2. Notation; combinatorial analysis.

History and archaeology make us conscious of a large number of "systems of notation"; their initial goal was to attach to each individual whole number (up to a limit which depends on the needs of the application) a name and a written representation, made up of a combination of a restricted number of symbols, accomplished by following more or less regular laws. By far the most frequent procedure consists in splitting up the whole numbers into sums of "successive units" $b_1, b_2, \ldots, b_n, \ldots$, of which each is a multiple by a whole number of the preceding one; and if in general $b_n / b_{n-1}$ is taken to be equal to the same number $b$ (the "base" of the system, most often 10), many exceptions to this rule are noticed, as with the Babylonians, where $b_n / b_{n-1}$ is sometimes equal to 10, sometimes to 6 [232], and in the chronological system of the Mayas, where $b_n / b_{n-1}$ is equal to 20 except for $n = 2$, and where $b_2 / b_1 = 18$ [228]. As for the corresponding written representation, it must indicate the number of "units" $b_i$ of each order $i$; in many systems (such as with the Egyptians, the Greeks and the Romans), successive multiples $k.b_i$, (where $k$ varies from 1 to $(b_{i+1}/b_i) - 1$) are represented by symbols which depend at the same time on $k$ and on $i$. A first and important improvement consisted in representing all the numbers $k.b_i$ (for the same value of $k$) by the same symbol: it is the principle of the "notation of position", where the index $i$ is indicated by the fact that the symbol representing $k.b_i$ appears in the ith position in the sequence of "slices" making up the number being represented. The first system of this kind is found among the Babylonians, who, without doubt already 2000 years before Christ, denote by the same sign all the multiples $k.60^{\pm i}$ corresponding to arbitrary values of the exponent $i$ ([232], pp. 93-109). The inconvenience of such a system resides of course in the ambiguity of the symbols used, so long as nothing indicates that the units of a certain order may be absent, in other words, so long as the system is not completed by the introduction of a "zero". One sees however the Babylonians making do without such a symbol for the greater part of their history; in fact they only use such a "zero" in the last two centuries before Christ, and still only in the interior of a number; until then, the context alone had to specify the significance of the number under consideration. Only two other systems use a "zero" systematically : that of the Mayas (in use, it would appear, already from the beginning of the Christian era [228]), and our actual decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number (and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contributions of the Hindus ([78], v. I). Of course, once the principle of the notation of position has been acquired, it was easy to extend it to any base whatever; the discussion concerning the merits of different "bases" proposed since the XVIIth century is a matter of the techniques of numerical Calculation and should not be tackled here. We limit ourselves to remarking that the operation that serves as foundation to these systems, the so-called "Euclidean" division, does not appear before the Greeks, and no doubt goes back to the first Pythagoreans, who made it the essential tool of their theoretical Arithmetic (see p. 85).

The general problems of enumeration, grouped under the name of "combinatorial Analysis", do not seem to have been tackled before the last centuries of classical Antiquity: only the formula $\binom{n}{2} = n(n-1)/2$ is attested to in the IIIrd century of our era. The Hindu mathematician Bhaskara (XIIth century) knows the general formula for $\binom{n}{p}$. A more systematic study is to be found in a manuscript of Levi ben Gerson, at the beginning of the XII-Ith century: he obtains the recurrence formula allowing the calculation of the number $V_n^p$ of arrangements of $n$ objects $p$ at a time, and in particular the number of permutations of $n$ objects; he states also rules equivalent to the relations $\binom{n}{p} = V_n^p / p!$ and $\binom{n}{n-p} = \binom{n}{p}$ ([311], v. VI, pp. 64-65). But this manuscript seems to have been ignored by his contemporaries, and its results have only been rediscovered little by little by the mathematicians of the following centuries. Among the later advances, let us pick out that Cardan proves that the number of non-empty subsets of a set of $n$ elements is $2^n - 1$; Pascal and Fermat, setting up the calculus of probability, rediscover the expression for $\binom{n}{p}$, and Pascal is the first to observe the relationship between these numbers and the binomial formula: this latter seems to have been already known to the Arabs from the XIIIth century, the Chinese in the XIVth century, and it was rediscovered in the West at the beginning of the XVIth century, as well as the method of calculation by recurrence known as the "arithmetical triangle", that is normally attributed to Pascal ([311], v. VI, pp. 35-38). Finally Leibniz, around 1676, obtains (without publishing it) the general formula for "multinomial coefficients", rediscovered independently and published 20 years later by de Moivre.