# Euclid's Elements

## Introduction

Euclid’s “Elements” form a set of 13 books dedicated to the fundamentals and logical and systematic development of geometry. It is the masterpiece of Greek Mathematics and for centuries it was the set geometry text in many schools. It is, like Newton’s “Principia” or Maxwell’s books about electrodynamic theory, or Darwin’s “The origin of species”, a book of synthesis. It is the first book of geometric foundation and its style and organization have been the model for later mathematical work.

It is not a practical handbook or a set of useful rules for calculating or measuring like the Egyptian or Babylonian documents of early times. It is a logical structure which responds to Plato’s geometry :

“The geometricians have in view practice only, and are always speaking, in a narrow and ridiculous manner, of squaring and extending and applying and the like-- they confuse the necessities of geometry with those of daily life; whereas knowledge is the real object of the whole science.”

The base which Euclid used to build his geometry is a set of definitions, postulates and common notions, called axioms by some authors. At the beginning, the definitions are 23 even if later some others are introduced. These definitions have the function of naming the elements with which geometry will be built. Some examples are :

• A point is that of which there is no part.
• And a line is a length without breadth.
• And the extremities of a line are points.
• A straight-line is (any) one which lies evenly with points on itself.
• And a surface is that which has length and breadth only.
• And the extremities of a surface are lines.
• A plane surface is (any) one which lies evenly with the straight-lines on itself.
• Parallel lines are straight-lines which, being in the same plane, and being produced to infinity in each direction, meet with one another in neither (of these directions).

Then it continues with the five postulates :

I. Let it have been postulated to draw a straight-line from any point to any point.
II. And to produce a finite straight-line continuously in a straight-line.
III. And to draw a circle with any center and radius.
IV. And that all right-angles are equal to one another.
V. And that if a straight-line falling across two (other), straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side) (sic).

Finally he enunciates some common notions whose number is variable depending on the versions. Some of these notions are :

• Things equal to the same thing are also equal to one another.
• And if equal things are added to equal things then the wholes are equal.
• And if equal things are subtracted from equal things then the remainders are equal.
• And things coinciding with one another are equal to one another.
• And the whole [is] greater than the part.

With the previous five postulates and the common notions he attempts to build geometry (Topic:Euclidean_Geometry). For modern criticism the system has many defects, however all the defects are insignificant compared with the extraordinary merit of having constructed a deductive science from a set of disperse concepts which constituted the previous math.

## Reference

1. The Republic, Book VII. Literature Project. Translated by Benjamin Jowett.
2. EUCLID’S ELEMENTS OF GEOMETRY. The Greek text of J.L. Heiberg (1883–1885) from Euclidis Elementa, edit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883–1885. Edited, and provided with a modern English translation, by Richard Fitzpatrick.