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Nicolas Chuquet's notations for the powers of the unknown (1484) : more than a powerful tool ?

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Abstract

Nicolas Chuquet is best known as an algebrist. He was a “master of algorism” in Lyons at the end of the fifteenth century and his work has been influenced by the relations he kept up with the commercial environment. In the algebraical part of the Triparty en la science des nombres, Chuquet proposes new notations and a new vocabulary for what he calls “rigle des premiers”. What were his sources? Has he been a creator? What was the mathematical status he assigned to his “rule”? Chuquet’s book has not been learned but by a few people who did not spread his notations (among them, E de la Roche chose to come back to italian tradition). It is clear that his new notations, because of the immediate possibility of generalization they allowed, were a powerful tool for the algebraical reckoning. But beyond that, we will focus on their mathematical involvements as well as their limits, in the context of the emergence of symbolic algebra.
1
Maryvonne Spiesser
Toulouse Mathematics Institute, France
maryvonne.spiesser@math.univ-toulouse.fr
Nicolas Chuquet’s notations for the powers of the unknown (1484) : more
than a powerful tool ?
1. Introduction
We don’t know much about Chuquet’s life, except that he was a Bachelor in medicine,
1
and
spent several years in Lyons at the end of the Century (he probably died around 1488). In the
1480 tax register, he is mentioned as an escripvain, which seems to mean a copist or/and a
writing master. A few years later, he appears as a master of algorism.
2
The treatise he wrote is kept in a single manuscript, and called by Chuquet himself
Triparty en la science des nombres. The copy was completed in Lyons, in 1484. The
manuscript was not read but by a few persons. It was the main source of Étienne de la Roche
when the latter wrote his Arismetique in 1520. The Triparty disappeared for nearly four
centuries and was rediscovered and partially edited by Aristide Marre in 1880.
3
Chuquet is usually presented as an algebrist, which gives a wrong idea of his work and
thought. Actually, more widely, Chuquet wrote a three part treatise about the “Science of
numbers”.
In the algebraical part, he gives new clever notations as well as a new vocabulary. I will
examine the mathematical involvements these notations induced in the treatise, as well as
their limits. But first I will sum up the content of the Triparty.
4
2. Contents of the Triparty and influences
The first part (fol. 2-45) is a practical arithmetic, an algorism, dealing with the Indian-arabic
numeration, the operations on whole and fractional numbers, the rules of three and other main
arithmetical rules in use at the time. In this part, Nicolas Chuquet followed the tradition of
French commercial arithmetical treatises, in the composition and in the choice of examples.
As a master of algorism, he knew perfectly this tradition, he had read works from his
predecessors (he cites one name) and, while living in Lyons, a famous commercial and
financial place, he must have been acquainted with merchants and bankers. But we must note
an important difference with the typical commercial arithmetics : all the examples use abstract
1
« Fait par Nicolas Chuquet parisien, bachelier en médecine », Triparty, Paris, BnF, ms fds fr. 1346, fol. 147.
2
Hervé L’Huillier, Éléments nouveaux pour la biographie de Nicolas Chuquet, Revue d’histoire des sciences
29 (1976), p. 347-350.
3
Michel Chasles pointed out this lost work in a paper presented to the French Academy of Science in 1841
(Histoire de l’algèbre, Comptes rendus de l’Académie des sciences, 12, n.2, p. 752).
4
Here are some studies about Chuquet : Hervé L’Huillier, Nicolas Chuquet, la Géométrie, première
géométrie algébrique en langue française, 1484, Paris, Vrin, 1979 ; Graham Flegg, Cynthia Hay, Barbara Moss
(eds), Nicolas Chuquet, Renaissance Mathematician, Dordrecht/Boston/Lancaster, D. Reidel Publ. Company,
1985 ; Cynthia Hay (ed), Mathematics from Manuscript to Print, 1300-1600, Oxford, Clarendon Press, 1988 ;
Maryvonne Spiesser, L’œuvre de Nicolas Chuquet dans le contexte des savoirs mathématiques de la fin du XV
e
siècle, Histoire littéraire de la France, t. 43, fasc. 1 (2005), 129-172 ; Maryvonne Spiesser, L’algèbre de Nicolas
Chuquet dans le contexte français de l’arithmétique commerciale, Revue d’histoire des mathématiques, fasc. XII-
1 (2006), p. 7-33.
2
numbers : no unit, no commercial or recreational problem. Applications are postponed at the
end of the text.
The second part (fol. 45v-81v, La seconde partie de ce livre tracte des racines et nombres
composez) deals with irrational numbers : how to extract approximate roots, how to reckon
with irrational abstract numbers. Chuquet uses interesting notations to write the root of a
composite expression
5
, and clearly states that irrationals are numbers, as well as integers or
fractions.
The third part (fol. 83-147) is the exposition of the “rule of the thing” (the Italian Regola
della cosa), which he called “rigle des premiers”. This expression is not easy to translate, as
Chuquet doesn’t give any explanation about the choice of the adjective “premier”, used as a
name, which corresponds to an unknown quantity to the first power. We will translate
“premiers” by “firsts”, to avoid falsifying the mathematician’s thought. Chuquet presents his
rule as the most efficient one in the science of numbers, that’s to say, the most efficient
among the rules of arithmetic, chiefly, the rules of three and of false positions : “Ceste rigle
est la clef, l’entree et la porte des abismes qui sont en la science des nombres”.
6
Parts two and three are the most original and interesting ones. This is why Chuquet is famous
for his “algebra”. But, we must keep in mind that this word is never used by the
mathematician who always employs the word “rule” : the “rule of firsts” is a practical rule to
solve problems in the field of numbers, a rule which allows to solve a wider class of problems
than the rules of three or false positions.
Three supplements follow the Triparty, as applications of the “science of numbers”. Most
of the sources are borrowed from former commercial treatises. The first section (fol. 148-210)
is a mix of problems, many of them being solved algebraically. The second one (fol. 211-
262v) is a practical geometry, which owes a lot to Chuquet’s predecessors (we can found a
very close text in a more ancient manuscript
7
). The third one is the application of the science
of numbers to commercial matters. In the last two supplements, a few problems are solved
thanks to algebraical methods.
We will now focus on the third part of the Triparty, the “rule of firsts”.
3. La règle des premiers
This “rule” raises a lot of questions. It appears as an original contribution, bringing more
advanced material than any other work of the time. There is indeed no evidence that an earlier
work could have been a source of inspiration for the mathematician. As far as Chuquet copied
other treatises in the other parts of the Triparty, we can wonder if he is the creator of his
algebra or if he took it from some lost papers, for instance Italian ones. He, himself, refers
only once to previous algebrists as “the Ancients”. But, this only means that he has read about
algebra. I will speak as if he was an innovator, but, of course, we can’t be sure.
3.1 Chuquet’s rigourous scheme
The third book of the Triparty is well organized in three principal parts, each of them being
divided in chapters. The first part, according to Chuquet’s words, is an introduction to the
following ones. It is interesting for our purpose, as Chuquet sets here the material he will need
5
See M. Spiesser, L’œuvre de Nicolas Chuquet… (cf n. 4).
6
Triparty, fol. 83v : « This rule is the key, entrance and gate of the abysses which are in the science of
numbers. »
7
Traicté de la praticque de geometrie et de la composition du quadrant, Cesena, Bibl. Malatestiana, ms S-
XXVI-6, fol. 300r-330r.
3
afterwards. It encloses definitions, notations and operations on what he calls “differances”
(our monomials and their inverse, ax
p
, with p integer). It is important to underline that
Chuquet considers these “differances” as numbers (as well as integers, rationals or
irrationals). The criterion to be a number is an operational one : “differances” are numbers
because it is possible to apply them the operations of arithmetic.
In the second part, Chuquet explains how to write the equation of a problem and how to
reduce equations. He gives a list of four types of equations (“quatre canons”) and the
algorisms to solve them.
The third part is the application of the four canons” to the resolution of problems on
numbers.
3.2 Chuquet’s terminology and notations
To Chuquet, the word “first” is equivalent to the ancient “thing” (“Les Anciens ont appellé
choses ce que je nomme premiers”). His notations are built as one can see in the following
examples :
12
1
(12 with exponent 1) must be read “twelve firsts” and we would write it 12x
1
.
12
3
is “twelve thirds”, for 12x
3
. 12
3
is a “differance”, 12 is the number and 3
is its
denomination. A number “without denomination”, like 12, can also be noted with the
exponent zero : 12
0
.
Chuquet also introduces negative exponents and negative coefficients, but without
explaining in a general way what links a
p
and a
-p
.
8
To summarize, using our modern
notations, he considers all the expressions ax
p
, where
a is any real number and p is any
positive, negative or null integer.
Here is a glimpse of Chuquet’s notations :
It provides a very advanced and probably baffling material in the context of late XV
th
century mathematics. Etienne de la Roche, greatly influenced by Chuquet, as I said, makes
here a step backwards : he explains Chuquet’s notations, but rarely uses them, and he hushes
up every passage or exercise of the Triparty including negative exponents.
3.3 Two important remarks
First, I chose to represent 12
3
(“twelve third”) by 12x
3
. But both are not equivalent. Chuquet
does not use or create a sign for the unknown, as our x”. He does not use either different
8
“Encore il advient aucunesfoiz que les denominacions sont notees et entendues estre moins, combien que
leur nombre soit plus et aulcunesfoiz moins […] comme 12 premiers moins que l’on peult ainsi noter 12
1m
.” [“It
may happen that the denominations are noted and considered as minus, their number being plus or minus [...]
like 12 firsts minus, which can be noted 12
1m
”] (Triparty, fol. 84v).
4
signs for the different powers of the unknown, as German cossists did, for instance. The
unknown disappears behind its coefficients, and is only defined by the exponents with the
adjectives first, second, third, etc. This process is very similar in its spirit to Bombelli’s or
Stevin’s one in the following century.
9
Finally, the concept of unknown, as we understand it,
seems to be far from Chuquet’s thought.
10
Chuquet is aware of the great improvement his notations open in mathematics. First, they
allow to name any power (any “differance”). He criticizes the “Ancients” who use a proper
name (and a special notation) for each power of the unknown, like “chose”, “champ”, “cube”,
“champ de champ”. He writes : “Telles denominacions ne sont pas souffisans pour fournir a
toutes differances de nombres veu qu’elles sont innumerables” [such denominations are not
sufficient to express every difference of number because they are innumerable]. Italian
masters of abbacus often used this type of construction, writing co. for cosa, ce. for census,
etc., as Luca Pacioli in the Summa
11
:
I think one of the main reasons why Chuquet can break with the tradition and give us a
general notation for any power is that he sets himself free from the traditional geometrical
representation of the three first powers. He just mentions it as a secondary remark.
12
He does
not link either the successive roots to geometry : his vocabulary and notations for nth-roots
are thought and built on the same pattern.
4. Mathematical involvements of Chuquet’s notations
Having described these notations, we can now consider what they involve and how enabling
they are in mathematics. I’ll focus on three points.
First, it brings the possibility to calculate easily with any power. It is clear by comparison
with the other notations in use, which don’t give immediatly the rank of the exponent and
don’t offer any possibility of generalization to any power of the unknown.
9
Rafael Bombelli, Algebra, Bologna, 1572 ; Simon Stevin, L’Arithmétique, Leiden, 1585.
10
To emphazize this idea, let’s say that, when calculating the solution of the equation
bax
p
=
, Chuquet
always writes : divide b by a
p
instead of “divide b by a”.
11
Luca Pacioli, Summa de arithmetica geometria proportioni et proportionalita, Venise, 1494, p. 67v.
12
On the contrary, Antic Roca writes in his Arithmética (1564) : Cosa es rayz o lado de un quadrado
equilaterale [...]. Censo es un numero quadrado [...]. Cubo es un numero cubico ygualmente alto, ancho, y
largo.” [Cosa is a root or the side of a square... Censo is a square number... Cubo is a cubical number equally
high, wide and long].
5
As a consequence, and this is the second point I want to emphasize, the relation between
the addition [resp. subtraction] of the exponents and the multiplication [resp. division] of the
expressions ax
p
, with p an integer, is more obvious :
“Commant on peult multiplier une differance de nombre en soy ou par une aultre [...] Il
convient multiplier nombre par nombre et denominacion avec denominacion se doit
adjouster. Et par ainsi, qui multiplie secondz par tiers […], il en vient quintz, et tiers par
quartz il en vient 7
es
et quartz par quartz il en vient 8
es
, et ainsi des aultres.”
13
According to Chuquet, the very reason of this rule is to be found in “un secret qui est es
nombres proporcionalz” [“a secret laying in proportional numbers”].
The comparative table between the powers of 2 and their positive exponents is well known,
but the rule is valid for zero and negative exponents as well. Chuquet could have given the
sole relation : ax
p
×
bx
q
= abx
p+q
for p and q positive, negative or null integers. He does not
do so ; maybe because, as I already said, he does not link explicitly and in a general way the
division by x
p
and the multiplication by x
-p
.
The last and important point is that Chuquet joins in four types all the equations reducible
to the first or second degree, with eventually the extraction of a nth-root. It seems to be an
important step towards abstraction and generalization : an infinity of equations are joined
together because their common feature is to be solvable thanks to the same algorism. These
four types the four canons involve two or three terms. With our modern notations, this
leads to :
ax
n
= bx
n+p
ax
n
+ bx
n+p
= cx
n+2p
ax
n
= bx
n+p
+ cx
n+2p
ax
n
+ cx
n+2p
= bx
n+p
.
In the three last canons, the terms are said “equally distant”. It means that we have a
geometrical sequence (x
n
, x
n+p
, x
n+2p
are in a continuous proportion), or an arithmetical one if
considering the exponents. Chuquet notes (has found out ?) that the equal distance is the
condition to be able to use the well known algorisms for the resolution of quadratic equations.
Further more, he makes no difference between successive powers like x
n
, x
n+1
, x
n+2
and any
“equally distant differences”, ie x
n
, x
n+p
, x
n+2p
.
By comparison, note that many Italian abbacus masters consider a list of 22 equations (“the
22 rules”) from the first up to the fourth degree, all reducible to the second or first degree.
14
Further on, in the examples, Chuquet considers equations up to the degree 10 ; and finally
notes that it could be as high as one likes, with consecutive degrees or not, so long as the
equation we get belongs to one of the former canons. To me, his ability to “compact” all the
usual cases in four canons has a great deal to do with his advanced notations.
At the end of his work, he leaves all the other types of equations to the sagacity of the
mathematicians to come.
We can note indeed that Chuquet does not break entirely with the ancient Arabic tradition,
because he does not break with the tradition of positive coefficients and because he does not
13
“To multiply a ‘difference’ [ax
p
] by a ‘difference’, the right thing to do is to multiply number [coefficient]
by number and to add denomination [exponent] to denomination. And thus, if you multiply seconds by thirds, it
comes out fifths, and thirds by fourths, it comes out 7
ths
and fourths by fourths, it comes out 8
ths
.” Triparty, fol.
86r.
14
Raffaella Franci, Trends in fourteenth Century Italian Algebra, Oriens-occidens, n° 4 (2002), p. 81-105, p.
85.
6
allow himself to set a polynomial equal to zero (though he currently employs negative and
null exponents).
5. Conclusion
Chuquet’s work was settled in the merchant’s culture and its practical mathematics. This can
explain why he did not develop theoretical explanations, or why he never gave a
demonstration of his rules, prefering notes and numerous examples. But this “non euclidean”
style in use among practicioners must not hide the great qualities and innovations of the
treatise. Chuquet’s Triparty is logically built : first, it deals with integers and brocken
numbers ; then irrationals and finally “differances”. His judicious and clever vocabulary and
notations for the successive roots and for the successive powers come from a very similar
idea, and allow a more synthetical approach of the algebraical reckoning. In Chuquet’s
opinion, the science of number is free from geometrical references, the first three powers (as
well as the first three roots) are not different from the following ones, which are infinite,
because there is an infinity of integers. Furthermore, Chuquet has made a great mental and
mathematical step by considering negative and null exponents and by extending the
operations to these quantities, even if he did not succeed entirely to get free from the tradition
and the historical restrictions concerning these quantities. Unfortunately, as I said, the
Triparty did not circulate much, and it seems that Chuquet’s ideas did not spread.
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