# Nicolas Chuquet

### Quick Info

Born
1445
Paris, France
Died
1488
Lyon, France

Summary
Nicolas Chuquet was a French mathematician who wrote the earliest French algebra book.

### Biography

Very little is known about Nicolas Chuquet's life and for a long time nothing about it was known except that he was the author of the very remarkable book Le Triparty en la science des nombres . Let us say immediately that the date of birth we give for him is a pure guess, and his place of birth is unknown but probably was Paris. The dates and place of his death are somewhat more certain. Let us now explain what is known, first looking at the information in his book.

The Triparty ends with the following:-
And thus to the honour of the glorious Trinity ends this book, which on account of these three principal parts I call Triparty. And also, because it was made by Nicolas Chuquet, Parisian, Bachelor of Medicine, I call it the Triparty of Nicolas in the science of numbers, which was begun, continued and finished at Lyon on the Rhone, the year of salvation 1484.
He describes himself as a Parisian and says that he is a Bachelor of Medicine but this does not prove he was born in Paris, only that he had lived most of his life there. We also learn that he must have come to Lyon several years before 1484 to have had the time to write this work. His claim to be a Bachelor of Medicine is interesting since this degree would not have qualified him to practise as a doctor, being only a first step towards that stage. Although historians have searched for his name in the records of French medical schools, no evidence has been found. It would at least be reasonable to guess that he set out to qualify as a doctor but changed direction.

Until the work of Hervé L'Huillier published in 1976, reference [16], we knew nothing more about Chuquet's life. We now know that he moved to Lyon in around 1480 since his name appears in the Lyon tax registers of the time. For example in the registers of 1485 and 1487 he is described as "Nicolas Chuquet, algoriste". L'Huillier writes [16]:-
We read in the register of taxes collected for the city in 1480 that on 4 June a certain "Master Nicolas, escripvain" paid his due, 3 sous 4 deniers; he lived in a portion of the city located on the Empire side and which is described: "From the 'Porte des Freres Mineurs' along by the 'rue de la Grenette' towards 'le Muton'." This is a very easy place to locate on old maps, and it corresponds to the current rue de la République. Master Nicolas appears in all the registers which remain for this part of the city until 1485; he is taxed about two livres for the emperor and one for the city. In the register of taxes for the city in 1485 appears in the same street and with the same neighbours "Nicolas Chuequet, algorist", taxed at 10 sous. The following register, relating to the year 1487, contains the indication "Nicolas Chuquet, algorist, 15 sous". The following year (1488), the tax register for the emperor indicates "the heirs Nicolas Chuquet, algorist, 23 sous".
The description of Chuquet as an 'escripvain' means that he earned his living as a copyist or writing master. The authors of [2] explain how Chuquet's occupations as given by these tax records fit in to the Lyon scene of that time:-
In the later fifteenth century Lyon was a cosmopolitan city with a highly active commercial community and a flourishing spice market. The fairs, of which the spice market was a significant part, had been established in 1464 under the royal protection of Louis XI. These fairs stimulated the growth of Lyon as an important centre of banking. and the city was also beginning to develop a printing industry. There was a need for the copying of legal and commercial documents, especially on the part of those who were less familiar with the French language, now increasingly being used in place of Latin. As the various commercial activities increased, there was a further need for masters skilled in the art of computation using the Hindu-Arabic numerals and capable of communicating that art to others. It would appear therefore that Chuquet initially set up in business in Lyon around 1480 to supply the need for copying, translating and drawing up commercial and legal documents. As his skill as a master of algorithm became known, he was increasingly in demand for his mathematical abilities, so that by 1485 he could be described solely as an algorist.
Evidence of Italian links in his writings could indicate that he had visited Italy, but it is equally possible that these links come from the large Italian community living in Lyon at the time Chuquet lived there. These Italian links are significant since at this time Italy was the place where most mathematical progress was taking place.

Although Chuquet's text Le Triparty en la science des nombres is a remarkable work, sadly it had relatively little influence on the development of algebra and the number systems since it was lost until Aristide Marre rediscovered it and published it in 1880. Chuquet's book is the earliest French algebra book although for some time La Roche was thought to have written the first French algebra. Chasles pointed out, in 1841, that La Roche's work held this distinction, but also pointed to a lost work of Chuquet as being earlier. When Chuquet's manuscript was found it was seen at once that La Roche had copied large parts of Chuquet's Le Triparty en la science des nombres . The manuscript of Le Triparty en la science des nombres even contained annotations in La Roche's handwriting. We are left wondering, therefore, what connection La Roche had with Chuquet. Although we have no definitive answer to this, nevertheless Hervé L'Huillier supplies some interesting connections in [16]:-
Another important piece of information is in the tax records. We know, because Aristide Marre had already noted it, that the manuscript of Chuquet's works ... belonged to another French mathematician, the Lyonnais Estienne de La Roche. According to the registers, his father, then himself from 1483, lived in rue Neuve, a place easy to find on old maps, and which is located two hundred steps from rue de la Grenette; thus Chuquet and Estienne de La Roche lived for eight years in very close proximity. It is impossible that they did not know each other. What was the nature of their relationship? It's hard to say. According to tax records, Estienne de La Roche's father, Pierre Villefranche, died in 1483; his son, who was not named for several years, was quite young at that time. Under these conditions, one can suppose that Chuquet taught him arithmetic and the other mathematical sciences; perhaps even, if one gives to "escripvain" the meaning of tutor, it is he who taught him to read and write. These are hypotheses that in the current state of my research I do not want to be more specific about.
Let us now look at the contents of Le Triparty en la science des nombres but, before doing so we had better put the work into context. Chuquet finished it in 1484, ten years before Luca Pacioli published Summa de arithmetica, geometria, proportioni et proportionalita . In 1484 mathematics books were just starting to be printed, the first being Euclid's Elements two years earlier. Chuquet's book would remain in manuscript for 400 years. Of French mathematicians that Chuquet would be familiar with, the most significant was Nicole Oresme who predates Chuquet by approximately 100 years. L'Huilleur has shown that Chuquet had access to at least one manuscript of Oresme. Work of the Arabic mathematicians had been transmitted to Europe beginning with Fibonacci in 1202, with which Chuquet is familiar, but it does not seem possible that Chuquet could possibly have known about more recent Arabic work such as al-Kashi's The Key to Arithmetic (1427).

Le Triparty en la science des nombres covers arithmetic and algebra. It was not printed however until 1880 so, as we said above, was of little influence. The first part deals with arithmetic and is in four chapters. The first two chapters are mainly concerned with work on whole numbers and fractions. He begins by introducing Hindu-Arabic numbers in a similar way to Fibonacci in Liber abaci . Chuquet writes:-
To number is to represent symbolically by common figures or to express with spoken words the number understood from understanding. In order to know how to number and to make use of this science, it is necessary to know the ten figures in this art by means of which one can write and express in figures any number - these figures are: 0, 9, 8, 7, 6, 5, 4, 3, 2, 1. Of these, the first from the right is worth or represents one. The second, moving to the left, is worth two, the third three, next four and so on continuing to the tenth which by itself has no value or represents nothing. Occupying a place, however, it changes the value of those that follow it, and for this reason it is called cipher or null or figure of no value.
...

Note that in this art the figures which are in the right most place are said to be and understandably are called first. The next figure moving to the left is called second, and the next following it called third, and next fourth and so on continuing without end. A further thing to know is that the first in order is worth once its value and the second is worth ten times its value. If a figure is third, it represents one hundred times its value, if fourth one thousand times
...
Notice here that Fibonacci has only nine figures while Chuquet has included zero and so has ten. Chuquet also seems to be treating zero as both a "no value" number and as an empty place holder. We say more about this below.

The third chapter of the first part deals with progressions, proportion etc. The fourth chapter sets out the rules for solving arithmetical problems. He gives standard methods such as 'the rule of three' and 'the rules of single false position' and double false position'. Finally rules for solving intermediate problems and, he correctly claims, an original rule of intermediate numbers, is given.

The second part of the Triparty deals with the extraction of roots. He works with what he calls compound roots such as, to quote one of his examples in modern notation, $\sqrt{14 + \sqrt{180}}$. He takes care not to use the geometrical terms "square" and "cube" but rather "second" and "third". This is quite deliberate since the geometric terms are restrictive; you cannot add a square to a cube, negative powers do not make geometric sense, etc. Chuquet wants to be free so that he can deal more broadly with numbers.

The third part of the Triparty deals with algebra. He calls it the rule of "premiers", where in our terminology today we would translate it as the rule of $x$, or the rule of the unknown. Chuquet does not claim originality here, but says that earlier writers have worked with the "thing" or "cosa". He writes:-
The Ancients called a thing what I call premiers ... . The seconds they named them squares ... . The thirds are called cubics ... . And the fourths they call them squares of squares ... . And there are hardly any more that are higher. Such denominations are not allowed so as to provide all the differences in numbers as they are innumerable.
Note that Chuquet uses 'denomination' where we would say 'exponent'. His notation is as follows. He writes $1^{1}$ for what we would write as $x$ or $x^{1}, 1^{3}$ for $x^{3}$, and $2^{5}$ for $2x^{5}$. We see expressions such as $m. 12^{2}$, which we would write as $-12x^{2}$, and remarkably, $m. 12^{2m.}$ which is $-12x^{-2}$. Negative indices arises when he divides $72^{1}$ by $8^{3}$ obtaining $9^{2m.}$, in our notation, $72x$ divided by $8x^{3}$ is $9x^{-2}$. There are also numbers such as
$Rc 12^{2} + 3^{1}$ which is $\sqrt{12x^{2} + 3x}$.
So, in this work, negative numbers, used as coefficients, exponents and solutions, appear for the first time. For an example of negative solutions, he gives a problem about the purchase of pieces of cloth of two qualities at 11 and 13 crowns each, which algebraically leads to the system
$x + y = 15, 11x + 13y = 160$.
You can easily solve these and get, as Chuquet does, $x = 17\large\frac{1}{2}\normalsize$ pieces and $y = -2\large\frac{1}{2}\normalsize$ pieces. He then says that according to common usage, the problem is impossible since it leads to an "impossible" amount of one of the pieces of cloth. He says, however, that one can accept the solution and goes on to justify it by making the negative quantity correspond to a debt. See [13] for Chuquet's justification of his solution to this problem in terms of debts.

In fact Chuquet's idea of number is remarkably broad, and includes 1, irrational numbers and, surprisingly, his premiers, so $12x$ is also a number for him as are his seconds, say $12x^{2}$, thirds such as $12x^{3}$, and so on. He writes:-
Number as it is expedient to us is taken here widely, not so much as only as a collection of many units, but also either 1, or part and parts of 1, as is any fraction. Any number is understood and considered in many ways. One and the first way is that we can consider a number as a discrete quantity or as a number simply taken without any denomination, or whose denomination is 0, and so from now these numbers will have 0 above them for their denomination in this way $12^{0}$ and $13^{0}$ etc. Secondly, each number is considered as the first number of a continuous quantity, which is also called a linear number. Such numbers will be noted by affixing a unit above them such as $12^{1}, 13^{1}, 20^{1}$, etc. Thirdly, every number is considered as a second number, or simply as a square number, and such numbers are marked by 2 in this way: $12^{2}, 13^{2}, 20^{2}$, etc.
Thus, he says that, in our notation, $12x^{0} = 12, 13x^{0} = 13$, so he has $a^{0} = 1$ for any number $a$. This is the first time such an expression appears.

The sections on equations include quadratic equations which he solves by completing the square.

Other things of interest is Chuquet's book is a graphical proof that 220 and 284 are amicable numbers, i.e. each is the sum of the factors of the other. See [9] for more information. He also discusses error checking in addition, subtraction, multiplication and division by "casting out", what we today would call modular arithmetic. There is nothing new here but Chuquet's discussion is interesting since he seems to consider the more common types of error made in these calculations. He writes:-
There are several kinds of proofs such as the proof by 9, by 8, by 7, and so on by other individual figures down to 2, by which one can prove and examine addition, subtraction, multiplication and division. Of these only the proof by 9, because it is easy to do, and the proof by 7, because it is even more certain than that by 9, are treated here. ... There is a difference between proof by 9 and proof by 7, for the proof by 9 can easily be carried out by adding up the figures, but not in the proof by 7. The proof by 7 is wrong less often than that by 9 because 7 has less in common with numbers than 9.
For a discussion of Chuquet's "casting out" see [8].

Finally let us look at how La Roche produced the first French printed work on algebra, Larismethique  (1520), by using the manuscript of Chuquet's Le Triparty en la science des nombres . Aristide Marre, in publishing the first printed version of Chuquet's work in 1880 wrote of La Roche:-
Without being accused of injustice or exaggeration one could say that he appropriated the work of Nicolas Chuquet, that he purely and simply copied the 'Triparty' in a host of places, that he suppressed some of the most important passages, especially in the algebra, that he shortened or lengthened others, in order to compose his 'Larismethique', vastly inferior to the 'Triparty', and finally that, if for four centuries Nicolas Chuquet and his work have remained in the shadows, it is above all to him [de la Roche] that we must attribute the prime cause.
This harsh view of La Roche is, however, challenged by Albrecht Heeffer in [14]:-
The importance of 'Larismethique' of de La Roche, published in 1520, has been seriously underestimated. One reason for the neglect is related to the inscrutable way he is referred to. Buteo and Wallis called him Stephanus à Rupe de Lyon. Other obscure references, such as Gosselin calling him Villafrancus Gallus have been overlooked by many commentators. His influence can be determined in several works that do not credit him but use problems or definitions from the 'Larismethique'. However, most damaging for its historical assessment was Aristide Marre's misrepresentation of the 'Larismethique' as a grave case of plagiarism. Marre discovered that the printed work of 1520 by Estienne de la Roche contained large fragments that were literally copied from Chuquet's manuscript of the 'Triparty'. Especially on the 'Appendice', which contains the solution to a large number of problems, Marre writes repeatedly that it is a literal copy of Chuquet. However, he fails to mention that the structure of the text of de la Roche, his solution methods and symbolism differs significantly from Chuquet. De la Roche introduces several improvements, especially with regards to the use of the second unknown. We provide an in-depth comparison of some problems solved by the so-called 'regle de la quantite' by Chuquet with those of de la Roche. We further report on the surprising finding that Christoff Rudolff's solution to linear problems by means of the second unknown in his 'Behend vnnd Hubsch Rechnung' of 1525 depends on Chuquet and de la Roche. As it is generally considered that algebra was introduced in Germany through Italy this provides a new light on the transmission of algebraic knowledge from France to the rest of Europe.

### References (show)

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