Algebra by Another Name ? Interpretations of Book II of Euclid's Elements John Little Holy Cross Summer Mathematical Seminar June 25, 2014

Goals for this talk To discuss a bit of the content of Euclid's Elements, in particular some propositions from Books I and II (Mostly for entertainment value) to look at a rather heated controversy from the late 1970's regarding the interpretation of what Euclid was up to here To convince you that history of mathematics and mathematics itself are not really the same thing – can actually be at cross- purposes at times

Historical background on Euclid Not much is known about Euclid personally – no firm dates of birth or death, place of birth, etc. Career in Alexandria (Egypt). Proclus (~450 CE): “This man lived in the time of the first Ptolemy (i.e. around 300 BCE); for Archimedes, who followed closely on the first Ptolemy makes mention of Euclid …. He is therefore younger than Plato's circle but older than Eratosthenes and Archimedes …. In his aim he was a Platonist, …, whence he made the end of the whole Elements the construction of the so-called Platonic figures.”

His best known surviving work – The Elements The earliest known complete manuscripts date from about 900 CE – about 1200 years after Euclid's death(!) (Other earlier fragments too.) For a long time, most editions derived from a version with commentary by a later Alexandrian mathematician named Theon from about 400 CE. In 1808, an earlier version (codex Vat. gr. 190) was discovered in the Vatican Library in Rome, with not too many differences – text was remarkably stable!

Pages from Vatican Euclid – hibit/d-mathematics/images/math01.jpg

One of the most influential mathematics textbooks We think there were other comparable books of Elements before Euclid's. According to Proclus one was written by Theudios, for instance. None of them survive! Euclid quickly superseded all those predecessors and “competitors” and put them “out of business” – became the standard compendium of basic mathematics that Greek and later students learned from for the next years.

What is in the Elements? Organized into 13 books Books I – IV, VI : plane geometry Books VII – IX: number theory Books V, X: abstract theory of proportions, commensurability Books XI – XIII: solid geometry, construction of Platonic solids The Vatican codex Vat. gr. 190 doesn't contain Book XIII, interestingly

Book I – triangles, parallels, area in the plane What Euclid accomplished with the Elements was a tightly structured deductive presentation of mathematics starting from explicitly stated unproved starting assumptions At start of Book I: A collection of Definitions (maybe added later) 5 “Common Notions” (also called Axioms) 5 Postulates (assumptions about geometry)

The 5 Common Notions 1.Things that are equal to the same thing are equal to one another. 2.If equals be added to equals, the wholes are equal. 3.If equals be subtracted from equals, the remainders are equal. 4.Things that coincide with one another are equal to one another. 5.The whole is greater than the part.

The First Four Postulates 1.(It is possible) to draw a straight line from any point to any point. 2.(It is possible) to produce any finite straight line continuously in a straight line. 3.(It is possible) to describe a circle with any center and distance. 4.All right angles are congruent to one another.

The Fifth Postulate 5. If a straight line falling on two straight lines makes the angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

The culmination of Book I Book I, Proposition 47. In a right triangle, the square on the hypotenuse is equal [in area] to the sum of the squares on the two other sides. The “Theorem of Pythagoras,” but stated in terms of areas. (Note: not as the algebraic identity c² = a² + b² for the side lengths) The proof Euclid gives has to rank as one of the masterpieces of Greek mathematics (at least), although it is far from the simplest possible proof The thing that is truly remarkable is the way the proof uses just what has been developed in Book I of the Elements.

Euclid's proof, construction Let the right triangle be ABC with right angle at A Construct the squares on the three sides (Proposition 46) and draw a line through A parallel to BD (Proposition 31)

Euclid's proof, step 1 <BAG + <BAC = 2 right angles, so G,A,C are all on one line (Proposition 14), and that line is parallel to FB (Proposition 28) Consider FBG

Euclid's proof, step 2 Proposition 37 implies areas of FBG and FBC are equal

Euclid's proof, step 3 AB = FB and BC = BD since ABFG and BCED are squares <ABD = <FBC since each is a right angle plus <ABC Hence BFC and ABD are congruent (Proposition 4 – SAS)

Euclid's proof, step 4 Proposition 37 again implies BDA and BDM have the same area. Hence square ABFG and rectangle BLMD have the same area (twice the corresponding triangles – Proposition 41)

Euclid's proof, conclusion A similar argument “on the other side” shows ACKH and CLME have the same area Therefore BCED = ABFG + ACKH. QED, or rather, in Greek: ὅ περ ἔ δει δε ῖ ξαι (!)

Book II, Proposition 2 Proposition 2. If a straight line is cut in any way whatsoever, then (the sum of) the rectangles contained by the pieces and the whole line (is) equal to the square on the whole line.

Comments – Proposition 2 “Obviously true” – why did Euclid feel the need to state this explicitly and prove it? And didn't we just use this in the proof of Proposition 47 in Book I?? Best conjectural explanation I have read for this – K. Saito: In the diagram from Book I, Proposition 47 the subdivided square was “visible” in the figure; this proposition says the same fact will be true for the areas even if they are not part of a “visible square” (i.e. even if the rectangles arise differently in a construction, are not side by side, etc.)

Book II, Proposition 5 Proposition 5. If a straight line is cut into equal and unequal pieces, then the rectangle contained by the unequal pieces of the whole straight line, plus the square on the difference between the equal and unequal pieces, is equal to the square on half of the straight line. Diagram from

Euclid's proof (paraphrased) AB is the original line; C is the midpoint so AC = CB; D is another point dividing AB into unequal pieces Construct the figure as shown with AK = DB perpendicular to AB, and BF = CE = DG = CB also perpendicular to AB

Proof of Book II, Proposition 5, cont. The rectangle ADHK has sides equal to the unequal pieces of the original AB But the area of ADHK equals the area of the “gnomon” CBFGHL (olive color region):

Proof, concluded Moreover, CD = LH = EG, so LHGE (cyan region) is a square Hence “gnomon” plus the small square is equal to the square on CB = half of AB

Book II of the Elements Consists of 14 Propositions, again 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(!) Even more so than Book I, this collection of propositions is set up as a collection of “lemmas” for later use

“Geometric algebra?” Many of you are probably thinking right about now – 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 19 th and early 20 th 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 a certain type of quadratic equation occurring in many Old Babylonian problem texts (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 ( ἀ ριθμός) 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. The “shocking” was Unguru's characterization(!)

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. Besides, Archimedes was active about 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, polemical 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 rather 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!