430 14. EQUATIONS AND ALGORITHMS(see Parshall, 1988), the work was largely a compilation of the works of Leonardo
of Pisa, but it did bring the art of abbreviation closer to true symbolic notation.
For example, what we now write as ÷ — \/x^2 — 36 was written by Pacioli as
l.co.mRv.l.ce m36.Here co means cosa (thing), the unknown; ce means censo (power), and Rv is
probably a printed version of Rx, from the Latin radix, meaning root.^19 Pacioli's
work was both an indication of how widespread knowledge of algebra had become
by this time and an important element in propagating it. The sixteenth-century
Italian algebraists who moved to the forefront of the subject and advanced it far
beyond where it had been up to that time had all read Pacioli's treatise thoroughly.
6.4. Chuquet. The Triparty en la science des nombres by Nicolas Chuquet is
accompanied by a book of problems to illustrate its principles, a book on geometrical
mensuration, and a book of commercial arithmetic. The last two are applications
of the principles in the first book. Thus the subject matter is similar to that of
al-Khwarizmi's Algebra or Leonardo's Liber abaci.
There are several new things in the Triparty. One is a superscript notation
similar to the modern notation for the powers of the unknown in an equation. The
unknown itself is called the premier or "first." Algebra in general is called the rigle
des premiers or "rule of firsts." Chuquet listed the first 20 powers of 2 and pointed
out that when two such numbers are multiplied, their indices are added. Thus,
he had a clear idea of the laws of integer exponents. A second innovation in the
Triparty is the free use of negative numbers as coefficients, solutions, and exponents.
Still another innovation is the use of some symbolic abbreviations. For example,
the square root is denoted R^2 (R for the Latin radix, or perhaps the French racine).
The equation we would write as 3x^2 +12 = 9x was written .3.^2 p.12. egaulx a .9.^1.
Chuquet called this equation impossible, since its solution would involve taking the
square root of — 63.
His instructions are given in words. For example (Struik, 1986, p. 62), consider
the equation
R^2 A^2 pA^1 p.2lp.l egaulx a .100,which we would write
\/ix^2 +Ax + 2x + 1 = 100.Chuquet says to subtract .2^p.l from both sides, so that the equation becomes
fl^242 p.4^1 egaulx a .èèôç^^1.Next he says to square, getting
42 p.4^1 egaulx a 9801.m.396|p.4^2.Subtracting 4^2 from both sides and adding 396.1 to both sides then yields
4001 egaulx a .9801..Thus ÷ = 9801/400.
(^19) The symbol Ha; should not be confused with the same symbol in pharmacy, which comes from
the Latin recipe, meaning take.