Nicolas Chuquet

If Regiomontanus was the most influential mathematician of the fifteenth century, then Nicolas
Chuquet must be considered as the greatest mathematician of that century. Chuquet’s
brilliance was not recognised during his lifetime, and his greatest work
Triparty en la science
des nombres
was not published until 1880.

Chuquet was born in Paris, France in around 1445 and died in around 1500, but the exact
dates are unknown. All we know with some degree of certainty is that the
Triparty was written in
Chuquet had a degree in medicine and lived most of his life in Lyon where he practised

Triparty as a book was unique at the time and Chuquet seems to have been influenced by
very few of his predecessors. The only ones he mentions are Boethius and Campanus, though
it is likely that he was influenced to some extent by Fibonacci’s
Liber Abaci.

The work is divided into three parts. The first concerns computation with rational numbers and
includes an explanation of the Hindu-Arabic number system. The second concerns computation
with irrational numbers, specifically roots of numbers. Chuquet used a syncopated algebra,
where for example he used R)2 for the square root symbol.

The third part of the
Triparty is the most important and deals with equations. He called this the
‘Regle des premiers’, which is loosely translated as ‘Rule of the unknown’. Chuquet invented an
exponential notation.

He went on in the third part of the
Triparty to solve equations, including equations whose
solutions were negative, and on at least one occasion, zero.

Chuquet is also accredited with the first use of the names for large numbers such as billion,
trillion and quadrillion.

Chuquet‘s work was ahead of its time. Because of this and the fact that the
Triparty was not
published during his lifetime, his work had little influence on other mathematicians of the time.
One exception was Etienne de la Roche who published a book in 1520 using much of the same
material as that found in the
Triparty. Even though de la Roche mentioned Chuquet’s name, he
did not give him full credit and some historians of mathematics have suggested that he should
have been charged with plagiarism.

A contribution of Chuquet to mathematics

One contribution of Chuquet that is quite easy to understand is known as the Regle des
nombres moyens
(Rule of mean numbers), which is stated in modern notation as follows:

Regle des nombres moyens:
If a, b, c and d are positive numbers, then (a + b)/(c + d) lies between a/c and b/d

This is quite easy to prove. Again I am using modern notation, but Chuquet’s argument must
have gone something like this:

The actual values of a, b, c and d determine which of a/c and b/d is the greater. So, without any
loss of generality, let us assume b/d is the greater. Thus we have:

a/c < b/d
ad < bc (since all the numbers are positive)
ac + ad < ac + bc
a(c + d) < c(a + b)
a/c < (a + b)/(c + d)  (1)

Starting again, but proceeding in a slightly different way:

a/c < b/d
ad < bc (since all the numbers are positive)
ad + bd <  bc + bd
d(a + b) < b(c + d)
(a + b)/(c + d) < b/d  (2)

Now take equations (1) and (2) together:
a/c < (a + b)/(c + d) < b/d