Ramble, Part 3: What Euclid Did

[stoutfellow]

As dan_ad_nauseam has noted in comments, it is not at all clear how much, if any, of Euclid's Elements is original with him. The elementary plane geometry and the number theory probably date back to the Pythagoreans; the more advanced geometry rests on the work of Eudoxus; and the detailed analysis of incommensurables that makes up Book X and the solid geometry of Book XIII are generally ascribed to Theaetetus. As a mathematician, he probably does not stand in the first rank among his contemporaries.

Nonetheless, he is probably the most famous of the classical-era mathematicians; the Elements was regarded as the premier textbook of mathematics for well over two thousand years. As a mathematician, he may not have been outstanding, but as a methodologist - in terms of his influence on how mathematics is done - he stands very nearly alone. (Only Descartes and, perhaps, Cauchy even come close.) Under the cut is some discussion of his impact.

First and foremost, Euclid established the basic format for the presentation of mathematics. The concept of proof had been around for quite a while; Thales is usually credited with its invention. But it appears to have been Euclid (it may have been Eudoxus again) who came to grips with the fundamental problem, which is that proof proceeds from old truth to new truth - it must have raw material to begin. Euclid's solution was to lay out a small number of definitions and fundamental truths as a foundation. On that basis, a superstructure of theorems was raised, each carefully laid out and proven, each providing support for the theorems to follow. This basic pattern of definitions and assumptions, theorems, and proofs has remained intact ever since. Modern mathematicians do not generally proceed with the rigid arrangement that Euclid used (with a few exceptions, such as the Bourbaki group), but the underlying structure remains. (There have been some changes in how the structure is to be understood, but I'll leave discussion of that until we get to the nineteenth century.)

Secondly, Euclid's work solidified the dominant position of geometry within mathematics. There are parts of the Elements that would today be classed as belonging to algebra, but they are presented in geometric terms. For example, there is the problem posed in Book II, Proposition 11: To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment. This is easily recast in modern terms: given a quantity a, divide it into two pieces x and a-x so that ax=(a-x)² - in other words, solve a certain quadratic equation. Euclid and his successors dealt with many of what we would call quadratic, cubic, and quartic equations in this essentially geometric fashion.

Similarly, Euclid's discussion of number theory begins in Book VII with definitions like these:

• A number is a part of another, the less of the greater, when it measures the greater.
• A prime number is that which is measured by an unit alone.
• A composite number is that which is measured by some number.

Note the use of the word "measure" here, and note also that Euclid introduces Eudoxus's theory of proportionality in Book V - well before this point. The definitions of "square" and "cube", as applied to numbers, also appear at the beginning of Book VII. (I'll say more about the consequences of this geometrized algebra later.)

Thirdly, Euclid established the primary importance of straightedge-and-compass construction within geometry. That is, the acceptable constructions of geometric figures were those which involved only the drawing of straight lines between known points and of circles with given center and radius. This was not an arbitrary restriction; Euclid's axioms included assertions concerning the existence of such lines and circles, but no other curves were mentioned. After all, the axioms were (apparently) intended to be simple and obvious truths; lines and circles seem to have an elementary character that, e.g., ellipses and spirals lack. The certainty that Euclid sought, therefore, rested more comfortably on those simplest of curves.

For good or ill, Euclid's Elements established the structure within which mathematicians of the future were to work. Though we have abandoned some elements of that structure and repurposed others, it remains a (though no longer the) central part of mathematics, and it is that which justifies Euclid's stature in mathematical history.

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Comments

Re: Ramble, Part 3, What Euclid Did

[ndrosen.livejournal.com]

I have nothing much to say -- you know the history of mathematics better than I do -- but I do want to say thanks for posting these mini-lessons in mathematical history. They have an audience, although how large an audience, I can't be sure.

Re: Ramble, Part 3, What Euclid Did

[stoutfellow.livejournal.com]

Thank you. It's good to hear that.