# Euclid of Alexandria the mathematician

If one were content to write a biographical note about ** Euclid'**s life, then it would be very short: we know little or nothing about the man who can be considered the greatest professor of mathematics in history.

It is only thought that he studied at the school of Plato's successors in Athens, before settling in Alexandria, at the invitation of Ptolemy I.

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But since these assumptions are based on the writings of Proclus, which date back 9 centuries after Euclid, they are not reliable!

What is well known about Euclid are the works that have come to us signed with his name, including Data, and especially the 13 volumes of the Elements.

Furthermore, it is not clear how exactly Euclid relates to the knowledge he presents.

It seems that none of the results of the Elements is due to Euclid, and that his work consists of a re-examination of different notions exhibited by various mathematicians.

*The school*

No one can say with certainty whether Euclid was a historian of science, a school principal, and whether he wrote his works for teaching.

Or if he entrusted his writings to his students, who could have continued publishing under Euclid's name even after his death.

One could suppose that Euclid, in the manner of Nicolas Bourbaki, was the nom de plume of a versatile mathematician: several mathematicians writing the same treatise under a pseudonym.

#### Let's look at the Elements. This book is so important that it would be the second most published text in history after the Bible.

It is still critical today, because most of the math courses in college come directly from it.

The first 4 volumes are dedicated to flat geometry. Euclid then initiated the axiomatic method by constructing the geometry in the plane using axioms and postulates.

More clearly, Euclid demonstrates the theorems of plane geometry from propositions he postulates as true (of the type: two quantities equal to one third are equal to each other).

*The postulates*

In modern mathematical language, these propositions would be definitions in the theory that we are trying to build.

Thanks to this point of view, Euclid shows great rigor, very unusual for his time.

One of the postulates formulated by Euclid, the 5th postulate, also known as the parallel postulate, has been problematic for a long time.

#### It says that, through a point outside a straight line, one can carry one parallel to that line, and only one.

Until the nineteenth century, some thought that this postulate was too much, that is, that it was a theorem that could be deduced from the other axioms and postulates.

But the work of Gauss, Riemann and Lobachevsky showed then that it was possible to build other types of geometry, where this postulate was replaced by another one.

In hyperbolic geometry, through a point outside a straight line, an infinite number of lines pass parallel to this line.

### Works

What science owes to Euclid (Charles Renouvier)

"Geometry was fully and rigorously founded at the time when Euclid wrote in the form of a synthesis.

Starting from absolute principles, formally enunciated, and developing into theorems from which the solution of problems gradually emerged.

The famous treatise that has remained the basis of the teaching of this science until today.

This truly admirable book, which dates back to the first half of the first century B.C., and which would never have been attempted to be repeated -not long ago- if it were not for the pretext of a gap it offers for the study of reports in cases of immeasurability.

It will only be replaced and substituted on the day when the surveyors have managed to give themselves clear ideas, free from any contradiction, on the questions of the infinity of quantity, of the measure in its relation to number.

#### We cannot go into the details necessary to clarify this difficult subject here.

Let us look at only the three main points that are important for the study of the progress of science and the scientific spirit in antiquity.

The first point is that there is not a single word in Euclid's treatise that touches on philosophical considerations, that is, of a higher order of generality than the geometrical theme.

## The ideas

Therefore, science is regularly founded and separated; and yet there is no doubt that it is based on the principle of ideas: Euclid is linked with Plato's doctrine, not with his metaphysical doctrine, but with his logical doctrine, whose author is Socrates.

In a word, it takes as principles of geometry pure geometrical ideas, or as concepts: the limit point, without extension, the line without width, etc.

The second observation concerns the measurement of the geometric quantity. Euclid and, like him, his successors compared geometrically considered quantities with each other; they reasoned about their mutual relations of capacity, as far as they could discover.

#### If one of them was presented with the idea of proceeding in this way, he would certainly reject it in the first examination.

For the simple reason that most of the quantities referred to in the surveyor's study would be immeasurable with any unit, of the same class, that could be chosen to measure them all.

That therefore the measurements would generally be only approximate, the reasoning would not be rigorous and the results of the truths approximately.

This objection being irrefutable, or at least we did not know that modern mathematicians have set themselves an invincibly logical way to avoid it.

The result is, in favor of the ancients, a legitimate prejudice of rigor of mind in them, and of scientific correctness, from which we have gradually distanced ourselves.

## Algebra

On the other hand, they have left to the modern ones, as we will see later, the creation of algebra that, understood in the wide and complete sense of the word, is pure mathematics, and this invention is a second wonder of the human mind.

A kind of supergeometry, whose importance and merit are independent of the philosophical exegesis that scholars seem to be still far from reaching.

The third observation refers to the famous "postulate of parallelisms".

Euclid conceived the plan of a science whose proposals would be rigorously demonstrated on the basis of definitions, which are geometrical ideas, on some common notions and on the least possible number of postulates.

It cannot be said that it has philosophically clarified the question of the nature of the postulates, their foundation and their number (why precisely these postulates, and in what capacity should they be chosen and limited?)

#### But modern surveyors cannot be said to have answered this question either.

Be that as it may, Euclid demonstrated his genius by assigning to the geometrical doctrine of parallelism and similar figures a postulate.

For this doctrine, throughout centuries, has remained at the point where it has been, except that recently it has been deemed convenient to question the truth of the postulate.

It could only have been then by denying at the same time, by hypothesis at least, the whole body of a geometry on which philosophy and science have always relied, and which is called Euclidean.

*Let's finish with a quick description of the other Element books: *

- Book V: dedicated to the theory of Eudox proportions, to the immeasurable numbers.
- Book VI: Similarities of the Plan.
- Books VII, VII, and IX: arithmetic, around gcd and prime numbers.
- Book X: quadratic algebraic numbers, discovered by the Pythagoreans.
- Books XI, XII and XIII: geometry in space, with volume of habitual solids, and study of regular polyhedrons.

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