Euclid lived in the time of the Egyptian king

Ptolemy I, who had been a general in the army of

Alexander the Great. Alexander’s conquests had

taken him to Egypt, where he founded the city of

Alexandria on the Nile Delta in 332 B.C. The metropolis

flourished, accumulating half a million

residents and becoming the world’s seat of scientific

and literary scholarship. Among its chief attractions

was the Alexandrian Library, which

boasted more than 500,000 Greek manuscripts

and translations.

Euclid,who probably received his mathematical

education in Athens from followers of PLATO, was

among the scholars drawn to Alexandria’s offerings,

and he established a school of mathematics

there. It is said that one of his pupils asked Euclid

what advantage he would gain by learning geometry.

Euclid instructed his slave to give the boy a

threepence, “since he must make gain out of what

he learns.”

The world’s most famous geometer did not discover

or invent the laws of geometry, as his theorems

and proofs came from the existing body of

Greek knowledge. Rather, his genius lay in organizing

and presenting them in a logical fashion.

Stoicheia, or Elements, totaling 13 books, is Euclid’s

masterpiece. The first six books cover plane geometry

(straight lines, intersection of lines, angles),

and the last three cover solid geometry (pyramids,

cones, cylinders, spheres). The middle books address

such subjects as ratios, proportions, magnitudes,

and prime numbers.

Euclid begins Elements with 25 definitions of

points, lines, plane surfaces, circles, parallels, and

other terms. This is followed by a list of five postulates,

which assume that it is possible, for instance,

to draw a line from one point to another; and five

axioms, such as “Things which are equal to the

same things are also equal to one another” and

“The whole is greater than the part.” From these

fundamentals, the first proposition—“On a given

finite straight line to construct an equilateral triangle”—

is demonstrated. From these elements

plus the proven first proposition, the second

proposition may be demonstrated, and so on. Each

new proposition can be traced to the previously

proven propositions on which it is based, all the

way back to Euclid’s initial descriptions of assumptions

and self-evident truths. There are a

total of 465 propositions. Ptolemy once wondered

whether there was a shorter way to reach these invaluable

conclusions, to which Euclid replied,

“There is no royal road to geometry.”

In addition to Elements, Euclid published several

other less-famous works. Fallacies, no longer

extant, provides methods for detecting illogical

conclusions; Data proves that if certain magnitudes

in a geometric figure are given, other magnitudes

may be deduced; Figures shows how to

divide figures proportionally; Surface-loci concerns

curved surfaces; Conics studies cones; Optics addresses

visual perspective; Elements of Music is

based on Phythagorean theory; and in his pioneering

work Phenomena, Euclid applies spheric geometry

to astronomy.

Through the centuries, Euclid’s Elements has

appeared in more than 2,000 different versions.

According to William Dunham, author of Journey

through Genius (1991),“This work had a profound

impact on Western thought as it was studied, analyzed,

and edited for century upon century, down

to modern times. It has been said that of all books

from Western civilization, only the Bible has received

more intense scrutiny than Euclid’s Elements.”

Great minds from Archimedes and CICERO

to Isaac Newton, Napoleon, and Lincoln have

studied this classic work, and it remains the definitive

geometry text in many classrooms.

English Versions of Works by Euclid

Euclid: The Thirteen Books of the Elements. Translated

by Sir Thomas L. Heath. New York: Dover Publications,

1908.

Euclid’s Phenomenon: A Translation and Study of a

Hellenistic Treatise in Spherical Astronomy. Translated

by Robert S. Thomas. Edited by J. L.

Berggren. New York: Garland, 1996.

Works about Euclid

Artmann, Benno. Euclid: The Creation of Mathematics.

New York: Springer-Verlag, 1999.

Mlodinow, Leonard. Euclid’s Window: The Story of

Geometry from Parallel Lines to Hyperspace. New

York: Simon & Schuster, 2002.