- EUROPE^429
given by Euclid in Book 10 of the Elements, that is, to construct a line of this length
using ruler and compass. In working on this question, Leonardo made two impor-
tant contributions to algebra, one numerical and one theoretical. The numerical
contribution was to give the unique positive root in sexagesimal notation correct to
six places. The theoretical contribution was to show by using divisibility properties
of numbers that there cannot be a rational solution or a solution obtained using
only rational numbers and square roots of rational numbers.
6.2. Jordanus Nemorarius. The translator and editor of the book De numeris
datis (On Given Numbers), written by Jordanus Nemorarius, says (Hughes, 1981, p.
11) "It is reasonable to assume... that Jordanus was influenced by al-Khwarizmi's
work." This conclusion was reached on the basis of Jordanus' classification of
quadratic equations and his order of expounding the three types, among other
resemblances between the two works.
De numeris datis is the algebraic equivalent of Euclid's Data. Where Euclid
says that a line is given (determined) if its ratio to a given line is given, Jordanus
Nemorarius says that a number is given if its ratio to a given number is given.
The well-known elementary fact that two numbers can be found if their sum and
difference are known is generalized to the theorem that any set of numbers can be
found if the differences of the successive numbers and the sum of all the numbers is
known.^17 In general, this book contains a large variety of data sets that determine
numbers. For example, if the sum of the squares of two numbers is known, and the
square of the difference of the numbers is known, the numbers can be found. The
four books of De numeris datis contain about 100 such results. These results admit
a purely algebraic interpretation. For example, in Book 4 Jordanus Nemorarius
writes:
If a square with the addition of its root multiplied by a given
number makes a given number, then the square itself will be given.
[Hughes, 1981, p. 100]^18Where earlier mathematicians would have proved this proposition with examples,
Jordanus Nemorarius uses letters representing abstract numbers. The assertion is
that there is only one (positive) number ÷ such that x^2 + ax = â, and that ÷ can
be found if a and â are given.
6.3. The fourteenth and fifteenth centuries. The century in which Nicole
d'Oresme made such remarkable advances in geometry, coming close to the cre-
ation of analytic geometry, was also a time of rapid advance in algebra, epitomized
by Antonio de' Mazzinghi (ca. 1353 1383). His Trattato d'algebra contains some
complicated systems of linear and quadratic equations in as many as three un-
knowns (Franci, 1988). He was one of the earliest algebraists to move the subject
toward the numerical and away from the geometric interpretation of problems.
In the following century Luca Pacioli wrote Summa de arithmetica, geometrica,
proportioni et proportionalita (Treatise on Arithmetic, Geometry, Proportion, and
Proportionality), which was closer to the elementary work of al-Khwarizmi and
more geometrical in its approach to algebra than the work of Mazzinghi. Actually
(^17) This statement is a variant of the epanthema (blossom) of Thymaridas.
(^18) This translation is my own and is intended to be literal; Hughes gives a smoother, more
idiomatic translation on p. 168.