Link to Physics 109N Home Page and Other Topics
Proclus Diadochus, AD 410-485.
(From his book: Commentary on Euclid's Elements I)
We must next speak of the origin of geometry in the present world
cycle. For, as the remarkable Aristotle tells us, the same ideas
have repeatedly come to men at various periods of the universe.
It is not, he goes on to say, in our time or in the time of those
known to us that the sciences have first arisen, but they have
appeared and again disappeared, and will continue to appear and
disappear, in various cycles, of which the number both past and
future is countless. But since we must speak of the origin of
the arts and sciences with reference to the present world cycle,
it was, we say, among the Egyptians that geometry is generally
held to have been discovered. It owed its discovery to the practice
of land measurement. For the Egyptians had to perform such measurements
because the overflow of the Nile would cause the boundary of each
person's land to disappear. Furthermore, it should occasion no
surprise that the discovery both of this science and of the other
sciences proceeded from utility, since everything that is in the
process of becoming advances from the imperfect to the perfect.
The progress, then, from sense perception to reason and from reason
to understanding is a natural one. And so, just as the accurate
knowledge of numbers originated with the Phoenicians through their
commerce and their business transactions, so geometry was discovered
by the Egyptians for the reason we have indicated.
It was Thales, who, after a visit to Egypt, first brought this
study to Greece. Not only did he make numerous discoveries himself,
but laid the foundation for many other discoveries on the part
of his successors, attacking some problems with greater generality
and others more empirically. After him Mamercus the brother of
the poet Stesichorus, is said to have embraced the study of geometry,
and in fact Hippias of Elis writes that he achieved fame in that
study.
After these Pythagoras changed the study of geometry, giving it
the form of a liberal discipline, seeking its first principles
in ultimate ideas, and investigating its theorems abstractly and
in a purely intellectual way.
[He then mentions several who developed this abstract approach
further: Anaxagoras, Hippocrates, Theodorus, etc.]
Plato, who lived after Hippocrates and Theodorus, stimulated to
a very high degree the study of mathematics and of geometry in
particular because of his zealous interest in these subjects.
For he filled his works with mathematical discussions, as is well
known, and everywhere sought to awaken admiration for mathematics
in students of philosophy.
[He then lists several mathematicians, including Eudoxus and Theatetus,
who discovered many new geometric theorems, and began to arrange
them in logical sequences-this process culminated in the work
of Euclid, called his Elements (of geometry) about 300
BC. ]
Euclid composed Elements, putting in order many of the
theorems of Eudoxus, perfecting many that had been worked out
by Theatetus, and furnishing with rigorous proofs propositions
that had been demonstrated less rigorously by his predecessors
… the Elements contain the complete and irrefutable
guide to the scientific study of the subject of geometry.
Most Greek writers accept Heroditus' claim that Egyptian geometry began with land measurement for tax purposes, but Aristotle, writing a century later, had a more academic, and perhaps less plausible, theory of the rise of geometry:
"..the sciences which do not aim at giving pleasure or at the necessities of life were discovered, and first in the places where men first began to have leisure. That is why the mathematical arts were founded in Egypt, for there the priestly class was allowed to be at leisure."
However, as Thomas Heath points out in A History of Greek Mathematics, one might imagine that if this were true, Egyptian geometry "would have advanced beyond the purely practical stage to something more like a theory or science of geometry. But the documents which have survived do not give any grounds for this supposition; the art of geometry in the hands of the priests never seems to have advanced beyond mere routine. The most important available source of information about Egyptian mathematics is the Papyrus Rhind, written probably about 1700 BC, but copied from an original of the time of King Amenemhat III (Twelfth Dynasty), say 2200 BC."
Heath goes on to give details of what appears in this document: areas of rectangles, trapezia and triangles, areas of circles given as (8d/9)2, where d is the diameter, corresponding to pi equal to 3.16 or so, about 1% off. There are approximate volume measures for hemispherical containers, and volumes for pyramids.
Heath's point is that this is all very impressive-and useful-so early in history, but it is not abstract in the sense that the later development of geometry by the Greeks was. Furthermore, by the time of Archimedes, the abstract Greek approach had in fact led to far more accurate expressions for all these areas, volumes, and so on.