- THE ARl ÔÇ Ì Ε TIG A OF DIOPHANTUS 411
to indicate addition, each term followed by the corresponding number symbol (for
which the Greeks used their alphabet). Terms to be added were placed first, sepa-
rated by a pitchfork (rtl) from those to be subtracted. Heath conjectured that this
pitchfork symbol is a condensation of the letters lambda and iota, the first two let-
ters of a Greek root meaning less or leave. Thus what we would call the expression
2x^4 - x^3 - 3x^2 + 4x + 2 would be written ÄíÁâòä Ì â rh ÊíÜÄíç.
Diophantus' use of symbolism is rather sparing by modern standards; he of-
ten uses words where we would use symbolic manipulation. For this reason his
algebra was described by the nineteenth-century German historian of mathematics
Nesselmann as a transitional "syncopated" phase between the earliest "rhetorical"
algebra, in which everything is written out in words, and the modern "symbolic"
algebra.^3
1.3. Determinate problems. The determinate problems in the Arithmetica re-
quire that one or more unknown numbers be found from conditions that we would
nowadays write as systems of linear or quadratic equations. The 39 problems of
Book 1 and the first ten problems of Book 2 are of these types. Some of these
problems have a unique solution. For example, Problem 7 of Book 1 is: From a
given unknown number subtract two given numbers so that the remainders have a
given ratio. In our terms, this condition says÷ — a = m(x — b),where ÷ is unknown, a and b are the given numbers, and m is the given ratio. Since
it is obvious that m > 1 if all quantities are positive and a <b, Diophantus has no
need to state this restriction.
Similarly, Problem 15 of Book 1 asks for two numbers (x and y, we would say)
such that for given numbers á and 6 the ratios ÷ + á : y — a and y + b : ÷ — b are
equal to two given ratios r and s.
The symbolic notation of Diophantus extended only as far as the unknown
and representations of sums, products, and differences. He had no way of forming
mathematical expressions containing the phrases "a given number" (a and 6 above)
and "a given ratio" (r and s above). As a result, he could explain his methods
of solution only by using a particular example, in the present case taking á = 30,
r = 2, b = 50, s = 3. He then assumed that y = ò + 30 and ÷ = 2ò — 30, so that the
first equation was satisfied automatically and the second became ò+80 = 3(2ò —80).
Here it is very easy to recognize the explicit manipulation of formal expressions,
leading to the discovery of the unknown number. This manipulation of expressions
is characteristic of algebraic technique.
Some of the problems that are determinate from our point of view may have no
positive rational solutions for certain data, and in such cases Diophantus requires
a restriction on the data so that positive rational solutions will exist. For example,
Problem 8 of Book 1 is to add the same (unknown) number to two given numbers
so that the sums have a given ratio. This problem amounts to the equation÷ + á = m(x + 6).(^3) Nesselmann is quoted by Jacob Klein (1934-36, p. 146). In the author's opinion, there is not
much for Diophantus to be transitional between, since little is known of his algebraic predecessors,
and later algebraists wrote everything out in words. Jacob Klein seems to share these reservations.