# 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 1484.

Chuquet had a degree in medicine and lived most of his life in Lyon where he practised medicine.

The *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 e.g. for 5x^{2} he used the notation .5.^{2} and for 8x^{-3} he used .8.^{3.m.} He was probably the first mathematician to use negative exponents.

By recognising the rule of exponents that the sum of exponents corresponds to the product of the powers (e.g 2^{3} × 2^{4} = 2^{7}) and producing a list for powers of 2, Chuquet effectively produced the first log tables.

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 **

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