## 10 Jan Of all books from Western civilization, only the Bible has received more intense scrutiny than Euclid’s Elements. (WILLIAM DUNHAM) | Part B’

This work had a profound impact on Western thought as it was studied, analyzed, and edited for century upon century, down to modern times.

We can think, for instance, of a circle as being a totally curvilinear, and thus quite intractable, plane figure. But, if we inscribe within it a square, and then double the number of sides of the square to get an octagon, and then again double the number of sides to get a 16-gon, and so on, we will find these relatively simple polygons ever more closely approximating the circle itself. In Eudoxean terms, the polygons are “exhausting” the circle from within.

This process is, in fact, precisely how Archimedes determined the area of a circle.

It is to Eudoxus that he owed this fundamental logical tool. In addition, Archimedes credited Eudoxus with using the method of exhaustion to prove that the volume of “any cone is one third part of the cylinder which has the same base with the cone and equal height,” a theorem that is by no means trivial. The reader familiar with higher mathematics will recognize in the method of exhaustion the geometric forerunner of the modern notion of “limit,” which in turn lies at the heart of the calculus. Eudoxus’ contribution was a significant one, and he is usually regarded as being the finest mathematician of antiquity next to the unsurpassed Archimedes himself.

It was during the latter third of the fourth century B.C. that Alexander the Great emerged from Macedonia and set out to conquer the world.

His conquests carried him to Egypt where, in 332 B.C., he established the city of Alexandria at the mouth of the Nile River. This city grew rapidly, reportedly reaching a population of half a million in the next three decades.

Of particular importance was the formation of the great Alexandrian Library that soon supplanted the Academy as the world’s foremost center of scholarship.

At one point, the facility had over 600,000 papyrus rolls, a collection far more complete and astounding than anything the world had ever seen.

Indeed, Alexandria would remain the intellectual focus of the Mediterranean world through the Greek and Roman periods until its final destruction in A.D. 641 at the hands of the Arabs.

Among the scholars attracted to Alexandria was a man named Euclid, who came to set up a school of mathematics. We know very little about his life either before or after his arrival on the African coast, but it appears that he received his training at the Academy from the followers of Plato.

Be that as it may, Euclid’s influence was so profound that virtually all subsequent Greek mathematicians had some connection or other with the Alexandrian School.

What Euclid did that established him as one of the greatest names in mathematics history was to write the Elements.

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.

The highly acclaimed Elements was simply a huge collection divided into 13 books-of 465 propositions from plane and solid geometry and from number theory.

Today, it is generally agreed that relatively few of these theorems were of Euclid’s own invention. Rather, from the known body of Greek mathematics, he created a superbly organized treatise that was so successful and so revered that it thoroughly obliterated all preceding works of its kind.

Euclid’s text soon became the standard. Consequently, a mathematician’s reference to 1.47 can only mean the 47th proposition of the first book of the Elements.

Down through the centuries, over 2000 editions of the Elements have appeared, a figure that must make the authors of today’s mathematics textbooks drool with envy.

As noted, it was highly successful even in its own day.

After the fall of Rome, the Arab scholars carried it off to Baghdad, and when it reappeared in Europe during the Renaissance, its impact was profound.

Part C’ follows

**Journey Through Genius: THE GREAT THEOREMS OF MATHEMATICS **

**William Dunham**

Image Α: https://en.wikipedia.org/wiki/Euclid%27s_Elements

Image Β: https://pixels.com/featured/euclid-jazzberry-blue.html?product=art-print

Image C: https://cdn.britannica.com/46/8446-050-BC92B998.jpg