The Story of Euclid

Without looking at any geometry textbook or reference, ask yourself the following fundamental questions:

• What is a point?
• What is a line?
• What is a surface?

Let us be honest: no easy answer come to our minds. The problem is that they, more than being geometry questions, are philosophic questions; specially the definitions of point and line.

However, a very long time ago, a Greek geometer of the name Euclid settled the answer not only to those simple questions, but to hundreds more.

In an elegant and concise way he stated the following axioms or postulates:

• A point is that of which there is no part.
• A line is a length without breadth
• The extremities of a line are points
•  A straight line is the one which lies evenly with points on itself
• A surface is that which has length and breadth only
• The extremities of a surface are lines
• A plane surface is the one which lies evenly  with the straight lines on itself

But those statements are just the beginning of an herculean work in the foundations of mathematics that has not only survived for more than two millennia, but has became essential in the teaching of geometry.

The Elements of Geometry of Euclid

W. B. Frankland's book about Euclid is also another monumental work. It is a complete book about the trail through history of Euclid's Elements of Geometry.

A complete copy of the Elements is something of a rarity at the present day: there are "so many worlds, so much to do," that printers have no time to print, nor readers to read, more than a selected portion. If by chance a complete copy is picked up, it will probably prove to be more than complete, for in it will be fifteen books instead of Euclid’s thirteen. The last two are works of supererogation on the part of an editor or two after the Christian epoch: to be precise, the fourteenth and fifteenth books were added by Hypsicles and perhaps Theon, both of Alexandria, the one a century, the other half a millennium after the composition of the genuine thirteen. It may be taken for granted that if thirteen was regarded as an unlucky number by Alexandrian students, fifteen was thought no better. Nowadays, it is very unusual for a scholar to familiarize himself with more than the first six, eleventh, and twelfth books.

From Euclid's Elements of Geometry we can construct, or solve almost every elementary geometry problem we can find in High School level or first year textbooks. For example, Euclid graciously solved problems and propositions like:

• Construct an equilateral triangle on a given finite straight-line.
• If a triangle has two angles equal to one another then the sides subtending the equal angles will also be equal to one another.
• Cut a given rectilinear angle in half.
• If two straight-lines cut one another then they make the vertically opposite angles equal to one another.
• In any triangle, the greater side subtends the greater angle.
• In right-angled triangles, the square on the side subtending the right-angle is equal to the (sum of the) squares on the sides containing the right-angle.
• To find the center of a given circle.

Those are easy samples of Euclid's geometry problems commonly found in modern textbooks. However we seldom hear from Euclid the man, Euclid the geometer, Euclid the mathematician. Once more readers have the opportunity to read about Euclid and his masterworks from an inspiring author and historian of Greek mathematicians.

Download The Story of Euclid now!

Note:

A Greek-English copy of The Elements of Geometry of Euclid can be download for free from the University of Texas.