 2 8

1 1 8 16

2 2 4 8 ✓

4 4 2 4 ✓

8 9 2 ✓

Compare this with the ‘modern’ method:

x +

x = 19 =⇒

x = 19 =⇒ x =

7 ·19

133

128 + 5

= 16

Are the any beneﬁts to the Egyptian approach?

Geometry Problem 48 in the Rhind papyrus involves using an

octagon to approximated the area of a circle. A square of side 9

is drawn, where each side is split into thirds and the four corner

squares are cut in half. The area of the octagon is then

81 −18 = 63

Since the area of the circle is

81π

, this amounts to the approximation

π ≈

= 3.11111 . . .

Problem 50 compares the same circle to a square of side 8: essen-

tially

81π

≈ 64 which corresponds to π ≈

256

= 3.16049 . . .

No explanation is given as to what inspired these methods, nor whether the scribes understood that

these were only approximations.

Other problems computed/approximated areas and volumes of trian-

gles, quadrilaterals, boxes, cylinders, and truncated pyramids. Again,

these were worked examples without general formulæ. The Egyp-

tians even had a notion of cotangent which they called the seked, use-

ful for describing and calculating slopes.

seked =

= cot θ

Exercises Most of these problems are taken from the ‘ofﬁcial’ textbook.

1. Use Egyptian techniques to multiply 34 by 18 and to divide 93 by 5.

2. Use Egyptian techniques to multiply 28 by 1 + 2 + 4 (problem 9 of the Rhind papyrus).

3. A part of the Rhind papyrus table for division by 2 reads as follows:

2 ÷11 = 6 + 66, 2 ÷13 = 8 + 52 + 104, 2 ÷23 = 12 + 276

The calculation of 2 ÷ 13 is given below, where the right hand side is a modern rendering, and

the terminal ‘2’ indicates that the last entry in the right-hand column is indeed 2:

1 13

2 6 2

4 3 4

8 1 2 8

52 4

104 8

8 52 104 1 2 4 8 8

Perform similar calculations for 2 ÷11 and 2 ÷ 23.

(If you want the exact results from the papyrus, you’ll need to work with

: denote this by 3)

4. Draw a picture of 5 loaves to help describe how the Egyptians might have divided them be-

tween seven people.

5. Use the method of false position to solve problem 28 of the Rhind papyrus:

A quantity and its 2/3 are added together, and from the sum 1/3 of the sum is sub-

tracted, and 10 remains. What is the quantity?

6. Calculate a quantity such that if it is taken two times along with the quantity itself, the sum

comes to 9 (problem 25 of the Moscow papyrus).

7. (a) Find all ways in which

can be written as a sum of reciprocals (a ≤ b ∈ N).

(b) Repeat your calculation for

whenever p is an odd prime.

2 Babylon/Mesopotamia

Babylon was an ancient city located near modern-day

Baghdad, Iraq. The term also serves as a shorthand

for the many Mesopotamian

empires/civilizations

dating back at least to 3000 BC: Sumeria, Akkadia,

Babylonia, etc.

Babylonians used cuneiform (wedge-shaped) script,

typically indentations on clay tablets. Most recovered

tablets date from the time of Hammurabi (c. 1800 BC)

or the Seleucid dynasty (c. 300 BC) which ruled after

the conquests of Alexander the Great.

Mathematical tablets are of two main types: tables

of values (multiplication, reciprocals, measures) and

worked problems.

Sexagesimal (base-60) Positional Enumeration Our modern decimal system is base-10 positional.

This means two things, both of which are easy to see with reference, say, to the number 3835.

• The symbol 3 represents both 3000 and 30: the meaning of a symbol depends on its position.

• The position of a symbol denotes the power of 10 by which it should be multiplied. Thus

3835 = 3 ·10

+ 8 ·10

+ 3 ·10

+ 5 ·10

Positional enumeration makes for efﬁcient calculations and easy representation of numbers of vastly

different magnitude. Contrast with the difﬁculty of performing calculations using (non-positional)

Egyptian hieroglyphic notation.

One of the key Babylonian contributions to mathematical history is the creation of (arguably) the

ﬁrst positional system of enumeration, dating to at least 2000 BC. Rather than our ten symbols 0–

9, Babylonians used only two: roughly ∨ for 1 and ∢ for 10, likely made by the same stylus. Any

number up to 59 could be written with combinations of these symbols, e.g.,

53 =

∢∢∢

∢∢

∨∨∨

The picture shows a typical cuneiform representation. Larger numbers were represented base-60.

For instance, the sexagesimal decomposition of 3835 is

3835 = 1 ·60

+ 3 ·60

+ 55 ·60

which the Babylonians would have written

∨ ∨∨∨

∢∢∢

∢∢

∨∨∨

∨∨

(∗)

Rather than using cuneiform, we’ll instead write 1, 3, 55;

‘Between two rivers,’ namely the Tigris and Euphrates. As indicated on the map, these rivers formed the backbone of

the fertile crescent, a region of early civilization, farming, crop and animal domestication.

Just as, for us, ‘3’ might mean 30, 3000 or even

1000

, for the Babylonians ∨ could mean 1, 60, 3600,

216000, or fractions such as

3600

depending on its position. There was no symbol for zero (as a

placeholder) until very late in Babylonian history, nor any sexagesimal point, so determining position

on ancient tablets can be difﬁcult. For instance, rather than 3835, (∗) might instead have represented

60 + 3 +

= 63

or 60

+ 3 ·60

+ 55 ·60 = 230100

To make things easier to read, we use commas to separate terms and, if necessary, a semicolon to

denote the sexagesimal point. Thus

23, 12, 0; 15 = 23 ·60

+ 12 ·60 +

= 83520

Why base-60? There are many theories, but we cannot be certain. Here are some ideas.

• The Babylonians might have combined two systems (base-10 and base-12) inherited from older

cultures.

• Since 60 has many proper divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30), more numbers have exact

representation than with decimal arithmetic: for instance

= ; 20, as a terminating sexagesimal,

is much simpler than the decimal 0.33333.

• As proliﬁc astronomers and astrologers, the Babylonians might have chosen 60 as a divisor of

360, approximately the number of days in a year. Our modern usage of degrees-minutes-seconds

for angle, hours-minutes-seconds for time, and the standard zodiac are all of Babylonian origin.

Indeed Babylonian units of measure often used factors of 60 for magnitude similarly to how

modern science uses 1000 (e.g., joules → kilojoules → megajoules).

This sort of historical question is rarely answerable in a satisfying way. Likely no-one ‘decided’ to

use base-60; like most cultural issues, it likely happened slowly and organically, without fanfare.

Basic Sexagesimal Calculations Addition and subtraction would have been as natural to the Baby-

lonians as decimal calculations are to us. For instance, we might write

21, 49

+ 3

, 37

25, 26

(in decimals 1309 + 217 = 1526)

Note how we carry 60 just like we are used to doing with 10 in decimal arithmetic: 49 + 37 = 1, 26.

Multiplication is signiﬁcantly harder. To mimic our familiar long-multiplication process would re-

quire memorizing up to the 59 times table! For small factors this might have been ﬁne. For larger

factors there is evidence of the Babylonians using two representations of a product in terms of squares

xy =



(x + y)

− x

−y





(x + y)

−(x −y)



Tablets consisting of tables of squares greatly aided the computation of large products. For instance,

31 ×22 =



−9





46, 49 −1, 21





45, 28



= 11, 7 + 15 = 11, 22 (= 682)

Fractions & Division As we’ve already seen, the Babylonians also represented non-integers using

sexagesimals. Tables of reciprocals

were used to quickly evaluate division using multiplication!

m ÷n = m ×

For example

= 0; 3, 20 =⇒

= 23(0; 3, 20) = 1; 9 + 0; 7, 40 = 1; 16, 40

This works nicely provided n has no prime divisors other than 2, 3 or 5, since any such

will be

an exact terminating sexagesimal.

Approximations were used for other reciprocals; a scribe would

choose a nearby denominator with an exact sexagesimal and state that the answer was approximate

≈

= 11(0; 2) = 0; 22

For more accuracy, one could choose a larger denominator. For instance, if a scribe wanted to divide

by 11, they might observe that 11 ·13 = 143 ≈ 144, from which

144

= 0; 0, 25 =⇒

≈

144

= 0; 5, 25

Scribes were explicit in acknowledging the approximation by stating, say, “11 does not divide.” Re-

member that a single digit in the second sexagesimal place means only

3600

, so even the most de-

manding application doesn’t require many terms (the above is 99.3% accurate!). The denominators

in some of these reciprocal tables were enormous, so far greater accuracy was often possible.

Another table listed all the ways an integer < 10 could be multiplied exactly to get 10.

1 10 5 2

2 5 6 1 40

3 3 20 8 1 15

4 2 30 9 1 6 40

We omit the commas for separation and the sexagesimal point as they did not exist. Moreover 7 is

missing since

(and thus

) is not an exact sexagesimal. It should be clear from the table that

= 1; 40 and

600

= 1, 6; 40

In the latter case, note that 600 = 10 ·60 would be written the same as 10, so this amounts to moving

the sexagesimal point in

= 1; 6, 40.

Analogous to the fact that

has a terminating decimal if and only if n has no prime divisors other than 2 or 5.

Being rational,

= 0.09090909 . . . = 0; 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, . . . has a periodic sexagesimal expansion as can

be found using a pocket-calculator:

= 5 +

5 ·60

= 27 +

3 ·60

= 16 +

4 ·60

= 21 +

, . . .

Linear Systems of Equations These could be solved by a combination of the method of false posi-

tion (guess and modify as per the Egyptians) and the consideration of homogeneous equations. For

instance, here is a (suitably modernized) Babylonian approach to solving the system

(

3x + 2y = 11

2x + y = 7

1. Choose one equation, say the second, and set

x =

y. Solve this (say using false position) to

obtain

x =

y =

= 2; 20.

2. Since (d, −2d) is the general solution to the homogeneous equation 2x + y = 0, all solutions to

the second equation have the form x =

x + d and y =

y −2d. Substitute into the ﬁrst equation:

11 = 3



+ d



+ 2



−2d



= 11 +

−d =⇒ d =

3. Finally compute x =

x + d =

= 3 and y =

y −2d =

−

= 1.

Step 2 should should remind you of the ‘nullspace’ method from modern linear algebra: all solutions

to the matrix equation ( 2 1)

(

)

= 7 have the form









+ n

where





is some particular solution (here x

= y

) and n =



−2d



lies in the nullspace of the

(row) matrix (2 1).

The Yale Tablet (YBC 7289) and Square-root Approximations One of the most famous tablets con-

cerns an approximation to

√

2. YBC stands for the Yale Babylonian Collection which contains over

45,000 objects. YBC 7289 is shown below alongside an enhanced representation of the numerals.

The tablet depicts a square of side 30 and labels the diagonal in two ways:

• 1; 24, 51, 10 is an approximation to

√

2, an underestimate by roughly 1 part in 2.5 million!

• 42; 25, 35 is an approximation to the diagonal when the side is 30.

The Babylonians more often used the simpler approximation 1; 25 = 1.41666 . . . which is still very

close. Given the impractical accuracy of YBC 7289, it is reasonable to ask how it was found. No-one

knows for certain, but two methods are theorized since both were used to solve other problems. It

should be stressed that no Babylonian proofs of these approaches are known.

1: Square root approximation

√

±b ≈ a ±

. This is essentially the linear approximation from

elementary calculus. If one chooses a rational number a whose square is close to 2, then the

error will also be small. For instance:

•

√

2 =

√

1 + 1 ≈ 1 +

= 1; 30 (a = 1)

•

√

2 =





≈

2/9

8/3

= 1; 25 (a =

= 1.3333 . . .)

•

√

2 =





≈

1/25

14/5

= 1; 24, 51, 25, 42, 51, 25, 42, . . . (a =

= 1.4)

2: Method of the Mean It may be checked (Exercise 11) that any sequence deﬁned by the recurrence

relation a

n+1





converges to

√

2. We apply this when a

= 1.

= 1, a

= 1; 30, a

= 1; 25

577

408

= 1 +

169

408

= 1; 24, 51, 10, 35, 17, . . .

665857

470832

= 1; 24, 51, 10, 7, 46 . . .

It seems incredible that any ancient culture would have bothered to go as far as this to obtain

the observed accuracy.

The same approach can be used to approximate other roots. For example, we can approximate

√

11 via a

n+1

) and a

= 3:

= 3; 20, a

199

= 3; 19, a

79201

23880

= 3; 18, 59, 50, 57, 17, . . .

Quadratic Equations The above methods were applied to solve general quadratic equations. A

question might be phrased as follows:

I added twice the side to the square; the result is 2, 51, 40. What is the side?

In modern language, we want the solution to x

+ 2x = 2 ·60

+ 51 ·60 + 40 = 10300.

Questions such as these were solved using templates, typically as worked examples. The above

problem requires the template for solving x(x + p) = q where p, q > 0. Since the Babylonians did not

recognize negative numbers, the other types of quadratic equation (x

= px + q, etc.) had different

templates.

To make things a little easier, we apply their approach to the simpler equation x

+ 4x = 2:

Set y = x + p (y = x + 4) and decouple the equation:

(

xy = q

y − x = p

(

xy = 2

y − x = 4

Use this to solve for x + y:

4xy + (y −x)

= p

+ 4q 4xy + (y −x)

= 4

+ 4 ·2

(y + x)

= p

+ 4q (y + x)

= 24

x + y =

+ 4q x + y =

√

24 ≈ 4; 54

where the square-root was approximated using one of the earlier algorithms, e.g.

√

24 =

−1 ≈ 5 −

= 4; 54

Since x + y and x −y are now known, we have a linear system which is easily solved:

x =

+ 4q − p

x ≈ 0; 27

The method of completing the square and the quadratic formula are at least 4000 years old!

While we’ve written this abstractly, in practice scribes would be copying from a particular example

of the same type. There were no abstract formulæ and everything was done without the beneﬁt

modern notation. There was moreover typically no written commentary to explain the method; often

all historians have to work with is a single column of numbers!

Note also that the template only found the positive solution; the Babylonians had no notion of nega-

tive numbers. Amazingly, they were able to address certain cubic equations similarly.

Pythagorean Triples The Plimpton 322 tablet (also at Yale) lists a large number of Pythagorean

triples (albeit with some mistakes). Due to the strange manner of encoding, it took scholars a long

time to realize what they had.

As an example, line 15 describes the Pythagorean

triple 53

= 45

+ 28

• The ﬁrst entry 1; 23, 13, 46, 40 is the exact sex-

agesimal for





• The second entry is 28.

• The third entry is 53.

• The last two entries indicate line number 15.

The ﬁrst three (interesting) entries are therefore



(

)

, b, c



where c

= a

+ b

. Since the table is

broken on the left side it is possible that a missing column explicitly mentioned a.

It is not known how the table was completed, though the ﬁrst column exhibits a descending pattern

that provides clues to its construction. One theory is that a scribe found rational solutions to the

equation v

= 1 + u

(equivalently (v + u)(v −u) = 1) by starting with a choice of v + u and using a

table of reciprocals to calculate v −u.

To revisit our example, if v + u =

= 1; 48, then

v −u =

v + u

= 0; 33, 20

We therefore have a linear system of equations in u, v whose solutions are

v = 1; 10, 40 =

, u = 0; 37, 20 =

We investigate this further in Exercise 7. The Plimpton tablet has been the source of enormous schol-

arship; look it up!

Geometry The Babylonians also considered many geometric problems. They used both π ≈ 3

and π ≈ 3

to approximate areas of circles. They had calculations (both correct and erroneous)

for the volume of a frustrum (truncated pyramid). They also knew that the altitude of an isosceles

triangle bisects its base, and that the angle in semicircle is a right-angle (Thales’ Theorem). None

of these statements were presented as theorems in a modern sense; we merely have computations

and applications that make use of these facts. We simply do not know the depth of Babylonian

understanding of such concepts.

Summary

• Sexagesimal positional enumeration. No zero. Fractions also used sexagesimal representation.

• More advanced than Egyptian mathematics but still practical/non-abstract. Perhaps Babylo-

nian mathematics only appears more advanced because we have so much more evidence: thou-

sands of tablets versus only a handful of papyri. Like Egypt, we have worked examples without

abstraction or statements of general principles.

• Some distinction (‘does not divide’) between approximate and exact results.

• Some geometry, but algorithmic/numerical methods predominate.

Exercises There is no single correct way to do Babylonian calculations. Play with the ideas and use

modern notation to get a feel for things without torturing yourself.

1. Convert the sexagesimal values 0; 22, 30, 0; 8, 6, 0; 4, 10 and 0; 5, 33, 20 into ordinary (modern)

fractions in lowest terms.

2. (a) Multiply 25 by 1, 4 (b) Multiply 18 by 1, 21.

(Either compute directly (long multiplication) or use the difference of squares method on page 8)

3. (a) Use reciprocals to divide 50 by 18. (b) Repeat for 1, 21 divided by 32.

4. Use the Babylonian method of false position to solve the linear system

(

3x + 5y = 19

2x + 3y = 12

5. (a) Convert the approximation

√

2 ≈ 1; 24, 51, 10 to a decimal and verify the accuracy of the

approximation on page 10.

(b) Multiply by 30 to check that the length of the diagonal is as claimed.

6. Babylonian notation is not required for this question.

(a) Use the square root approximation (pg. 11) with a =

to ﬁnd an approximation to

√

(b) Taking a

= 3, apply the method of the mean to ﬁnd the approximation a

√

7. Recall that v

= 1 + u

in the construction of the Plimpton tablet.

(a) If v + u = α, show that u =

(α −α

−1

) and v =

(α + α

−1

(b) Suppose v + u = 1; 30 =

. Find u, v and the corresponding Pythagorean triple.

(d) Repeat for v + u = 2; 05 =

. This is line 9 of the tablet.

8. Solve the following problem from tablet YBC 4652. I found a stone, but did not weigh it; af-

ter I subtracted one-seventh, added one-eleventh (of the difference), and then subtracted one-

thirteenth (of the previous total), it weighed 1 mina (= 60 gin). What was the stone’s weight?

(The meaning of the problem isn’t completely clear: make your best guess!)

9. Solve the following problem from tablet YBC 6967. A number exceeds its reciprocal by 7. Find

the number and the reciprocal.

(In this case, two numbers are reciprocals if their product is 60)

10. For this question it is helpful to think about the corresponding facts for decimals.

(a) Explain the observation on page 9 regarding which reciprocals n have a terminating sexa-

gesimal. Can you prove this?

(b) Find the periodic sexagesimal representation of

and use geometric series prove that you

are correct.

11. For this question, look up the AM–GM inequality and remind yourself of some basic Analysis.

(a) Suppose (a

) is a sequence satisfying the recurrence a

n+1

). Prove that a

≥

√

whenever n ≥ 2.

(b) Prove that lim a

√

12. (a) The Babylonians used an approximation of the form A ≈

for the area of a circle in

terms of its circumference. To what approximation for π does this correspond?

(b) Use a Babylonian method to show that

√

3 ≈

lar arcs built from a circumscribed equilateral triangle. If the

arc-length is a, use Babylonian approximations to prove that

the area of the bulls-eye is ≈

. What, approximately, are its

dimensions (width/height)?

(Use as much modern trigonometry as you like!)

3 Ancient Greek Mathematics

3.1 Overview of Ancient Greek Civilization & Early Philosophy

Ancient Greek civilization dates from around 800 BC, centered on the peninsula between the Adriatic

sea (to the west) and the Aegean (to the east) that forms the mainland of the modern Greek state.

Ancient Greek culture was decentralized, consisting of semi-independent city-states connected by

trade. Accomplished sea-farers, they extended their reach round the northern and eastern coasts

of the Mediterranean, from Iberia (Spain) to the Black Sea, Anatolia (western Turkey), and Egypt,

building and capturing a network of city-states and trading outposts.

Philip of Macedon ‘uniﬁed’ the Greek peninsula just before his death in 336 BC. His son, Alexander

‘the Great,’ launched a massive campaign conquering Persia, Egypt, Babylon, and western India be-

fore his own death in Babylon in 323 BC. Alexander left provincial governors to manage captured

territory; some of these structures endured for centuries (Egypt’s Ptolemaic dynasty), whereas oth-

ers were overthrown after only a few years (parts of India). While Alexander’s conquests did not

produce a long-lasting centralized Greek empire, they were effective at expanding the reach of Greek

cultural practice and philosophy, and brought external ideas into the Greek tradition.

The core part of Greek territory was absorbed by the Roman empire around 146 BC. In line with

typical Roman practice, those Greeks who agreed to accept Roman governors and taxation became

Roman citizens.

As such, the Greek culture of inquiry and scholarship was left largely intact under

Roman rule. Greek culture and learning was central to the later Byzantine (eastern Roman) empire

(centered on Byzantium/Constantinople/Istanbul), which lasted until Constantinople was captured

by the Islamic Ottomans in 1453. For centuries prior, Islamic scholarship had itself been signiﬁcantly

inﬂuenced by the ancient Greeks; the main consequence of the fall of Constantinople for knowledge

transfer was to encourage the exodus of scholars to Rome, helping to fuel the nascent European

Renaissance.

Greek Territory c. 500 BC

While this might sound reasonable, resistance wasn’t a realistic option. ..

Greek mathematics is part of a much wider development of science and philosophy encompassing a

change of emphasis from practicality to abstraction. One reason for this was the Greek blending of

religion/mysticism with natural philosophy: a desire to describe the natural world while preserving

the perfection/logic in the gods’ design.

Early Greek inquiry into natural phenomena was encouraged through the personiﬁcation of nature

(e.g., sky = man, earth = woman). By 600 BC, philosophers were attempting to describe such phe-

nomena in terms of natural predictable causes and structures. For example, some viewed matter as

being comprised of the ‘four elements’ (ﬁre, earth, water, air) combined in the correct proportions.

While the system of the world was seen as divinely-designed, explanations relying on the whims of

the gods were discouraged.

While the Greeks certainly used mathematics for practical purposes, philosophers idealized logic

and were unhappy with approximations. This led to the development of axiomatics, theorems and

proof, concepts for which there is scant pre-Greek evidence. The ancient Greek language is indeed the

source of three words of critical importance:

Mathematics From mathematos (µαθήµατoς), meaning knowledge or learning; the term covered es-

sentially anything that might be taught in Greek schools.

Geometry Literally earth-measure, a combination of two terms:

Gi (γη) Dates from pre-5

century BC, meaning land, earth or soil. Capitalized (Γη) it could refer

to the Earth (as a goddess).

Metron (µέτρoν) A weight or measure, a dimension (length, width, etc.), or the metre (rhythm) in

music.

Theorem From theoreo (θεώρέω), meaning ‘I contemplate/consider.’ In a mathematical context this

become theorema (θεώρηµα): a proposition to be proved.

Ancient Greece had several schools, mostly private and open only to men. Typically arithmetic was

taught until age 14, followed by geometry and astronomy until age 18. The most famous scholars

of ancient Greece were the Athenian trio of Socrates, Plato and Aristotle,

whose writings became

central to the Western/Islamic philosophical tradition. Plato’s Academy in Athens was a model for

centuries of schooling; the centrality of geometry to the curriculum was evidenced by the famous

inscription above its entryway: “Let none ignorant of geometry enter here.”

Ancient Greek Enumeration

The Greeks had two primary forms of enumeration, both dating from around 800–500 BC.

In Attic Greek (Attica = Athens) strokes were used for 1–4, with larger numerals using the ﬁrst letter

of the words for 5, 10, 100, 1000 and 10000. For example,

• Πεντε (pente) is Greek for ﬁve, whence Π denoted the number 5.

• Δεκα (deca) means ten, so Δ represented 10.

• Η (hekaton), X (khilias) and M (myrion/myriad) denoted 100, 1000 and 10000 respectively.

• Larger numbers were written using combinations of these symbols, similarly to both Egyptian

hieroglyphs and (the later) Roman numerals: e.g., ΧΗΗΠ|| = 1207.

Each taught his successor, with the birth of Socrates to the death of Aristotle covering 470–322 BC.

Ionic Greek (Ionia = mid Anatolian coast) numerals used the Greek

alphabet, an approach possibly copied from Egyptian hieratic enu-

meration. Larger numbers used a left subscript mark (like a comma)

to denote thousands and/or M (with superscripts) for 10000 as in

Attic Greek. For example,

35298 = ,λ,εσϟη =

M,εσϟη

The ancient Greek alphabet included three archaic symbols ϛ ϟ ϡ

(stigma, qoppa, sampi), with which you’re likely unfamiliar.

1 α 10 ι 100 ρ

2 β 20 κ 200 σ

3 γ 30 λ 300 τ

4 δ 40 µ 400 υ

5 ε 50 ν 500 ϕ

6 ϛ 60 ξ 600 χ

7 ζ 70 o 700 ψ

8 η 80 π 800 ω

9 θ 90 ϟ 900 ϡ

The Greeks also used Egyptian fractions, denoting reciprocals with an accent over the symbol: e.g.,

θ =

. The use of Egyptian fractions dominated in Europe well into the middle ages.

Ionic Greek enumeration has persisted, with few changes, into modern times, although Hindu–

Arabic numerals are also in common usage. Eventually a bar was placed over numbers to distinguish

them from words (e.g., ξθ = 89), while modern practice is to insert a keraia (similar to an apostrophe)

at the end of a number: thus 35298 = ͵λ͵εσϟηʹ.

Both the Attic and Ionic systems were suitable for record-keeping but terrible for calculations! Later

Greek mathematicians adapted the Babylonian sexagesimal system for calculation purposes, helping

cement its modern use of in astronomy, navigation and time-keeping.

Exercises 3.1. 1. State the number 1789 in both Attic and Ionic notation.

2. Represent

as a sum of distinct unit fractions (Egyptian style). Express the result in (Ionic)

Greek notation.

(The answer to this problem is non-unique)

3. For tax purposes, the ancient Greeks would approximate the area of a quadrilateral ﬁeld by

multiplying the averages of the two pairs of opposite sides. In one example, the two pairs of

opposite sides were given as

a =

opposite c =

, and,

b =

opposite d = 1

where the lengths are in fractions of of a schonion, a measure of approximately 150 feet. Find the

average of a and c, the average of b and d, and thus the approximate area of the ﬁeld in square

schonion. The taxman then rounds up the answer to collect a little more tax!

3.2 Pre-Euclidean Greek Mathematics

The publication of Euclid’s Elements (c. 300 BC) forms a natural breakpoint in Greek mathematics,

since is subsumed much of what came before it. In this section, we consider the contributions of sev-

eral pre-Euclidean mathematicians. There are very few sources for Greek mathematics & philosophy

before 400 BC, so almost everything is inferred from the writings and commentaries of others.

Thales of Miletus (c. 624–546 BC) One of the ﬁrst people known to state abstract general principles,

Thales was a trader based in Miletus, a city-state in Anatolia. He travelled widely and was likely

exposed to mathematical ideas from all round the Mediterranean. Here are some statements at least

partly attributed to Thales:

• The angles at the base of an isosceles triangle are equal.

• Any circle is bisected by its diameter.

• A triangle inscribed in a semi-circle is right-angled (still known as Thales’ Theorem).

Thales’ major development is generality: his propositions concern all triangles,

circles, etc. The Babylonians & Egyptians had examples of such results, but

we have little indication that they viewed these abstractly. In accordance with

the Greek idea of a theorem (‘to look at’), Thales’ reasoning was almost cer-

tainly pictorial. As an example of contemporary geometric reasoning, by 425 BC

Socrates could describe how to halve/double the area of a square by joining the

midpoints of edges.

Pythagoras of Samos (c. 572–497 BC) Like Thales, Pythagoras travelled widely, eventually settling

in Croton (southeast Italy) where he founded a school lasting 100 years after his death. Plato is

believed to have learned much of his mathematics from a Pythagorean named Archytas.

The Pythagoreans practiced a mini-religion whose ideas lay outside the mainstream of Greek soci-

ety.

One of their mottos, “All is number,” emphasised their belief in the centrality of pattern and

proportion. The following quote

gives some ﬂavor of the Pythagorean way of life.

After a testing period and after rigorous selection, the initiates of this order were allowed

to hear the voice of the Master [Pythagoras] behind a curtain; but only after some years,

when their souls had been further puriﬁed by music and by living in purity in accordance

with the regulations, were they allowed to see him. This puriﬁcation and the initiation

into the mysteries of harmony and of numbers would enable the soul to approach [be-

come] the Divine and thus escape the circular chain of re-births.

The Pythagoreans were particularly interested in musical harmony and its relationship to number.

They are credited with relating musical intervals to the ratios of lengths of vibrating strings:

• Identical strings whose lengths are in the ratio 2:1 vibrate an octave apart.

• A perfect ﬁfth corresponds to the ratio 3:2.

• A perfect fourth corresponds to the ratio 4:3.

The use of these ratios to tune musical instruments is still known as Pythagorean tuning.

For instance, most of our knowledge of Socrates comes from the voluminous writings of Plato and Aristotle. The

earliest known Greek textbook/compilation (Elements of Geometry) was written around 430 BC by Hippocrates of Chios; no

copy survives, though most of its material probably made it into Book I of Euclid.

They were vegetarians, believed in the transmigration of souls, and accepted women as students; controversial indeed!

Van der Waerden, Science Awakening pp 92–93

Theorems 21–34 in Book IX of Euclid’s Elements are Pythagorean in origin. For instance:

Theorem. (IX .21) A sum of even numbers is even.

(IX .27) Odd less odd is even.

The Pythagoreans also studied perfect numbers, those which equal the sum of their proper divisors

(e.g., 6 = 1 + 2 + 3), and they seem to have observed the following famous result.

Theorem (IX. 36). If 2

−1 is prime then 2

n−1

−1) is perfect.

They moreover considered square and triangular numbers (

m(m + 1)) and tried to express geomet-

ric shapes as numbers, in service of their belief that all matter could be formed from basic shapes.

Incommensurability and the Pythagorean Theorem As with other ancient cultures, the only num-

bers in Greek mathematics were positive integers. These were used to compare lengths/sizes of objects.

Deﬁnition. Lengths are in the ratio m : n if some sub-length divides exactly m

times into the ﬁrst and n times into the second.

Two lengths are commensurable if some sub-length divides exactly into both.

Ratio 3 : 2

While modern mathematics has no problem with irrational ratios (e.g., the diagonal of a square to its

side is

√

2 : 1), this conﬂicted with the core Pythagorean belief that any two lengths were commensurable.

Identifying lengths with real numbers (underlined), we restate their assertion in modern language:

∀m, n ∈ R

, ∃ℓ ∈ R

, ∃a, b ∈ N, such that m = aℓ and n = bℓ

This is complete nonsense, for it insists that every ratio of real numbers m : n = a : b is rational!

The Pythagorean commensurability supposition stems from their basic tenets: all is number (includ-

ing length ratios), and the design of the gods is perfect (number means positive integer). The dis-

covery of incommensurable ratios produced something of a crisis; a possibly apocryphal story states

that a disciple named Hippasus (c. 500 BC) was set adrift at sea as punishment for its revelation.

By 340 BC, however, the Greeks were happy to state that incommensurable lengths exist.

Theorem (Aristotle). If the diagonal and side of a square are commensurable, then odd numbers

equal even numbers.

Inferred proof. In Socrates’ doubled-square, suppose that side : diagonal = a : b; these are integers!

Assume at least one of a or b is odd, else the common sub-length may be doubled.

The larger square is twice the smaller, whence the square numbers have ratio

: a

= 2 : 1

It follows that b

is even and thus divisible by 4. But then a

is also even, whence

both a, b are even. Whichever of a, b was odd is also even: contradiction!

a a

Note the similarity of this argument to the modern proof of the irrationality of

√

While there is no evidence that the Pythagoreans ever provided a correct proof of their famous The-

orem, one argument possibly attributable to them relied on the ﬂawed notion of commensurability.

‘Proof’ of the Pythagorean Theorem. Label the right-triangle a, b, c and

drop the altitude to the hypotenuse c as shown. Let d be the part of

the hypotenuse under a. Similar triangles tell us that

a : d = c : a =⇒ a

: ad = cd : ad =⇒ a

= cd

Thus the square on a has the same area as the rectangle below the

d-side of the hypotenuse. Repeat the calculation on the other side to

obtain b

= c(c −d), and sum to complete the proof.

Since the only numbers were integers, a, b, c, d denote integer multiples

of an assumed common sub-length. This restriction destroys the gen-

erality of the argument, since most triangles cannot be so labelled.

Book I of the Elements is plainly organized so as to provide a rigorous repair of the above argument

which did not depend on the ﬂawed notion of commensurability. With our modern understanding

of real numbers and continuity, there is nothing wrong with the above.

Theaetetus of Athens (417–369 BC) Theaetetus is likely the source for much of the most difﬁcult

parts (Books VII & X) of the Elements. His essential deﬁnition of (in)commensurability comes from

applying what is now known as the Euclidean algorithm to line segments.

Deﬁnition. Let a > b be lengths/segments.

Repeatedly apply the division algorithm:

a = q

b + r

< b (∃q

∈ N

and a length r

< b)

b = q

+ r

< r

= q

+ r

< r

, etc.

We say that a and b are commensurable if the algorithm terminates: some remainder r

divides exactly

into r

n−1

. Otherwise a and b are incommensurable.

Ratios are equal a : b = c : d precisely when the sequences of quotients in the algorithm are equal.

If a and b are commensurable, then r

is their greatest common sub-length. If we write a = ar

and

b = br

for some integers a, b and rewrite the algorithm in the modern fashion, the result is the

standard Euclidean algorithm computation of gcd(a, b) = 1.

Example 1 37 : 13 = 148 : 52 since we obtain the same sequence of quotients (2, 1, 5, 2):

37 = 2 ·13 + 11 148 = 2 ·52 + 44

13 = 1 ·11 + 2 52 = 1 ·44 + 8

11 = 5 ·2 + 1 44 = 5 ·8 + 4

2 = 2 ·1 8 = 2 ·4

Thus ‘a is longer than b’ is a statement about lengths, not numbers. Only the quotients q

need be integers.

Example 2 We sketch a proof that the side AB and diagonal AC of a regular pentagon ABCDE are

incommensurable.

1. Prove that △BAG is isosceles (just count angles!).

2. Take a =

and b =

. The ﬁrst line of the

algorithm reads

so we write a = q

b + r

where r

and q

= 1.

3. Since

, the second line of the algorithm reads

so that we again have a quotient of q

= 1.

4. Appealing to congruent isosceles triangles △DCG

∼

△EHF we see that

is the

diagonal of the interior regular pentagon. The third line of the algorithm is therefore the same

as the ﬁrst: we are back to considering the ratio of the diagonal to the side of a regular pentagon.

The algorithm therefore continues forever with all quotients being 1.

Example 3 In modern language, the diagonal to the side of a square is the incommensurable ratio

√

2 : 1. We apply Theaetetus’ algorithm:

√

2 = 1 ·1 + (

√

2 −1)

1 = 2 ·(

√

2 −1) + (3 −2

√

Observe that 3 −2

√

2 = (

√

2 −1)

, whence the second line reads 1 = 2x + x

. The following lines in

the algorithm may therefore be obtained by repeatedly multiplying by x:

x = 2x

+ x

, x

= 2x

+ x

, etc.,

resulting in a never-ending sequence

of quotients: 1, 2, 2, 2, 2, 2, . . .

Eudoxus of Knidos (c. 390–337 BC) Eudoxus was arguably the most proliﬁc pre-Euclidean mathe-

matician. Apart from attending and perhaps teaching at Plato’s academy, he is famous for explaining

how to calculate with ratios of lengths (segments). For example:

Deﬁnition. A : B > C : D if there exist positive integers m, n such that mA > nB and mC ≤ nD.

At ﬁrst glance it appears as if Eudoxus is telling us how to compare rational numbers; if A, B, C, D are

integers, we see that

≥

which is trivially satisﬁed by taking m = D and n = C. To Eudoxus however, A, B, C, D could also

be interpreted as segments. Building on the work of Theaetetus, his mathematics told the Greeks how

to approximate incommensurable ratios with rational ratios.

If you’re interested in number theory, investigate the relationship of Theaetetus’ algorithm to continued fractions. . .

Examples 1. To see that 13 : 3 > 17 : 4, simply choose m = 4 and n = 17 to obtain

4 ·13 = 52 > 51 = 3 ·17

2. We show that the side : diagonal of a square is greater than 1 : 2;

equivalently, the diagonal is less than twice the side.

In the picture, side : diagonal = A : B. Choosing m = D = 2 and

n = C = 1, we see that

mA = 2A > B = nB (diag > side of large square)

In modern language, this is merely

√

Zeno of Elea (c. 450 BC) Zeno’s arguments have provided philosophical fodder for millennia. Sev-

eral illustrate the essential difﬁculties of the inﬁnite and the inﬁnitesimal that lie at the heart of the

controversy around the development of calculus. Here are two of the most famous:

Achilles and Tortoise Achilles chases a Tortoise. After time t

, Achilles reaches the Tortoise’s starting

position, but the Tortoise has moved on. After another time t

, Achilles reaches the Tortoise’s

second position; again the Tortoise has moved. In this manner Achilles spends t

+ t

+ ···

in the chase. Zeno’s paradoxical conclusion is that Achilles never catches the Tortoise.

This paradox may be resolved (see Exercise 8), at least assuming both Achilles and the Tortoise

travel at constant speeds: even though it be split into inﬁnitely many subintervals of time, the

total duration of the chase can expressed as the (ﬁnite) value of a convergent inﬁnite series.

Arrow paradox An arrow is shot from a bow. At any given instant the arrow doesn’t move. If time is

made up of instants, then the arrow never moves.

This time Zeno debates the the idea that a ﬁnite time period can be considered as a sum of

inﬁnitesimal instants. The same difﬁculty is central to integration.

Constructions and Geometry By the mid 5

century BC, Greek mathematicians were solving ge-

ometric problems using ruler-and-compass (peg-and-cord) constructions. This approach could have

come to Greece from India, or might have arisen organically. Constructions were based on three

rules, which became the ﬁrst three postulates (axioms) of Book I of Euclid’s Elements.

1. Two points may be joined by a straight line segment.

2. Any segment may be extended indeﬁnitely.

3. Given a center and radius, one may draw a circle.

Theorems were often stated as problems: e.g., to bisect a given angle. A proof provided ﬁrst a construc-

tion, then an argument justifying that the construction really had solved the problem.

By the time of Euclid, the Greeks knew how to construct an equilateral triangle, a square and a regular

pentagon in a given circle.

Example We construct a regular pentagon in a given circle.

1. Draw perpendicular diameters AB and CD and bisect OA at M.

2. Draw an arc centered at M with radius

. Let N be the in-

tersection of this arc with OB.

3. Draw an arc centered at C with radius

. Let R be the inter-

section of this arc with the original circle.

4. Move

around the circle to create a regular pentagon.

A purely geometric proof of the validity of this construction is too difﬁcult for us, but you can easily

check it using a calculator, Pythagoras’ and the cosine rule: if the circle has radius 2, then

= (

√

5 −1)

+ 2

= 10 − 2

√

5 = 2

+ 2

−2 ·2 ·2 cos 72°

Construction problems have motivated mathematicians ever since. In 1796, Gauss (then 19) con-

structed a regular 17-gon. A classiﬁcation of constructable regular polygons took until 1837.

Theorem. A regular n-gon is constructable if and only if n = 2

···F

where F

, . . . , F

are distinct

primes of the form 2

)

+ 1.

After the 17-gon, the next prime-sided constructable n-gon has 257 = 2

+ 1 sides!

By 400 BC, the Greeks were referencing the second and third impossible constructions of antiquity:

1. Trisecting a general angle.

2. Doubling (the volume of) a given cube.

3. Squaring a circle (construct a square with the same area as a given circle).

It wasn’t until the advent of ﬁeld theory in the 1800s that such were proved to be impossible using

ruler-and-compass constructions.

Summary Several of the mathematical techniques in this section are difﬁcult, and the results tech-

nical. It isn’t important to become proﬁcient with all of these ideas! Instead play with them to help

develop an appreciation of two overarching points:

1. Even before Euclid, the focus of Greek mathematics was more abstract and less practical than

other ancient cultures (Egyptians, Babylonians, Chinese, etc.), largely due to the inﬂuence of

wider Greek philosophy and religion. The modern liberal arts ideal of learning for its own

sake—to celebrate the beauty of knowledge and to expand the mind—is, to a large extent, a

Greek inheritance.

2. The ancient Greeks pondered fundamental mathematical questions and concepts—number ver-

sus length, continuity, irrationality, inﬁnitesimals, constructions—ideas that have stimulated

mathematical research ever since. These particular issues would not rigorously be resolved un-

til the 1800s when luminaries such as Gauss, Cauchy and Riemann developed modern analysis

and algebra.

Theorem IV. 11 of the Elements presents a less practical construction. Ours follows from Theorem XIII. 10: if a regular

pentagon, hexagon and decagon are inscribed in a circle, then their sides form a right-triangle.

“You can’t square that circle” is now a metaphor for something that can’t be done.

Exercises 3.2. 1. Construct ﬁve Pythagorean triples using the formula



−1



where n is

odd. Construct ﬁve more using the formula





−1,





+ 1



where m is even.

2. Suppose 2

−1 = p is prime (its only positive divisors are itself and 1). List the positive divisors

of 2

n−1

−1) and hence prove Theorem IX. 36.

3. Draw a picture with dots to show that eight times any triangular number plus 1 makes a square,

and that any odd square diminished by 1 becomes eight times a triangular number. That is:

(a) 8 ·

m(m + 1) + 1 is a perfect square.

(b) If n is odd, then n

−1 = 8 ·

m(m + 1) for some m.

4. Find a construction (using the ruler-and-compass constructions) to bisect a given angle, and show

that it is correct.

5. Sketch a construction inscribing a regular hexagon in a circle.

(Assume you can construct an equilateral triangle on a given segment—Thm I. 1 of Euclid, pg. 27)

6. (A line-doubling paradox) One line has twice the length of another and so has more points.

However, there is a bijective correspondence between the points on these lines; the two lines

therefore have the same number of points.

Explain the second observation. How can you resolve the paradox?

7. The cycle of ﬁfths is a musical concept stating that twelve perfect ﬁfths equals seven octaves

(pg. 18). State this claim numerically, and show that it is a contradiction.

(Hint: two strings are seven octaves apart if their lengths are in the ratio 2

: 1)

8. We use modern language to resolve Zeno’s paradox of Achilles and the Tortoise. Suppose

Achilles travels at speed v

, the tortoise at speed v

< v

, and that the tortoise starts a distance

d ahead of Achilles.

(a) Prove that t





for each positive integer n.

(b) Compute

∑

∞

n=0

using the geometric series formula from calculus.

motion of Achilles relative to the tortoise.

9. Use Theaetetus’ deﬁnition of equal ratios to prove that 46 : 6 = 23 : 3.

10. (Hard) A line of length 1 is divided at x so that

1−x

. Prove that 1 and x are incommensu-

rable. Indeed, show that 1 : x is the same as diagonal : side of a regular pentagon.

(Hint: the ﬁrst line of the algorithm is 1 = x + x

. . . )

11. (Hard) Let a > b and c be positive lengths. Use Eudoxus’ deﬁnition to prove that c : b > c : a.

(Hint: let n be the smallest integer such that n(a −b) ≥ c; its existence is the “archimedean property”)

3.3 Euclid and the Elements

Euclid worked in the Library of Alexandria, named for the Greek gen-

eral Alexander the Great who conquered Egypt in 323 BC. The Library

was constructed around 320 BC as a means of organizing the knowl-

edge of the world and for the demonstration of Greek power. Al-

though it was seriously damaged on several occasions, the Library re-

mained a center of scholarship until around AD 500. Below is a map of

the city around AD 400: note the size and centrality of the Library.

It is hard to argue against Euclid’s Elements (c. 300 BC) as the most inﬂuential mathematics text ever

produced. Likely a compilation of earlier mathematical work rather than a pure original, it was

edited and added to over the centuries, eclipsing and subsuming other works. Particular import

were the edits of Theon of Alexandria (c. AD 400) and his daughter Hypatia, both proliﬁc scholars in

their own right. Due to edits such as these, the precise contents of the original are unknown.

Extant fragments date to around AD 100. The earliest (almost) complete copy is from the the 9

century; written in Greek and held at the Vatican, it is missing some of the edits of Theon & Hypatia,

thus demonstrating that multiple versions were in circulation.

Earliest Fragment c. AD 100 Full copy 9

Until the mid 20

century, some version of the Elements would have been used as a high-school

textbook in most western and middle-eastern countries. Many editions and variations have been

produced, four of which are shown below:

Latin translation, 1572 High School textbook, 1903

Pop-up edition, 1500s Color edition, 1847

You can download Byrne’s color edition here (very large ﬁle!). It is notably different from many

earlier editions: it contains a much longer list of deﬁnitions, inserts many more axioms, and relabels

propositions 4 and 5 as axioms (pages xviii–xxiii). The picture on the cover page is Euclid’s proof of

Pythagoras’ Theorem (Book I, Thm. 47).

A Brief Overview of the Elements

The Elements consists of thirteen books covering two- and three-dimensional geometry, computa-

tions and number theory. Whether some version of every book was written by Euclid himself is

unknown.

The key feature of the Elements is its axiomatic presentation. Each book begins with a list

of axioms/postulates and deﬁnitions and proceeds to prove theorems deduced from these. This ax-

iomatic method is essentially universal in modern mathematics, and its advent is fundamentally what

sets Greek mathematics apart from everything that came before.

We brieﬂy discuss Book I, then give some ﬂavor of the remainder of the text with a few example

results. Several examples of material from later books were mentioned in the previous section.

Book I Consists of 48 theorems, culminating with Pythagoras’ and its converse. It seems likely that

Euclid organized Book I with the goal of proving this important result in a rigorous manner: recall

(pg. 20) how the Pythagorean ‘proof’ relied on the erroneous notion of commensurability. Here are

the postulates from Book I: the ﬁrst three are what deﬁne ruler-and-compass constructions (pg. 22).

P1 Given any two points, a straight line can be drawn between them

P2 Any line may be indeﬁnitely extended

P3 Given a center and a radius, a circle may be drawn

P4 All right angles are equal to each other

P5 If a straight line crosses two others so that the angles on the same side make less than two right

angles, then the two lines meet on that side of the original.

The ﬁfth postulate is awkwardly phrased. An equivalent modern statement is Playfair’s axiom:

There is at most one parallel through a given point not on line.

For centuries, mathematicians tried to prove that this postulate was a theorem of the others until; it

was eventually shown to be necessary with the advent of hyperbolic geometry in the 1800s. Euclid’s

refusal to use the parallel postulate until Theorem 29 suggests he understood this awkwardness.

Theorems are generally presented as problems: a pictorial construction is provided, then Euclid proves

that the construction really solves the problem. Here is Euclid’s ﬁrst theorem.

Theorem (I. 1). Problem: To construct an equilateral triangle on a given segment.

Proof. Given construct two circles (P3)

Join one of the circle intersections to the endpoints of the original segment (P1)

The result is an equilateral triangle; indeed the three sides are congruent, for

are radii of a common circle, as are

It is not even certain that Euclid was a single person as opposed to a ﬁgurehead or a name for a collective. It is hardly

surprising that we know so little about someone who lived 2300 years ago. The same questions are sometimes raised about

William Shakespeare who lived only 400 years ago!

After this Euclid proceeds to establish several well-known results. Since this isn’t a geometry class,

we’ll omit most of the details. You can ﬁnd more of these here, in Byrne, or elsewhere.

Thm I. 4 Side angle side: if triangles have two pairs of congruent sides and the angles between them

are also congruent, then the remaining sides and angles are congruent.

Thm I. 15 Vertical angles: if two lines meet, then the opposite angles made are congruent.

Thms I. 27 & 29 Angles and parallels: if a line falls on two other lines, then the two lines are parallel

if and only if the alternate angles are congruent (α

∼

β in the picture).

Thm I. 15 Thms I. 27/29 Thm I. 32 Thm I. 41

Thm I. 32 Angle sums in a triangle: if one side of a triangle is protruded, the exterior angle equals

the sum of the opposite interior angles.

In the picture α + β = δ; in modern language α + β + γ = 180°.

Thm I. 41 A parallelogram and triangle on the same base and with the same height have area in the

ratio 2:1.

The last two results of Book I are Pythagoras and its converse.

Theorem (I. 47). The square on the hypotenuse of a right-triangle has area equal to the sum of the

areas of the squares on the remaining sides.

Proof. Given a right-angle at A, drop the perpendicular from A across

to L.

△FBC and □ABFG share the same base BF and height AB.

By Thm I. 41,

area(□ABFG) = 2 area(△BCF)

Similarly (base BD, height BO)

area(□BOLD) = 2 area(△ABD)

Side-Angle-Side (Thm I. 4) =⇒ △ABD

∼

△FBC; the trian-

gles have the same area, and so

area(□ABFG) = area(□BOLD)

Similarly area(□ACKH) = area(□OCEL).

LD E

The converse (Thm I. 48) is Exercise 3.

Book II Geometric solutions to problems that would now be treated using algebra. Much of this is

attributable to the Pythagoreans.

(Thm II. 11) A segment may be divided so that the rectangle contained by the whole and one

of the sub-segments is equal to the square on the remaining sub-segment.

We rephrase this in modern language. Suppose the given segment AB has length a, our goal is

to ﬁnd H on AB such that

= x and x

= a(a −x). Euclid is providing a geometric solution

to a quadratic equation!

1. Construct □ABDC on AB

2. Let E be the midpoint of AC and connect EB

3. Extend AC and lay off EF

∼

EB on AC extended

4. Construct □AFGH and drop the perpendicular to I

5. To ﬁnish: prove that area(□AFGH) = area(□BDIH)

The solution x =

√

5−1

a means that AH : HB is the

golden ratio.

a − x

Book III Theorems regarding circles and tangency. Most of this material

likely came from Hippocrates.

(Thm III. 31) Thales’ Thm: a triangle in a semi-circle is right-angled.

Book IV Construction of regular 3, 4, 5, 6 and 15-sided polygons inscribing and exscribing a circle.

Also likely the work of Hippocrates.

Book V Ratios and magnitudes

a la Eudoxus.

(Thm V. 11) If a : b = c : d and c : d = e : f then a : b = e : f

Book VI Ratios of magnitudes applied to geometry (similarity results).

(Thm VI. 4) Triangles with equal angles have corresponding sides proportional.

(Thm VI. 8) The altitude from the right angle of a right triangle divides it into two triangles

similar to each other and the the original.

(Thm VI. 31) Corrected Pythagorean proof (pg. 20) of Pythagoras’ using the proportions of Eu-

doxus and Thm VI. 8.

Book VII Divisibility and the Euclidean algorithm. Probably due to the Pythagoreans and Theaete-

tus.

Book VIII Number progressions, geometric sequences. Possibly due to studies in music by Archy-

tas (a Pythagorean who taught Plato mathematics).

Book IX Number Theory: even/odd + perfect numbers.

(Thm IX. 20) There are inﬁnitely many primes.

Book X Discussion of commensurable and incommensurable ratios. Long and difﬁcult, possibly

derived from Theaetetus.

Book XI Solid geometry (lines/planes in 3D).

(Thm XI. 28) A parallelepiped is bisected by its diagonal plane.

Book XII Ratios of areas and volumes (Eudoxus).

(Thm XII. 2) The areas of circles are in the same ratio as the squares on their diameters.

Book XIII Construction of regular polyhedra inside a sphere and their classiﬁcation.

(Thm XIII. 10) If a regular pentagon, hexagon and decagon are inscribed in the same circle,

then their sides form a right-triangle.

One could study the Elements and its inﬂuence for a lifetime and not be done! Hopefully this very

brief overview convinces you why the book had such a profound impact on mathematics.

Exercises 3.3. 1. Prove Thales’ Theorem (III. 31) (pg. 29).

(Hint: start by joining the center of the circle to the apex of the triangle. . . )

2. Use the picture to provide a proof of Thm I. 32: the sum of the three

interior angles of a triangle is equal to two right angles.

Show that the proof depends on Thm I. 29, and therefore on the

parallel postulate.

(This isn’t quite the same as Euclid’s argument)

3. Suppose that the square on side BC of △ABC has the same area

as the sum of the squares on the other sides AB, AC. As in the

picture, draw a perpendicular AD

∼

AB.

(a) Explain why DC

∼

BC.

(b) Hence conclude that △ABC is right-angled at A.

4. Prove Thm III. 3: A diameter of a circle bisects a chord if and only if it is perpendicular to the

chord.

5. Verify that Euclid’s construction for Thm II. 11 really does solve the given problem.

(You can use modern algebra!)

6. Draw a semi-circle with diameter 9 + 5 = 14. Solve the equation

geometrically, by

constructing a vertical line whose length is x.

7. Show that areas of similar segments of circles are proportional to the squares of the length of

their chords.

(You may assume that areas of circles are proportional to the squares on their diameters and can use

modern algebra/trigonometry if you wish)

3.4 Archimedes of Syracuse

Archimedes (287–212 BC) is arguably the greatest ancient mathematician. Syracuse is on the island

of Sicily at the foot of the Italian peninsula; at the time of Archimedes’ birth this was a Greek city-

state, though under threat from the expanding Roman Empire. Archimedes famously helped defend

Syracuse against the Romans using catapults, though he ultimately died at their hands after the city

fell. He is believed to have travelled to Alexandria in his youth and perhaps studied with scholars at

the library, including Eratosthenes (pg. 34).

Archimedes’ genius was practical not just mathematical. Beyond his anti-Roman catapults, he is

credited with a large number of inventions and technical innovations, including Archimedes’ screw,

still used in modern irrigation systems to elevate water. He is acknowledged as the founder of hydro-

statics, where Archimedes’ principle states that an object immersed in water loses weight equal to that

of the displaced water. A famous story recounts Archimedes using this to detect whether a smith

had used all the gold he had been given in the manufacture of a crown.

Example A cube with side length 20 cm ﬂoats such that the water-

line is halfway up the cube. By Archimedes’ principle, the weight of

the cube is the same as that of the volume of displaced water:

20 ×20 × 10 = 4000 cm

which has a weight (mass) of roughly 4 kg.

20 cm

Levers Archimedes made great study of levers, both for practical purposes and as a method of

calculation.

Given masses M

, M

located distances r

, r

from a pivot, Archimedes states:

• The lever balances ⇐⇒ M

: M

= r

: r

• The lever rotates clockwise ⇐⇒ M

: M

< r

: r

• The lever rotates counter-clockwise ⇐⇒ M

: M

> r

: r

In modern terms we’d compare the torques τ

= M

and τ

= M

. Since torque requires the

multiplication of non-numerical quantities, Archimedes would instead have considered this using

Eudoxus’ theory of proportions.

For example, to ﬁnd the mass M

required to balance a lever given M

= 12 lb, r

= 4 ft and r

= 3 ft,

Archimedes would have observed that

: M

= 4 : 3 =⇒ M

= 16 lb

The Method: is Archimedes the founder of calculus? A previously unknown work of Archimedes

was discovered in 1899. As an amazing application of the lever principle, Archimedes makes an ar-

gument that looks remarkably like modern calculus; he could be claimed to be its earliest practitioner

by 1800 years! The method was outlined in a letter to Eratosthenes and includes part of an argument

for proving Archimedes’ favorite theorem, a picture of this result was engraved on his tomb.

Theorem. A cone, hemisphere and cylinder with the same base and height have volumes in the

ratio 1 : 2 : 3. Using modern formulæ, if the height is r, then the volumes are

πr

: πr

Here is a modernized version illustrating Archimedes’ approach.

Suppose the ‘base’ is a disk with radius 1, remove the hemisphere

from the cylinder and place the cone beneath. Compare the cross-

sections the same distance y from the apex of the cone.

• The circular cross-section of the cone has radius y whence its

area is proportional to the square on the radius: πy

• The upper annular cross-section has area proportional to the

difference of the squares on the radius of the cylinder and on

the distance x. By Pythagoras’ the cross-sectional area is

π(1

− x

) = π(1 − (1 − y

)) = πy

The cross-sections are therefore in balance with respect to a vertical lever whose pivot lies at the

center of the picture. Archimedes concludes that the cone and the upper-ﬁgure are in balance: that is

cone

= V

cylinder

−V

hemisphere

Combining this with the fact that V

cylinder

= 3V

cone

(e.g., Exercise 7d) gives the desired result.

Here is another argument of Archimedes’ with a suggestion of calculus. A disk comprises inﬁnitely

many concentric circles, the circumference of each being proportional to its radius. ‘Unwind’ these

circles to obtain a triangle; one side is the radius of the disk, the other its circumference. The area of

a circle is therefore that of a triangle with sides the radius and circumference of the circle: A =

rc.

The Method includes several of these calculus-like discussions. While efﬁcient, Archimedes felt that

his approach didn’t constitute a proof and provided alternative arguments elsewhere in his writings.

The essential problem is this;

Can we really say that an area equals its cross-sectional lines? Or that a volume equals its

cross-sectional areas? Lines have no width so if we add them up we have no area. If they

have width, then inﬁnitely many of them have inﬁnite area.

These are really variations of Zeno’s paradoxes (pg. 22) regarding inﬁnitesimals and indivisibles!

Archimedes’ arguments would be resurrected in the early 1600s by Cavalieri and Galileo as the de-

velopment of calculus gathered pace. The same duality of presentation characterised this later devel-

opment: Newton and others found the inﬁnitesimal approach efﬁcient, but felt the need to present

geometric proofs to convince readers that their results weren’t mere trickery.

It is tempting to imagine what might have happened if Archimedes’ method had been accepted and

preserved as part of the Greek canon; if calculus had been developed 1800 years earlier, how might

this have affected technological development? Would the space-race have happened in AD 500?!

Quadratures Archimedes also approximated areas and arc-lengths of various ﬁgures using limit-

like argumentation. Here is how he approached the area/circumference of a circle.

1. Inscribe a regular hexagon in a circle (of radius 1 say) and compute its perimeter (6).

2. Halve each angle to obtain a regular dodecagon: compute its perimeter (12

2 −

√

3).

3. Repeat the angle-halving process: Archimedes did this with 24-, 48- and 96-gons to obtain an

increasing sequence of perimeters bounded above by the circumference of the circle (2π).

4. Repeat the same calculation with circumscribed polygons to obtain a decreasing sequence of

over-estimates.

5. Using 96-sided polygons allowed Archimedes to obtain the estimate 3

< π < 3

Archimedes’ halving process relied on an induction step, an ap-

proximation of which we mimic here. Suppose we have an

isosceles triangle with equal legs 1, altitude d

, and chord 2h

We halve the angle to ﬁnd the new altitude d

n+1

and chord 2h

n+1

Everything follows from three applications of Pythagoras’:

1 = d

+ h

(2h

n+1

)

= h

+ (1 −d

)

1 = d

n+1

+ h

n+1

Expanding and cancelling, we obtain

n+1

(1 + d

), h

n+1

= 1 − d

n+1

(1 −d

)

n+1

1 − d

Since d

√

and h

, we may compute the entirety of both sequences:

2 +

√

3, d

2 +

√

3, . . . d

2 +

2 + ··· +

2 +

√

2 −

√

3, h

2 −

2 +

√

3, . . . h

2 −

2 + ··· +

2 +

√

where the n

terms have n copies of the digit 2 under the square-root. The circumference and area

of the 6 ·2

-sided polygon inscribed in the circle are therefore

= 12 · 2

, A

= 6 · 2

n−1

These sequences increase to 2π and π respectively. For a 96-sided polygon, Archimedes would have

had to approximate

= 12 · 2

= 96

2 −

2 +

√

3 > 6.282 =⇒ π > 3.141 > 3

Other Highlights of the later Greek Period: 300 BC–AD 500

We’ll consider ancient astronomy, including Greek contributions, in the next chapter. Here are a few

of the other developments of the late Greek period and some historical context.

• Eratosthenes (276–194 BC) grew up in Cyrene (c. 500 miles west of Alexandria in modern-day

Libya) and moved to Alexandria in adulthood to become its librarian. He is credited with a

simple algorithm for ﬁnding primes: the Sieve of Eratosthenes.

– List the integers n ≥ 2.

– Leave 2 and delete all its multiples.

– Leave 3 and delete its multiples.

– Repeat ad inﬁnitum: each time one reaches a number, leave it and delete its multiples.

– The remaining list contains all the primes.

• Apollonius (225 BC) writes an eight-volume book on conic sections building on earlier work of

Menaechmusus (350 BC).

• By 146 BC the Greek empire had fallen under Roman rule. Alexandria remained important.

Educated Greeks still spoke and wrote in Greek rather than (Roman) Latin. For context, Julius

Caesar ruled Rome around this time (died 44 BC).

• Heron (AD 75) proves the formula

s(s − a)(s −b)(s − c) for the area of triangle, where s =

(a + b + c) is the semi-perimeter. This was likely known to Archimedes; Heron’s work was a

compilation of earlier mathematics.

• Around AD 100 the Neopythagorean’s worked in Alexandria, studying music, philosophy, and

number, with the intent of reviving the teachings of Pythagoras.

• Around AD 400, Theon and Hypatia produce the most widely-read edition of Euclid’s Elements

as well as improving upon several earlier mathematical topics.

• In AD 395 the Roman empire split into eastern and western parts centered on Rome and Byzan-

tium/Constantinople. The western empire rapidly declined under the pressures of corruption

and barbarian attacks, collapsing completely by AD 500. Alexandria experienced riots and a

bloody power-struggle (Hypatia was murdered by a mob in 415) and the library of Alexandria

was severely damaged and possibly destroyed at this time. In 642, Alexandria was captured

by the new Islamic caliphate. Much of the material in the library survived by being copied and

transported to various places of learning; particularly Constantinople and Baghdad. For the

next 600 years, the knowledge of Alexandria was largely a mystery to (western) Europe.

Exercises 3.4. 1. If a weight of 8 kg is placed 10 m from the pivot of a lever and a weight of 12 kg is

placed 8 m from the pivot in the opposite direction, toward which weight will the lever incline?

Answer using Archimedes’ language.

2. Use Eratosthenes’ Sieve to ﬁnd all the primes < 100.

3. (a) Prove Heron’s formula as follows.

i. Let h be the altitude and x the base of the left-hand right-

triangle. Apply Pythagoras’ to the two right-triangles to

show that

x =

+ b

−c

ii. Substitute in h

= b

−x

to ﬁnd h in terms of a, b, c and thus deduce Heron’s formula.

(b) Find the area of a triangle with sides 4, 7 and 10.

4. Suppose C

is the circumference of a 6 ·2

-sided inscribed polygon in a unit circle. Show that

the circumference of the corresponding circumscribed polygon is C

5. Use the modern formula A =

ab sin C to prove that, for any k ∈ N

k sin

2π

< π < k tan

Moreover, explain why both sides converge to π.

6. Instead of modern algebra, Archimedes used several geometric lemmas to help ﬁnd the areas

of polygons inscribed in and circumscribing circles. Here is one; prove it!

Let OA be the radius of a circle and AC be tangent to the circle at A. Let D lie on AC such that

OD bisects ∠COA. Then

and

(Hint: draw a picture and let T be the intersection of the circle and

OC)

7. Archimedes used a geometric series approach to evaluate the area inside a parabola. Use mod-

ern algebra for this question.

(a) Suppose y = a + bx + cx

is the equation of a parabola.

If P, Q, R have x co-ordinates in an arithmetic sequence

x − ϵ, x, x + ϵ, show that the area of △PQR is A =

;

independent of x!

(b) With reference to the picture, explain why the areas of the

labelled triangles satisfy A

parabola bounded by PR is

(d) How might this result have been applied to show that the volume of a cone equals

that

of a cylinder with the same base and height?

4 Ancient Astronomy & Trigonometry

Throughout history, astronomy has been (and remains) an important driver of mathematical devel-

opment. In particular, early trigonometry (triangle-measure) was largely developed to facilitate astro-

nomical computations, for which there are many practical beneﬁts: for instance,

Calendars The phases of the moon (whence month), the seasons, and the solar year are paramount.

Without an accurate calendar, 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). However, for most of human history the two were insepa-

rable. In light-polluted modern cities, it is hard to imagine the signiﬁcance the night sky held for

our ancestors, even a couple of centuries ago. Almost all religions imbue the heavens with mean-

ing; understanding and predicting heavenly movements provided a massive historically 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 (page 37).

Such alignments are also 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 pre-1000 BCBabylon. A tablet dated to 686 BC de-

scribes 60–70 constellations and stars with aspects familiar to modern astrologers, including

Taurus, Leo, Scorpio and Capricorn. During the same period Chinese and Indian astronomers

developed different systems of constellations.

• 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.

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).

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 Solstices (≈ 21

June/December) The north pole is maximally tilted towards/away

from the sun. Solstice comes from Latin meaning ‘sun stationary.’ The location of the ris-

ing/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 Cancer/Capricorn) are

the lines of latitude where the sun is directly overhead at noon on one solstice.

Vernal/Autumnal Equinoxes (≈ 21

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.

Astronomical measurements must be 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

); 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.

Modern astronomers typically measure latitude and longitude (declination/right-ascension) with respect to the earth’s

equatorial plane rather than the ecliptic. Such equatorial co-ordinates are ﬁrst known to have been introduced by Hipparchus

of Nicaea (page 40). Right-ascension is measured in hours-minutes-seconds rather than degrees, where 24 hours = 360°,

though modern scientiﬁc practice is to use decimals rather than sexagesimal minutes and seconds.

The Circumference of the Earth

One of the earliest problems in astronomy is to ﬁnd the size of the earth. Eratosthenes of Cyrene

(c. 200 BC, page 34) performed one of the ﬁrst accurate estimations by measuring 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

′

·360° at Alexandria.

• The circumference of the earth is therefore approxi-

mately 50 ·5000 = 250, 000 stadia.

rays

sun’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, and the celestial sphere rotates around it once per day.

• The planets lie on concentric spherical shells centered on the earth.

When such assumptions are tested by observation, two major contradictions appear:

Variable brightness The apparent brightness of heavenly bodies, particularly planets, is non-constant.

Retrograde motion Planets mostly follow the east-west motion of the heavens, though sometimes they

are seen to slow down and reverse course.

If planets move at constant speed around circles centered on the earth, how can these observations

be explained? Attempts to produce accurate models while preserving spherical/circular motion led

to the development of new mathematics.

One early approach is due to Eudoxus of Knidos (c. 370 BC, page 21), who 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, with the outermost sphere being that of the ﬁxed stars. The

motion generated by such a model

is highly complex. Eudoxus’ approach is capable of producing

retrograde motion, but not the variable brightness of stars and planets.

The link is to a very nice ﬂash animation of Eudoxus’ model that would have been far beyond Eudoxus’ ability to

visualize and measure.

Epicycles & Eccentric Orbits Apollonius of Perga (2

C. BC) is most

famous for his study of conic sections, but is relevant here for developing

two models of solar/planetary motion.

In the 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 might have been complex and im-

practical, but was more in line with the assumptions 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 these if you’ve played with the

toy Spirograph. An observer at the center sees the apparent bright-

ness 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 arose.

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’ proposal: parallax.

distant stars

nearer star

Earth and Sun

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.

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).

The astronomical unit of one parsec is the distance to a star exhibiting one arc-second (

3600

°) of parallax: roughly 3.3

light-years or 3 ×10

km, an unimaginable distance to anyone before the scientiﬁc revolution. Since the nearest star to our

sun, Proxima Centauri, lies 4.2 light years = 0.77 parsecs away, the rejection of Aristarchus’ hypothesis is understandable!

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 accurately compute chords of circles. His chord tables are acknowledged as the

earliest lists 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. In modern language

crd α = 2r sin

crd α

crd(180

◦

−α)

Hipparchus chose a circle with circumference 360° (in fact he used 60 · 360 = 21600 arc-minutes),

whence r =

21600

2π

≈ 57, 18; (base-60). Note that this is sixty times the number of degrees per radian.

His chord table was constructed starting with two obvious values:

crd 60° = r = 57, 18; crd 90° =

√

2r = 81, 2;

Since (Thales Theorem) the large triangle is right-angled, the Pythagorean theorem can be used to

obtain chords for angles 180° − α. In modern language

crd(180° − α) =

(2r)

−(crd α)

= 2r

1 −sin

(α/2) = 2r cos

Pythagoras was again used to halve and double angles in an approach analogous to Archimedes’

quadrature of the circle (page 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

= crd(180° −2α). Since ∠BDA = 90°,

it follows that BD is parallel to OM and so

crd(180° − 2α)

Now apply Pythagoras to △CMD:

(

crd α

)



crd 2α





r −

crd(180° − 2α)



(

)

(crd 2α)

+ r

−r crd(180° −2α) +

crd(180° − 2α)

(crd 2α)

+ r

−r crd(180° −2α) +

(4r

−(crd 2α)

)

= 2r

−r crd(180° −2α) = 2r

−r

−(crd 2α)

In modern notation this is one of the double-angle trigonometric identities!

sin

= 2r

−2r

cos α ⇐⇒ cos α = 1 −2 sin

One radian is the angle subtended by an arc equal in length to the radius of the circle. Hipparchus essentially does this

in reverse: the circumference is ﬁxed so that degree now measures both subtended angle and circumferential distance.

Example To calculate crd 30°, we start with crd 60° = r. Then

crd 120° =

−r

√

3 r

=⇒ crd 30° =

−r crd(180° −60°) =

−

√

2 −

√

3 r

In modern language this yields an exact value for sin 15°:

crd 30° = 2r sin 15° =⇒ sin 15° =

2 −

√

Continuing this process, we obtain crd 150° =

2 +

√

3 r, whence

(crd 15°)

= 2r

−r crd 150° =



2 −

2 +

√



=⇒ crd 15° =

2 −

2 +

√

3 r

Again translating: sin 7.5° =

2 −

2 +

√

In similar fashion, Hipparchus computed the chords of 7.5°, 15°, . . . , 172.5°, in steps of 7.5°. Of course

everything was an estimate since he had to rely on repeated approximations for square-roots. All

Hipparchus’ original work is lost, so we know of his approach only by reference. In particular, while

the above method is probably due to Hipparchus, 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

° 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.

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)

Syene (page 38) is almost exactly on the Tropic of Cancer.

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

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 (this last probably courtesy of Menelaus c. AD 100). It contains our best evi-

dence as to the accomplishments of Hipparchus and describes his calculations. The Almagest formed

the basis of Western/Islamic astronomical theory well into the 1600s.

Ptolemy’s Calculations Ptolemy used several innovations to compute more chords at greater accu-

racy 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

10 −2

√

Halving/Doubling Angles Ptolemy used what was probably Hipparchus’ method:

crd

α = 2r

−r crd(180° −2α) = 60



120 −crd(180° − 2α)



crd(180° − α) =

(2r)

−crd

α =

120

−crd

approximating square-roots to the desired accuracy. For example,

crd 30° =

60( 120 −crd 120°) =



120 −60

√



= 60

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 multiple-

angle/addition formulæ.

Interpolation The observation α < β =⇒

crd β

crd 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 (

°). 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.

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 must have been a gargantuan task, one for which Ptolemy likely had much assis-

tance.

Ptolemy (Ptolemaeus) is a Greek name, while Claudius is Roman, reﬂecting the changing cultural situation in Egypt.

This is 1 +

= 1.0472222 . . . = 120 sin

1.00003625...

°, an already phenomenal level of accuracy.

How did Ptolemy know exact values for crd 36° and crd 72°? Everything is in Euclid’s 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 =

√

5 −1

In a circle of radius 60, this gives the exact value

crd 36° = 60x = 30(

√

5 −1)

◦

2. Now let AD = y be the side of a regular pentagon inscribed in the same circle. Applying

Pythagoras, we see that







1 − x



= x

Since x

= 1 − x, this multiplies out to give Euclid’s result

= 1

+ x

from which we obtain the exact value

crd 72° = 60y = 30

10 −2

√

While these values were geometrically precise, Ptolemy used sexagesimal approximations to square-

roots to obtain the values stated in his tables:

crd 36° = 37; 4, 55 crd 72° = 70; 32, 3

He must have used a far higher degree of accuracy in order to obtain similarly accurate values for

other chords.

Angle-addition & the Multiple-angle Formula Computation of crd(α ± β) was facilitated by ver-

sions of the multiple-angle formulæ 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.

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:

△ABE ∼ △DBC △ABD ∼ △EBC

The proof follows immediately: since

and

, we have

= (

)

Corollary. If α > β, then

120 crd(α − β) = crd α crd(180° − β) − crd β crd(180° − α)

In modern language, divide through by 120

to obtain

sin

α − β

= sin

cos

−sin

cos

Proof. If

= 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 modern multiple-angle formulæ for sin(α ± β) and cos(α ± β).

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).

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

10 −2

√

5 is known, and

crd 108° = crd(180° −2 ·36°) = 120 −

crd

36° = 30(1 +

√

we see that

crd 42° =

120



10 −2

√

5 ·60

2 +

√

3 −60

2 −

√

3 ·30(1 +

√



= 15



10 −2

√

5 ·

2 +

√

3 − (1 +

√

2 −

√



≈ 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 tells us 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)

10 + 2

√

≈ 0.32491

This is precisely cot 72°, though Ptolemy had no such notion.

◦

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

in terms of the sine function. What

facts about

sin 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

cos

(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)

5 Ancient Chinese Mathematics

Documented civilization Dates from c. 3000 BC. Until around

AD 200, China refers roughly to the area in the map: a small

fraction of the modern state’s territory north of the Yangtze

and centered around the Yellow river.

Earliest mathematics Oracle Bone enumeration dates from the

Shang dynasty (c. 1600–1046 BC) commensurate with the

earliest known Chinese character script. Most information

on the Shang comes from later commentaries, though origi-

nal oracle bones have been excavated, particularly from the

ancient capital Anyang. Astronomy, the calendar and trade

were dominant drivers of mathematical development.

Zhou dynasty (1046–256 BC) and the Warring States period (475–221 BC) Many mathematical texts

were written, though most have been lost; their content must be inferred from later commen-

taries. Rapid change created pressure for new systems of thought and spurred technological

development. Feudal lords employed philosophers, of whom the most famous was Confucius

(c. 500 BC).

Technological developments included the compass (for navigation) and the use of

iron in warfare.

Later history and expansion Between the 221 BC victory of the Qin Emperor Shi Huang Di

and the

forced abdication (at the age of 6) of the last Qing Emperor Puyi in 1912, China was ruled by

a succession of dynasties. By the end of the Qing, Chinese territory had expanded to roughly

its modern borders. The Chinese Civil War (1927–1949) resulted in victory for the communists

under Mao Zedong and the foundation of the modern Chinese state. While this simple de-

scription might suggest a long calm in which culture and technology could develop in comfort,

in reality the empire experienced many rebellions, schisms and ﬂux, often exacerbated by the

changing whims of emperors and later leaders.

Transmission of knowledge East Asia (modern China, Korea, Japan, etc.) is geographically sepa-

rated from other areas of early civilization by tundra, desert, mountains and jungle. During

the Han dynasty (c. 200 BC–AD 220) a network of trading routes known as the silk road was

established, connecting China, India, Persia and Eastern Europe; the Great Wall was in part

constructed to protect these trade routes. Geographical separation meant that trade was lim-

ited, and there is little evidence of mathematical and philosophical ideas making the journey

until many centuries later. For instance, there is no evidence of sexagesimal notation being

used in China, suggesting that Babylonian and Greek astronomy did not travel eastwards be-

yond India. Similarly, certain eastern mathematical ideas such as matrix-style calculations saw

no analogue in the west until many centuries later. There are, however, indications that early

decimal calculations in India may have been inspired by the Chinese counting board approach.

On balance, it seems reasonable to conclude that Chinese and Mediterranean mathematics de-

veloped essentially independently.

Confucius was an adviser to Lu, a vassal state of the Zhou. Confucianism emphasises stability and unity as a counter

to turmoil. Taoism, the competing contemporary philosophical system, is more comfortable with change and adaptation.

Very loosely these were the conservative and progressive political philosophies of their day.

Sun Tzu’s military classic The Art of War dates from this time.

Famous for book-burning, rebuilding the great walls, and for the Terracotta Army of Xi’an.

Early Mathematical Texts

Zhou Bi Suan Jing (The Mathematical Classic of the Zhou Gnomon

and the Circular Paths of Heaven)

The oldest suspected Chinese mathematical work was likely compiled some time in the period

500–200 BC. Largely concerned with astronomical calculations, it was presented in the form

of a dialogue between the 11

century Duke of Zhou (of I Ching fame) and Shang Gao (one

of his ministers, and a skilled mathematician). It contained perhaps the earliest statement of

Pythagoras’ Theorem as well as simple rules for fractions and arithmetic.

Suanshu Shu (A Book on Arithmetic) Compiled around 300–150 BC, it covered topics such as fractions,

the areas of rectangular ﬁelds, and the computation of fair taxes.

Jiu Zhang Suan Shu (Nine Chapters on the Mathematical Arts) Written between 300 BC and AD 200, this

the most famous ancient Chinese mathematical text. Many topics are covered, including square

roots, ratios (false position and the rule of three

), simultaneous linear equations, areas and

volumes, right-angled triangles, etc. The Nine Chapters was hugely inﬂuential, in part due to

the detailed commentary and solution manual to its 246 problems written by Liu Hui in AD 263.

Several of our examples below come from Liu’s work.

These texts typically involved worked examples with wide application. There is no notion of ax-

iomatics on which one could construct a modern-style proof.

The Gao Gu Unsurprisingly, the Chinese do not attribute Pythagoras’ Theorem to the Greeks: its

name instead refers to the shorter and the longer of the two non-hypotenuse sides of the triangle.

Here is an early mention of the gao gu. Is this a

‘proof’? Is it a claim about all right-triangles, or

merely an observation of the triple (3, 4, 5)? It can

be made rigorous (see Exercise 4), but it is unclear

whether this was the intention of the author.

Another example describes how to ﬁnd the diameter of the circle

inscribed in a right-triangle with gao 8 and gu 15. A picture was

drawn and the answer stated:

d =

2 ·8 · 15

8 + 15 + 17

= 6

Here is a modern explanation. Given a = 8 and b = 5, the hypotenuse is c =

√

+ 15

= 17, and

the area of the large triangle is the sum of three smaller triangles, each having height

ab =

a ·

d +

b ·

d +

c ·

d =⇒ d =

2ab

a + b + c

Again we ask: is this a general method or an example?

Gnomon: “One that knows or examines.” Also refers to the elevated piece of a sun/moondial.

Given equal ratios a : b = c : d, where a, b, c are known, then d =

The Bamboo Problem Here is another problem from the Nine

Chapters, as depicted in Yang Hui’s 1262 Analysis of the Nine Chap-

ters.

A bamboo has height 10 chi. It breaks and the top touches the

ground 3 chi from the base of the stem. What is the height of the

break?

In modern language: if a, b, c are the sides of the triangle with

hypotenuse c, we know b + c = 10 and a = 3; we want b. The

solution given is

b =



10 −



chi

Think about why!

The Out-In Principle Liu made many other contributions to

mathematics, including estimating π in a manner similar to

Archimedes. He made particular use of the out-in principle for

comparing area and volume:

1. Area and volume are invariant under translations.

2. If a ﬁgure is subdivided, the sum of the areas/volumes of

the parts equals that of the whole.

These are essentially axioms for area/volume in Euclidean ge-

ometry. For instance, Liu gave the argument shown in the pic-

ture in justiﬁcation of the gao gu: the large square is subdivided

and the in pieces A

, B

, C

translated to new out pieces A

, B

, C

to assemble the required squares.

Liu extended the out-in principle to analyze solids, comparing the volumes of four basic solids:

• Cube (lifang)

• Right triangular prism (qiandu)

• Rectangular pyramid (yangma)

• Tetrahedron (bienuan)

These could be assembled to calculate the volume of, say, a truncated pyramid:

Chinese Enumeration

The ancient Chinese had two parallel systems of enumeration. Both are essentially decimal.

Oracle Bone Script and Modern Numerals The earliest Chinese writing, oracle bone script, dates

from around 1600 BC. The numbers 1–10 had distinct symbols, as did 20, 100, 1000 and 10000. These

were decorated to denote various multiples. Some examples are shown below.

Given all the possibilities for decoration, the system is complex and more advanced than other con-

temporary systems. Modern Chinese numerals are a direct descendant of this script:

Observe the similarity between the expressions for the ﬁrst 10 digits. The second image denotes 842,

where the second and fourth symbols represent 100’s and 10’s respectively (literally eight hundred four

ten two). A zero symbol is not required as a separator: one could not confuse 205 (two hundred ﬁve)

with 250 (two hundred ﬁve ten). The system is partly positional: for instance the symbol for 8 can also

mean 800 if placed correctly, but only if followed by the symbol for 100.

Rod Numerals and the Counting Board The second dominant form of enumeration dates from

around 300 BC and was in wide use by AD 300. Numbers were denoted by patterns known as zongs

and hengs: zongs represent units, 100’s, 10000’s, etc., while hengs were for 10’s, 1000’s, 100000’s, etc.

Rod numerals were immensely practical—in extremis they could easily be scratched in the dirt! More

commonly, short bamboo sticks or counting rods—of which any merchant would carry a bundle—

would be used in conjunction with a counting board: a grid of squares on which sticks could be placed

for ease of calculation. This technology facilitated easy trade and gave rise to several methods of

calculation which will seem familiar. There was no need for a zero in this system as an empty space

did the job. Variations of the rod numeral system persisted in China morphing into the Suzhou system

which can still be found in some traditional settings.

Basic Counting Board Calculations Addition and subtraction are straightforward by carrying and

borrowing in the usual way. The smallest number was typically placed on the right. Multiplication

is a little more fun. Here we multiply 387 by 147.

3 8 7

1 4 7

Arrange rods: we use modern numerals for clarity

3 8 7

4 4 1

1 4 7

3 ×147 = 441, note the position of 147

8 7

4 4 1

1 1 7 6

1 4 7

Delete 3, move 147 and multiply: 8 × 147 = 1176

8 7

5 5 8 6

1 4 7

Sum rows

5 5 8 6

1 0 2 9

1 4 7

Delete 8, move 147 and multiply: 7 × 147 = 1029

5 6 8 8 9

1 4 7

Sum rows: in conclusion, 387 ×147 = 56889

The algorithm is just long-multiplication, starting with the largest digit (3) instead of the units as is

more typical in Western education.

Division is similar to long-division. To divide 56889 by 147 one might have the following sequence

of boards

5 6 8 8 9

1 4 7

−→

5 6 8 8 9

1 4 7

−→

3 8

1 2 7 8 9

1 4 7

−→

3 8 7

1 0 2 9

1 4 7

In the ﬁrst two boards, 147 goes 3 times into 568.

In board 3, we subtract 3 × 147 from 568 to leave 127, shift 147 one place to the right, and observe

that 147 goes 8 times into 1278.

In the ﬁnal step we have subtracted 8 ×147 from 1278 to leave 102, before shifting 147 to its ﬁnal

position on the right. Since 147 divides exactly seven times into 1029, we are done.

There is nothing stopping us from dividing when the result is not an integer; one simply continues

as in long-division, with fractions represented as decimals.

Simultaneous linear equations The coefﬁcients of a linear system were placed in adjacent columns

and then column operations were performed. The method is identical to what you learn in a linear

algebra class, but with columns rather than rows. Here is an example.

(

3x + 2y = 7

2x + y = 4

−→

3 2

2 1

7 4

−→

1 2

1 1

3 4

−→

1 1

1 0

3 1

−→

0 1

1 0

2 1

−→ x = 1, y = 2

This matrix method was essentially unique to East Asia until the 1800s.

Euclidean algorithm The counting board lent also itself to the computation of greatest common

divisors, which were used for simplifying fractions. Here is the process applied to

35 35 35 14 14 7

91 56 21 21 7 7

At each stage, one subtracts the smaller number from the larger. Once the same number is in each

row you stop. You should recognize the division algorithm at work! Since gcd(35, 91) = 7, both

could be divided by 7 to obtain

in lowest terms.

Negative numbers There is a strong case that the Chinese are the oldest adopters of negative num-

bers, though these were not originally thought of as such. Different colored rods were used to denote

a deﬁciency in a quantity, commonly when balancing accounts. The Nine Chapters describes using red

and black rods in this manner. This practice was known by AD 1, roughly 500 years before negative

numbers were used in calculations in India. It is possible that there was some transference of this

idea from China to India.

Music, Mysticism and Approximations Like the Pythagoreans, the ancient

Chinese were interested in music and pattern for mystical reasons. While the

Pythagoreans delighted in the pentagram, the Chinese created magic squares

(grids whose rows, columns and diagonals sum to the same total) as symbols

of perfection.

8 3 4

1 5 9

6 7 2

3 ×3 magic square

The notion of equal temperament in musical tuning was ﬁrst ‘solved’ in China by Zhu Zaiyu (1536–

1611), some 30 years before Mersenne & Stevin established the same idea in Europe. This required

the computation of the twelfth-root of 2 which Zhu found using approximations for square and cube

roots:

√

2 =

√

Zhu’s approximation was correct to 24 decimal places! Indeed the Chinese emphasis on practicality

meant that they often had the most accurate mathematical approximations of their time:

• Approximations to π including

√

10,

355

113

377

120

. Most accurate in the world from 400–1400.

• Methods for approximating square and cube roots were found earlier than in Europe. Ap-

proximations to solutions of higher-order equations similar to the Horner–Rufﬁni/Newton–

Raphson method were also discovered earlier.

• Pascal’s triangle ﬁrst appeared in China around 1100. It later appeared in Islamic mathematics

before making its way to Europe.

Two Famous Problems

We ﬁnish with two famous Chinese problems. The ﬁrst is known as the Hundred Fowl Problem and

dates from the 5

century AD. It was copied later in India and then by Leonardo da Pisa (Fibonacci)

in Europe, thus demonstrating how some Chinese mathematics travelled westwards.

If cockerels cost 5 qian (a copper coin), hens 3 qian, and 3 chicks cost 1 qian, and if 100

fowl are bought for 100 qian, how many cockerels, hens and chicks are there?

In modern language, we want non-negative integers x, y, z satisfying

(

5x + 3y +

z = 100

x + y + z = 100

The stated answers are (4, 18, 78), (8, 11, 81), (12, 4, 84) while the solution (0, 25, 75) was ignored.

Finally we consider the Chinese Remainder Theorem for solving simultaneous congruence equations.

This result dates from the 4

century AD, after which it travelled to India where it was described

by Bhramagupta, and thence to Europe. This example comes from Qin Jiushao’s Shu Shu Jiu Zhang

(Nine Sections of Mathematics, 1247).

Three thieves stole three identical vessels ﬁlled with rice, but whose exact capacity was

unknown. The thieves were caught and their vessels examined: the quantities left in each

vessel were 1 ge, 14 ge and 1 ge respectively. The thieves did not know the exact quantities

they’d stolen. The ﬁrst used a horse ladle (capacity 19 ge) to take rice from the ﬁrst vessel.

The second used a wooden shoe (17 ge) to take rice from his vessel. The third used a bowl

(12 ge). What was the total amount of rice stolen?

In modern language, the capacity x of each vessel satisﬁes

x ≡ 1 ( mod 19), x ≡ 14 (mod 17), x ≡ 1 (mod 12)

The given answer, x = 3193 ge, represents the smallest possible capacity of each vessel, with all other

solutions being congruent modulo 19 · 17 · 12 = 3876, as you should be able to conﬁrm if you’ve

studied number theory! The total amount of rice stolen is then

(x −1) + (x −14) + (x −1) = 3x −16 = 9563

Since congruence equations are simply underdetermined linear equations

x ≡ 1 ( mod 19) ⇐⇒ ∃y ∈ Z such that x = 1 + 19y

solutions to both of these problems can be effected using counting board methods.

Exercises 5. 1. Verify the result of the Bamboo problem.

2. Solve the Hundred Fowl Problem by substituting z = 100 − x − y in the ﬁrst equation and ob-

serving that x must be divisible by 4.

3. Use a counting board method to:

(a) Solve the linear system

(

8x + y = 28

3x + 2y = 17

(b) Multiply 218 ×191.

4. Turn Zhao Shuang’s pictorial argument from the

Arithmetical Classic of the Gnomon into a proof of

Pythagoras’ Theorem.

5. Solve problem 24 of chapter 9 of the Nine Chapters.

A deep well 5 ft in diameter is of unknown depth (to the water level). If a 5 ft post is

erected at the edge of the well, the line of sight from the top of the post to the edge of

the water passes through a point 0.4 ft from the lip of the well below the post. What

is the depth of the well?

6. Solve problem 26 of chapter 6 of the Nine Chapters.

Five channels bring water into a reservoir. If only the ﬁrst channel is open, the reser-

voir ﬁlls in

of a day. The second channel by itself ﬁlls the reservoir in 1 day, the third

channel in 2

days, the fourth in 3 days, and the ﬁfth in 5 days. If all the channels are

opened together, how long will the reservoir take to ﬁll?

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

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

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.

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

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

, 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

ı (Pakistan) in 1881,

which has been carbon-dated to the 3

or 4

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

century.

The examples below are better understood than the Bakhsh

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

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.

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

-13

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

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

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

C.,

Bhaskaracharya stated:

If you were to divide by zero you would get a number that was “as inﬁnite as the god

Vishnu.”

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

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).

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

°, 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.

This is also the root of the word sinus meaning bay or gulf (e.g., in your nose).

Exercises 6.1. 1. The Mahavedi (pg. 57) contains ﬁve Pythagorean triples; ﬁnd them.

2. To simplify square root expressions, Bhaskara used the formula

a +

√

b =



a +

−b





a −

−b



Prove Bhaskara’s formula and use it to simplify

2 +

√

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 =

GE. Then MN is the

radius of the desired circle.

Show that if AB = s and MN = r, then

2 +

√

Show that this implies a value for π equal to 3.088311755.

4. Solve the following problem of Mah

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.

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

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

–13

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

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.

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.

A series of religious–military campaigns 1096–1291 with the goal of wresting control of the Holy Land, particularly

Jerusalem, from Islam.

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.

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

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

+ 7x = 4 −2x

=⇒ 3x

+ 7x = 4

Al-muq¯abala: x

+ 7x = 4 + 5x =⇒ x

+ 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

(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

+ 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

+ ax = b

where a, b > 0: here a is the number of ‘roots,’ and b the total ‘amount.’

• Halve the number of roots



2 =



• Multiply by itself



4 =



• Add to the total amount



64 =

+ b



• Take the root of this



8 =

+ b



• Subtract half the number of roots



6 =

+ b −



Al-Kw

arizm

ı essentially constructs the quadratic formula =

−a+

√

+4b

, while the pictorial justiﬁca-

tion is Euclid’s (Elements, Thm II. 4). The geometry should be obvious: the original square (x

) has

been increased by four of its roots; the algorithm is simply ‘completing the square’ (x + 2)

= 64.

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).

Many words beginning al- are of Arabic origin (alkali, albatross, etc.), as are others that have been latinized (elixir).

Al-kit¯ab al-mukhtasar f¯ı his¯ab al-˘gabr wa’l-muq¯abala.

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

+ y = 1 then x =

√

u K

amil essentially substitutes y =

1+x

3+x

into the quadratic (with solution y =

√

5−1

). 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

+ 5x = 28 and 2x

+ 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

• 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.

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.

6.3 Islamic Mathematics II: Spherical Trigonometry and the Qibla

Late 8

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

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:

versin θ = 1 − cos θ

Al-B

ır

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

ı moves towards the modern consideration of

trigonometry in terms of triangles rather than circles.

sin α

cos α

versin α

crd 2α

tan α = cot β

sec α = csc β

Trigonometric tables with improved accuracy over Ptolemy were created for all these ‘functions.’

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°

Versed sine refers to the measurement of a length in a reversed direction (perpendicular) to that of sine.

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

• 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

⌢

sin

⌢

sin

⌢

sin

⌢

sin

⌢

sin

⌢

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

⌢

sin

⌢

sin

⌢

sin

⌢

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

= sin ∡COK = sin ∡COB = sin

⌢

= sin ∡COL = sin ∡COA = sin

⌢

The usual sine formula for plane triangles says

sin α =

sin

⌢

sin

⌢

The same ratio is obtained for △ADE.

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

⌢

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

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

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 (

and

), Ptolemy says

=⇒

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.

The great-circle arc-lengths on the sphere may be found from straight-line distances via the usual

chord relations: e.g.,

= crd

⌢

LM= 2 sin

⌢

Ptolemy’s theorem now becomes a relation between arc-lengths

sin

⌢

= sin

b −c

+ sin

⌢

sin

⌢

By bisecting α we obtain two pairs of right-triangles; al-Waf

a’s the-

orem tells us that

sin

⌢

sin c

sin

⌢

sin b

=⇒ sin

⌢

= sin

b −c

+ sin

sin c sin b (∗)

To complete the proof we apply the multiple-angle formulæ

sin

(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

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).

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

A = 1 − sin

A . . .)

Sine rule (Al-Waf

sin A

sin a

sin B

sin b

sin C

sin c

Cosine rule (Al-B

ır

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

≈

and cos

≈ 1 −

, which recover the ﬂat (Euclidean geometry)

versions of these statements.

Examples. 1. On a sphere of radius 1, an equilateral triangle has side length

. Splitting it in half

creates two right-triangles with adjacent

and hypotenuse

. The angles in the triangle are

therefore

α = cos

−1

tan

= cos

−1

≈ 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

= cos

2000

cos

4000

+ sin

2000

sin

4000

cos 45°

= cos

cos 1 +

√

sin

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.

Exercises 6.3. 1. A right-isosceles triangle on the surface of a unit sphere has equal legs of length

. 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

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

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.

6. On a sphere of radius r, Pythagoras’ Theorem may be stated

cos

= cos

cos

(∗)

where c is the hypotenuse and a, b the other side-lengths. Use the

Maclaurin series cos x =

∞

∑

n=0

(−1)

(2n)!

to expand (∗) to degree 4.

Suppose a, b are constant so that c is a function of r. Prove that

lim

r→∞

= a

+ b

. 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

What does Pythagoras (∗) say for this triangle?

(Hint: What fraction of the sphere is covered by the triangle?)

7 The Renaissance in Europe

7.1 Fibonacci and Notational Development

Between the fall of Rome in 476 and the early renaissance

(c. 1100) came Europe’s dark ages. The

one-dimensional view is that this was a time of limited learning and technological progress; as ever,

the reality was more complex. Learning persisted in monasteries, though new researches took second

place to a conservative focus on preserving the wisdom of the ancients.

By 1100, the small shifting kingdoms that had characterised Europe were starting to come together

as more stable feudal states.

The maps below show the major feudal societies of medieval Eu-

rope (France, England, Holy Roman Empire, Poland, Austria) consolidating. While it would have

meant little to the average peasant, a large stable nation is able to produce and support a larger elite

population with the time and money to pursue education, and fund libraries and universities. By

the 1700s, the borders of western Europe are largely recognizable; political and social organization

had expanded so that a much larger proportion of the population—though small in comparison to

today—were able to take advantage of and contribute to the growth of knowledge. The European

renaissance is often contrasted with a decline in Islamic power, but again the story is more complex:

Islam retreats from Spain, while the Ottoman Empire becomes dominant in south-east Europe.

Between 900 and 1300, the population of Europe roughly tripled to around 100 million. Trade in-

creased, along with which came knowledge.

Learning (and land) came via wars with Islam (the

Crusades, Spain, etc.). It helped that Islamic scholars so venerated the Greeks; Europeans could tell

themselves that they were merely ‘reclaiming’ ancient knowledge which had been ‘stolen’ by their

cultural and religious enemies. The fall of Constantinople to Mehmet in 1453 marks both the high

point of Islamic conquest and the start of the decline of Islamic scientiﬁc dominance. Many intellec-

tuals ﬂed Constantinople—where the Byzantines still preserved much of Alexandria’s learning—for

Rome, helping to further fuel developments. With powerful enemies to the east, Europeans began

travelling greater distances by sea,

beginning the colonial era of global European empire.

Literally rebirth. Dates vary by location and discipline (Italy vs. France, art vs. philosophy, etc.) but a wide net would

encompass the 12

to 17

centuries.

An arrangement where powerful landowners could demand service (rent, food, warriors) from their tenants.

A particularly important trading hub was Venice, from where Marco Polo (1254–1324), perhaps the most famous trader

of the period, travelled the silk road to China.

Christopher Columbus (born Genoa 1451) famously ‘discovered’ America in 1492 while looking for sea routes to Asia.

Scientiﬁc and philosophical progress was spurred by the translation of works from Arabic and an-

cient Greek into Latin, with the ﬁrst universities being formed to teach this canon: Bologna 1088,

Paris 1150, and Oxford 1167. The typical student was a young man of wealth who had been privately

tutored in grammar, logic & rhetoric (the trivium). At university he would study the Greek-inﬂuenced

quadrivium (geometry, astronomy, arithmetic & music). While Islamic improvements were incorpo-

rated, scholars gave pre-eminence to the Greeks: Euclid for geometry, Aristotle for logic/physics,

Hippocrates/Galen for medicine, Ptolemy for astronomy. Early universities were often funded by

the Church and ‘research’ was more likely to involve the justiﬁcation of biblical passages using Aris-

totle than the conduct of experiments.

Leonardo da Pisa (Fibonacci c. 1175–1250) Fibonacci

likely ﬁrst encountered the Hindu–Arabic

numerals while trading with his father in North Africa. He was impressed by the ease of calculation

they afforded and is the ﬁrst European known to use them (contemporary Europeans used Roman

numerals and Egyptian fractions). Fibonacci’s 1202 text Liber Abaci was written to instruct traders

in their use. The ﬁrst page below explains how to compute with decimal fractions, with the two

columns at the bottom of the page showing how to repeatedly multiply 100 (and then 10) by the

fraction

. Thus:

100, 90, 81,

72 (= 72.9),

65 (= 65.61),

59 (= 59.049), etc.

Note how the fractional part is written backwards on the left, using a bar to separate numerator and

denominator; the Indians wrote fractions without a bar and it is thought Islamic scholars inserted it

for clarity in the 1100s. The second picture is of Fibonacci’s famous sequence: read top-to-bottom 1, 2,

3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Amongst other inheritances from the Hindu–Arabic tradition,

Fibonacci is the ﬁrst known European to work with negative numbers, provided these represented

deﬁciencies or debts in accounting.

The name was given to Leonardo by French scholars of the 1800s: ﬁlius Bonacci means son of Bonacci.

Algebraic Notation and Development

To a modern reader, the most obvious mathematical development of the renaissance is notational.

Fibonacci’s fractional notation was cutting-edge for the 1200s but essentially everything else except

numbers and fractions was written in sentences. The next 500 years slowly brought notational im-

provements which eventually allowed algebra to eclipse geometry as the primary language of rea-

soning. Here is a brief summary.

Italian Abacists, 14

C. This group continued Fibonacci’s advocacy for the Hindu–Arabic system

against the traditional use of Roman numerals, and also for the use of accompanying algo-

rithms. Their approach was highly practical and largely for conducting trade. Here is a typical

problem described by the group:

The lira earns three denarii a month in interest. How much will sixty lire earn in eight

months?

The Abacists introduced shorthands and symbols for unknowns and certain mathematical op-

erations. Cosa (‘thing’) was used for an unknown. Censo, cubo and radice meant, respectively,

square, cube and (square-)root. These expressions could be combined, for example ‘ce cu’ (read

censo di cubo) referred to the sixth power of an unknown (x

)

= x

Luca Pacioli, Italy late 1400s. Introduced p, m (piu, meno) for plus and minus. For example 8m2 de-

noted eight minus two.

Nicolas Chuquet, France 1484. His text Triparty en la science des nombres borrowed Pacioli’s p and m,

and introduced an R-notation for roots. For instance R

7 meant

√

7, while

4 −

√

2 would be written R

4mR2

The underline indicates grouping (modern parentheses).

Christoff Rudolff, Vienna 1520s. Introduced symbols similar to x and ζ for an unknown and its

square. He had other symbols for odd powers and produced tables showing how to multiply

these. The words he used show Italian and French inﬂuence: algebra in the German-speaking

world was known as the art of the coss (German for thing). Rudolff also introduced ± symbols

as algebraic operations, though these had been used for around 30 years as a preﬁx denoting an

excess or deﬁciency in a quantity (proﬁt/loss in accounting). A period denoted equals and he is

also credited with the ﬁrst use of the square-root sign

√

, which is nothing more than a stylized

r. This would be written in front of a number to denote a root, e.g.

√

23 rather than

√

23.

Robert Recorde, England 1557. Introduced the equals sign in The Whetstone of Witt, asserting, ‘No

two things are more equal than a pair of parallel lines.’

Francois Vi`ete, France 1540–1603. Before Vi

ete, mathematicians typically described how to solve

particular equations algorithmically via examples (e.g. x

+ 3x = 14 rather than x

+ bx = c),

expecting readers to change numbers to ﬁt a required situation. Vi

ete pioneered the modern

use of abstract constants, using letters to represent both unknowns and constants.

Simon Stevin, Holland 1548–1620. De Thiende (The Tenths) demonstrated how to calculate using

decimals rather than fractions. Stevin arguably completed the journey whereby the concept

of number subsumed that of magnitude, asserting that every ratio is a number. This increased

the application of algebra by permitting the description of any geometric magnitude.

William Oughtred, England 1575–1660. Introduced × for multiplication, though he often simply

used juxtaposition. Oughtred combined Vi

ete’s general approach (abstract constants) with

symbolic algebra. For instance, to solve a quadratic equation A

+ BA + C = 0, where A

means ‘A-squared’ (A-quadratum in Latin) and B, C are constants, he’d write the quadratic for-

mula as

A =

√

−C : −

In Oughtred’s notation, colons were parentheses.

Thomas Harriot, England 1560–1621. Made several steps towards modern notation including juxta-

position for multiplication and a modern encompassing root-sign. For example,

cccc + 27aa

√

2 + b meant

+ 27a

√

2 + b

Ren´e Descartes, France/Holland 1596–1650. Used exponents for powers (a

, a

) and solidiﬁed the

convention of using letters at the end of the alphabet (x, y, z) for unknowns and those at the

beginning (a, b, c) for constants.

While modern mathematics uses many specialized symbols (∅, ⇒, e, π, etc.), basic notation is essen-

tially unchanged from the mid 1600s. This is not to say that all mathematicians uniformly used the

most modern notation: for instance, papers of Leonhard Euler (1700s) used Harriot’s juxtaposition

notation for exponents, and some published works from the late 1800s still wrote equations in words.

It is also worth mentioning in this context Gutenberg’s 1439 invention of the printing press, which

naturally aided the dissemination of all learning. The relative ease of production meant a great

increase in the availability of written material, but also in the rejection/abandonment of some older

texts when money could not be found to make an updated printed edition.

Exercises 7.1. 1. Repeatedly divide 10 by 5 a total of ﬁve times (to

), expressing the results using

Fibonacci’s notation.

2. What would Nicolas Chuquet have meant by the expression 4pR

7 mR5?

3. How would William Oughtred have expressed the solution to the quadratic equation A

BA + C? What about Thomas Harriot?

4. I am owed 3240 ﬂorins. The debtor pays me 1 ﬂorin the ﬁrst day, 2 the second day, 3 the third

say, etc. How many days does it take to pay off the debt?

5. (A problem of Antonio de Mazzinghi) Find two numbers such that multiplying one by the

other makes 8 and the sum of their squares is 27.

(Hint: let the numbers be x ±

√

7.2 Polynomials: Cardano, Factorization & the Fundamental Theorem

As an example of contemporary algebraic notation, we consider Girolamo Cardano’s 1545 Ars Magna

(Great Art or the Rules of Algebra), in which he describes how to solve quadratic and cubic equations.

The example on the right (start of Caput V, page 9 of the linked

pdf), is the beginning of Cardano’s description of how to solve

+ 6x = 91, quadratum & 6 res aequalie 91 (square and 6 things

equals 91), by completing the square. He employs several single-

letter abbreviations but still writes in sentences and provides a

pictorial justiﬁcation. In modern algebra, the argument is nothing

more than completing the square:

+ 6x = 91 =⇒ x

+ 2 ·3x + 3

= 91 + 3

=⇒ (x + 3)

= 100

=⇒ x + 3 = 10

=⇒ x = 7

where the picture justiﬁes 7

+ 2 ·21 + 3

= 10

The quadratic algorithms were well-established by this time; on

the following page of his text, we ﬁnd a picture from the work

of al-Khw

arizm

ı (Exercise 6.2.2), after which comes more obvious

mathematical notation. Even though Rudolff’s ± and

√

were in

use, Cardano wrote almost everything in words augmented with

fractional notation and Pacioli’s p and m.

As was typical for the time, Cardano describes negative solutions

as ﬁctitious; he even writes the square root of −15 at one point,

though only to mention its absurdity. He also follows the Islamic

approach of solving a concrete problem of each type rather than

proceeding abstractly, though observe the gratia exempli (“for the

sake of an example”) as acknowledgement that the general prob-

lem x

+ ax = b may be solved identically.

It is for the solution of cubic (and quartic) equations that Cardano

is most famous. Below we describe, in modern notation, Car-

dano’s method for solving the cubic equation x

+ bx = c where

b, c > 0, though we stress (again) that Cardano only gave examples

not a general formula.

Let u, v satisfy u

−v

= c and uv =

. Then x = u −v is seen to solve the cubic.

+ bx = (u −v)

+ b( u −v) = u

−3u

v + 3uv

−v

+ b( u −v)

= (u

−v

) + (u −v)(b −3uv) = c

However u and v also satisfy

+ v

)

= (u

−v

)

+ 4(uv)

= c

+ 4





so we obtain a system of linear equations in the unknowns u

, v

which are easily solved:







+ v

+ 4





−v

= c

=⇒











u =









v =









−

=⇒ x = u −v =









−









−

Examples 1. We apply Cardano’s method to x

+ 6x = 7, which has the obvious solution x = 1.

(

−v

= 7

uv =

= 2

=⇒ (u

+ v

)

= 7

+ 4 ·2

= 81 =⇒ u

+ v

= 9

=⇒ u

(9 + 7) = 8, v

(9 −7) = 1

=⇒ x = u −v =

√

8 −

√

1 = 2 −1 = 1

2. This time we solve x

+ 3x = 14.

(

−v

= 14

uv =

= 1

=⇒ (u

+ v

)

= 14

+ 4 ·1

= 200 =⇒ u

+ v

= 10

√

=⇒

(

(10

√

2 + 14) = 5

√

2 + 7 = (

√

2 + 1)

= 5

√

2 −7 = (

√

2 −1)

=⇒ x = u −v = (

√

2 + 1) − (

√

2 −1) = 2

As this shows, Cardano’s formula might produce an ugly expression for a simple answer—of

course, the cube root of 5

√

2 ±7 is the tip of everyone’s tongue!

3. The equation x

+ 3x = 10 may be solved using Cardano’s method, though in this case the

answer is just ugly.

(

−v

= 10

uv =

= 1

=⇒ (u

+ v

)

= 10

+ 4 ·1

= 104 =⇒ u

+ v

= 2

√

=⇒ u

√

26 + 5, v

√

26 −5

=⇒ x = u −v =

√

26 + 5 −

√

26 −5 ≈ 1.6989

Being unable or unwilling to work directly with negative numbers, Cardano modiﬁed his method

to solve other cubics such as x

+ c = bx, and moreover described how to remove a quadratic term

from a cubic using what is now known as the Tschirnhaus substitution (x = y −

+ ax

+ bx + c =



y −



+ a



y −



+ b



y −



+ c = y

−

y + ··· (∗)

Cardano’s student, Lodovico Ferrari, extended the method to solve quartic equations in terms of the

solution of a resultant cubic.

Negative Solutions and Complex Numbers

By the late 1500s, mathematicians were mentioning negative solutions to equations—these were usu-

ally described as ﬁctitious, or false roots, but this didn’t stop them from being investigated. Rafael

Bombelli (1526–1572, Rome) even introduced a notation for complex numbers, described their alge-

bra, and showed how they could be used to ﬁnd solutions to any quadratic or cubic equation. For

instance, he wrote 4 + 3i and 4 − 3i as follows:

4 p di m 3, read ‘quattro piu di meno tre’ (four plus of minus three), and,

4 m di m 3, ‘quattro meno di meno tre.’

Given Bombelli’s belief in the ﬁctitiousness of complex numbers, the effort he expended in their

honor is extraordinary: this is a prime example of pure abstraction, math for the sake of math!

In modern language, the three roots of Cardano’s cubic x

+ bx = c are

u −v, ζu −ζ

v, ζ

u −ζv

where ζ = e

2πi/3

−1+

√

is a primitive cube root of unity. Together with the Tschirnhaus substitu-

tion (∗), Cardano’s formula therefore solves all cubic equations.

Examples 1. Returning to one of our previous examples, if x

+ 3x = 14, then u =

√

2 + 1 and

v =

√

2 −1, from which the three solutions are

u −v = 2

ζu − ζ

v = (

√

2 + 1)

−1 +

√

−(

√

2 −1)

−1 −

√

= −1 +

√

u −ζv = (

√

2 + 1)

−1 −

√

−(

√

2 −1)

−1 +

√

= −1 −

√

2. To ﬁnd a solution to x

+ 3x

= 3, we perform the Tschirnhaus substitution x = y −

= y − 1

before applying Cardano’s method:

−3y

+3y −1 + 3(y

−2y + 1) = 3 =⇒ y

−3y = 1 (b = −3, c = 1)

=⇒

(

−v

= 1

uv = −

= −1

=⇒ (u

+ v

)

= 1

+ 4(−1)

= −3

=⇒ u

+ v

√

3i =⇒ u

(

√

3i + 1), v

(

√

3i −1)

=⇒ x = u −v −1 =

√

3i + 1

−

√

3i −1

−1

This is ugly! If you know Euler’s formula and you choose a compatible pair of cube roots (you

need uv = −1), this will evaluate to one of three real numbers: the positive solution is in fact

x = 2 cos 20° −1 ≈ 0.8794.

Factorization & the Fundamental Theorem of Algebra

By the late 1500s, Vi

ete’s abstraction allowed him to streamline Cardano’s methods. He also investi-

gated the relationship between the coefﬁcients of a polynomial and its roots. For Vi

ete the roots had

to be positive, but later improvements by Thomas Harriot and Albert Girard (1629) applied this to

all polynomials with any roots. For instance, if ax

+ bx + c = 0 has roots r

, r

, then

−r

)



a(r

+ r

) + b



= a(r

−r

) + b(r

−r

) =



+ br

+ c



−



+ br

+ c



= 0

Provided the roots are distinct,

we conclude that

= −(r

+ r

= −r

−

= −r

+ (r

+ r

= r

These are the quadratic version of what are known as Vi

ete’s formulas; other versions exist for every

degree. Their use amounts to an early form of factorization.

Example To solve 3x

−2x −1 = 0, ﬁrst spot that r

= 1 is a root. By Vi

ete’s formulas,

+ r

= −

=⇒ r

= −

or alternatively r

= −

=⇒ r

= −

Think about the relationship between this approach and factorization!

A nice side-effect is a method for obtaining the quadratic formula by solving a pair of simultaneous

equations analogous to Cardano’s cubic approach:







+ r

= −

−r

+ r

)

−4r





−4

=⇒ r

, r

= −





−4

ete’s formulas were central to the later development of Galois Theory (1830) and the Abel–Rufﬁni

theorem regarding the insolubility of quintic and higher-degree polynomials.

The remaining key results regarding solutions of polynomials also appeared around this time:

Fundamental Theorem of Algebra Including multiplicity, a degree n polynomial has n complex

roots. Girard offered the ﬁrst version in 1629, though a complete proof didn’t appear until

the work of Argand, Cauchy and Gauss in the early 1800s.

Factor Theorem In 1637, Descartes proved that p(r) = 0 ⇐⇒ p(x) is divisible by x −r, essentially

via long-division.

For instance, to ﬁnd the roots of p(x) = x

+ 2x

−13x + 10, Descartes would observe:

• p(1) = 0 =⇒ x − 1 is a factor; use long-division to obtain p(x) = (x −1)(x

+ 3x −10).

• q(x) = x

+ 3x −10 has q(2) = 0; divide to get q(x) = (x −2)(x + 5).

• The roots of p(x) are therefore 1, 2 and −5 (Descartes called this last a false root).

We’ll return to Descartes in the context of Analytic Geometry in the next chapter.

In fact the formulas hold even when r

= r

Exercises 7.2. 1. (a) Find Vi

ete’s formulas for the polynomial p(x) = x

+ ax

+ bx + c with roots

, r

; that is, ﬁnd the coefﬁcients a, b, c in terms of the roots.

(Hint: Multiply out p(x) = (x −r

)(x −r

). . . )

(b) Solve x

− 6x

+ 9x − 4 = 0 using Girard’s method: ﬁrst, determine one solution by in-

spection, then use Vi

ete’s formulas for the cubic to investigate the relationship between

the remaining roots.

2. (a) Apply Cardano’s method to the equation x

+ 6x = 20.

(Hint: to ﬁnish, compute (1 +

√

)

(b) If b, c > 0, Cardano’s method ﬁnds a single positive solution to x

+ bx = c. Explain why

such an equation always has exactly one real solution which is moreover positive.

3. Prove that if t is a root of x

= cx + d, then

c −

and r

−

c −

are both roots of x

+ d = cx. Use this to solve x

+ 3 = 8x.

4. Consider the cubic equation aaa −3raa + ppa = 2xxx (as written by Harriot). Show that the

substitution a = e + r reduces this to an equation without a square term.

As an example, reduce the equation aaa − 18aa + 87a = 110 to a cubic in e without a square

term. Find all three solutions in e and therefore ﬁnd the solutions to the original equation in a.

5. Find all roots of the cubic p(x) = 2x

−3x

−3x + 2 using Descartes’ factor theorem.

6. Use Cardano’s method (with Tschirnhaus substitution and complex numbers!) to ﬁnd the so-

lutions to the equation

+ 4 = 3x

Verify that you get the same solutions using Descartes’ factor theorem.

(Hint: all solutions are integers!)

7. (If you are very comfortable with complex numbers) Use Euler’s formula to verify that the

equation x

+ 3x

= 3 has the positive solution x = 2 cos 20° −1. It also has two negative real

solutions: ﬁnd them!

7.3 Astronomy and Trigonometry in the Renaissance

Distinctly European progress in trigonometry came courtesy of Johannes M

uller (a.k.a. Regiomon-

tanus,

1436–1476). De Triangulis Omnimodius (Of all kinds of triangles, 1463) provided an axiomatic

update of Ptolemy’s Almagest and its Islamic improvements. Though the title refers to triangles,

his approach remains circle-based (chords and half-chords). Regiomontanus was a renowned as-

tronomer; his tracking of a comet from late 1471 to spring 1472 provided controversial evidence that

objects could move between the, supposedly ﬁxed, heavenly spheres of ancient Greek theory.

Domenico Novara (1454–1504), Regiomontanus’ student, inherited much of his unpublished work.

He became a student of Pacioli in Florence and an astronomer at the University of Bologna, though

he is now perhaps best known as adviser to a young Pole, Nicolaus Copernicus (1473–1543), who

studied in Bologna from 1496 with the ostensible intent of joining the priesthood. . .

Copernicus concluded that Ptolemy’s geocentric model could not be

reconciled with astronomical observation. De revolutionibus orbium ce-

lestium (On the revolutions of the heavenly spheres), published a year after

his death, describes how to compute within a heliocentric (sun-centered)

model. This, Copernicus believed, was the obvious solution to the

problem of retrograde motion that had plagued the ancient Greeks.

The animation demonstrates Copernicus’ solution: with the sun at the

center, the retrograde motion of Mars and Jupiter are easily explained.

The outer circle represents the ‘ﬁxed stars’ against which the motion

of the planets are observed.

Copernicus’ work is now described as a revolution, though it was not perceived so at the time. De

revolutionibus was dedicated to the Pope, welcomed by the Church, and used by Vatican astronomers

to aid in calculation. The difﬁculty and narrow readership of his work made it unthreatening to

contemporary dogma. Copernicus did not present heliocentrism as reality nor did he advocate for

overturning long-held beliefs. Within a century, however, the Copernican theory had found its bull-

dog in Galileo, and conﬂict between science and the Church became unavoidable.

Trigonometry is ﬁnally about triangles!

Georg Rheticus (1514–1574) deﬁned trigonometric functions purely

in terms of triangles, referring to the perpendiculum (sine) and ba-

sis (cosine) of a right-triangle with ﬁxed hypotenuse. Rheticus was

a student of Copernicus and helped posthumously to publish his

work.

In 1595, Bartholomew Pitiscus ﬁnally introduced the modern term

with his book Trigonometriæ, in which he purposefully sets out to

solve problems related to triangles. The picture is the title page from

the second edition (1600—MDC in Roman numerals). Both Rheti-

cus and Pitiscus had problems which look very familiar to modern

readers, such as solving for unknown sides of triangles.

His grand-sounding name is a latinization of his birthplace K

onigsberg (King’s Mountain), Bavaria, Germany.

Copernicus wasn’t the ﬁrst to suggest such a model. Several ancient Greek scholars embraced heliocentrism, with

Aristarchus of Samos (c. 310–230 BC) credited with its ﬁrst presentation. However, Aristarchus’ views were rejected by the

mainstream Greeks, and it is likely Copernicus never encountered his work.

Example (Pitiscus) A ﬁeld has ﬁve straight edges of

lengths 7, 9, 10, 4 and 17 in order. The distance from the

ﬁrst to third vertex is 13 and from the third to ﬁfth is 11.

What is the area of the ﬁeld?

The problem can be solved easily using Heron’s formula,

but Pitiscus opts for trigonometry. We give a modernized

version that depends on applying the law of cosines to the

three large triangles.

cos α =

+ 13

−9

2 ·7 · 13

137

182

cos β =

+ 13

−11

2 ·17 · 13

337

442

cos γ =

+ 11

−10

2 ·4 · 11

The values of α, β, γ and therefore the altitudes a, b, c of the three major triangles could be read off a

table, or found exactly using Pythagoras’:

a = 7 sin α = 7

1 −cos

α =

182

−137

√

1595,

b = 13 sin β =

√

81795, c = 4 sin γ =

√

255

The total area is easily computed:

A =

(13a + 17b + 11c) =



√

1595 +

√

81795 + 5

√

255



≈ 121.4

Kepler’s Laws

Johannes Kepler (1571–1630) was a student of Tycho Brahe

for the last two years of Brahe’s life. He

inherited Brahe’s position, decades-worth of astronomical data, and his philosophy on the impor-

tance of observation-based theory. Kepler also embraced the mystical Pythagorean view that nature

reﬂects harmony, a belief that partly drove his scientiﬁc pursuits. To Kepler, any observation of a nat-

ural, simple ratio was something of great import. For example, in observing that the daily movement

of Saturn at its furthest point from the sun was roughly 4/5 of that at its nearest point, his temptation

was to assume that ‘roughly’ must be ‘exactly.’

Thanks to Brahe, Kepler had data on roughly thirteen orbits of

Mars and two of Jupiter. From these data, he posited three laws.

1. Planets move in ellipses with the sun at one focus.

2. The orbital radius sweeps out equal areas in equal times. In

the picture, the sectors all have the same area and the planet

moves more slowly the further it is from the sun.

3. The square of the orbital period is proportional to the cube of

the semi-major axis of the ellipse: T

∝ a

Tycho Brahe (1546–1601) was a Danish astronomer who worked for Austro-Hungarian Emperor Rudolph II in Prague

for 25 years, producing a wealth of accurate astronomical measurements. While these helped burnish the Copernican

theory, he is also known for his 1572 observation of a nova (a new star that later disappeared, now understood to be the

death of a distant star), and then a comet in 1577; both provided yet more evidence for the changeability of the heavens.

Kepler’s laws are empirical observations rather than the result of mathematical proof. However, the

process of their discovery demonstrates Kepler’s tremendous mathematical abilities.

Starting point Kepler began by assuming the essential correctness of the Copernican model in that

all planets exhibit uniform circular motion round the sun.

Orbital Estimation Kepler’s data told him the direction to each

planet, but not the distance, though his Copernican assump-

tion allowed him to estimate the relative distance.

For exam-

ple, using the direction from Earth to Mars at equally spaced

times and by drawing circles of different radii for possible or-

bits of Mars, he was able estimate an orbit where Mars’ mo-

tion would also be uniform. This required an enormous num-

ber of trigonometric calculations: each measurement of plan-

etary longitude/latitude relative to Prague had to be converted

to measurements relative to the sun. Everything was subject

to errors of estimation.

Modifying the model Kepler altered his model to reﬂect Earth’s slightly non-circular orbit. He ﬁrst

tried an equant model (in the style of the ancient Greeks), offsetting the center of the orbit

slightly from the sun. Despite this, he failed to ﬁt his data for Mars to pure circular motion.

The First Law Abandoning circles, Kepler now permitted planets to move in ovals. He decided to

approximate Mars’ orbit with an ellipse and set out to calculate its parameters, stumbling on

an almost perfect match when the sun was placed at a focus. The geometric signiﬁcance of the

focus was exactly the natural beauty sought by Kepler. Having now established the ﬁrst law

for Mars, he repeated the exercise for the other known planets (Mercury, Venus, Earth, Jupiter,

Saturn) as well as he could given his inferior data.

The Second Law Kepler’s second law followed an inﬁnitesimal argument based on inspired guess-

work. To ﬁt his elliptical orbits, planetary velocity was non-constant, appearing inversely pro-

portional to the distance from the sun (v =

). Kepler used this to approximate the area of a

sector swept out by the radius vector: over a small time-interval ∆t, a planet travels a distance

v∆t and thus sweeps out an approximate triangle of area

rv∆t =

k∆t. In modern language,

this is the conservation of angular momentum. Kepler had no justiﬁcation beyond the fact that

it seemed to ﬁt the data. In particular, he did not know why a planet should move more slowly

when further from the sun.

The Third Law This was stated with very limited analysis. Given the relative sizes of each orbit and

its period, some inspired guesswork allowed him to observe that

is approximately the same

value for each.

Kepler’s discoveries were revealed over many years in several texts, and his magnum opus Epitome

astronomiæ Copernicanæ was published in 1621. Within a century Issac Newton had provided a math-

ematical justiﬁcation of Kepler’s laws based on the theory of calculus and his own axioms: an inverse

square law for gravitational acceleration and his own three laws of motion.

Following Ptolemy, Brahe thought the Earth-Sun distance was around 1/10 of its true value. Kepler thought this an

underestimate by at least a factor of three. In 1659, Christiaan Huygens found the distance to an accuracy of 3%.

A Religious Interlude: Protestantism, the Counter-Reformation and Calendar Reform

In 1563, Pope Gregory began the Catholic Church’s push-back against the spread of Protestantism,

the counter-reformation. Of particular interest to science and mathematics was the newly created In-

dex Librorum Prohibitorum, a list of books contradicting Church doctrine. This was also a response

to the new technology of mass-printing, which made disseminating controversial new ideas easier.

Heliocentrism came quickly under attack: Kepler’s book was banned immediately upon publica-

tion in 1621. However, his location far from Rome meant that Kepler and his ideas were relatively

safe. The ultimate result of Gregory’s crackdown was the slow ceding of scientiﬁc power to northern

(Protestant) Europe where papal diktat had no effect.

In contrast to the anti-scientiﬁc book-banning fervor of the counter-reformation, Pope Gregory is

also famous for shepherding an astounding scientiﬁc achievement: calendar reform. By 1500, as-

tronomers knew the solar year to be roughly 11

minutes shorter than the 365

days of the Julian

calendar.

For 1200 years, Easter had been decreed to be the Sunday after the ﬁrst full moon after

the vernal equinox (March 20

/21

), but by 1500 the equinox was happening 10–11 days earlier. The

impetus to correct the date of Easter meant that calendar reform and astronomical modelling were

now an important Church project.

A century of effort

resulted in the Gregorian calendar (designed by Aloysius Lilius and Christopher

Clavius). Gregory imposed the new calendar in all Catholic countries in 1582. The 10 day deﬁcit was

corrected by eliminating October 5

–14

1582. To prevent the error re-occurring, the computation

of leap-years was also changed: centuries are now leap-years only if divisible by 400, thus 1600 was

a leap year, but 1900 was not. The Gregorian calendar is astonishingly accurate, losing only one day

every 3000 years. Since it emanated from Rome, many Protestant parts of Europe took decades if not

centuries to adopt the new calendar. The Eastern Orthodox Church still computes Easter using the

Julian calendar, which is now 13 days behind the Gregorian.

Galileo Galilei (1564–1642)

Based in northern Italy, Galileo was close to the center of Church power; unlike Copernicus and

Kepler, he openly challenged its orthodoxy. While undoubtedly a great mathematician, he is more

importantly considered the father of the scientiﬁc revolution for his reliance on experiment and ob-

servation. He famously observed Jupiter’s moons with a telescope of his own invention, noting that

objects orbiting an alien body was counter to Ptolemaic theory. Skeptics, when shown this image,

preferred to assert that it must be somewhere inside the telescope!

In 1632 Galileo published Dialogue Concerning the Two Chief World Systems, a Socratic discussion be-

tween three characters: Salviati argued for Copernicus, Simplicio was for Ptolemy, and Sagredo was

an independent questioner. The character of Simplicio was provocatively modeled on conserva-

tive philosophers who refused to consider experiments and bore a notable resemblance to the Pope.

Salviati almost always came out on top and Simplicio was made to appear foolish. The text resulted

in Galileo’s conviction for heresy; all his publications, past and future, were banned, and he spent

the remainder of his life under house arrest.

Martin Luther’s Ninety-ﬁve Theses (1517) is generally considered the start of the Protestant Reformation. Europe saw

several gruesome religious wars over the next 150 years as various countries broke from Catholicism and Rome.

Named for Julius Caesar, the Julian year has 365 days with a leap-day added every four years.

Pope Sixtus IV tried to recruit Regiomontanus to the cause in 1475, though the mathematician died ﬁrst. Copernicus

was among those invited to consider proposals in the early 1500s, though he distanced himself, perhaps because he knew

that his developing heliocentric ideas would not be accepted.

Despite Church efforts, Galileo’s works continued to be distributed by his supporters and he contin-

ued working. His most important scientiﬁc text, Discourses Concerning Two New Sciences (materials

science and kinematics) was smuggled out of Italy to be published in Holland in 1638. In this book

he resurrects his characters from Two Chief World Systems and famously refutes Aristotle’s claim that

heavier objects fall more rapidly than lighter ones.

Here are two results from this text.

Theorem. If acceleration is uniform, then the average speed is the average of the initial and terminal

speeds.

Proof. Galileo argues pictorially.

In essence, CD is the time-axis, increasing downward. Velocity is measured hor-

izontally from the time-axis to the uniformly sloped line AE, of which I is the

midpoint.

The distance travelled is the area between AE and the time-axis, which plainly

equals the area of the rectangle between GF and the time-axis. The velocity cor-

responding to GF is the average of those corresponding to A and E.

Here is the calculation using modern algebra. Suppose the object has velocity

as it passes A and v

as it passes v

, and that t =

. Then the distance

travelled is

t = v

t + area(△ABE) = v

t +

−v

)t =

+ v

whence v

is the average of the initial and ﬁnal speeds.

Corollary. A falling object dropped from rest will traverse distance in proportion to time-squared,

: d

= t

: t

This is Galileo’s version of the well-known kinematics formula d =

Proof. Let d

, d

, v

represent the distances travelled and the speeds of the dropped body at times

and t

. Since acceleration is uniform,

: v

= t

: t

By the Theorem,

0 + v

, and d

whence

: d

= v

: v

= t

: t

Supposedly by dropping weights off the leaning tower of Pisa, though take this story with a pinch of salt. . .

Galileo follows this by decomposing the motion of a projectile into horizontal (uniform speed) and

vertical (uniform acceleration) components, thereby proving that projectiles follow parabolic paths.

Galileo covered several other important mathematical topics, some of which we’ll mention when we

discuss calculus. While his mathematical ideas were cutting-edge for the time, it is his insistence

on testing theory against data that makes him a true revolutionary. By 1600 very few of Aristotle’s

easy-to-refute claims had been rejected due to experimental testing; the hostility Galileo provoked

by doing so perhaps explains why. This is the core of the scientiﬁc revolution: primacy is given to

experiment and observation over ancient ‘wisdom,’ whatever the source.

Galileo was ﬁnally cleared of heresy by the Catholic Church in 1992.

Exercises 7.3. 1. Compute today’s date in the Julian calendar and explain your calculation.

2. (A problem of Copernicus) Given the three sides of an isosceles

triangle, to ﬁnd the angles.

Suppose AB and AC are the equal legs of the triangle. Circum-

scribe a circle around the triangle and draw another with center A

and radius AD =

AB.

(a) Why is Copernicus introducing the second circle?

(b) Explain why the ratio of each of the equal sides to the base of

△ABC equals that of the radius AD to the chord DE.

= 10 and

= 6, use modern trigonometry

to ﬁnd the three angles of the original triangle.

D E

3. Verify that Heron’s formula gives the same solution to Pitiscus’ problem (pg. 78).

4. Given that Earth’s orbital period is 1 year and that the mean distance of Mars from the sun is

1.524 times that of Earth, use Kepler’s third law to determine the orbital period of Mars.

5. According to Kepler’s second law, at what point in a planet’s orbit will it be moving fastest?

6. Galileo states the following.

A projectile ﬁred at an angle α = 45° above the horizontal at a given initial speed

reaches a distance of 20,000. Then, with the same initial speed it will reach a distance

of 17,318 when α = 60°, or α = 30°.

Check this statement: if you want a challenge, try to do without the standard Physics formulæ

and instead use ratios!

8 Analytic Geometry and Calculus

8.1 Axes and Co-ordinates

Modern mathematics is almost entirely algebraic: we trust equations and the rules of algebra more

than pictures. For example, modern mathematics considers the expression (x + y)

= x

+ 2xy + y

to follow from the laws (axioms) of algebra:

(x + y)

= (x + y)(x + y) (deﬁnition of ‘square’)

= x(x + y) + y(x + y) (distributive law)

= x

+ xy + yx + y

(distributive law twice more)

= x

+ 2xy + y

(commutativity)

For most of mathematical history, this result would have been viewed

geometrically as in Euclid’s Elements (Thm II. 4):

The square on two parts equals the squares on each part plus

twice the rectangle on the parts.

The proof was a simple picture.

We’ve seen how algebra and algebraic notation were slowly adopted in renaissance Europe. While its

utility for efﬁcient calculation was noted, algebra was not initially considered acceptable for proof and

calculations would be justiﬁed geometrically. From our modern viewpoint this seems backwards: if a

student today were asked to prove Euclid’s result, they’d likely label the ‘parts’ x and y, before using

the algebraic formula at the top of the page to justify the result! Of course each line in the algebraic

proof has a geometric basis:

• Distributivity says that the rectangle on a side and two parts equals the sum of the rectangles

on the side and each of the parts respectively.

• Commutativity says that a rectangle has the same area if rotated 90°.

Modern mathematics has converted geometric rules into algebra and largely forgotten their geomet-

ric origin! This slow movement from geometry to algebra is one of the great revolutions of mathe-

matical history, completely changing the way mathematicians think. More practically, the conversion

to algebra allows easier generalization: how would one geometrically justify an expression such as

(x + y)

= x

+ 4x

y + 6x

+ 4xy

+ y

Euclidean Geometry is synthetic: based on purely geometric axioms without formulæ or co-ordinates.

The revolution of analytic geometry was to marry algebra and geometry using axes and co-ordinates.

Modern geometry is primarily analytic and it is now rare to ﬁnd a mathematician working solely in

synthetic geometry—algebra’s domination of Euclidean geometry is total! The critical step in this

revolution was made almost simultaneously by two Frenchmen. . .

Pierre de Fermat (1601–1665) Mathematics was Fermat’s pastime rather than his profession, though

this didn’t prevent him making great strides in several areas such as probability, analytic geometry,

early calculus, number theory and optics.

Some of Fermat’s fame comes from his enigma, with most

of what we know of his work coming in letters to friends in which he rarely offers proofs. He would

regularly challenge friends to prove results, and it is often unknown whether he had proofs himself

or merely suspected a general statement. Being outside the mainstream, his ideas were often ignored

or downplayed. When he died, his notes and letters contained many unproven claims. Leonhard

Euler (1707–1783) in particular expended much effort proving several of these.

Fermat’s approach to analytic geometry was not dissimilar to that of Descartes which we shall de-

scribe below: he introduced a single axis which allowed the conversion of curves into algebraic

equations. We’ll return to Fermat when we discuss the beginnings of calculus in the next section.

Ren´e Descartes (1596–1650) In his approach to mathematics, Descartes is the chalk to Fermat’s

cheese, rigorously recording everything. His deﬁning work is 1637’s Discours de la m´ethode. . .

While

enormously inﬂuential in philosophy, Discours was intended to lay the groundwork for investigation

within mathematics and the sciences—Descartes ﬁnishes Discours by commenting on the necessity

of experimentation in science and on his reluctance to publish due to the environment of hostility

surrounding Galileo’s prosecution.

The copious appendices to Discours contain Descartes’ scientiﬁc

work. It is in one of these, La G´eom´etrie, that Descartes introduces axes and co-ordinates.

We now think of Cartesian axes and co-ordinates as plural, but

both Fermat and Descartes used only one axis. Here is a sketch

of their approach.

Draw a straight line (the axis) containing two ﬁxed points la-

belled 0 (the origin) and 1. All points on the axis are identiﬁed

with numbers x (originally only positive).

A curve is described as an algebraic relationship between x and

the distance y from the axis to the curve measured using a family

of parallel lines intersecting both.

−2 −1 0 1 2 3

Neither Descartes nor Fermat had a second axes, though their approach implicitly imagines one,

the measuring line through the origin. It therefore makes sense for us to speak of the co-ordinates

(x, y); the modern terms abscissa (x) and ordinate (y) date from shortly after the time of Descartes. It

wasn’t long before a second axis orthogonal to the ﬁrst was instituted (Frans van Schooten, 1649), an

approach that quickly became standard.

Fermat was wealthy but not aristocratic, attending the University of Orl

eans for three years where he trained as a

lawyer. You’ve likely encountered his name in relation to two famous results in number theory:

Fermat’s Little Theorem p prime =⇒ x

≡ x (mod p) for all integers x.

Fermat’s Last Theorem If n ∈ N

≥3

, then x

+ y

= z

has no integer solutions with xyz = 0. Fermat is not believed to have

proved this beyond a special case (n = 4), with a complete proof not appearing until the 1990s.

. . . of rightly conducting one’s reason and of seeking truth in the sciences. The primary part of this work is philosophical and

contains his famous phrase cogito egro sum (I think therefore I am).

At this time, France was still Catholic. Descartes had moved thence to Holland in part to pursue his work more freely.

In 1649 Descartes moved to Sweden where he died the following year.

Example 1 The previous picture shows some of the ﬂexibility in Descartes’ approach. The curve

y = x

+ 1 is drawn, where the ‘y-axis’ is inclined 60° to the horizontal. To recognize the curve in a

more standard fashion, we can perform a change of co-ordinates. Suppose P = (x, y) with respect to

the slanted axes and (X, Y) with respect to the usual orthogonal axes.

−2 −1 0 1 2 3

x, X axes

Y axis

y axis

origin

60°

The second picture shows that

(

X = x + y cos 60° = x +

Y = y sin 60° =

√

For any point on the curve,

√

3X −Y =

√

3x =⇒ (

√

3X −Y)

= 3x

= 3(y −1) = 3



√

Y −1



=⇒ 3X

−2

√

3XY + Y

−2

√

3Y + 3 = 0

which is an implicit equation for a parabola.

Example 2 Analytic geometry affords easy proofs of many results that are signiﬁcantly harder in

Euclidean geometry. For instance, here is the famous centroid theorem.

Choose axes pointing along two sides of a triangle with the origin at

one vertex.

If the two axes-side lengths are a and b, then the third side has equa-

tion bx + ay = ab or y = b −

x and the midpoints of the sides have

co-ordinates as in the picture. Now compute the point 1/3 of the

way along each median: for instance



, 0



(0, b) =

(a, b)

One obtains the same result with the other medians, whence all three

meet at a common point G, the centroid of the triangle.

(0, 0) (a , 0)

(0, b)

(

, 0)

(0,

)

(

)

This ability to choose axes to ﬁt the problem is a critical advantage of analytic geometry, largely dispens-

ing with the tedious consideration of congruence in synthetic geometry.

This really is a parabola, just rotated! This may be conﬁrmed by computing its discriminant: a non-degenerate quadratic

curve aX

+ bXY + cY

+ linear terms is a parabola if and only if b

−4ac = 0.

Descartes used his method to solve problems that had proved more difﬁcult synthetically, such as

ﬁnding complicated intersections of curves. As we’ll see in the next section, such arguments would

often rely on his Factor Theorem (pg. 75). Given the novelty of his approach, Descartes typically

gave geometric proofs of all assertions to back up his algebraic work. However he also saw the

future, stating that once several examples were done it was no longer necessary to draw physical

lines and provide a geometric argument, the algebra was the proof.

Exercises 8.1. 1. Assume that xy = c represents a hyperbola with asymptotes the x- and y-axes.

Show that xy + c = rx + sy also represents a hyperbola, and ﬁnd its asymptotes.

2. Determine the locus of the equation b

−2x

= 2xy + y

(Hint: add x

to both sides and remember that ‘axes’ do not have to be orthogonal. . . )

3. We describe a method whereby Descartes constructed the product of two lengths.

Let

−→

BC and

−→

BD be rays forming an acute angle at B.

Our goal is to multiply

Suppose

= 1, where A lies on

−→

BD.

Join AC and draw DE parallel to AC so that E ∈

−→

BC.

Prove that

is the product of

and

Similarly, given lengths

and

, construct a segment whose length is the quotient

4. Here is a geometric justiﬁcation of Descartes for the solution of the equation x

= ax + b

where a, b > 0.

• Let OQ and PQ be perpendicular with lengths

and b respectively.

• Draw the circle centered at O with radius

• Draw and extend the line

←→

OP.

• The solution is x =

(a) Prove that Descartes’ construction indeed recovers a solution.

(b) Show that the other solution to the equation x

= ax + b

is negative (false to Descartes).

How is it visible in the picture?

+ ax = b

5. Following the work of Islamic mathematicians such as Omar Khayyam, Fermat could describe

the solutions to certain cubic equations as the intersection of two conics. For example, to solve

+ bx

= bc (b, c > 0), he would introduce a new variable y by setting both sides equal to bxy.

The (positive) solution is then the x-solution to the system of equations

(

+ bx = by

xy = c

Sketch these curves explicitly for the example x

+ 4x

= 24.

8.2 The Beginnings of Calculus

At the heart of calculus is the relationship between velocity, displacement, rate of change and area.

• The instantaneous velocity of a particle is the rate of change of its displacement.

• The displacement of a particle is the net area under its velocity-time graph.

To state these principles requires graphs. Analytic geometry makes computation much easier (rate

of change means slope). Once it appeared in the early 1600s, the rapid development of calculus was

arguably inevitable. However, many of the basic ideas long predate Descartes and Fermat.

In the context of the above principles, the Fundamental Theorem of Calculus is intuitive: complete

knowledge of displacement (from some starting point) is equivalent to complete knowledge of ve-

locity. The modern statement is more daunting, though familiar:

Theorem. 1. If f is continuous on [a, b], then F(x) :=

f (u) du is continuous on [a, b], differen-

tiable on (a, b), and F

′

(x) = f (x).

2. If F is continuous on [a, b] with integrable derivative on (a, b), then

′

(x) dx = F(b) − F(a).

The triumph of the modern version is its abstraction and wide applicability: we’ve gone way beyond

considerations of velocity ( f ) and displacement (F). The challenge of teaching

and proving the Fun-

damental Theorem lies in developing and understanding what is meant by continuous, differentiable

and integrable. The quest for good deﬁnitions of these concepts is the story of analysis in the 17-1800s.

We begin with some older considerations of the velocity and area problems.

The Velocity Problem pre-1600

The modern ideas of uniform/average velocity are straightforward:

Measure how far an object travels in a given time interval and divide one by the other.

While several ancient Greek mathematicians considered this (and uniform acceleration), the chal-

lenge of considering a ratio of two unlike quantities (distance : time) proved difﬁcult to surmount.

Around 1200, Gerard of Brussels resurrected this approach as a deﬁnition of velocity, though it was

not considered a numerical quantity in its own right.

Gerard was credited in the 1330s by the Oxford/Merton Thinkers as inﬂuencing their investigations

of instantaneous velocity, a much more difﬁcult issue. They offered the following deﬁnition and made

the ﬁrst statement of the ‘mean speed theorem,’ though both are vague and logically dubious.

Deﬁnition. The velocity of a particle at an instant will be measured as the uniform velocity along

the path that would have been taken by the particle if it continued with that velocity.

This is really a convoluted idea of inertial motion.

Theorem. If a particle is uniformly accelerated from rest to some velocity, it will travel half the

distance it would have traveled over the same interval with the ﬁnal velocity.

Calculus students can easily be taught the mechanics of calculus (the power law, chain rule, etc.) without having any

idea of its meaning; witness the power and curse of analytic geometry and algebra!

For centuries it was thought that Galileo was the ﬁrst to state such ideas (compare his falling body

discussion, pg. 81), but the Oxford group beat him by 250 years. They had no algebra with which to

prove their assertions and essentially only asserted examples.

In the 1350’s, Nicolas Oresme (Paris) considered velocity geometrically by (essentially) drawing

velocity-time graphs. As we saw previously, this is essentially the approach taken by Galileo; it

is also an early version of axes. A major difference is that Galileo married mathematics to observation;

uniform acceleration for Galileo was precisely the motion of a falling body.

A proper deﬁnition of instantaneous velocity is difﬁcult because it requires limits, measuring average

velocity over smaller and smaller intervals. You are in good company if you ﬁnd this challenging:

Zeno’s arrow paradox is essentially an objection to the very idea of instantaneous velocity! Even if

one accepts the concept, its direct measurement, even in modern times, is essentially impossible.

The Area Problem pre-1600

We’ve previously seen two situations in which calculus-like methods were used to describe areas.

• Archimedes (sec. 3.4) computed/approximated the area of a circle and inside a parabola using

inﬁnitely many triangles. His ‘cross-section’ approach to ﬁnding area and volume also seems

modern, though this work remained unknown until 1899.

• Kepler (pg. 79) argued for his second law (equal areas in equal times) using inﬁnitesimally small

triangles to approximate segments of an ellipse. He also applied this method to several other

problems, crediting Archimedes with the approach.

The modern notion of Riemann sums is just a special case of approximating an area using small

rectangles: the philosophical challenge is again the notion of limits and inﬁnitesimals.

In an early antecedent, Oresme describes how to compute

the distance travelled by a particle whose speed is constant

on a sequence of intervals. For example:

Over the time interval



n+1



a particle travels at speed

1 + 3n. How far does it travel in 1 second?

Oresme drew boxes to compute areas and obtain

d =

∞

∑

n=0

(1 + 3n)



−

n+1



∞

∑

n=0

1 + 3n

n+1

= 4

Similar to Archimedes, the inﬁnite sum was evaluated by

spotting two patterns:

0 1

···+

n+1

= 1 −

n+1

+ ···+

n+1

= 1 −

n + 2

n+1

Oresme had neither our notation nor our (limit-dependent) concept of an inﬁnite series! He also

worked with similar problems for uniform accelerations over intervals. These are not true Riemann

sums, nor are they physical, for a particle cannot suddenly change speed!

For instance, radar Doppler-shift (as used to catch speeding motorists) requires measuring the wavelength of a radar

beam, which essentially compute the average velocity over a very small time interval.

Calculus `a la Fermat & Descartes

The advent of analytic geometry allowed Fermat and Descartes to turn the computation of instanta-

neous velocity and related differentiation problems into algebraic processes. The velocity of an object

is now identiﬁed with the slope of the displacement-time graph, which can be computed using vari-

ations on the modern method of secant lines. We discuss their competing methods.

Fermat’s method of adequation Consider the function p(x) =

−12x + 19; the goal is to ﬁnd the minimum, which we know

to be located at the x-value m = 2.

Fermat argues that if x

, x

are located near m such that p(x

) =

p(x

), then the polynomial p(x

) − p(x

) (which equals zero!)

is divisible by x

− x

. Indeed

0 =

p(x

) − p(x

)

− x

−12x

+ 19 − x

+ 12x

−19

− x

)(x

+ x

−12)

− x

= x

+ x

−12

p(x)

0 1 2 3

Since this holds for any x

, x

with p(x

) = p(x

), Fermat claims it also holds when x

= x

= m

(note the assumption of continuity!), and he concludes

−12 = 0 =⇒ m = 2

By considering values of x near m, it is clear that Fermat really has found a local minimum. We

recognize the idea that the slope of the tangent line is zero at local extrema.

This approach dates from the 1620s and is similar to earlier work of Vi

ete. Fermat later alters his

method by considering values p(x) and p(x + e) for small e (x is ‘adequated by e’). The difference

p(x + e) − p(x) is more easily divided by e without factorizing. Compared with the above, we obtain

0 =

p(x + e) − p(x)

+ 3x

e + 3xe

+ e

−12x −12e + 19 − x

+ 12x −19

e + 3xe

+ e

−12e

= 3x

−12 + 3xe + e

Fermat ﬁnished by setting e to zero and solving for x as before. Observe the derivative p

′

(x) =

−12 and how e plays the role of h in the modern expression

′

(x) = lim

h→0

p(x + h) − p(x)

Fermat’s method works for any polynomial, where the limit deﬁnition of derivative requires no more

than simple evaluation at h = 0. Fermat also extended his method to cover implicit curves and their

tangents.

Descartes’ method of normals Descartes and Fermat are known to have corresponded regarding

their methods. Descartes indeed seems to have felt somewhat challenged by Fermat, and engaged

in some criticism of his approach. Descartes’ method (in La G´eom´etrie) relies on circles and repeated

roots of polynomials in order to compute tangents.

In this example, we compute the slope of the curve y =

(10x − x

) at the point P = (4, 6).

0 4

Let N = (4 + ν, 0) be where the normal to the curve intersects the x-axis.

Draw a circular arc radius

r centered at N. If r is close to n, the circle intersects the curve in two points Q, R near to P. The line

←→

QR plainly approximates the tangent at P.

The co-ordinates of Q, R may be found by solving algebraic equations: substituting y =

(10x − x

)

into the equation for the circle results in an equation with two known roots, namely the x-values of

Q and R. By the factor theorem,

(



x −(4 + ν)



+ y

= r

y =

(10x − x

)

=⇒ (x −Q

)(x − R

) f (x) = 0

where f (x) is some polynomial (in this case quadratic). Rather than doing this explicitly, Descartes

observes that if the radius r is adjusted until it equals n, then Q and R coincide with P and the above

equation has a double-root:

(



x −(4 + ν)



+ y

= n

y =

(10x − x

)

=⇒ (x − P

)

f (x) = (x −4)

f (x) = 0

Factorization can be done by hand using long-division (note that ν and n are currently unknown!):

substituting as above, we obtain

0 = x

−20x

+ 116x

−32(4 + ν)x + 16(4 + ν)

−16n

= (x −4)

−12x + 4) + 32(3 −ν)x + 16(12 + 8ν + ν

−n

)

Since the remainder 32(3 − ν)x + 16(12 + 8ν + ν

− n

) must be the zero polynomial, we conclude

that ν = 3. By similar triangles, the slope of the curve at P is therefore

−y

At the time, ν was known as the subnormal and t the tangent.

Fermat and Area The previous methods permit differentiation, albeit inefﬁciently. Fermat also ap-

proached the area problem, in a manner similar to Oresme. Here is an example where we ﬁnd the

area under the curve y = x

between x = 0 and x = a.

Let 0 < r < 1 be constant. The rectangle on the interval

[ar

n+1

, ar

] touching the curve at its upper right-corner has

area

= (ar

− ar

n+1

) ·(ar

)

= a

(1 −r)r

The sum of the areas is therefore

∞

∑

n=0

= a

(1 −r)

∞

∑

n=0

(1 −r)

1 − r

1 + r + r

+ r

Setting r = 1 recovers the area under the curve:

aarar

···

(ar)

(ar

)

This is non-rigorous by modern standards and again implicitly invokes limits

by setting r = 1.

Regardless, Fermat is able to establish the power law

dx =

n+1

for any positive integer n.

Italian Calculus in the 17

Century: the Area and Volume Problems

In contrast to the work of Fermat and Descartes, contemporary Italian scholars were more focused on

integration problems. Here is Galileo’s classic ‘soup bowl’ problem, where he compares the volume

between a hemisphere and a cylinder to that of a cone.

Galileo observed

that the cross-sectional areas on both sides are equal (to πy

). Since all cross-

sections are equal, so must be the volumes. Unfortunately for Galileo, he couldn’t sufﬁciently address

two philosophical objections:

The zero-measure problem If cross-sections are ‘equal,’ then the top cross-sections state that a circle

‘equals’ a point.

Inﬁnitesimals sum to the whole? Can we claim that equal cross-sections imply equal volumes?

It was Galileo’s advocacy on these points that ﬁrst gained him notoriety with the Church. His later

evangelism for the Copernican theory merely rekindled old animosities.

For Fermat, r =

would have been rational, and he’d have set m = n at the end as in his adequation method

As did Archimedes 1900 years earlier (pg. 32), though Galileo was unaware of it.

Bonaventura Cavalieri (1598–1647) Cavalieri, a student of Galileo and a Jesuat scholar, gave a more

thorough discussion of indivisibles in 1635. He is particularly remembered for Cavalieri’s principle:

If geometric ﬁgures have proportional cross-sectional measure at every point relative to a

line, then the ﬁgures have measure in the same proportion.

Galileo’s soup bowl is an example of this reasoning, where the

‘line’ is any vertical.

Another classic example involves sliding a stack of coins or a

deck of cards; the volume of the slanted coin stack equals that

of the cylinder.

Extending his principle, Cavalieri inferred the power law

dx =

n+1

, giving reasonable

arguments for n = 1 and 2. Here is a sketch of his approach for n = 2.

Draw a cube of side x inside a cube of side a. This consists of three

congruent pyramids with base a

and eight a.

Consider the pyramid apex O and whose base is the square face near-

est the viewer. The cross-section of this pyramid at position x has

area x

. In Cavalieri’s language, the pyramid is ‘all the squares;’ in

modern language

dx =

Cavalieri also used his method (Book IV, Prop 19 of Geometria Indivisibilis) to calculate the area en-

closed in an Archimedean spiral. Suppose B moves at constant speed along a line OA rotating at

constant speed about O. In polar co-ordinates B traces a curve r = kθ for 0 ≤ θ ≤ 2π (we take k = 1).

= r, then the arc inside the spiral has length

ℓ(r) = 2πr ·

2π − θ

2π

= r(2π − r)

Imagine the arc as a noodle which, when cut at B and allowed to fall,

forms the line CD. The area in the spiral therefore equals that within the

parabola which (thanks to Archimedes and Cavalieri himself) is

that of

the largest triangle that can ﬁt inside, namely

(2π) ℓ(π) =

y = r(r −2π)

Cavalieri did this slightly differently, his parabola being drawn in a rectangle and the difference

subtracted from a triangle, but the above is easier to visualize.

Cavalieri did not court controversy like Galileo, being well-aware of the contentious nature of indi-

visibles and taking great pains to distinguish ‘all the lines/squares’ of a ﬁgure from the ﬁgure itself.

Geometria Indivisibilis was dense and difﬁcult, so it wasn’t easy to challenge. Cavalieri therefore re-

mained relatively safe, even as his political rivals (the Jesuit order) worked hard to stamp out the

‘dangerous’ study of indivisibles.

Evangelista Torricelli (1608–1647) A contemporary of Galileo and Cavalieri, Torricelli made several

applications of Cavalieri’s principle and advocated for its careful use.

If the sides of a rectangle are in the ratio 2 : 1, so also are indicated

red and blue segments. In Cavalieri’s language, ‘all the lines’ of the

red triangle are twice ‘all the lines’ of the blue triangle! Torricelli ob-

serves that the red triangle cannot have twice the area of the blue, since

they are congruent and points out that Cavalieri’s principle has been

misapplied: the cross-sections were not measured with reference to a

common line.

In modern language,

x dx =

2y dy are equal via the substitution x = 2y. The point is that the

inﬁnitesimals are also in the same ratio dx : dy = 2 : 1.

Another of Torricelli’s examples offers a seeming paradox.

The hyperbola with equation z =

is rotated around the z-

axis. A cylinder centered on the z-axis with radius x lying

under the surface has surface area

A = circumference · height = 2πxz = 2π

Underneath the graph at x, Torricelli draws a circular disk

with area 2π. Since the area of this disk is independent of

x, he argues that the volume under the original surface out

to radius a equals the volume of the solid cylinder:

V = 2πa

Torricelli argues that this is a correct use of Cavalieri’s

principle since the cylindrical ‘cross-sections’ and the cir-

cular cross-sections are both measured with respect to the

same line (the x-axis).

This is precisely the method of volume by cylindrical

shells that we learn in modern calculus:

V =

2πx ·

dx = 2πa

The conundrum is that the surface is inﬁnitely tall! How

can we justify the idea that it lies above a ﬁnite volume?

Galileo, Cavalieri and Torricelli mark the end of 400 years of Italian dominance of in science and

mathematics dating back to Fibonacci. Their ideas were too controversial to thrive so close to the

center of Church power. The center of European science and philosophy therefore moved northward.

The English and French (protestant) reformations of the 1500s together with developing ideas of

reformed government

meant that Northern Europe provided more fertile ground for new ideas.

For instance Hobbes’ Leviathan written during the English Civil War (1642–1651) was a plea for the constraint of abso-

lute monarchical power. The War itself indeed proved decapitatingly effective at reining in a King. . .

Exercises 8.2. 1. Find the maximum of p(x) = 5 + x −2x

using Fermat’s ﬁrst method.

2. Use Fermat’s second method (“+e”) to ﬁnd the maximum of bx −x

. How might Fermat decide

which of the two solutions to choose as his maximum?

3. Justify Fermat’s ﬁrst method of determining maxima and minima by showing that if M is a

maximum of p(x), then the polynomial p(x) − M always has a factor (x − a)

, where a is the

value of x giving the maximum.

4. Use Descartes’ method of normals to compute the slope of the curve y = x

at (a, a

5. Suppose the surface of a sphere of radius r is subdivided into inﬁnitesimal regions of equal

area. Following Kepler, use the formula for the volume of a cone (

base·height) to ﬁnd the

relationship between the volume V of the sphere and its surface area A.

6. (A problem of Kepler) Show that the largest circular cylinder that can be inscribed in a sphere

is one in which the ratio of diameter to altitude is

√

2 : 1.

(Hint: Relate the problem to ﬁnding the maximum of the function f (x) = x −

, for which you can

use modern calculus)

7. We consider a version of Gregory St. Vincent’s 1647 approach to the area under the hyperbola

xy = 1.

(a) If 1 < a < b and r > 0 as in the picture, explain why

the areas A and B under the curve satisfy the same

inequalities

b − a

< A, B <

b − a

(Since [a, b] may be subdivided into arbitrarily many subin-

tervals, the areas A and B are therefore equal)

0 1

a b ar br

(b) If A(x) is the area under the hyperbola between 1 and x, explain why A satisﬁes the loga-

rithmic identity

A(x

) = A(x

) + A(x

)

Why are you not surprised by this?

(For simplicity, assume 1 < x

< x

)

8. Consider two copies of triangle with sides a, b arranged

into a rectangle. Argue that ‘all the lines’ of the rectangle

are twice ‘all the lines’ of the triangle.

(In modern language,

b dy = 2

y dy)

9. Repeat Cavalieri’s analysis of the spiral (pg. 92) to ﬁnd the area inside one revolution of the

curve r = kθ for any k > 0.

8.3 Calculus in the late 1600s

By the second half of the 17

century, the mathematical center of Europe had moved northwards, to

France, Germany, Holland and Britain. In this section we present some of this work, culminating in

the efforts of Newton and Leibniz.

Hendrick van Heuraet (1634–1660) Working in Holland, van

Heuraet studied Descartes’ and argued that the arc-length of a

curve described by a function y equals the area under the curve

z =

where n is the ‘normal’ curve. His method appeared in

Frans van Schooten’s 1659 version of Descartes’ La Geometrie. To

see why this should make sense, recall Descartes’ method of nor-

mals and observe that the ratio n : y equals that of ds : dx in a

differential triangle.

His most famous example involved calculating the arc-length of

the curve y

= x

. By Descartes’ method,

(

(x − a −ν)

+ y

= n

= x

=⇒ 0 = x

+ x

−2(x + ν)x + (a + ν)

−n

= (x − a)

(x + 2a + 1) + (3a

−2ν)x

+ ν

+ 2aν − 2a

−n

Arc-length

equals

Area

Since the remainder must be zero, we conclude that ν =

, from which

n =

+ y

+ a

=⇒ z =

a + 1

The arc-length from x = 0 to a is therefore the area under the parabola z

x + 1 between the same

limits. By the usual Archimedean

-triangle approach, we see that

Arc-length =





+ a



z(a) −

·1





a +



3/2

−

James Gregory (1638–1675) Hailing from Aberdeen, Scotland, Gregory studied in Italy with Stefano

Angeli, a pupil of Torricelli, before returning to Scotland where he became chair of mathematics at

St. Andrews, and then Edinburgh.

Gregory repeats van Heuraet’s work relating the length of a curve to the area under another, before

considering whether the process can be reversed: given a curve z(x), can we ﬁnd a curve y(x) such

that the arc-length of y is given by the area under z? In modern language, given z, ﬁnd y such that

1 + y

′2

dx =

z dx

which we’d view as solving the ODE

√

−1. Gregory’s solution was to deﬁne y to be the area

under the curve

√

−1 from x = 0 to a. This is essentially part 1 of the fundamental theorem: if

you want something whose slope (derivative) is given, deﬁne it to be the area under the curve!

Isaac Barrow (1630–1677) Like Gregory, Barrow also studied mathematics in Italy (and France),

before returning to England to become the inaugural Lucasian Professor of Mathematics

at Trinity

College Cambridge. Barrow’s work remained predominantly geometric; he stated proving geometric

versions of both parts of the fundamental theorem, though credited Gregory with part of the argu-

ment. In a precursor of Newton’s work, he also modiﬁed Fermat’s algorithm for differentiation. For

example, here is how Barrow would have found the slope of the curve x

+ 2xy

= c at a point (x, y):

• Replace x and y with x + e and y + a respectively and expand:

+ 2ex + e

+ 2xy

+ 4axy + 2a

x + 2ey

+ 4eay + 2ea

= c

• Delete everything from the original equation x

+ 2xy

= c and every expression containing

two or more of the terms e, a:

2ex + 4axy + 2ey

= 0

• Rearrange: the slope is the ratio a : e = −x −y

: 2xy

This is implicit differentiation, where dx = e and dy = a. Note again the essential difﬁculty with

these algorithmic approaches to calculus: the inﬁnitesimal quantities e, a are necessary for the calcu-

lation, but most of them are discarded when no longer useful! Can we really calculate with objects

that must simultaneously exist and be zero?

Isaac Newton (1642–1727)

The caricature of Newton is of an obsessive genius—difﬁcult to get along with, but with a phenom-

enal ability to concentrate on problems. One possibly apocryphal story describes how he continued

lecturing to an empty room even after no-one had turned up!

We are mostly interested in Newton’s mathematics,

though his wider fame comes from its wide applica-

tion, particularly to gravitation. In Philosophiæ Natu-

ralis Principia Mathematica (1686), Newton applied his

three laws of motion

and the machinery of calcu-

lus to prove the relationship between Kepler’s laws

of planetary motion (pg. 78) and an inverse-square

gravitational force. The Principia is Newton’s ﬁrst

published work involving calculus, though many

of its results and his method appear to have been

worked out 20 years previously, just after completing

his undergraduate studies and while Cambridge was

closed due to the 1665–6 plague epidemic.

Kepler’s 1st law =⇒ inverse-square force

One of the most prestigious world-wide academic positions in theoretical Physics, in large part due to the fame of its

second incumbent: Issac Newton. Later chairs include Paul Dirac and Stephen Hawking.

This is at the heart of Bishop George Berkeley’s (after whom the Californian city and university are named) famous

1734 objection to calculus; that inﬁnitesimals are merely the “ghosts of departed quantities.”

I. Inertial motion; II. F = ma; III. Equal-and-opposite forces. Consider these as the axioms of Newtonian mechanics.

Newton’s geometric presentation was typical for the time. He comments on how calculations are

more efﬁcient using indivisible methods, but that the ‘hypothesis of indivisibles seems somewhat

harsh’ (he likely wants to counter the impression that his method is philosophically shaky). Newton’s

approach makes it hard for modern readers to extract calculus algorithms; indeed the Principia is not

a calculus textbook and it is not really possible to learn calculus directly from it. Nevertheless, it

contains notions of many standard concepts, for instance:

Limits/continuity Book I, Lemma I: “Quantities, . . . which in any ﬁnite time converge continually

to equality, and before the end of that time approach nearer to each other than by any given

difference, become ultimately equal.”

One can see modern ideas appearing (e.g., ∀ϵ > 0,

a −b

< ϵ =⇒ a = b) though there is a

long way to go! What, for instance does approach mean?

Product Rule In the following pages, Newton argues for the product rule by augmenting the sides

of a rectangle.

If a, b are inﬁnitesimal changes in the sides of a rectangle with sides A, B, then the moment of

mutation (inﬁnitesimal change) of the generated rectangle AB is the quantity aB + bA. In more

familiar language,

(A + a)(B + b) − AB ≈ aB + bA

where Newton ignores the double-inﬁnitesimal quantity ab, exactly as did Barrow and others

before him.

Power Law In the same pages, Newton asserts the general power law for rational exponents

. . . the moment of any power A

will be

n−m

in what seems like a non-rigorous appeal to induction based on the product rule. In reality,

Newton established this using inﬁnitesimal arguments; we’ll see one of his methods for this

shortly.

In contrast to the mostly synthetic presentation in the Principia, Newton made great use of algebra

in his private calculations and correspondence with friends. Some of these private works were pub-

lished many years later. We discuss some of his methods in what follows.

Fluxions and Fluents Newton’s main language for calculus (he had several!) referred to time-

dependent quantities x, y as ﬂuents and their derivatives as ﬂuxions, denoted using dots

y; the

modern notation y

′

comes from this.

Anti-derivatives were denoted by placing an accent directly

over a quantity:

x is a ﬂuent of which x is the ﬂuxion. Newton had several algorithms for computing

ﬂuxions, often variants of those of previous mathematicians: for instance, here he ﬁnds the relation-

ship between the ﬂuxions of ﬂuents x, y satisfying x

+ 3xy

+ y = 5.

• Rearrange as a polynomial in x: thus x

+ (3y

)x + (y −5) = 0.

• Multiply terms by a decreasing arithmetic sequence (e.g. 2, 1, 0) and the entire expression by

(2x + 3y

)

• Repeat for y, using the same arithmetic sequence; in this example, 2 corresponds to x

, so we start

with 3 for y

(3x)y

+ 0y

+ y + (x

−5) = 0 ⇝ (9xy

+ y)

• Sum these expressions, set equal to zero and rearrange for the required ratio:

(2x + 3y

)

x + (9xy

+ y)

y = 0 =⇒

= −

2x + 3y

9xy

+ y

The arithmetic sequence encodes the power law for derivatives, and the result is exactly what you’d

expect from modern implicit differentiation.

The Binomial Series Also discovered during the plague years, but ﬁrst appearing in a private letter

of 1676, is Newton’s discovery of the binomial series

(1 + x)

= 1 +

∞

∑

k=1

α(α −1)(α − 2) ···(α − k + 1)

which allowed Newton to expand expressions such as

(1 + x)

1/2

= 1 +

x −

−

128

+ ···

Newton’s version was only for fractional exponents and is more difﬁcult to read:

P + PQ

= P

AQ +

m −n

BQ +

m −2n

CQ +

m −3n

DQ + ··· (∗)

Newton wrote exponents using juxtaposition, and A, B, C, D, . . ., meant ‘the previous term’: thus

= P

m/n

, A = P

m/n

and B =

AQ =

m/n

Q. In more modern language, (∗) reads

(P + PQ)

m/n

= P

m/n

Q +

m(m −n)

m/n

m(m −n)(m −2n)

m/n

+ ···

A ﬂuent is a ‘ﬂowing’ quantity: to us, a smooth function, though this was not formally deﬁned. To be in ﬂux is to be

changing, hence ‘rate of change.’ Newton’s dot-notation (‘pricked letters’) persists in the modern ﬁeld of dynamics: for

instance

r = −

r is the differential equation arising from Newton’s second law together with the inverse-square law for

gravitation (note the double dot for the second derivative).

Newton’s ‘proof’ wouldn’t pass modern muster. His discoveries were largely the result of some

inspired pattern-spotting! Several examples were explicitly veriﬁed by multiplying out or using

long-division. For instance the series for

√

1 + x may be obtained

1 + x = (1 + ax + bx

+ cx

+ ···)

= 1 + 2ax + (a

+ 2b)x

+ (2c + 2ab)x

+ ···

=⇒ a =

, b = −

= −

, c = −ab =

, etc.

=⇒

√

1 + x = 1 +

x −

+ ···

Newton did not work through the full theory of inﬁnite series as you would encounter in a typical

undergraduate analysis course. For instance:

•

√

1 + x is well-deﬁned for x ≥ −1, yet the resulting series only converges when −1 ≤ x ≤ 1.

What happens in general: must a series converge to the original/generating function?

• Is it legitimate to differentiate and integrate power series term-by-term as if they are polynomi-

als? Answer: (mostly) yes, though it was 150 years before Cauchy, Weierstraß and others could

rigorously conﬁrm this.

The Power Law & the Fundamental Theorem Newton’s ability to expand expressions as inﬁnite

series was essential to his arguments. Here is one of his arguments to prove the power law.

1. Assume that the area under a curve y is given by a function

z =

+ 1

m + n

m+n

2. If x is increased by an inﬁnitesimal amount o, then the new area

under the curve is found using the binomial series

z + oy =

m + n

(x + o)

m+n

m + n

m+n



1 +



m+n

m + n

m+n



1 +

m + n

(m + n)m

+ ···



3. Following Barrow, Newton cancels the terms in the original equation, divides by o, and throws

out all remaining o-terms. The result is

y =

m + n

m+n

m + n

= x

m/n

Note the link-up with the fundamental theorem, which is intuitively obvious when y is a

‘ﬂowing’ (continuous) quantity: the additional area is approximately an inﬁnitesimal rectan-

gle oy = dz with base o = dx and height y, whence

dz = y dx ↭

y(t) dt = y(x)

Using similar approaches, Newton produced one of the ﬁrst tables of integrals, listing much of what

you’d ﬁnd inside the covers of an undergraduate calculus textbook! He also obtained power series

for trigonometric and logarithmic functions, partly following the work of Gregory and others. By

combining his approach with the power law, he was able to efﬁciently integrate and differentiate an

enormous variety of functions.

Gottfried Wilhelm Leibniz (1646–1716)

Leibniz hailed from Leipzig, southwest of Berlin, in what was then part of the Holy Roman Empire.

His initial studies were in philosophy, following his professor father. Though he eventually became

a diplomat and then counsellor to the Duke of Hanover, his taste for advanced mathematics was

fueled during his 1672–76 sojourn in Paris, where he was introduced to van Shooten’s expansion of

Descartes’ geometric ideas, in particular the concept of the differential triangle (page 95) which was

already in use by others such as Pascal and Barrow.

The familiar notations for derivatives

and integrals

y dx come from Leibniz. Very loosely, here

is its origin and how it relates to the fundamental theorem. Suppose that

, z

), (x

, z

), . . . , (x

, z

), x

< x

< ··· < x

describe a sequence of points along a curve z(x) deﬁned on an interval [x

, x

]. One may then form

the sequences of differences and sums of the ordinates z



δz





−z

, z

−z

, . . . , z

−z

n−1





∑





, z

+ z

, . . . , z

+ ···+ z



Two relationships between sums and differences are immediate:

1. The difference sequence of the sums returns the original sequence:



∑





, z

, . . . , z



2. The sum of the difference sequence is the net change in the ordinate:

∑



δz



= (z

−z

) + (z

−z

) + ··· + (z

−z

n−1

) = z

−z

Leibniz’s notation arises from viewing a curve as an inﬁnite sequence of points: he writes dz for the

inﬁnitesimal differences and

for the sum of inﬁnitely many inﬁnitesimal objects. Observation 2

becomes

dz = z(x

) − z(x

): the sum of the inﬁnitesimal changes in z is its net change. Indeed, if

we assume that z(x) describes the area

under a curve y(x), the fact that dz = y dx recovers part 2 of

the fundamental theorem of calculus:

y dx = z: the area under the curve is the sum of its inﬁnitesimal increments.

Observation 1 makes sense once we apply it to a ‘sequence’ of inﬁnitesimals z

⇝ y dx, whence it

becomes part 1 of the fundamental theorem:

1. d



y dx



= y dx, or alternatively,



y dx



= y: the rate of change of the area

function is the ordinate.

Notation is chosen to match Newton’s discussion of the power law (page 99).

100

While we are happy to refer to inﬁnitesimals and inﬁnite sums, Leibniz was more cagey in his pub-

lications out of fear of criticism. He referred to each dx as an arbitrary (if small) ﬁnite line segment,

and therefore—like every other contemporary practitioner of calculus—fails to get to grips with the

essential paradoxes involved. Regardless, he and his followers became adept at manipulating differ-

ential expressions. For instance, if y = z

+ 2z, Leibniz might compute

dy = (z + dz)

+ 2(z + dz) −z

−2z = 3z

dz + 3z(dz)

+ (dz)

+ 2 dz

= (3z

+ 2z) dz

where the (dz)

and (dz)

terms are discarded due to their (relatively) inﬁnitesimal size. Using

such approaches Leibniz justiﬁed general formulas such as the linearity of derivatives, the product,

quotient and power rules. Even the chain rule is easy in Leibniz’s notation: given y(x) =

√

1 + x

Leibniz might perform a substitution u = 1 + x

and observe that

dy = d

√

u =

√

u + du −

√

u =

u + du −u

√

u + du +

√

1 + x

Like Newton, Leibniz worked extensively with power series. As a ﬁnal example here is his compu-

tation of that for sine. In contrast to Newton’s approach,

Leibniz derived the well-known second-

order ODE satisﬁed by sine.

In a circle of radius 1, let s be the polar angle and y = sin s the ordinate.

By considering the differential triangle, we see that

1 − y

=⇒ (1 − y

)(ds)

= (dy)

(∗)

Leibniz supposes that inﬁnitesimals ds describing the arc are constant

and applies d again (with the product rule):

−2y (dy)(ds)

= 2(dy)d(dy) =⇒

d( dy)

(ds)

= −y





1 − y

He then writes y(s) = sin s = c

+ c

s + c

+ c

··· as a power series. Since at s = 0, y = sin s = 0

and ds = dy (by ∗), Leibniz sees that c

= 0 and c

= 1. He now differentiates twice and equates

coefﬁcients:

d( dy)

(ds)

= 2c

+ 6c

s + 12c

+ 20c

+ 30c

+ ··· = −y = −s − c

−c

−···

=⇒ 0 = c

= c

= ··· , c

= −

, c

= −

120

, . . .

to obtain the familiar series

y = sin s = s −

120

+ ··· = s −

−

+ ···

While not precisely the same as modern calculus—in particular, the differentials dx, dy, ds are sep-

arate quantities—this should all seem very familiar! The computational efﬁciency of such notation

super-charged mathematics and its applicability to real-world problems.

Newton ﬁrst found a series for arcsine before computing its inverse.

We’ve bracketed everything. When these are removed, we obtain the standard Leibniz notation for second derivatives!

101

Exercises 8.3. 1. Show that to ﬁnd the length of the arc of the parabola y = x

one needs to

determine the area under the hyperbola y

−4x

= 1.

2. Use Barrow’s a, e method to determine the slope of the tangent line to the curve x

+ y

= c

3. Calculate a power series for

1−x

by using long-division.

4. Use the binomial series to obtain a power series expression for ln(1 + x), which Newton knew

to describe the area under the curve

1+x

5. Using his calculus, Newton was able to extend older methods of approximation.

• Suppose f (c) = 0 and that a

is an initial approximation to c.

• The tangent line at



, f (a

)



has equation

y = f (a

) + f

′

)(x − a

)

which intersects the x-axis at a

:= a

−

f (a

)

′

)

• Iterate to obtain a sequence (a

) that (typically) converges to c:

n+1

= a

−

f (a

)

′

)

lim

n→∞

= c

f (x)

(a) If f (x) = x

−c, show that Newton’s method is the Babylonian method of the mean.

(b) Use Newton’s method to solve the equation x

−2 = 0 to a result accurate to eight decimal

places. How many steps does this take?

6. Calculate the relationship of the ﬂuxions in the equation x

− ax

+ axy −y

= 0 using multi-

plication by the progression 4, 3, 2, 1. Compare to what happens if you use the progression 3,

2, 1, 0. What do you notice?

7. Use Leibniz’s differential triangle to argue that

x = cos s and dx = −y ds

Where does the negative sign come from? Hence ﬁnd the standard power series representation

for cosine and conclude that the rate of change (derivative) of sine is cosine.

8. Given a curve y(x) with y(0) = 0, Leibniz’s transmutation theorem relates the area under y and

the curve z(x) = y − x

obtained by considering the y-intercepts of the tangent lines to the

original:

y dx =



ay(a) +

z dx



If x

= y

, ﬁnd z and use the transmutation theorem to ﬁnd the area

y dx.

How does the transmutation theorem relate to integration by parts?

102