428 14. EQUATIONS AND ALGORITHMS6.1. Leonardo of Pisa (Fibonacci). Many of the problems in the Liber abaci re-
flect the routine computations that must be performed when converting currencies.
These are applications of the Rule of Three that we have found in Brahmagupta
and Bhaskara. Many of the other problems are purely fanciful. Leonardo's indebt-
edness to Arabic sources was detailed by Levey (1966, pp. 217-220), who listed 29
problems in the Liber abaci that are identical to problems in the Algebra of Abu
Kamil. In particular, the problem of separating the number 10 into two parts sat-
isfying an extra condition occurs many times. For example, one problem is to find
÷ such that 10/x + 10/(10 - x) = 6±.
The Liber quadratorum. The Liber quadratorum is written in the spirit of Dio-
phantus. The resemblance in some points is so strong that it would be very strange
if Leonardo had not seen a copy of Diophantus. This question is discussed by the
translator of the Liber quadratorum. (Sigler, 1987, pp. xi-xii), who notes that strong
resemblances have been pointed out between the Liber quadratorum and al-Karaji's
Fakhri, parts of which were copied from the Arithmetica, but that there are also
parts of the Liber quadratorum that are original. The resemblance to Diophantus
is shown in such statements as the ninth of its 24 propositions: Given a nonsquare
number that is the sum of two squares, find a second pair of squares having this
number as their sum. Leonardo's solution of this problem, like that of Diophantus,
involves a great deal of arbitrariness, since the problem does not have a unique
solution.
One advance in the Liber quadratorum is the use of general letters in an argu-
ment. Although in some proofs Leonardo argues much as Diophantus does, using
specific numbers, he becomes more abstract in others. For example, Proposition 5
requires finding two numbers the sum of whose squares is a square that is also the
sum of the squares of two given numbers (Problem 9 of Book 2 of Diophantus). He
says to proceed as follows. Let the two given numbers be .a. and .6. and the sum of
their squares .g.. Now take any other two numbers .de. and .ez. [not proportional
to the given numbers] the sum of whose squares is a square. These two numbers
are arranged as the legs of a right triangle. If the square on the hypotenuse of this
triangle is .g., the problem is solved. If the square on the hypotenuse is larger than
.g., mark off the square root of .g. on the hypotenuse. The projections (as we would
call them) of this portion of the hypotenuse on each of the legs are known, since
their ratios to the square root of .g. are known. Moreover, that ratio is rational,
since they are the same as the ratios of .o. and .b. to the hypotenuse of the original
triangle. These two projections therefore provide the new pair of numbers. Being
proportional to .a. and .6., which are not proportional to the two numbers given
originally, they must be different from those numbers. This argument is more con-
vincing, because more abstract, than proofs by example, but the geometric picture
plays an important role in making the proof comprehensible.
The Flos. Leonardo's approach to algebra begins to look modern in other ways
as well. In one of his works, called the Flos super solutionibus quarumdam ques-
tionum ad numerum et ad geometriam vel ad utrumque pertinentum (The Full
Development^16 of the Solutions of Certain Questions Pertaining to Number or
Geometry or Both, see Boncompagni, 1854, P- 4), he mentions a challenge from
John of Palermo to find a number satisfying ÷^3 + 2x^2 + lOx = 20 using the methods
(^16) The word flos means bloom, and is used in the figurative sense of "the bloom of youth." That
appears to be its meaning here.