Chapter 1
Pre-Hellenic Antiquity
This very short chapter is intended only to give an overview of some of the first
geometric ideas which arose in various civilizations before the influence of the sys-
tematic work of the Greek geometers. So pre-Hellenic should be understood here as
“before the Greek influence”.
From this “pre-Hellenic antiquity”, we know of various works due to the Egyp-
tians and the Babylonians. Indeed, these are the only pre-Hellenic civilizations
which have produced written geometric documents that have survived to the present
day.
It should nevertheless be mentioned here that some works in China and India—
posterior to the golden age of geometry in Greece—are considered by some histori-
ans as “pre-Hellenic” in the sense of being “absent of Greek influence”. But not all
specialists agree on this point. Therefore we choose in this book to mention these
developments at their chronological place, after the Greek period.
1.1 Prehistory
Prehistory is characterized by the absence of writing. In those days, the transmission
of knowledge was essentially oral. But nowadays, we no longer hear those voices.
Therefore prehistory remains as silent about geometry as it is about all other aspects
of human life. The best that we can do is to rely on archaeological discoveries and
try to interpret the various cave pictures and objects that have been found.
The first geometric pictures date from 25000 BC. They already indicate some
mastering of the notions of symmetry and congruence of figures. Some other objects
of the same period show evidence of the first arithmetical developments, such as the
idea of “counting”.
Particularly intriguing is the picture in Fig. 1.1: it seems to be evident that the
prehistoric artist did not just want to draw a nice picture: he/she wanted to emphasize
some mathematical discovery. Indeed, looking at this picture, we notice at once that:
• doubling the side of the triangle multiplies the area by 4; tripling the side of the
triangle multiplies the area by 9;
F. Borceux, An Axiomatic Approach to Geometry, DOI 10.1007/978-3-319-01730-3_1, 1
© Springer International Publishing Switzerland 2014
2 1 Pre-Hellenic Antiquity
Fig. 1.1
• counting the number of small triangles on each “line” we observe that
1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9.
The oldest written documents that we know concerning geometry already mention
the corresponding general results:
• multiplying the lengths by n results in multiplying the areas by n2 ;
• the sum of the n first odd numbers is equal to n2 .
To what extent was the prehistoric artist aware of these “theorems”? We shall prob-
ably never know.
A tradition claims that the origin of arithmetic and geometry is to be found in
the religious rituals of our ancestors: they were fascinated by the properties of some
forms and some numbers, to which they attributed magical powers. By introduc-
ing such magical forms and numbers into their rituals they might perhaps draw the
benediction of their gods.
Another tradition, reported by Herodotus (c. 484 BC–c. 425 BC) presents geome-
try as the precious daughter of the caprices of the Nile. Legendary Pharaoh Sesostris
(around 1300 BC; but probably a compound of Seti and Ramesses II) had, claims
Herodotus, distributed the Egyptian ground between “the” (by which we understand
“some few privileged”) inhabitants. The annual floods of the Nile valley, the origin
of its fertility but also of many dramatic events, made it necessary to devise practical
methods of retracing the limits of each estate after each flood. These methods were
based on triangulation and probably made use of some special instances of Pythago-
ras’ theorem for constructing right angles. For example, the fact that a triangle with
sides 3, 4, 5 has a right angle seems to have been known at least since 2000 BC.
But the Nile valley certainly does not have the hegemony of early developments
of mathematics, not even in Africa: the discovery in 1950 of the Ishango bone in
Congo, dating from 22000 BC, is one of the oldest testimonies of some mathemat-
ical activity. And various discoveries in Europe, India, China, Mesopotamia, and
so on, indicate that—at different levels of development—mathematical thought was
present in many places in the world during antiquity.
However, up to now, modest prehistory has unveiled very little of its personal
relations with geometry.
1.2 Egypt 3
Fig. 1.2
Fig. 1.3
1.2 Egypt
The oldest mathematical papyrus which has reached us is the so-called Moscow pa-
pyrus, most likely written around 1850 BC. But our main knowledge of Egyptian
mathematics during high antiquity comes from a more extended papyrus copied by
the scribe Ahmes around 1650 BC. These papyri contain the solutions to many arith-
metical and geometrical problems whose elaboration, according to Ahmes, dates
back to 2000 BC.
The Moscow papyrus is also called the Golenischev Mathematical Papyrus, after
the Egyptologist Vladimir Golenishchev who bought the papyrus in Thebes around
1893. The papyrus later entered the collection of the Pushkin State Museum of Fine
Arts in Moscow, where it remains today. The Ahmes papyrus is often referred to
as the Rhind papyrus, so named after Alexander Henry Rhind, a Scottish antiquar-
ian, who purchased the papyrus in 1858 in Luxor, Egypt. The papyrus was appar-
ently found during illegal excavations on the site of the mortuary temple of Pharaoh
Ramesses II. It is kept at the British Museum in London.
For example, Problem 51 of the Ahmes papyrus shows that
The area of an isosceles triangle is equal to the height multiplied by half of the base.
The explanation is a “cut and paste” argument as in Fig. 1.2. Cut the triangle along
its height; reverse one piece, turn it upside down and glue both pieces together; you
get the rectangle on the right.
An analogous argument is used in Problem 52 to show that
The area of an isosceles trapezium is equal to the height multiplied by half the sum of the
bases.
See Fig. 1.3, which is again “a proof” in itself. At least, it is a “proof” in the spirit
of the time: in any case, an argument based on congruences of figures.
However, let us stress that the Egyptians did not have a notion of what a “the-
orem” or a “formal proof” is, in the mathematical sense of the term. In particular,
they did not make a clear distinction between a precise result and an approximative
4 1 Pre-Hellenic Antiquity
one. For example, one finds in Egyptian documents the following strange rule to
compute the area of a quadrilateral:
The area of an arbitrary quadrilateral is obtained by multiplying the half-sums of the pairs
of opposite sides.
This is in clear contradiction with Problem 52 in the Ahmes papyrus: the area of an
isosceles trapezium. But this did not seem to disturb anybody. Even more amazing
is the corollary (presented as such) inferred from this general “rule”:
The area of a triangle is equal to the half of one side multiplied by the half sum of the other
two sides.
This is again in contradiction with Problem 51 in the Ahmes papyrus, but the biggest
surprise is elsewhere:
The Egyptians were able to consider a triangle as a special case of quadrilateral: a quadri-
lateral having one side of length zero.
This is an abstraction process which we would not have expected in those days.
Both papyri also consider the area of a circle. In Problem 50 of the Ahmes pa-
pyrus, it is claimed that
A circle whose diameter is 9 units has the same area as a square whose side is 8 units.
In view of the now well-known formula for the area of a circle, this yields the value
256
π= ≈ 3.16
81
. . . which is not that bad! As a matter of comparison, the Bible (in the first book
of Kings, VII-23) gives the value 30 for the circumference of a circle of diameter
10 . . . that is a value π = 3. It does not seem that the Egyptians were aware of the
existence of a unique “quantity” π to be used for all circles, whatever their size.
Later we shall discuss what the nature of such a “quantity” π could have been in
antiquity (see Sect. 2.6).
On the other hand the Egyptians had discovered the relation (an exact relation,
this time) between the length and the area of a circle.
The area of a circle is to its length as the area of the square constructed on the diameter is
to its perimeter.
In modern algebraic notation, if R is the radius of the circle, this means
πR 2 (2R)2
= .
2πR 4(2R)
The Egyptians also knew how to compute the volume of a pyramid:
1
× base × height.
3
We do not know how they discovered this formula, but we can easily imagine how
they would have made use of it.
Problem 56 of the Ahmes papyrus also investigates the “similarity” of triangles.
1.3 Mesopotamia 5
Fig. 1.4
Two right angled triangles having their respective sides proportional have their correspond-
ing angles equal.
In this problem (see Fig. 1.4), the angles are measured by what we call today their
“cotangent”. Such a result was important for the construction of pyramids, in order
to keep the slope constant.
1.3 Mesopotamia
Let us now leave the Nile valley and jump to the valleys of the Euphrates and
the Tigris, in Mesopotamia. Thus we leave behind the hieroglyphs and switch to
cuneiform writing (as early as 3000 BC), most often carved on clay tablets instead
of papyrus: and this ensured a much better conservation through the centuries.
The Babylonians were exceptional algebraists and astronomers. They were able
to solve the equations of degree one or two, and even some equations of higher
degrees. We inherited from them the sexagesimal division of time and angles. Some
tablets are also trigonometric tables, giving the values of the secants of the angles.
But Mesopotamia, like Egypt, did not really distinguish between exact and ap-
proximate results. A tablet (see [1]) gives the approximate areas of the first seven
regular polygons. As far as the circle is concerned, it is claimed that the perimeter
of the regular hexagon inscribed in a circle (= six times the radius) is equal to 24
25 of
the circumference. In modern notation
24
6R = 2πR
25
which yields the value
25
π= = 3.125.
8
Another tablet claims that the volume of a truncated cone or pyramid is obtained
as the half (sic) sum of the base multiplied by the height! Egyptians had the correct
formula (one third instead of one half). As far as the development of geometry is
concerned we can assume there was little communication between the two civiliza-
tions.
As a matter of fact, the Babylonians were making extensive use of Pythagoras’
theorem, at least one millennium before Pythagoras was born. On one tablet, one
finds the following problem:
6 1 Pre-Hellenic Antiquity
Fig. 1.5
A ladder is leant along a wall. The top of the ladder glides of three units along the wall while
the foot of the ladder moves nine units on the ground. What is the length of the ladder?
Another tablet explains how to find, using “Pythagoras’ theorem”, the apothem of a
chord inscribed in a circle.
Contrary to what was thought for a long time, the development of geometry in
Mesopotamia was at least comparable, and maybe even superior, to the development
of geometry in Egypt during antiquity. In particular, the systematic use of Pythago-
ras’ theorem contrasts with its explicit absence in the Egyptian papyri.
The Babylonians also knew that
Theorem An angle inscribed in a half circle is necessarily a right angle.
The Egyptians were unaware of this fact, which is generally attributed to Thales,
who lived a millennium later. How did the Babylonians find and justify this result?
We do not know. Perhaps they made empirical observations of the following kind
(see Fig. 1.5):
• Let ABC be the angle inscribed in a half circle.
• Construct B , the point symmetric to B with respect to the centre O of the circle.
• This yields four isosceles triangles, forming two equal pairs.
• Thus the four angles at A, B, C, B are equal.
• By symmetry, the opposite sides of the quadrilateral ABCB are equal.
• The diagonals of this quadrilateral ABCB are equal as well (and equal the di-
ameter of the circle).
This was certainly sufficient reason to convince them that the quadrilateral is a rect-
angle.
1.4 Problems
1.4.1 Show that a cube is the union of three equal pyramids. This yields at once the
formula for the volume of one of these pyramids.
1.5 Exercises 7
1.4.2 Consider a pyramid whose base is a square and whose summit projects or-
thogonally on the center of the base (like the pyramids constructed by the Egyp-
tians). By a “cut and paste” argument, infer the formula giving the volume of such
a pyramid.
1.4.3 Prove that the “ladder problem” of the Babylonians (see Sect. 1.3) has in-
finitely many solutions.
1.5 Exercises
1.5.1 By a “cut and paste” argument, infer the formula for the area of a parallelo-
gram.
1.5.2 By a “cut and paste” argument, infer the formula for the area of an arbitrary
triangle.
1.5.3 By a “cut and paste” argument, infer the formula for the area of an arbitrary
trapezium.
