Algebra by Another Name Algebra by Another Name? Interpretations of Book II of Euclid's Elements John Little MONT 107Q February 17, 2017

Book II of the Elements Consists of 14 Propositions, leading to a “punchline” Book II, Proposition 14. To construct a square [with area] equal to [that of] any given rectilinear figure In traditional terms – a “problem” rather than a “theorem” – gives an explicit construction, proves that the results are correct “quadrature” in its original sense(!) A collection of propositions is set up as a collection of “lemmas” for later use

“Geometric algebra?” Isn't Proposition 2 “just” a geometric version of the distributive law for multiplication over addition: Write AC = x, CB = y, then (x + y)(x + y) = (x + y)x + (x + y)y Couldn't you also prove Proposition 5 using those ideas too? Perhaps write AD = x and DB = y and assume x > y Then claim is: x y + ((x – y)/2)² = ((x + y)/2)² And of course the answer is, yes if we are thinking about the underlying logical relationships!

But is that what Euclid meant by this? But another question to ask here is: Was this Book II of the Elements a sort of “algebra in geometric form” for the Greeks? Many well-known historians of mathematics in 19th and early 20th centuries thought so H. Zeuthen, P. Tannery, O. Neugebauer, B. L. van der Waerden, … T. L. Heath (translator of most commonly- used English version of Euclid): Book II contains “... the geometric equivalent of the algebraical operations … “

C. Boyer, from “Euclid of Alexandria” “It is sometimes asserted that the Greeks had no algebra, but this is patently false. They had Book II of the Elements, which is geometric algebra and served much the same purpose as does our symbolic algebra. There can be little doubt that modern algebra greatly facilitates the manipulation of relationships among magnitudes. But it is undoubtedly also true that a Greek geometer versed in the fourteen theorems of Euclid's 'algebra' was far more adept in applying these theorems to practical mensuration than is an experienced geometer of today.”

Boyer, cont. “Ancient geometric 'algebra' was not an ideal tool, but it was far from ineffective. Euclid's statement (Proposition 4), 'If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments,' is a verbose way of saying that (a + b)² = a² + 2ab + b²”

O. Neugebauer's view Provocatively, Neugebauer even noted that Proposition 5 is equivalent to (i.e. can be restated with the same algebraic relation as) a step-by-step procedure for solving the type of quadratic equation occurring in many Old Babylonian problem texts like YBC 6967 (dating to about 1800 BCE) He suggested that Book II of the Elements might record a sort of “technology transfer” from the Babylonian tradition into Greek mathematics, but recast in typically Greek geometric form

Sabetai Unguru's critique S. Unguru, On the need to rewrite the history of Greek mathematics, Archive for History of Exact Sciences 15 (1975/76), 67—114. Forcefully refutes “geometric algebra” as a correct description of Book II of Euclid Unguru's main point: it's geometry pure and simple; Greek mathematics did not have any of the apparatus of symbolic algebra Rejects and even ridicules Neugebauer's proposed “Babylonian connection” because no explicit evidence exists for it

Unguru's argument, summarized Attempting to “explain” Euclid this way is perniciously wrong from the historical point of view because it uses modern concepts that are a false description of a fundamentally different understanding of mathematics (“conceptual anachronism,” or “Whig history” – presents the past as leading inevitably to the present): In symbolic algebra, variables are effectively placeholders for numerical values, but for the Greeks, the idea of number (ἀριθμός) always referred to “counting numbers” (positive integers)

Unguru, continued So, Euclid never used a numerical value as a measure of length or area and used different methods in the “arithmetical books” VII – IX of the Elements Moreover, depending on how we label different lengths in figures, the algebraic “translation” can end up being quite different For example, Book II, Proposition 5 could also be written as (x + y)(x – y) + y² = x² if we make x = AC, y = CD Which algebraic version was Euclid thinking of? Unguru's answer: NONE of them!

If Unguru wanted to start a war, he succeeded! From Unguru's article – a fairly typical example of the tone – really quite extraordinary(!) “ … history of mathematics has been typically written by mathematicians … who have either reached retirement age and ceased to be productive in their own specialties or become otherwise professionally sterile … the reader may judge for himself how wise a decision it is for a professional to start writing the history of his discipline when his only calling lies in professional senility …”

A bitter academic controversy When Unguru's article appeared, one of the mathematicians/historians Unguru had savaged, B. L. van der Waerden, was still alive (his dates: 1903 – 1996). You can imagine how well he liked that passage from Unguru's article! He published a rejoinder – a defence of his point of view in the same issue of the Archive for History of Exact Sciences – “A defence of a 'shocking' point of view”, 199 – 210.

H. Freudenthal – another response “What is algebra and what has it been in history?” Archive for History of Exact Sciences 16 (1976/77), 189 – 200. Argues that there is indeed algebra in Greek mathematics, using examples from Archimedes But of course, a historian would say “I thought we were talking about Euclid. Archimedes was active about 50-70 years after Euclid's time, … “

André Weil “weighs in” “Who betrayed Euclid? Extract from a letter to the editor,” Archive for History of Exact Science 19 (1978/79), 91 – 93: Essentially asks: “who was responsible for allowing such a trashy article to be published? What is happening to the quality of this journal?” And gets in a nice ad hominem attack: “ … it is well to know mathematics before concerning oneself with its history … ”

A “tempest in a teapot?” This may all strike you as a nasty but silly disagreement over a minor issue. But it points out a fundamental difference between doing mathematics and doing history of mathematics (as history). As mathematicians, recognizing logical connections between old and new work and making reinterpretations is a part of what we do. When apparently different things are logically the same, just expressed in different ways, we can treat them as the same(!) So we are always looking for those equivalences – finding them can represent an advance in our understanding!

A “tempest in a teapot?” And (as Unguru insinuated in his own nasty way) van der Waerden, Freudenthal, Weil were certainly all primarily mathematicians who had eminent research records and then turned to writing history later in their professional careers Not surprising that they had the “habits of mind” and point of view of working mathematicians, not historians! In particular, to put words in their mouths: “if it's logically equivalent to algebra, then it's a geometric form of algebra”

A “tempest in a teapot?” But for intellectual historians, it's not so much logical equivalences that matter – it's particular features, differences! Each culture, era, scientific school, etc. is a unique and separate thing – their first and most important job is to understand Greek mathematics on its own terms, not on our terms A fundamentally different way of thinking; can see Unguru and van der Waerden, Freudenthal, Weil talking past each other without really understanding the others' points because they aren't approaching the question from the same place.

Winners and losers? In many ways, have to say Unguru was the “winner” here A small but growing corps of historians of mathematics (as distinct from mathematicians doing history) now exists and they take Unguru's view for the most part Even mathematicians doing history have to be sensitive to these issues to have their work accepted these days

Since the controversy – whose vindication? From a historian's point of view it's also interesting to ask how understanding of Euclid's Book II evolved over time More recent work (see Corry, Archive for History of Exact Sciences (2013)) – as the Elements passed through the Medieval Islamic world and Renaissance Europe (where our symbolic algebra was developed), the connection between geometry and algebra was realized and alternate algebraic proofs started to be presented for the propositions in Book II (!) “Geometric algebra” probably does do a good job of describing later understanding!