Making up Numbers

A History of Invention in Mathematics

EKKEHARD KOPP
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 CHAPTER 1

Arithmetic in Antiquity

The monuments of wit survive the monuments of power.

Sir Francis Bacon, Essex’s Device, 1595

Summary

In this chapter the focus is on two ancient civilisations: Babylonian and

Greek. Our evidence for the former comes from a large number of sun-
dried clay tablets (found in modern-day Iraq) that were only deciphered less
than a century ago. By contrast, the mathematics and philosophy developed
in the Greek city states (notably Athens) and surrounding territories, well
over 2000 years ago, have underpinned Western civilisation ever since the
Renaissance. The content of the thirteen books of Euclid’s famous Elements
of Geometry dominated Western school mathematics well into the twentieth
century, usually giving school pupils their first experience of mathematical
proofs. It remains a beacon of mathematical achievement in antiquity.
In Babylonian arithmetic, on the other hand, we find the first truly po-
sitional number system, essentially equivalent to our decimal system, al-
though its base was 60 rather then 10. Traces of this system remain in our the
division of an hour into 60 minutes, each of which has 60 seconds, for exam-
ple. We begin the chapter with a brief glimpse of the ways in which this sex-
agesimal number system was used in the area around the Tigris-Euphrates
valley to solve a variety of practical problems, notably including quadratic
equations.
Mathematical development in Ancient Greece is traced back to Pythago-
ras of Samos (c.570-c.495 BCE), who was both a philosopher and a mathe-
matician. Very little survives of the work of the influential quasi-religious
Pythagorean sect he founded, except in occasional accounts by later com-
mentators, of whom Plato (c.428-c.348 BCE) and Aristotle (384-322 BCE) are
perhaps the most reliable. This chapter explores the group’s philosophi-
cal claim that ‘All is Number’ and the arithmetical techniques that led them
to remarkable insights, such as the famous Pythagoras theorem, but also into
logical difficulties. Their influence on the later work of the Athenian school
around Plato, much of it preserved in Euclid’s Elements, can be seen the lat-
ter’s Books VII-IX and in an exhaustive study of incommensurables in Book X.
c Ekkehard Kopp CC BY 4.0 https://doi.org/10.11647/OBP.0236.01
14 1. ARITHMETIC IN ANTIQUITY

Remaining with arithmetic, the chapter closes with a brief look at the (much
later) Arithmetika of Diophantus (c.210-c.290).

1. Babylon: sexagesimals, quadratic equations

Historical research relies on written records as its primary source of ev-

idence. For this reason I omit mention of tallying or counting with sticks
that precedes the earliest written records. Written records from early civil-
isations in China or India used materials that were not easily preserved,
so that direct evidence of their work is scarce.1 The best-preserved records
from early civilisations are found on Egyptian papyri and hieroglyphs and
on Babylonian clay tablets.
Most Babylonian tablets stem from the Old Babylonian period (1830-1501
BCE), others from the Seleucid period of the last three or four centuries BCE.
A considerable number of mathematical clay tablets has been discovered.
Some contain various tables of numbers, others describe recipes for solv-
ing specific numerical problems. Many are thought to have been used in
schools training scribes for Babylonian society, which was probably an elite
profession, open to a select few.
The tablets were inscribed in cuneiform script with a wedge-shaped sty-
lus as shown in Figure 3—the name derives from cuneus, the Latin term for
‘wedge’—and dried in the sun. The extent of their mathematical sophisti-
cation only became clear when cuneiform script was fully deciphered in the
1930s, much of it by the Austrian-American mathematician Otto Neugebauer
(1899-1990), [34]. Earlier historians of mathematics had paid more attention
to Egyptian geometry and arithmetic, although its impact on later mathe-
matical development is perhaps less significant. For this reason Egyptian
mathematics will not be considered here.2
The Babylonian number system combined 60 as the number base to-
gether with symbols for tens and units. For digits up to nine, the number
was marked by that number of vertical wedges, and the number of multiples
of 10 was marked similarly by up to five horizontal (or tilted) wedges. This
enabled them to display numbers 1, 2, ..., 59. We call such a number system
sexagesimal, just as we use the term decimal for our usual (base 10) numbers,
or binary (also dyadic) when using the base 2 (as in modern computing). The
reason for the Babylonians’ choice of 60 is not known, but the fact that 60 =

1An account of Chinese mathematics and astronomy can be found in Volume 3 of Joseph
Needham’s multi-volume work Science and Civilization in China. See also Chinese Mathematics,
A concise history by Li Yan & Du Shiran, (translated by J.N. Crossley and A.W.-C. Lun), Oxford,
Oxford Science Publications, 1987, and the article ‘Chinese Mathematics’, by Joseph Dauben,
in the volume edited by V.J. Katz et al.: The Mathematics of Egypt, Mesopotamia, China, India, and
Islam: A Sourcebook. Princeton, Princeton University Press, 2007.
2
For Egyptian mathematics see (e.g.): A. Imhausen, Mathematics in Ancient Egypt. A Con-
textual History, Princeton, Princeton University Press, 2016.
 1. BABYLON: SEXAGESIMALS, QUADRATIC EQUATIONS 15

Figure 3. 10329 in cuneiform script

2 × 2 × 3 × 5 has more divisors (in fact, twelve: 1, 2, 3, 4, 5, 6, 10, 12, 20, 30, 60)
than 10 = 2 × 5 (which has only four: 1, 2, 5, 10) may have been a factor in
this choice.
The key observation, nearly 4000 years ago, was that, once symbols for
1, 2, 3, ..., 59 had been decided upon (and executed with no more than five
horizontal and nine vertical wedge strokes), all other (whole) numbers could
be understood with these symbols. To write numbers outside the range 1
to 59, the Babylonians used a positional (or place-value) system, breaking up
the numbers according to successive powers of 60 and separating these by
a space, as in Figure 3, which shows the number
10329 = 2 × (60)2 + 52 × (60)1 + 9 × (60)0 .
(The final term is simply 9, since n0 = 1 for any n. This follows from the
power law na × nb = na+b , using b = −a.)
The spaces between each group of wedges indicate the relative power
of 60 that each group occupies. Here we have implicitly assumed that we
are dealing with a whole number.
However, the positional system was also used to include sexagesimal frac-
tions. For example, the numbers
52 9
2 + 52 × (60)−1 + 9 × (60)−2 = 2 + +
60 3600
9
2 × 60 + 52 + 9 × (60)−1 = 172 +
60
would be written exactly as the number given in Figure 3. As no space was
left at the end of a number, its absolute size often had to be inferred from the
problem under discussion, although in some tablets the number would be
followed by a word indicating what power was intended for the final group
of wedges. More seriously, in the Old Babylonian tablets there is no symbol
for 0 to indicate the absence of a power (as would be needed in 7209 =
2 × (60)2 + 9, for example), although some texts appear to indicate this by
leaving an extra internal space.
By the time of the second major set, dating from the Seleucid period (the
last four centuries BCE) the second ambiguity had been removed. The oc-
currence of zero was now indicated by a space marked with two small oblique
wedges, showing that that particular power of 60 is ‘skipped’. To write down
7209, the scribe would now replace the central group of wedges in Figure
16 1. ARITHMETIC IN ANTIQUITY

3 (denoting 52 × 60) by two oblique wedges, rather than simply omitting

it without further indication. However, this practice appears only to have
been used when zero occurs in a intermediate position, such as in this ex-
ample. It was not used at the end of a number, so the absolute size of the
number would continue to be deduced from the context of the particular
problem.
Despite its peculiarities, the Babylonians could use their system to add,
multiply, subtract and divide numbers in much the same way as we do with
decimal notation, and to treat fractional parts of the numbers in exactly the
same way as the integral parts. This was a major notational and conceptual
advance.
It is convenient to use Neugebauer’s notation to express the sexagesimal
system in our decimal symbols. For example, the above number 2 × 60 +
52 + 9 × (60)−1 = 172.15 is written by Neugebauer as 2, 52; 9. The powers of
60 are separated by commas, where Babylonians would use spaces instead,
and a semicolon separates the fractional from the integral part.
A large proportion of the cuneiform tables that have been found contain
arithmetical tables, listing, in sexagesimal form, squares, cubes, reciprocals
and even square and cube roots of numbers. They probably served in the
ancient schools for scribes as the precursors of the books of logarithmic ta-
bles that were prevalent in our secondary schools until a few decades ago,
before being replaced by electronic calculators and computers.3
The cuneiform tables were necessarily incomplete: they dealt only with
regular sexagesimals, i.e. numbers that could be expressed simply in sexa-
gesimal form. This was not possible for certain fractions, such as the recip-
rocal of 7, for example. In a tablet containing a typical table of reciprocals
one usually finds two columns, and the two numbers in the same row al-
ways have 60 as their product. But immediately following the row listing
the numbers 6 and 10 (the ‘reciprocal’ of 6 is 60 6 = 10) we find the numbers
8 and 7; 30, which represents 60 8 = 7 1
2 , written to base 60 as 7 + 3060 . The
row that would contain 7 and its reciprocal is simply omitted. The reason
for this is clear: 60
7 cannot be written as a finite sexagesimal – when using
‘long division’, as in 60 30
8 = 7 + 60 (in Neugebauer’s notation: 7; 30), the ra-
tio 7 cannot be expressed as a sum of the form a601 + (60)
60 a2 an
2 + ... + (60)n for

any finite sequence of numbers (ai )i≤n of the numbers {1, 2, ..., 59}, since all
remainders are non-zero.
The same problem arises in our familiar decimal notation: at school we
all meet infinite ‘recurring’ decimal expansions such as 31 = 0.33333.... and
1
7 = 0.142857142857.... Decimal notation (that is, dividing 1.000000... by 7)
requires the second of these to begin with the finite sum

3The invention and role of logarithms will be discussed in Chapter 3.

 1. BABYLON: SEXAGESIMALS, QUADRATIC EQUATIONS 17

1 4 2 8 5 7
+ + + + + .
10 (10)2 (10)3 (10)4 (10)5 (10)6
The numerators 1, 4, 2, 8, 5, 7 of the six terms repeat indefinitely, the sum
1
of these terms is multiplied by (10) 6k for k = 0, 1, 2, ..., and the results are

then summed. Thus the expression in decimal notation even of simple ratio-
nal numbers often leads to summing an indefinite number of terms.4 Notice
that, in the sexagesimal system, 17 provides the only ‘irregular’ reciprocal
among numbers below 10, wheras in the decimal system the reciprocals of
3, 6, 7, 9 are all ‘irregular’!
In practice, and in modern computers, we handle this problem by using
‘rational approximation’: we terminate the expansion after a set number of
decimal places, giving us an approximation that is sufficiently close for our
purposes. The Babylonians used the same principle. Babylonian approxima-
tions of irregular sexagesimal reciprocals could easily be given with a high
degree of accuracy, as would be needed for calculations with large numbers,
for example in astronomy. The use of base 60 has the advantage that good
accuracy can be achieved in relatively few steps: for example, an error of
1 4 1
at most ( 60 ) = 12,960,000 (achieved after four steps) is usually negligible in
practice.
The tables of reciprocals allowed division to be carried out easily: taking
a
b as the product a × ( 1b ) would allow the scribe to ‘look up’ the reciprocal
of b in a table and multiply it by a, while interpreting the product in terms
of the correct powers of 60. Such techniques are well suited to handle arith-
metic with large numbers and can be applied very effectively in calculations
resulting from astronomical or navigational observations.
Going beyond reciprocals, cuneiform tablets have been found showing
that the Babylonians knew general methods√for approximating square roots.
A simple but effective method to estimate a is to guess a first approxima-
tion, say r1 . If its square exceeds a (we write this as r12 > a), we see that
as a second guess the ratio ra1 will be too small. The arithmetical average
of these two guesses, r2 = 21 (r1 + ra1 ), provides a better estimate, but will
again be too large, so that r22 > a.5 Now repeat this process, starting with
r2 in place of r1 , and
√ continue in this fashion. One quickly obtains a good
approximation to a.
In the collection held at Yale University, USA, the tablet today known
as Yale7289, which
√ dates from between 1800 and 1600 BCE, displays the ap-
proximation of 2 by 1; 24, 51, 10. This equals r3 if one starts with the over-
estimate r1 = 1; 30
√ (i.e. r1 = 1.5 in24decimal notation, which gives r12 = 2.25).
51 10
Approximating 2 by r3 = 1 + 60 + (60)2 + (60) 3 (which we would write

4We return to this issue in Chapter 7.

5See MM for a simple proof of this claim.
18 1. ARITHMETIC IN ANTIQUITY

√
Figure 4. Approximating 2

as 1.4142162963 in decimal form) is accurate to 5 decimal places. The tablet

is incomplete, and no workings are shown, but it seems plausible that the
scribe might have used the above method.
While Egyptian papyri provide evidence that the solution of linear equa-
tions (i.e. of the form ax−b = 0, with solution x = ab ) formed a standard part
of Egyptian mathematics, few Babylonian texts appear to deal with such
problems. Instead, many tablets include more complex problems that lead
to quadratic equations. A general solution procedure for quadratics is illus-
trated in several Old-Babylonians tablets, although they always deal with
specific numerical problems. A text from the early Hammurabi period, for
example, poses the problem of finding the side of a square, given that the
area less the side is 14, 30. This number is ambiguous, since we don’t have a
symbol for zero at this stage. We will read 14, 30 as (14 × 60) + 30 = 870, as
the Babylonians often preferred to start calculations with whole numbers.
If the side is x, the area of the square is x2 , and we must solve the equation
x2 − x = 870.

Using Neugebauer’s notation for numbers, we translate the scribe’s in-

structions as: Take half of 1, which is 30, and multiply it by 30, which is 15. Add
this to 14, 30 to get 14, 30; 15. This is the square of 29; 30, and the result is 30, the
side of the square.
To understand the quotation from the tablet, recall that we would write
1
2 as 0; 30 and 14 as 0; 15 instead of 30 and 15. The scribe probably found
these from a table of reciprocals, given as the numbers whose products with
2, respectively 4, come to 60. To follow his procedure we reconstruct the
general method in modern terminology. In the equation x2 − x = 870 the
coefficient of the term in x (the linear term) is −1, while 870 is the constant
term. Call these b and c respectively, so that the equation we seek to solve
is x2 + bx = c. Following the scribe’s instruction we now divide b by 2 and
square the result, obtaining ( 2b )2 , which we add to c. We then take the square
root (the words ‘square’ and ‘square root’ were used interchangably by the
Babylonians) of this sum (probably looking it up in a table) and and subtract
b
2 to find that r
b b
x = − + ( )2 + c.
2 2
This recipe can be derived from the following simple picture (although
we have no direct textual evidence of any geometric figures that may have
 2. PYTHAGORAS: ALL IS NUMBER 19

Figure 5. Solving quadratic equations

been drawn): the equation x2 +bx = c says that by adding the square of side
x to a rectangle with base b and height x, we obtain a given area c. To find
x, we cut the base b of the rectangle in half, then arrange these two thinner
rectangles on the square (one on top, one on the side, as in Figure 5, which
is taken from [44]). We ‘complete the square’, which has the new base x + 2b ,
and to keep the two sides of the equation equal we need to add the small
(black) square of side ( 2b ) to the area c. Taking the square root on both sides
q
yields x + 2b = ( 2b )2 + c. The numbers used by the scribe are b = −1,
c = 870. So ( 2b )2 + c = 870 14 = 3481
4 = ( 59 2 1
2 ) = 29 2 , as claimed. Now, to
find x, we subtract 2 to obtain x as the solution of our quadratic equation.6
b

Since b = −1, this means that we should add 21 to 29 12 and thus obtain 30,
as required.
It is important to emphasise that there is still much discussion amongst
historians of mathematics on the proper interpretation of cuneiform tablets.
The above discussion reflects one particular reconstruction. Nonetheless, it
is clear that the tablets portray a society in which significant mathematical
techniques were taught and used to solve relatively complex quantitative
problems.

2. Pythagoras: all is number

Speculations about the origins of various systems for counting continue

to occupy historians and philosophers today, and written evidence of such
musings has also been preserved in Ancient Greek texts, though even these
6In MM it is shown how this procedure leads very simply to the general formula for the
‘solution of the quadratic equation’ we all learn at school.
20 1. ARITHMETIC IN ANTIQUITY

are unlikely to have been the first to consider such questions. In the ancient
text Problems, attributed to Aristotle, he ponders the reasons why in his time
10 seemed to be used ‘universally’ as the base for number names:
Why do all men, whether barbarians or Greeks, count up to ten, and not up to
some other number, such as two, three, four or five, so that they do not go on to repeat
one of these and say, for example, ‘one-five’, ‘two-five’, as they say ‘one-ten’ [eleven],
‘two-ten’ [twelve]? Or why, again, do they not stop at some number beyond ten
and then repeat from that point? For every number consists of the preceding number
plus one or two, etc, which gives some different number; nevertheless ten has been
fixed as the base and people count up to that.7
He then lists some possible reasons that may provide insight into the fa-
miliar arithmetic of his time – which he attributes primarily to the Pythagore-
ans, followers of Pythagoras of Samos.
Is it because 10 is a perfect number, seeing as it comprises all kinds of number,
even and odd, square and cube, linear and plane, prime and composite? Or is it
because ten is the beginning of number, since ten is produced by adding one, two,
three, and four? Or is it because the moving bodies are nine in number? ..... Or is
it because all men had ten fingers....
Aristotle’s reference to nine ‘moving bodies’ could be an an allusion to
the astronomical system developed by the Pythagorean Philolaus (ca. 470-
385 BCE). This system was reported to postulate the existence of a ‘cen-
tral fire’ around which the earth and the eight celestial bodies visible to the
naked eye, namely the sun, moon, five planets and the ‘sky’ (the fixed stars),
would rotate. The earth would revolve about the central fire daily, the moon
monthly and the sun annually, thus explaining why sun and moon rise and
set. In order to arrive at the number 10 – which had special significance
for the Pythagoreans – Philolaus is said to have claimed the existence of a
‘counter-earth’, which he assumed to be situated directly opposite the Earth
from the ‘central fire’, also revolving about it daily, and which therefore al-
ways remained invisible to us!
I now consider ideas attributed to the Pythagoreans, as reported by later
commentators, a little further, not least to understand more about the ‘kinds
of number’ Aristotle refers to. Greek mathematics, in its various guises, has
been singularly influential in the development of the subject through the
ages. Let us start with the origins of Pythagorean arithmetic.

2.1. Ratios and musical harmony. No first-hand written records of the

discoveries of Pythagoras and his immediate followers survive today. Aris-
totle and his teacher Plato have a good deal to say – often highly critical and
sometimes obscure – about Pythagorean beliefs and mathematical achieve-
ments. Their testimony on Pythagoras, though coming a good century after

7T.L. Heath, Mathematics in Aristotle, Taylor and Francis, e-book, 2011.

 2. PYTHAGORAS: ALL IS NUMBER 21

the fact, is distinctly more reliable than are the much later and highly par-
tisan accounts produced by the so-called neo-Pythagoreans, who sought to
resurrect and expand the elaborate number mysticism that Pythagoras’ quasi-
religious sect had initiated.
Our focus is on the arithmetic of the Pythagoreans, rather than on their
mystical beliefs. Paradoxically, the major source for our understanding of
the techniques of Pythagorean arithmetic is a work that does not deal pri-
marily with arithmetic at all. It is the vastly influential treatise The Ele-
ments of Geometry (see e.g [21]), widely known simply as the Elements and
produced in the Egyptian port city Alexandria by the mathematician Eu-
clid.8 The thirteen books of this work comprise the most widely studied
mathematical text of all time, and were fundamental in shaping the subject
throughout more than two millennia.
In Aristotle’s Metaphysics we find a concise summary of Pythagoras’ es-
sential belief system:
in numbers, he thought that they perceived many analogies of things that exist
and are produced, more than in fire, earth, or water: as, for instance, they thought
that a certain condition of numbers was justice; another, soul and intellect, ... And
moreover, seeing the conditions and ratios of what pertains to harmony to consist in
numbers, since other things seemed in their entire nature to be formed in the likeness
of numbers, and in all nature numbers are the first, they supposed the elements of
numbers to be the elements of all things. (Arist. Met. i. 5.)
Here Aristotle refers to the speculations of Empedocles, who argued (ca.
450 BCE) that air, earth, fire and water made up the basic four elements
from which everything was constructed. Aristotle refers to three of those,
to contrast them with Pythagoras’ view that numbers are the basic building
blocks. Assigning numbers to various physical objects or concepts played a
significant part in Pythagorean number mysticism.
Although detailed ancient references to Pythagorean arithmetic are not
numerous, it is a widely held view that they concerned themselves exten-
sively with ratios, which we will interpret in terms of ratios of positive whole
numbers, i.e. positive fractions of quantities. Texts suggest that these explo-
rations were prompted by empirical evidence that simple ratios of string or
pipe lengths in musical instruments can produce harmonious sounds.9 The
Pythagoreans calculated that an octave must correspond to the ratio 2 : 1, a
fifth to 3 : 2, a fourth to 4 : 3 (we say ‘two-to-one’, three-to-two’, etc.).

8We know very little about Euclid himself. The fifth-century commentator Proclus tells us
that Euclid was active in Alexandria during the reign of Ptolemy I Soter, who ruled Egypt from
323 to 285 BCE. Euclid may have studied in Athens at Plato’s Academy, and later established a
substantial school in Alexandria. Most writers date the Elements as from around 300 BC.
9The most comprehensive translation of these ancient sources is found in the German text
Die Fragmente der Vorsokratiker by H. Diels and W. Kranz (6th ed.), Weidmann, Dublin, 1952.
22 1. ARITHMETIC IN ANTIQUITY

Their derivations, fortuitously preserved for us in various fragments

that appear as comments in another substantial work by Euclid, The Divi-
sion of the Canon (usually known by its Latin name: Sectio Canonis), seem
to be based on two underlying postulates which they took as not requiring
further proof (see [38]):
(i) musical intervals [the differences in pitch between two notes] can be
quantified by means of ratios of two (whole) numbers;
(ii) harmonic intervals [intervals pleasing to the ear when two notes are
played together, such as in the above examples] are characterised by ratios of
two forms: either n : 1 or (n + 1) : n, for some whole number n. Conversely, for
any whole number n, the ratio n : 1 produces a harmonic interval.
The Pythagoreans had observed experimentally that octaves and double
octaves are harmonic, while repeated fifths and fourths are not, and also
that following a fifth by a fourth (or vice versa) produces an octave. With
the postulates (i),(ii), Pythagorean music theory can be derived quite simply,
using the geometric mean G of two given quantities a, b. This is defined via
the proportion a : G :: G : b (in words: ‘a is to G as G is to b’). We represent
a
this by the identity G = Gb , so that G is the solution of the equation G2 = ab.
If the octave is given by the ratio ab , the double octave ac must satisfy cb =
b
a since each ratio represents an octave. So b is the geometric mean of c and
a. But then ac cannot have the form n+1 n . Whenever three quantities a, b, c are
in geometric proportion, we have c : b :: b : a, so that with c = n + 1, a = n,
we would obtain b2 = n(n + 1) Since n(n + 1) lies strictly between n2 and
(n+1)2 , it cannot be a perfect square. So b cannot be a whole number. Hence
postulate (ii) ensures that the double octave ac has the form m : 1.
Next, consider the fifth and fourth. Both are harmonic intervals, so the
form of their ratios must be either n : 1 or n + 1 : n. If either of them had the
form b : a = n : 1, their compound ratio c : a would be harmonic. But it was
observed empirically that double fourths and double fifths are not harmonic.
Hence the fifth and fourth must each have the form (n + 1) : n, (for different
n > 1) and their composition becomes the octave, as above. The simplest
numbers of the form n+1 3 4 3 4
n are 2 and 3 . Multiplying those provides 2 × 3 = 1 ,
2

so that the ratio 2 : 1 provides the octave.

The interval leading from the note a fourth up from the starting point to
the note a fifth up – described as a whole tone – became the principal unit in
the tonal scale. Since the ratios representing the fourth and fifth are multi-
plied when we add the intervals, subtraction of the intervals forces division
of the ratios, so that we obtain ( 32 )/( 34 ) = 98 as the ratio representing the
whole tone.
The earliest musical scale based on such simple numerical ratios is cred-
ited to Pythagoras himself, together with the discovery that the frequency
of a vibrating string is inversely proportional to its length. However, the
 2. PYTHAGORAS: ALL IS NUMBER 23

earliest reliable manuscripts on Pythagorean music theory stem from Philo-

laus, more than a century after Pythagoras. He derived the above ratios as
well as the three intervals making up the fourth (or tetrachord), which are
9 : 8, 9 : 8, 256 : 243. The adjustment to the final interval is required if
one starts with two whole notes, to ensure that the final interval takes us
to the ratio derived for the fourth: we need 98 × 98 × ab = 43 , which leads to
a 8 8 4 27 256
b = 9 × 9 × 3 = 35 = 243 . Such calculations led to what is today known
as the Pythagorean diatonic scale, which Plato adpoted in constructing the
‘world soul’ in his Timaeus.
The symbolic notation we have used in this reconstruction was not used
by the Pythagoreans. They and their successors did not perceive ratios as
‘numbers’ – this term was reserved for multiples, or what we would call
the positive whole numbers 2, 3, 4, .... Euclid’s Elements provide a strikingly
vague definition of ratio as: ‘a sort of relation in respect of size between two mag-
nitudes of the same kind’.
For the Pythagoreans, ratios were essentially a tool for comparing magni-
tudes, which were interpreted geometrically, as seen in Euclid’s works. Num-
bers enter the discussion as multiples that tell us how often, in a ratio A : B,
these quantities are ‘measured’ exactly by some (smaller) unit. Relating
magnitudes A, B to a pair (m, n) of whole numbers (which we would regard
as the fraction mn ) then signifies that the common unit ’goes exactly m times
into’ A and n times into B. Thus, in particular, for A and B to have a ratio,
these two quantities must necessarily be of ‘the same kind’: both are mu-
sical intervals, or whole numbers, or geometric lengths, areas, or volumes,
measured by a common unit. The unit itself is not regarded as a number in
the same sense as the ‘multiples’ 2, 3, 4, ...
However, the Pythagoreans could compare any two ratios A : B and
C : D, irrespective of whether these were ‘of the same kind’ (e.g. if A, B
were lines, while C, D were areas). These four quantities are in proportion if
the pair A, B, measured by some common unit (e.g. a length), generates the
same pair of numerical multiples (m, n) as does the pair C, D, when mea-
sured by some other common unit (e.g. an area). This Pythagorean theory
of proportions, largely preserved in Books VII to IX of Euclid’s Elements, was
central to their mathematical framework.
While the Pythagoreans discussed such arithmetical relationships ver-
bally, without symbolic notation, they made considerable progress in their
efforts to quantify musical relationships. Their triumphant conclusion was:
‘All is Number‘. By this they meant that all natural phenomena can be under-
stood in terms of the ratios of positive whole numbers. This turned out to
be a rather sweeping conclusion, as they themselves discovered! Nonethe-
less, their ideas mark an important step (and the earliest that has been pre-
served) in humanity’s attempts to describe natural phenomena systemati-
cally through quantitative analysis.
24 1. ARITHMETIC IN ANTIQUITY

Figure 6. Pythagoras’ Theorem

2.2. Pythagorean triples. For many, the name Pythagoras evokes mem-
ories of school mathematics, whether they be pleasant or painful! His fa-
mous theorem about the relationship of the sides of a right-angled triangle
is probably the best-known result of Greek mathematics.
Pythagoras’ Theorem
In any right-angled triangle the square on the hypotenuse is the sum of the
squares on the other two sides.
If we denote the lengths of the sides by a, b, c with c as the hypotenuse,
then this means that a2 + b2 = c2 .
A simple proof is illustrated in Figure 6, where we consider two ways
of dividing up the square with side (a + b). On the left, on each side mark
off lengths a, b in order, starting at top right and going clockwise, and join
points to produce four copies of the right-angled triangle with sides (a, b, c),
situated around a quadrilateral whose sides all have length c, ‘tilted’ through
the base angle θ between sides b and c of the triangle. At each vertex of this
quadrilateral we have angles θ and (90◦ − θ) in the triangles that meet there,
hence the remaining angle is a right angle, and therefore the tilted figure is
a square.
On the right mark off the length a in both directions from the top right
vertex, and similarly length b from the bottom left vertex, to construct squares
with sides a, b, meeting in a point. What remains are two copies of the rec-
tangle with sides a, b. The total area of the two rectangles equals that of the
four right-angled triangles with sides (a, b, c) on the left, as they have the
same base and height. Subtracting the triangles on the left leaves the square
on the hypotenuse, while subtracting the two rectangles on the right leaves
the squares on the legs of the triangle.
We have proved that a2 +b2 = c2 . (As Euclid would have put it: we have
taken equals from equals, so the remaining areas are equal.) This proof has
been called the ‘Chinese proof’ of the theorem, as it occurs in the ancient
Chinese text Chou Pei Suan Ching. Euclid’s Elements, Book I, has a quite
different proof.
 2. PYTHAGORAS: ALL IS NUMBER 25

This relationship between the sides of a right-angled triangle was well-

known to the Old Babylonians, who routinely made use of the theorem a
thousand years before Pythagoras was born. Various tablets use it in a vari-
ety of problems: on a tablet now in the British Museum (BM85196), a beam
of length 30 standing against a wall is said to have slipped from a verti-
cal position so that the top has slipped 6 units. The scribe asks how far
the lower end moved. Thus we have a right-angled triangle (a, b, c) with
hypotenuse c = 30 and leg b = 24 units. To find a the scribe computes
p
(30)2 − (24)2 = 18. The Babylonians applied this recipe, as one with gen-
eral validity, in varied practical settings; the modern notion of verifying its
validity diagrammatically stems from the later development of Greek ge-
ometry.
The best-known example displaying the depth of Babylonian under-
standing of the theorem is the tablet Plimpton 322 in the Yale collection,
which dates from 1800 BCE. Although now broken and incomplete, this
lists a considerable array of triples (a, b, c) of whole numbers, now com-
monly known as Pythagorean triples, which satisfy the equation a2 + b2 = c2 .
The simplest Pythagorean triples will be familiar: (3, 4, 5) uses the smallest
whole numbers possible, giving 32 + 42 = 52 (our example above multiplies
each side by 6). It is also easy to check that the triples (5, 12, 13), (8, 15, 17)
and (7, 24, 25) are Pythagorean.
In Plimpton 322, the scribe’s methodology in choosing his particular
triples still leads to lively discussions among historians, but there is no doubt
that he was familiar with very many such triples, including some with im-
pressively large numbers, and that he arranged them in a consistent pattern,
whose purpose we can only guess today.10
The two basic methods for generating Pythagorean triples were well
known in Ancient Greece. In our terms they are:
(a) if m > 1 is an odd number, then (m, 21 (m2 − 1), 21 (m2 + 1)) is a
Pythagorean triple;
(b) if m is an even number greater than 2, then (m, ( m 2 m 2
2 ) − 1, ( 2 ) + 1)
11
is a Pythagorean triple.
The influential fifth-century neo-Platonist commentator, Proclus, (while
not notable as a reliable source, and writing nearly a millennium later) at-
tributes (a) to Pythagoras himself and (b) to Plato. Both are easily checked

10See, for example, a trenchant rebuttal of earlier interpretations in Eleanor Robson: Nei-
ther Sherlock Holmes not Babylon: A Re-assessment of Plimpton 322, Historia Mathematica 28
(2001) 167-206. See [25] for an account of the tablet.
11For m = 2 the formula in (b) yields ( m )2 − 1 = 0, which leads to the trivial triple
2
(2, 0, 2), corresponding to the ‘triangle’ with base angle 0. Note also that the triples do not all
lead to distinct triangles. For m = 3 and m = 4 we obtain (3, 4, 5) from the first formula and
(4, 3, 5) from the second.
26 1. ARITHMETIC IN ANTIQUITY

by simple algebra: write out the squares in (a):

1 1 1 1
m2 + (m2 −1)2 = [4m2 +m4 −2m2 +1] = [m4 +2m2 +1] = [ (m2 +1)]2 .
4 4 4 2
proving that (a) is Pythagorean.
The proof of (b) is almost identical to the above and is left as a simple
exercise for the reader. Readers allergic to algebra or who dislike powers
higher than 3 (as did the Ancient Greeks) may safely skip these algebraic
arguments. We will focus instead on simple geometric techniques by which
the Pythagoreans may have derived these results.

2.3. Pebbles, triangles and squares. Four aspects of the arithmetic of

the Pythagoreans are widely accepted as tools that were available to them.
They
(i) used ‘pebble arithmetic’ for visual displays of number patterns,
(ii) regarded odd and even as ‘the two proper forms of number’,
(iii) used triangular, square and oblong numbers (for definitions see be-
low),
(iv) explored Pythagorean triples.
An ancient (apparently Babylonian) technique of using an L-shaped fig-
ure, which the Pythagoreans called a gnomon (or stick and shadow), to gen-
erate particular number patterns also seems to play a significant part in their
reasoning.
Books VII-IX of Euclid’s Elements contain arithmetical results that are
generally seen as exemplars of Pythagorean methods, although his proofs
use geometrical figures contructed by straightedge and compass rather than
diagrams consisting of groups of pebbles. (Here a straightedge is a ruler with-
out marked lengths.)
Closely following [26], we now reconstruct some of these techniques
in modern terms. Recall, however, that the visual character of early Greek
mathematics (later expressed so elegantly in Euclid’s Elements) meant that
only multiples of the chosen ‘unit’, and not the unit itself, were regarded as
actual numbers. In a geometric construction, a given length (area, volume)
will measure an arbitrarily chosen unit (length, area, volume) a certain num-
ber of times.
Beginning with pebble arithmetic, one can combine (i) and (iii) above to
define three kinds of figurate numbers reportedly used by the Pythagoreans
to represent different numbers (see Figure 7). A whole number is said to be:
Triangular if it is represented in triangular form, using rows of 1, 2, 3, ...
pebbles; 3, 6, 10, ... are examples.
Square if it is a perfect square, made up of equal numbers of pebbles in
each row/column; e.g. 4, 9, 16, ...
 2. PYTHAGORAS: ALL IS NUMBER 27

Figure 7. Figurate Numbers

Figure 8. Sum formulae

Oblong if it is a rectangle, with one more pebble in one direction than

the other, thus of the form n(n + 1); e.g. 6, 12, 20, ...
Larger triangular numbers can be built simply by adding more rows -
each row having one more pebble than the previous one. This immediately
begs the question: how do we find the sum 1 + 2 + 3 + ... + n? Does it help to
look at our pebble representation? The triangular number itself appears not
to provide immediate enlightenment. However, looking instead at the peb-
bles in an oblong number we can immediately find the answer—see Figure
8(a).
Drawing a diagonal (from just above the top right to just below the bot-
tom left pebble) we have divided our oblong number into two equal pieces.
But the area of the rectangle with sides n and (n + 1) units is obviously
n(n + 1), as represented by our oblong number. Each of the two equal trian-
gular pieces into which we have split this oblong number has n pebbles in its
bottom row, so the nth triangular number is one-half of the oblong number
n(n + 1). Summing over all the rows in the triangle, we obtain the familiar
formula for the sum of the first n numbers:
1
1 + 2 + 3 + ... + n = n(n + 1).
2
Square numbers give us immediate insight into the sum of the first n
odd numbers – see Figure 8(b). Successive square numbers can be built up
28 1. ARITHMETIC IN ANTIQUITY

Figure 9. Putting gnomons about the unit

by starting with one pebble and adding successive gnomons – here taken as
symmetrical L-shaped figures, the first having 3 pebbles, the next 5, then 7,
etc. To obtain a square with n pebbles on each side we need the last gnomon
to contain (2n − 1) pebbles (the corner pebble serves both sides). In other
words, we have demonstrated the identity
1 + 3 + 5 + ... + (2n − 1) = n2 .

These examples show that such summation formulae will have been
well within the range of Pythagorean arithmetic.

2.4. Pebbles and gnomons. Aristotle discusses the Pythagorean prac-

tice of constructing square numbers by ‘setting the gnomons round the unit’ -
see Figure 9(a). In [26] this is interpreted as follows: start with one pebble
at the centre, add a gnomon containing 3 pebbles below and to its right, fol-
low this by 5 pebbles above and to its left, and continue alternating in this
fashion, each time adding the next odd number of pebbles.
This idea leads one quite naturally to a ‘pebble proof’ that for odd M,
the triple
1 1
(M, (M 2 − 1), (M 2 + 1))
2 2
is Pythagorean: since M is odd, so is its square M 2 . Thus M 2 ± 1 are both
even, so all three the above are whole numbers. For any K, the difference
of squares (K + 1)2 − K 2 = 2K + 1 (which is also the number of pebbles in
the gnomon we would add to the square of side K to obtain the next one).
When M is odd we can therefore write as M 2 = 2K +1. Now K = 12 (M 2 −1)
is the side of the smaller square and K + 1 = 21 (M 2 + 1) is the side of the
larger square obtained by adding the gnomon M 2 to the smaller one. In
other words, we have three squares, with sides, respectively, given by
1 1
(M, (M 2 − 1), (M 2 + 1)).
2 2
 2. PYTHAGORAS: ALL IS NUMBER 29

The sum of the areas of the first two squares equals that of the third, so these
numbers form a Pythagorean triple.
For even M, similar ideas lead to the construction of Pythagorean triples
of the form
M M
(M, ( )2 − 1), ( )2 + 1).
2 2
For any even M, the square M 2 equals 4K for some K, and we can write
this as M 2 = (2K − 1) + (2K + 1). These are two successive gnomons, taking
us from the square with side (K − 1) to that with side (K + 1). In other
words,
(K + 1)2 − (K − 1)2 = 4K = M 2 ,
which again shows that the triple (M, ( M 2 M 2
2 ) − 1), ( 2 ) + 1) is Pythagorean.
Figure 9(b) shows this for K = 4, with the perimeter of the larger square
split into four equal parts.
There is no direct written evidence that such methods were actually em-
ployed by the Pythagoreans. The simple tools used here suggest that these
results were within their range. The above arguments have an advantage
over the purely algebraic proofs in that they do not involve powers greater
then 2, rather than using fourth powers. Since early Greek mathematicians
reasoned largely via geometric pictures, they had no use for powers beyond
cubes, as there are three spatial dimensions.
Although other interpretations of the Pythagoreans’ obsession with the
number 10 have been given (see e.g. [44]), the triangle representing 10 may
help us understand the quotation from Aristotle at the start of this section.
The triangular number 10, the tetractys, (shown in Figure 7) has four rows,
the top containing a single pebble (a point, dimension 0), the second two
pebbles (two points define a line, dimension 1), the third three pebbles (three
points define a triangle in the plane, dimension 2), and the bottom line has
four pebbles (which define a tetrahedron in space—a triangular pyramid whose
faces are four equilateral triangles, one serving as the base, with the other
three meeting at the top vertex—dimension 3). The number 10, the sum of
these four rows (the tetrad), thus represents the universe – while also serv-
ing as the unit for the dekad, the next higher order of counting (making 10
the base of the number system).
Figure 9(b) illustrates a simple result in what is today known as number
theory: any square whose sides consist of an odd number of pebbles can be
built from the unit (a single pebble) by adding pairs of consecutive gnomos
around the unit. Each such pair can be split—as we did in Figure 9(b)—into
4 equal pieces. Hence an odd square number always leaves remainder 1 (the
central pebble!) when it is divided by 4.
On the other hand, a square with even sides obviously divides into four
equal pieces, each having sides with length one-half that of the original. So
even squares are divisible by four.
30 1. ARITHMETIC IN ANTIQUITY

Today we would describe these facts by saying that a perfect square

leaves remainder 0 or 1 when divided by 4. This relationship is expressed
as: n2 ≡ 0 or 1 (modulo 4). In other words, when we divide 4 into a perfect
square, we will never get 2 or 3 as the remainder.
This has consequences for right-angled triangles whose sides are whole
numbers. It means that if C is even in a Pythagorean triple (A, B, C) then
so are A and B : if both A and B are odd, the squares A2 , B 2 each leave
remainder 1 upon division by 4, so the sum A2 + B 2 leaves remainder 2, so
the sum cannot equal C 2 , which is divisible by 4. If exactly one of A, B is
odd, then A2 + B 2 would be odd, while C 2 is even. So if A2 + B 2 = C 2 and
C is even, we are forced to conclude that A and B are even.
So: if the hypotenuse C of a right-angled triangle with integer sides
is even, then so are the other two sides. Next, suppose that not all sides
(A, B, C) are even. Then C cannot be even, so it must be odd, by the above.
If A, B were either both even or both odd, then A2 +B 2 would be even, hence
C would be even. So if any sides are odd, then C, and exactly one of A or B,
must be odd.
We summarise this as a result that will be useful in Section 3 below –
we will call it our
First Divisibility Lemma:
In a right-angled triangle whose sides (A, B, C) are whole numbers:
(i) if C is even, then are all three sides are even.
(ii) if any sides are odd, then C is odd, one of A, B is even and the other odd.

2.5. Side and diagonal. These ‘pebble proofs’ illustrate some of the
techniques probably available to the Pythagoreans for the analysis of var-
ious geometric shapes as well as number relationships. However, Aristotle
tells us that in this analysis they came across magnitudes that are incompati-
ble with their bold claim that ratios of whole numbers (formed by multiples
of a fixed unit), which were so useful in the analysis of musical harmony,
could explain all natural phenomena. Since no original records remain to
tell us how this came about, we again offer what constitutes one of several
plausible scenarios for this discovery, rather than historical fact.
Construct a square of side 2 (in whatever units you prefer) and divide
it into four unit squares by joining opposite midpoints of its sides. Divide
each unit square into two isosceles right triangles by drawing diagonals that
meet at the midpoints of the larger square. (See Figure 10).
Take any of the eight triangles. By Pythagoras’ theorem, the square on
its hypotenuse is 12 + 12 = 2. Thus the hypotenuse, which is a line segment
with length l, say) is the side of a square whose area is exactly double that
 2. PYTHAGORAS: ALL IS NUMBER 31

Figure 10. Side and diagonal

of a unit square.12 So what is this length l–is it representable by a number,

and if so, by what sort of number?
Aristotle answers this question with a logical argument showing that
l cannot be expressed as a ratio of two whole numbers. It is his favourite
example to illustrate proof by contradiction, an important proof technique we
will come across repeatedly. He starts by assuming that the claim he is try-
ing to prove is false, and shows that this must lead to a contradiction, so that
the claim cannot be false, hence must be true.
To justify this logical principle, Aristotle argued that a proposition must
either be true or be false. Thus either the proposition (P ), or its negation
(notP ), must hold. The assertion that these are the only possibilities is Aris-
totle’s famous principle of the ‘excluded middle’. Like the great majority of
modern mathematicians, I will side with him in this book, and utilise proof
by contradiction frequently to justify my claims – although, as we will see
in Chapter 10, this attitude is not quite universal.
Aristotle’s proof that the diagonal of the unit square cannot be expressed
as a ratio of two whole numbers goes as follows. Suppose that the relation-
ship between the side and diagonal of the unit square can be expressed as a
ratio of two whole
√ numbers. (In modern terms, this amounts to the asser-
tion that l = 2 is a rational number.) This would mean that we can write
l = ab for some whole numbers a, b with no common factors.
Multiplying both sides by b, then squaring the results, we would have
a2 = l2 b2 = 2b2 ,
since the square on the side with length l has area 2. If a were odd, then a2
would be odd, but in fact it equals 2b2 . Hence a must be even and can be
written as a = 2c for some whole number c. Then 2b2 = a2 = (2c)2 = 4c2 ,

12This fact is exploited famously in Plato’s dialogue Meno, where ‘Socrates’ teaches a
young slave how to construct a square with double the area of a given one, while arguing
that what the youngster was doing was not learning, but that he was simply ‘remembering’ a
true statement that he had known subconsciously all along.
32 1. ARITHMETIC IN ANTIQUITY

which, after cancellation, becomes

2c2 = b2 .
This is turn shows that b2 , and therefore b, is also even.
So a and b have common factor 2, contrary to our assumption that they
have no common factors. This contradiction shows that l = ab is impossible
for whole numbers a, b.
But this proof, used by Aristotle as an example that would be well-
known to his readers, seems rather unlikely as a method of discovery. In order
to begin the above proof, one would need to suspect, at least, that l cannot
be a ratio of two whole numbers. How might this possibility present itself
to an unsuspecting Pythagorean?
Well, we have seen that the diagonal cuts the unit square into isosceles
triangles, and our Pythagorean might wish to identify how such a triangle
would produce a Pythagorean triple by adopting a suitable (smaller) unit
length. The sides of the triangle would then be lengths expressible as whole
number multiples (A, A, C) of this unit length, since the two sides meeting
at the right-angle have equal length. Using part (i) of the First Divisibility
Lemma proved at the end of Section 2.4, we see that, if the hypotenuse C is
even then so is A, as in that case all three sides must be even. It that case
we can halve each side and retain an isosceles triangle. Continuing to do
this we must eventually arrive at an isosceles triangle whose hypotenuse C
is odd. But now (ii) of the same result tells us that one of the legs of the
triangle must be even and the other odd. But here the two legs have the
same length A! This is a contradiction, proving that the ratio of diagonal to
side in the unit square cannot be expressed as a ratio of whole numbers.
The fact that the side and diagonal of any square cannot simultane-
ously be multiples of the same unit means that they are not ‘co-measurable’
(or, more elegantly, commensurable) in terms of any chosen unit. They pro-
vide an example of two magnitudes (lengths) that are incommensurable. The
Pythagoreans would presumably have found this highly disturbing, as they
had no way of expressing the relationship between these two lengths by way
of a number, yet it is clear that one can easily construct such lengths.
Of course, this is just one possible (if plausible) reconstruction of ‘the
discovery of incommensurables’ by the Pythagoreans. There is a continuing
debate among historians whether the ratio of diagonal and side of a square
was actually the first quantity of this type to be considered. The fragmen-
tary nature of surviving ancient texts and commentaries allows a variety of
interpretations. Prominent among the alternatives put forward as the first
known incommensurable quantity is the ratio of the diagonal to the side
of a regular pentagon, often √ called the ‘golden section’ and given in mod-
ern terminology as 21 (1 + 5) = 1.61803399... (see also Chapter 2, Section
√
2.2). I will not take sides in such disputes, but will continue to use 2 as
 2. PYTHAGORAS: ALL IS NUMBER 33

Figure 11. Sides and areas in proportion

our prototype of incommensurables, since it is the familiar example cited

by Aristotle.
The existence of incommensurable magnitudes in geometry presents a
problem for a Pythagorean world view that asserts that all objects can be
measured in terms of ratios of whole numbers. As mentioned earlier, this
world view is still visible, two centuries later, in the definition of propor-
tionality that Euclid uses in Book VII of his Elements, where he reports and
develops arithmetical results attributed to the Pythagoreans.

In Book VII, Euclid says that the magnitudes A, B and C, D are in pro-
portion (recall that we denote this by A : B :: C : D) if A is ‘the same multiple,
part or parts’ of B as C is of D.13 In other words, if mA = nB for some natural
numbers m, n, then we need mC = nD for the two ratios to be in propor-
tion. In our language, this says that D must be the same ‘rational multiple’
of C as B is of A: if B = m m
n A then D = n C, and we describe the ratio as the
m
rational number n . For Euclid, on the other hand, ratio and proportion are
not about ‘numbers’, so he does not express the ratio in this way.
With the above definition the following basic fact about rectangles can
no longer be said to hold in general:
In a rectangle, a line parallel to one side divides the other side in the same pro-
portion as the resulting areas.
This simply means that A : B = AC : BC with A, B, C as in Figure 11.
This relationship between a ratio of line segments and the ratio of ar-
eas with these segments as their bases, while obviously true, can in some
circumstances become a meaningless statement in the Pythagorean world-
view: we need only take B as the chosen unit length and choose A as a
length
√incommensurable with this unit. In modern notation, we might take
A = √2 = C and B = 1. In that case the rectangles AC and BC comprise
2 and
√ 2 area units respectively, while the line
√ segments√A, B have lengths
of 2 and 1 linear units. But the claim that 2 : 1 = 2 : 2 makes no sense
13A more general definition (due to Eudoxus) of proportion is given in Book V of the Ele-
ments, but it is generally accepted that in Book VII Euclid follows the Pythagorean concepts.
34 1. ARITHMETIC IN ANTIQUITY

√
according to the above definition
√ of proportions, since
√ 2 is not ‘a multiple,
part or parts’ of 1 (nor 2 of 2), precisely because 2 = m n is impossible for
whole numbers m, n.
In his treatiseTopics, Aristotle highlights this difficulty clearly and sug-
gests that a way out would be to alter the definition of proportionality so that
it can encompass the above. We will see later how this problem was solved
conclusively, reportedly by Eudoxus (408-355 BCE). The new definition is
central to Book V of the Elements.
Despite the difficulties
√ posed by incommensurables, it clear that the
problem posed by 2 did not stop early Greek mathematics in its tracks,
although it may have been instrumental in shifting its focus decisively from
arithmetic to geometry, which appeared to be where a systematic study of
these newly discovered magnitudes could be undertaken.

3. Incommensurables

Plato, whose Academy was, in its time, the most influential philosophi-
cal school in ancient Athens, frequently phrased his writings in the form of
dialogues between ‘Socrates’ and other characters to drive home the main
tenets of his philosophy. In one such dialogue the mathematician Theaetetus
appears as a young man, relating how a Pythagorean lecture on incommen-
surables inspired him to make a significant breakthrough in identifying an
unlimited number of examples of such magnitudes. The dialogue is less
a historical account than a graphic lesson illustrating Plato’s philosophical
beliefs.

3.1. The Theodorus lesson. The passage describes, in the words of the
youthful Theaetetus, his reaction to a lesson given around 400 BCE by the
Pythagorean mathematician Theodorus of Cyrene, in which he demonstrated
the incommensurability of the square roots of 3, 5, 6, ...‘up to 17’ with the
unit.
It is worth reading an excerpt to gain insight into its strongly visual de-
scription of mathematical statements – the translation is from [23]:
THEAETETUS: Theodorus was writing out for us something about roots,
such as the roots of three or five, showing that they are incommensurable by the
unit: he selected other examples up to seventeen—there he stopped. Now as there
are innumerable roots, the notion occurred to us of attempting to include them all
under one name or class.
SOCRATES: And did you find such a class?
THEAETETUS: I think that we did; but I should like to have your opinion.
SOCRATES: Let me hear.
 3. INCOMMENSURABLES 35

THEAETETUS: We divided all numbers into two classes: those which are
made up of equal factors multiplying into one another, which we compared to square
figures and called square or equilateral numbers;—that was one class.
SOCRATES: Very good.
THEAETETUS: The intermediate numbers, such as three and five, and every
other number which is made up of unequal factors, either of a greater multiplied by a
less, or of a less multiplied by a greater, and when regarded as a figure, is contained
in unequal sides;—all these we compared to oblong figures, and called them oblong
numbers.
SOCRATES: Capital; and what followed?
THEAETETUS: The lines, or sides, which have for their squares the equilateral
plane numbers, were called by us lengths or magnitudes; and the lines which are the
roots of (or whose squares are equal to) the oblong numbers, were called powers or
roots; the reason of this latter name being, that they are commensurable with the
former [i.e. with the so-called lengths or magnitudes] not in linear measurement,
but in the value of the superficial content of their squares; and the same about solids.
SOCRATES: Excellent, my boys; I think that you fully justify the praises of
Theodorus, and that he will not be found guilty of false witness.
In summary, Theaetetus is here distinguishing between square numbers
(what we today call ‘perfect squares’) and all other positive whole numbers
(conveniently lumped together as ‘oblong’, since he is thinking only about
areas of rectangles). He argues that in the former case, where the area is
a perfect square, the sides (whose length is the square root of the area) are
commensurable with the unit, hence may, in true Pythagorean fashion, be
called ‘magnitudes’. In the latter case, however, the side of a square whose
area is an ‘oblong’ number (hence equals that of a non-square rectangle) is
commensurable with the unit ‘in square only’ and so is not a Pythagorean
magnitude – since it is not a ‘multiple, part or parts’ of the unit length.
As a simple example of this terminology, 3 is√ represented by a 3 × 1
rectangle; the side of a square with this area is 3. Although the area, 3
p 2 √
(= ( 3) ) is obviously commensurable with the unit,
√ the side 3 is not, as
(according to Theaetetus) Theodorus showed. So: 3 and the unit are ‘com-
mensurable in square only’. Euclid also adopts this terminology in Book X
of his Elements.
In other words, the square root of a positive whole number either is
itself a whole number (so the original number is a perfect square), or else
it is irrational (cannot equal the ratio of two whole numbers). Confusingly
for us, Theaetetus describes irrational square roots as ‘roots’ or ‘powers’ to
distinguish them from his ‘magnitudes’, but the distinction between the two
classes is clear nonetheless. The passage does not contain any indication of
the proof of Theaetetus’ bold claims.
36 1. ARITHMETIC IN ANTIQUITY

We state the key result announced by Theaetetus rather more succinctly

in modern terminology, as:
Theaetetus’ Theorem
√
(i) n is incommensurable with the unit unless n is a perfect square.
√
(ii) 3 n is incommensurable with the unit unless n is a perfect cube.
The second claim is contained in Theaetetus’ remarkably nonchalant –
almost throw-away – remark ‘and the same about solids’. Not surprisingly,
Socrates is impressed: ‘Excellent, my boys.’
Theaetetus’ √claims can be proved in a manner analogous to that given
by Aristotle for 2. However, rather than simply relying on the fact that a
whole number can be either odd or even, this proof (which we give in Chap-
ter 8) crucially makes use of what is now called the Fundamental Theorem of
Arithmetic, which we will discuss in Chapter 7.
We may wonder how Theodorus’ case-by-case analysis might have em-
ployed the limited techniques we ascribed earlier to the Pythagoreans; in
particular, whether this throws light on the reason he stopped at 17. This
question has been debated extensively among historians. Our account pro-
vides a brief glimpse of the reconstructions in [26].
The only tools to be used are: pebble arithmetic with figurate numbers,
the duality of odd and even, and Pythagorean triples.
Figure 12 displays three examples. We will consider (a) and (c) here.
See MM for (b).
Theodorus is reported as dealing with each square root individually
and he ‘stopped at 17’, which could suggest that he encountered some diffi-
culty with this case. We will see below why this might have been so.
Rather than look at each case in turn, we can use the duality between
odd and even, to consider the numbers 1, 2, 3, ..., 16 in four groups according
to their remainders when divided by 4 = 22 . In each group, the numbers
can be handled similarly.

√ First,
√ 4,√ are divisible by 4, with
8, 12, 16 √ √ 4, 16√as perfect squares, and
8 = 2 2, 12 = 2 3. So, dealing with 2 and 3 also deals with these
cases.
Next, 1, 5, 9, 13, 17 leave remainder 1. Of these, 1 and 9 are perfect squares.
Among
√ numbers before 17, this leaves only 5 and 13. We give the proof for
5 below.
Thirdly, 2, 6, 10, 14 leave remainder 2√
when divided by 4, so they have
the form 4N + 2 = 2(2N + 1). Of course, 2 has√already been done. Proofs
for the remaining cases are slightly longer – for 6 see MM.
 3. INCOMMENSURABLES 37

Figure 12. Theodorus’ lesson

Finally, 3, 7, 11, 15 leave remainder 3, so have the form 4N + 3. Here we

again just consider the first of these.
√ √
The ‘Theodorus proof’ given below for 5 (and in MM for 6) requires
a lemma on divisibility by 4.
Second Divisibility Lemma:
In a Pythagorean triple (A, B, C) where not all numbers are divisible by 4, the
only number divisible by 4 is either A or B.
This result follows readily from the duality of odd and even. A ‘pebble
proof’ can be found in MM.
√ √
Thus 3 and 5 provide typical examples to illustrate what may have
been Theodorus’ approach.
√ √
Begin with 3. If there are multiples of the unit, A, B, such that 3 :
1 = A : B, then the right-angled triangle
√ with legs A, B gives rise to a
Pythagorean triple (A, B, 2B), since ( 3)2 + 12 = 22 , so that A2 + B 2 =
(2B)2 . The two triangles in Figure 12(a) are similar, as corresponding sides
are in proportion. If A and B are both even, we can halve them successively
38 1. ARITHMETIC IN ANTIQUITY

until at least one of them is odd. But the hypotenuse (2B) will then still be
even, and, as in statement (i) of the First Divisibility Lemma (see the end of
Section 2.4), this makes all three sides even. Since
√ at least one side is odd,
this contradiction
√ shows that we cannot have 3 : 1 = A : B, proving that
(in our terms) 3 is not a rational number.
√
√Next, we consider 5 : 1 =√A : B. As 5 is odd, our Pythagorean triple
is ( 5, 21 (5 − 1), 21 (5 + 1)) = ( 5, 2, 3), and our similar triangle has sides
(A, 2B, 3B), (see Figure 12(c)). Now if B is even, so is 3B, which means
that all sides are even, and the ratio A : B is not in lowest terms. Hence we
can assume that B is odd, in which case 3B is odd, so 2B is the only even
side (by statement (ii) of the First Divisibility Lemma). This means that B is
even, since by the Second Divisibility Lemma, the only even side, 2B, must √
be divisible by 4. This contradicts our assumption that B is odd. Hence 5
cannot be rational.
Why does Theororus’ method fail at 17? Observe that it is the first non-
square number of the form 8N + 1. If we attempt a ‘Theodorus proof’ that it
is
√ incommensurable with the unit, we would, as usual, assume that A : B =
17 : 1. This will provide a triple (A, 8B, 9B), since 17 = 92 − 82 .
But now our ‘Theodorus’ methods will no longer provide the desired
contradiction: the hypotenuse, 9B, is even only if B is even, and then all
three sides will be even, so we can halve them repeatedly and confine our-
selves to the case where B is odd. But then, with the odd hypotenuse 9B,
statement (ii) of the first Divisibility Lemma tells us that one leg is even and
the other odd. This does not conflict with our triangle, since we now have A
odd, 8B even and 9B odd. An appeal to the Second Divisibility Lemma will
not help either: we know that not all sides are divisible by 4, as A is known
to be odd. But now the other leg of the triangle, 8B, is divisible by 4, while
the hypotenuse, 9B, is not. So the lemma cannot lead to a contradiction.
The two results we have relied
√ on so far are not conclusive. We need a
different analysis to deal with 17. But all we need is to consider divisibility
by 3. So here is another divisibility lemma:
Pythagorean triples and divisibility by 3 :
In a right-angled triangle with sides (A, B, C), if 3 divides the hypotenuse C,
then it also divides A and B.
For a ‘pebble proof’, consider a square number represented by peb-
bles. If its side is divisible by 3, then so is the square itself. Starting with
a square of side 3n, we build ever larger squares by adding gnomons: the
first gnomon has 2(3n) + 1 pebbles. The next gnomon is divisible by 3 (it
has 2(3n + 1) + 1 pebbles), so the resulting square also leaves a remainder 1
when divided by 3, while the third gnomon again produces a square with
side divisible by 3. Thus a perfect square cannot leave remainder 2 when
 3. INCOMMENSURABLES 39

divided by 3.14 Now suppose than A2 + B 2 = C 2 and that 3 divides C. If

neither of A, B is divisible by 3, then A2 + B 2 leaves a remainder of 2, which
is impossible, while if only one of them is divisible by 3, we cannot have
A2 + B 2 = C 2 , since C 2 is divisible by 3. We have shown that 3 divides both
A and B, which completes the proof of the lemma.
We apply this lemma to the triangle with sides (A, 8B, 9B). The hy-
potenuse 9B is divisible by 3, hence so are A and 8B. This means that B is
also divisible by 3, hence A, B have 3 as a common factor and are not in low-
est terms. This contradicts the choice of A and B, since we√always assume
them given in lowest terms. The contradiction √ shows that 17 : 1 = A : B
is not possible for whole numbers A, B, so that 17 is irrational, as claimed.
The above arguments suggest that a first step towards the ‘generalisa-
tion’ needed for the proof of the result announced by Theaetetus is to con-
sider divisibility by 3. This takes us past 17. To continue indefinitely, how-
ever, we must assume that the whole numbers represented by A, B have no
common factors at all, rather than simply avoiding 2 as a common factor, as
Theodorus does.

3.2. Euclid’s classification. This excursion into early Greek arithmetic

‘by reconstruction’ illustrates how the number concept can be widened pro-
gressively through the analysis of particular problems—even if, as in this
case, the mathematicians of the time reacted by refusing to regard the ‘new
magnitudes‘, the incommensurables, as numbers, and decided that they
should be studied by geometric methods instead.
The later books of Euclid’s Elements give a clear account of the degree to
which they succeded in providing such a classification. The straightedge-
and-compass constructions developed in the Elements, producing what we
would today describe as various combinations of square roots, are extensive.
Book X, by far the longest of the 13 books, is in large measure a compendium
of these techniques. It includes 115 propositions, almost a quarter of the 465
propositions contained in the Elements. Since the constructions presented
there are often quite complex, and have in any case been superseded by
algebraic descriptions of the quantities in question, we will not attempt any
proofs.
Euclid begins his definitions by distinguishing (just as Theaetetus had
done in our extract from Plato) between commensurable magnitudes (multi-
ples of the same unit) and magnitudes that are commensurable in square only
(i.e. in our terms, the ratio A : B does not equate to a rational number, but
A2 : B 2 = m : n for some natural numbers, giving equal areas, nA2 = mB 2 ).
We should recall that Euclid does not deal directly with numbers, but
with geometric magnitudes. So, saying that two lengths are commensurable

14In modern notation (see also Section 2.4): n2 ≡ 0 or 1 (modulo 3).

40 1. ARITHMETIC IN ANTIQUITY

in square only means that they are incommensurable, but the squares on them
‘are measured by the same area’, as above. He starts by assuming a fixed
assigned line (which serves as the unit of measurement) and calls a line rhētos
if it is commensurable with the assigned line (either in length or in square
only). He refers to an area commensurable with the square of the assigned
line by the same term. Following [13], we use the term expressible to translate
rhētos.
Expressible lines and areas (which are, at worst, square roots of ratio-
nal numbers in our language) represent the ‘easy case’ for Euclid; his main
interest is in classifying the inexpressible lines and areas. The first 18 propo-
sitions of Book X provide a detailed account of the properties of expressible
magnitudes.
The next eight deal with the first subclass of inexpressible magnitudes,
the medial: a medial area is equal to a rectangle with expressible sides com-
mensurable in square only, and the side of a medial square is called a medial
line. To express this in modern
√ notation, fix the length of the assigned line
as a. The lengths a and 2a (the side and diagonal again!) are incommen-
surable, but commensurable in square, since √ the squares√on these sides are
a2 and 2a2 . The rectangle√with sides a and 2a has area 2a2 , so the square
4
with this area has side 2a. So the whole collection of fourth roots forms
part of Euclid’s class of medial lines.
He next considers sums and differences of incommensurable
√ √ lengths
√
and areas. In our notation this includes surds like 2 ± 3 or 1 ± 5. He
shows that neither can be a medial. He calls the sum a binomial, the dif-
ference an apotome (terms still used in the sixteenth century by Cardano –
see Chapter 2) and subdivides each of the binomials and apotomes into six
different classes, analysing the relations between them and the medial.
In modern terminology what he deals with are sums and differences,
repetitions and other combinations of various irrational square roots. Alto-
gether, his classification amounts to the identification of 23 different classes
of incommensurables, all of which can be recovered as solutions of polyno-
mial equations with integer coefficients and of degree at most 4.
One motivation historians frequently cite for Euclid’s extensive classifi-
cation is that it includes, in particular, all the incommensurables needed for
the construction of the five Platonic solids—the tetrahedron, cube, octahedron,
dodecahedron and icosahedron—which Euclid achieves in the final books of the
Elements. It turns out that comparing the edges of these three-dimensional
figures, especially the last two, with the diameter of the sphere in which he
assumes them to be inscribed, requires much of the sophisticated analysis
Euclid develops in Book X.
 4. DIOPHANTUS OF ALEXANDRIA 41

4. Diophantus of Alexandria

While geometry remained the focus of most classical Greek mathemat-

ics, Archimedes, not all that long after Euclid, had a freer approach to the
number concept, as testified in his Sand-Reckoner and in his approximations
of particular irrationals, such as his remarkably accurate estimate that π lies
strictly between 3 10 10
71 and 3 70. .
More consistent moves away from the visual approach emerged during
the Hellenistic period, from the death of Alexander the Great in 323 BCE
to the battle of Actium in 31 BCE. This battle confirmed the dominance of
Octavian – the future Emperor Augustus – over the forces of Mark Antony
and Cleopatra. Early in this period the Egyptian port of Alexandria (estab-
lished by Alexander the Great in 331 BCE) became a pre-eminent centre of
learning. The city maintained this exalted status for nearly six centuries,
throughout much of the Roman Empire. Major fires – the first during Julius
Caesar’s invasion of Egypt in 48 BCE – led to the destruction of the large
Royal Library and the probable loss of the bulk of its estimated 400,000
manuscripts. Its daughter library, housed in the Serapeum (a pagan tem-
ple) survived and protected Alexandria’s status as a centre of learning for
another three centuries. It was finally destroyed in 391, following the Roman
Empire’s adoption of Christianity, under Emperor Constantine in 313.
Fortunately, significant parts (at least 6 out of a reported 13 volumes)
of one of the most influential mathematical works written near the end of
the period, the Arithmetica of Diophantus (ca. 210 to ca. 290) have survived.
Four further books were found in a ninth-century Arabic transcription in
1968; these are thought by some scholars to include translations of later
notes on Diophantus’ work made by Hypatia (ca. 370-415), the first known
female mathematician, who was killed by a Christian mob during conflicts
in Alexandria in 415.
Despite its title, the Arithmetica is, in effect, a substantial work in what
we might call algebra (although written more than five centuries before the
term itself existed). Diophantus invented a system of scribal abbreviations
(rather than a symbolic notation) to describe, analyse and solve various types
of equation. He also broke with Greek tradition by considering powers
higher than the third, removing the constraints imposed by geometric in-
terpretations. He listed categories of numbers—which he still described as
made up of some multiple of units, although, as we shall see, he accepted ratios
of these among the solutions of various problems he posed. His notation
for various species of number he particularly sought to investigate started
with squares (designated by capital delta, ∆) and cubes. He went on to de-
fine square-squares (fourth powers), indicated by ∆Υ ∆ (two deltas together
with a separating index) as well as higher powers up to cube-cubes. His no-
tation distinguished carefully between what we would today call variables
42 1. ARITHMETIC IN ANTIQUITY

and constants, and he discussed problems involving various equations, ex-

pressed verbally and solved using his abbreviated notation. Diophantus
consistently used symbols for the ‘unknown’ variable, but he usually as-
signed specific integer values to the constants he considered. He studied
systems of linear as well as quadratic equations (which, as we have seen,
have their origin in Babylonian times and also appear, in geometric guise,
in Book II of Euclid’s Elements), cubic equations and beyond, up to the sixth
degee (cube-cubes).
He was only interested in positive rationals as solutions (there is no ev-
idence that he accepted negative numbers), and frequently posed indetermi-
nate problems, with more unknowns than equations, where the conditions
imposed by the equations do not specify a unique solution. In complete
contrast to Euclid’s Elements, the Arithmetica contains no theorems, but con-
sists of a series of solved problems that display various methods by which
one may find two or more unknowns, usually under the requirement that
certain expressions of them result in perfect squares or perfect cubes.
For example, in Book II, Problem 8 asks the reader: To divide a given
square number into two squares.
In other words, we want to find (positive rational) x, y that satisfy x2 +
y 2 = b2 for some given number b. Typical of his methods is that Diophantus
chooses a specific example, b = 4, so that x2 +y 2 = 16. His general method of
solution seeks to ensure that either the quadratic term or the constant term
in the equation disappears. He first notes that if the first square is x2 , the
other is y 2 = 16 − x2 , and then says that y should be taken in the form ax − b
for some whole number a, again using y = 2x − 4 as his specific example.
In this case he obtains
16 − x2 = y 2 = (2x − 4)2 = 4x2 + 16 − 16x
and the constant term (16) cancels, so that, grouping like terms together,
5x2 = 16x.
Therefore x = 16 12
5 and y = 5 would solve the problem, and the two required
256
squares are 25 and 25 , whose sum is 400
144
25 = 16.
This example shows that, while apparently unwilling to consider con-
sider negative quantities as ‘numbers’, Diophantus nonetheless makes use
of an assertion that is commonly learnt in primary school today. This is the
familiar claim that ‘a minus times a minus is a plus’.
This is evident when he multiplies out (2x − 4)2 and obtains the term
(−4)(−4) = 16. Diophantus’ notation differs from ours: he invents a ‘sub-
traction sign’, which looks like a capital Greek letter lambda (Λ) with a ver-
tical line through the middle – it may be an abbreviation of lepsis, which
means negation. He has no symbol for addition, so terms after the sub-
traction sign are simply grouped together. Nevertheless, without specific
 4. DIOPHANTUS OF ALEXANDRIA 43

mention, he correctly applies the convention that multiplying out two terms
starting with a negation sign must result in a term without one. This is fully
in keeping with Aristotle’s principle of the excluded middle (cf. Section 2.5):
if the negation of a proposition is false, the proposition itself must be true;
that is, a proposition is the negation of its negation.
As we can see, Diophantus was satisfied with obtaining rational num-
bers as solutions for the problems he posed. He thus moved beyond the
restrictions imposed by classical Greek mathematicians, in effect regarding
rational numbers as numbers in their own right, for instance by accepting
rationals such as 256 144
25 and 25 as ‘squares’, which provide the solution of
3
the above problem II.8. In similar fashion, various rationals of the form m n3
would be included included among his ‘cubes’. However, he never articu-
lated a fully consistent system of adding or multiplying fractions; nor did he
use fractions explicitly when formulating his problems, or in arithmetical
operations to simplify the equations he set up to solve these problems.
In modern number theory, Diophantus’ name is invoked today for the
modern field of diophantine analysis, which was mostly inspired by the work
of Pierre de Fermat (1601-1665). In this field of research attention is focused
on the more difficult task of finding integer solutions to indeterminate prob-
lems.