Download Book II of the Elements

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 crosspurposes 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 –
http://www.ibiblio.org/expo/vatican.exhibit/ex
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superseded all those predecessors and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 2000+ 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 ᐃABC 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 ᐃFBG
Euclid's proof, step 2

Proposition 37 implies areas of ᐃFBG and ᐃFBC 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 ᐃBFC and ᐃABD are congruent (Proposition 4 – SAS)
Euclid's proof, step 4


Proposition 37 again implies ᐃBDA and ᐃBDM 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:
ὅπερ ἔδει δεῖξαι (!)
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 http://aleph0.clarku.edu/~djoyce/java/elements/elements.html:
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 19th and early 20th centuries thought so
H. Zeuthen, P. Tannery, O. Neugebauer, B.
L. van der Waerden, …
T. L. Heath (translator of most commonlyused 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 (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.
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 5070 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!