**Estienne de la Roche**'s family lived on the rue Neuve in Lyon and also owned property near the town of Villefranche. Because of where he lived when young, La Roche was sometimes known as 'Villefranche.' This is almost the total information about his life which is known. It comes from the Lyon tax register of 1493, which also describes de la Roche as a qualified master of 'argorisms.' This is a rather unusual spelling of 'algorisms,' but let us note that de la Roche uses the same spelling 'argorisms' in his book. De la Roche's date of birth is unknown, and the date we have given of 1470 is simply a guess based on the fact that he had to be old enough to be described as a qualified master of 'argorisms' by 1493. At the time when de la Roche lived in Lyon it was the commercial capital of France so it is natural that someone living there would become interested in the mathematics of the time relevant to commerce. La Roche taught commercial arithmetic in Lyon for 25 years. Clearly he was well thought of as a teacher of arithmetic since he was often called a 'master of ciphers.'

It is postulated by most historians the he learned mathematics from Nicolas Chuquet for he certainly had in his possession many manuscripts in Chuquet's hand making it appears that he was on good terms with Chuquet. This fact is known since a number of copies of Chuquet's manuscripts still exist which have marginal annotations in de la Roche's handwriting. See the end of this article for a description of the route the manuscripts took to reach us today.

De la Roche published *Larismetique nouellement composée* in 1520 which was considered an excellent arithmetic book with good notation for powers and roots. However in 1880 Aristide Marre published Chuquet's *Triparty*. It was immediately discovered that the first part of de la Roche's *Larismetique* is essentially a copy of Chuquet's *Algebra*. Florian Cajori, in volume 1 of his *A History of Mathematical Notations* published in 1928 writes:-

It was thought at this time that de la Roche was simply a plagiarist. However more recent work has come to a less harsh conclusion about de la Roche. It now appears that de la Roche was trying to teach important mathematics which was not available to the French public. For instance Barbara Moss argues [5]:-De la Roche mentions Chuquet in two passages, but really appropriates a great deal from his distinguished predecessor, without, however, fully entering into his spirit and adequately comprehending the work. It is to be regretted that Chuquet did not have in de la Roche an interpreter acting with sympathy and full understanding.

Let us now look at some examples of notation from de la Roche's... that the charge of plagiarism against Estienne de la Roche is largely an anachronism, and that the difference in emphasis may be explained by the readers he had in mind and by the climate of mathematical opinion, while the similarities are close enough to indicate how far his potential readership would have been receptive to a more faithful presentation of Chuquet's ideas. ... If de la Roche had been more faithful to Chuquet's notation and terminology and had used it consistently, and especially if this had given rise to some controversy in the literature, then Bombelli's work, at least, might have been enhanced. In this sense, and in this sense only, can de la Roche be held responsible for Chuquet's ideas remaining in obscurity.

*Larismetique nouellement composée*. He used the same notation as Chuquet for + and -, namely

*p*and

*m*with a ~ above each. He used several notations for roots, so where we would write √5 he would write

^{2}5 where there is a line through the right hand leg of the R. For the cube root he would write either

^{3}5 or

^{}5. This strange use of a square to denote the cube only appears in de la Roche, not being followed by anyone else. As well as

^{4}5 he would use H 5 for the fourth root of 5, and H

^{}5 for the sixth root.

Here is his description of numbers, powers and the unknown (a thing). We follow Cajori's translation:-

Here is an example of one of his problems with solution which we have put into modern notation:-And a number may be considered as a continuous quantity, in other words, a linear number, which may be designated a thing or as primary, and such numbers are marked by the apposition of unity above them in this manner121^{}or131^{}, etc., or such numbers are indicated also by a character after them, like12.p. or13.p. ... Cubes one may mark12.3^{}or13.3^{}and also1213^{}. or^{}.

De la Roche was writing a commercial arithmetic, and as such it was highly successful. He says that he is writing:-Next find a number such that, multiplied by its root, the product is10. Solution: Let the number be x. Then x multiplied by √x gives √x3^{}=10. Now, as one of the sides is a radical, multiply each side by itself. You obtain x3^{}=100. Solve. There results the cube root of100, i.e.3^{}√100is the required number. Now to prove this, multiply3^{}√100by6^{}√100. But first express3^{}√100as a sixth root, by multiplying100by itself, and you have6^{}√10000. This multiplied by6^{}√100gives6^{}√1000000, which is the square root of the cube root, or the cube root of the square root, or6^{}√1000000. Extracting the square root gives3^{}√1000which is10, or reducing by the extraction of the cube root gives the square root of100, which is10, as before.

He says of his own work that it is:-... with only a little addition to what I have found and experimented with in my time in practice: and of all this I have made a little tract entitled 'The arithmetic of Estienne de la Roche.'

such as:-... the flower of several masters, experts in the art, ...

This statement itself suggests that de la Roche was not trying to hide his dependence on others, for many mathematicians of this period would have made no acknowledgement to others at all. Indeed de la Roche took parts of Chuquet, parts of Pacioli and parts of Philippe Friscobaldi, a French banker who is considerably less well known as a mathematics writer than the other two. The unfortunate part is that de la Roche lacked any real skill in forming them into his teaching book.... Nicolas Chuquet from Paris, Philippe Friscobaldi from Florence, and Luca Pacioli from Burgo.

Aristide Marre's introduction to his 1880 edition of Chuquet's *Triparty* tells us how the manuscript which de la Roche owned (and annotated) came down to us today [2]:-

As de la Roche's birth date is unknown, so is his death date. It appears he was still alive when hisAfter it had been in the possession of de la Roche, it was bought by an Italian, Leonardo de Villa, according to a note written in Latin at the beginning of the work. It then passed into the famous library of Jean Baptiste Colbert(1619-1683), the finance minister of King Louis XIV who was also the founder of the French Academy ... In1732, some eight thousand volumes, formerly in Colbert's possession, passed into the royal library of King Louis XV, among which was Chuquet's manuscript. ... Today it is ... in the Bibliothèque Nationale.

*Larismetique nouellement composée*was published in 1520, but since Gilles Huguetan revised the work for publication in 1438, de la Roche must have died a few years before 1438. The given date of 1430 is simply a guess using this small amount of information.

**Article by:** *J J O'Connor* and *E F Robertson*