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Ancient Greek Algebra
What the Greeks did in the area of algebra is subtle , so
subtle that one may well debate whether they
did anything at all in this area. Apart from one tantalizing reference to a
problem that is algebraic in
nature, it is very difficult to find anything that deserves the name before the
time of Diophantus in the
third century CE. The one exception is a reference to a problem that was
apparently stated during the time
of Plato. All that is known about it is the following. In his commentary on
Nicomachus’ Introduction
to Arithmetic Iamblichus presents an obscurely worded technique that he ascribes
to one Thymaridas,
which the nineteenthcentury philologist Georg Heinrich Ferdinand Nesselmann
(1811–1881) was able to
elucidate. This technique solves the problem of determining a number of
quantities , given the sum of all of
them and the sum of a fixed one with each of the others. Iamblichus called this
technique the epanthema
(blossom, that is, full development) of Thymaridas.
The general focus of Euclid’s thirteen books is geometric,
with an interruption in Books 7–9 to discuss
number theory. To find algebra in this treatise one has to interpret the results
suitably. The main place to
look is in Book 2, which is a short book, containing only 14 propositions. The
first ten of these do look like
algebraic propositions clothed in geometric language. For example, Proposition 1
asserts what we would
call the distributive law for multiplication : If one of two lines is divided,
the rectangle on the two lines
is (the sum of ) the rectangle on each of the parts and the undivided line. We
would interpret this
proposition as saying but that is our
language, not Euclid’s. The “algebra” (if that is what
it is) gets more complicated. Proposition 5, for example, asserts that If a line
is divided into equal and
unequal parts, then the square on the equal parts equals (the sum of ) the
rectangle on the two parts
and the square on the segment between the two points of division. If the two
unequal parts are a and b,
we would rewrite this statement as the basic identity for solving quadratic
equations
which, as we have seen, was a fundamental principle used in the cuneiform
tablets. Proposition 9 asserts
what we would call the parallelogram law: If a line is divided into equal and
unequal parts, the (sum
of ) the squares on the unequal parts equals twice the (sum of ) the square on
the half and the square
on the segment between the two points of division. That is,
Up to
this point, it appears that the Greeks are merely using geometric language to do
what was done numerically
by the Mesopotamian mathematicians; and it is natural to suppose that they did
so because square roots fit
smoothly into geometry, but not so smoothly into arithmetic. The view that this
book is basically algebraic in
character is associated with the Danish historian of mathematics Hieronymus
Georg Zeuthen (1839–1920).
It must be said that these first ten propositions of Book 2 do lend themselves
to that interpretation.
The last four propositions are applications of these ten
propositions, however, and they do not suggest
algebra at all. Proposition 11 is a construction problem: to divide a line so
that the rectangle on the
whole and one of the parts equals the square on the other part, that is, to
divide the line in mean
and extreme ratio . This problem does not have any obvious algebraic source that
needed to be recast as
geometry. It does, however, use the previous propositions . Propositions 12 and
13 give what we now call
the law of cosines for obtuse and acuteangled triangles respectively, in the
form that the square on the
side opposite the obtuse angle exceeds the sum of the squares on the other two
sides by the rectangle on
one of the sides and the length of the projection of the other side on its
extension outside the obtuse angle.
For an acute angle the excess is a deficiency and the projection falls inside
the angle. The fourteenth and
final proposition of the book, which shows how to construct a square equal to
any polygon, also has a very
obvious geometric source and also makes use of the previous propositions.
We conclude that if Euclid was intending to do algebra
geometrically, he kept his intention well hidden.
His applications of the “geometric algebra” in Book 2 are topics of intrinsic
interest in geometry, not algebra
problems disguised as geometry, as is the case in many other cultures. On the
other hand, the “application”
problems in Book 6 are less obviously geometric in origin and do map nicely into
the problems of finding
two numbers (represented by lines in Euclid’s case) having a prescribed product
(rectangle) and sum or
difference, according as the problem is one of application with defect or
application with excess. Yet,
once again, Euclid states all these theorems in terms of parallelograms having a
prescribed angle, whereas
rectangles with square defect or excess would suffice to represent the algebraic
problems. The case that
Euclid’s geometry contains algebra in disguised form is difficult to make.
A somewhat better case can be made in the numbertheoretic
books, where at least numbers are reasoned
on as abstractions. The algebraic spirit is there, but the focus is on studying
the intrinsic properties
of numbers, not on finding numbers satisfying certain conditions in relation to
other numbers. For example,
in Proposition 12 of Book 7, it is proved that if two numbers are such that when
the smaller is continuously
subtracted from the larger the remainder is never equal to its predecessor until
the number 1 is reached, then
the numbers are relatively prime, that is, not both divisible by any integer
larger than 1. The proof consists of
showing that if a number divides both numbers, then it divides their difference;
and conversely if it divides
the difference and one of the two numbers , then it divides both numbers. The
assumption of a larger divisor
than 1 then leads to the impossibility that the number 1 is divisible by a
larger number. In the proof, the
numbers are called AB and as if they were
intended to be represented by line segments. The reasoning
does not depend on any such mental picture, yet the author seemed to have
thought it was needed. Was it
really such a radical, counterintuitive step to use an abstract symbol to
represent an unspecified number?
The first record of such a step being taken in Greek mathematics came at least
500 years after Euclid, in the
work of Diophantus discussed in the Chapter 14.
The PostEuclideans. Algebra is now universally used to
discuss conic sections in analytic geometry
and calculus courses, and some historians have argued that the work of
Apollonius himself is basically
algebraic. Certainly that is the impression one would get from reading the
translations of it given by Heath,
who took it on himself to save the modern reader time by translating everything
into modern algebraic
symbolism. One must be wary of seeming familiarity in historical documents,
however. As Jacob Klein
(1968, p. 5) says, “most of the standard histories attempt to grasp Greek
mathematics itself with the aid of
modern symbolism, as if the latter were an altogether external ‘form’ which may
be tailored to any desirable
‘content.’ ” Precisely! To translate in this case is to betray. It is true that
the locus description of a conic
section as given by Apollonius, which involves reference to two fixed, mutually
perpendicular lines, is
very easily translated into what we now call the equation of the curve . That
fact by no means implies that
Apollonius thought of them in that way. Indeed, all his complicated
Euclideanstyle proofs loudly proclaim
the exact opposite.
If we look at the work of other later geometers like
Heron , who does write prescriptions that amount
to formulas , it is still difficult to discern any attempt to determine (the
length of) an unknown line from
its position in a figure. Heron seems to be concerned with measurement but not
the solution of problems
leading to equations.
The ancient Greek writer most often credited with doing
algebra—indeed, the Renaissance mathematicians
credited him with creating it—was Diophantus. That verdict, however, is disputed
by Jacob Klein
(1899–1978) in very thorough and interesting study of Greek mathematics (Klein,
Jacob, 1968 ). This work
is discussed in Chapter 14 of the text in connection with the work of Diophantus.
Here we merely note
that in our discussion (Sect. 11.1 of the text) of what Pappus called analysis
we mentioned that the process
he described—imagining a construction completed and seeing how its properties
determine what it must
be—was precisely algebra when the object of contemplation was numerical. For
Pappus, of course, it was
not numerical, and hence his analysis was not algebra. That point lies at the
heart of Klein’s argument, as
we shall see in the next chapter.
Literature
Klein, Jacob, 1934–1936. Greek Mathematical Thought and the Origin of Algebra,
translated by Eva
Brann, MIT Press, Cambridge, MA, 1968.
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