Nicolas Chuquet (~1445-1488)  Historical Sketch

Nicolas Chuquet was born sometime in 1445 in Paris, and he died in Lyon, France, at about age 43. The exact dates of his birth and death don't appear to be known, and in fact not very much at all seems to be known about him. His bachelor degree was in medicine, though mathematics occupied much of his interest. He moved to Lyon at about age 35, where he described himself as an algoriste, a word that is today not in use, but surely it meant a person engaged in algorithmic work, i.e., mathematics at some level.<\p>

Chuquet gets credit for writing a truly significant manuscript in 1484, with the title Triparty en la science des nombres, which is a confusing title since party and triparty are not (today, at least) words in French. However, parte is French for the English part. So the title is taken to mean something like A three-part book on the science of numbers. In any case, it was a book that got around to the subject of algebra. In part one, Chuquet covers the Hindu-Arabic numeration system, zero, positive and negative numbers, fractions, order, averages, perfect numbers, progressions, etc. He even got around to the fact that

If

where a,b,c,d are positive, then

which is today associated with what are called Farey fractions. In part two is found a presentation of roots of numbers, the first use of the radical sign, even getting to compound roots such as

 

and the like, as well as indexed radicals such as

 

for cube roots. Part three is the algebra part, introducing what we today call unknowns, exponential notation, including for the first time negative exponents and the exponent 0, including the statement that x⁰ = 1, laws of exponents, quadratic equations, and roots of quadratic equations. His discussion of the powers of two amounted to what today are called logarithms base two. And he even got to what we now call imaginary numbers as solutions of equations. This 1484 manuscript did not appear in print until 1880.

There was some bit of ado concerning one of Chuquet's students, Estienne de La Roche, who taught arithmetic for some 25 years. and who published some of Chuquet's Triparty as being his own work. It was later discovered that La Roche borrowed from other authors as well, and acknowledged them namelessly as "masters, experts in the art."

Demonstrate with some examples, with a,b,c,d positive integers, that if

where a,b,c,d are positive, then

Can you prove this holds in general?