# What is the origin of the distinction between assignable and inassignable number?

Leibniz described his infinitesimals as being inassignable numbers in a number of texts, e.g., in his Cum Prodiisset that was analyzed in detail by H. Bos in a seminal text dating from the 1970s. The distinction between assignable and inassignable number seems to appear out of nowhere in Leibniz (as does the term infinitesimal itself). Historically is there an earlier source for this?

For some references, see :

See 5.1 Indivisibles and Infinitely Small Quantities, page 119-on, and see page 126 for Leibniz De Quadratura Arithmetica (ca.1676), regarding Theorem 6 :

that a curvilinear figure can be approximated with an arbitrary degree of accuracy by step-figures, in such a way that the difference between the area under the curve and the area of the step-figures can be made less than any assigned quantity.

The source can be found in Cavalieri geometry of indivisibles and the revival of the exhaustion method.

See : Gottfried Wilhelm Leibniz, The Early Mathematical Manuscripts of Leibniz (J.M.Child ed, 1920 - Dover reprint), Letter to Bernoulli (1703), page 19 :

I filled some hundreds of sheets with them [the results that I obtained] in that year [the year 1672]. These I divided into two classes of assignables and inassignables. Among assignables I placed everything I obtained by the methods previously used by Cavalieri, Guldinus, ToricelIi, Gregory St.Vincent and Pascal, such as sums, sums of sums, transpositions, "ductus," cylinders truncated by a plane, and lastly by the method of the center of gravity; and among inassignables I placed all that I obtained by the use of the triangle which I at that time called "the characteristic triangle".

Since I introduced not only the first differences, but also the second, third and other higher differences, inassignable or incomparable with these first differences [...].

See H.J.M.Bos, Differentials and Higher-Order Differentials in the Leibnizian Calculus (1974), pag.13-on : inassignable means incomparable ["less than any assigned quantity"].

It seems to me that is a new term introduced by Leibniz :

I agree with Euclid Book V, Def.5 that only those homogeneous quantities are comparable, of which the one can become larger than the other if multiplied by a number, that is, a finite number.

This is the case of an increment in a numerical (i.e. discrete) series.

Leibniz "extrapolate" the method of taking the sum of a series to a continuous quantity (i.e. a function) :

I feel that this method and others in use up till now can all be deduced from a general principle which I use measuring curvilinear figures, that a curvilinear figure must be considered to be the same as a polygon with infinitely many sides.

if $x$ or $y$ were not discrete terms, but continual terms, that is, not numbers whose differences are assignable intervals, but straight line abscissas increasing continually or by elements, that is, by inassignable intervals, so that the sequence of terms constitutes the figure, [...].

Thus, "infinitesimal" increments are inassignable quantity because they are incomparable to the quantity "increasing continually", because they violate Euclide's definition.

Finally, see Gottfried Wilhelm Freiherr von Leibniz, The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686 (Richard Arthur editor, 2001), page 89, from the ms Infinite Numbers (ca.1676) :

The circle - as a polygon greater than any assignable [quolibet assignabili maius] [...]. So when something is said about the circle we understand it to be true of any polygon such that there is some polygon in which the error is less than any assigned amount $a$ [error minor sit quovis assignato $a$] [...]. And so if certain polygons are able to increase according to some law, and something is true of them the more they increase, our mind imagines some ultimate polygon; and whatever it sees becoming more and more so in the individual polygons, it declares to be perfectly so in this ultimate one. And even though this ultimate polygon does not exist in the nature of things, one can still give an expression for it, for the sake of abbreviation of expressions. [See Leibniz's Principle of Continuity.]

• Mauro, Leibniz had two accounts for the calculus: one using infinitesimals, and one using Archimedean techniques (that today sound like an informal account of epsilontics). In Quadratura Arithmetica he used both. The sample you provide above seems to be an Archimedean technique. At any rate it does not mention the distinction between assignable and inassignable quantity that Leibniz mentions elsewhere. My question was not for references in Leibniz but rather sources for Leibniz's distinction in earlier authors if such exist. Dec 21 '15 at 20:03
• Thanks, that's a very interesting passage from the letter to Bernoulli. Incidentally, what the editor (Child) describes as a draft manuscript of an answer Nieuwentijt is misdated by Child who places it in the 1680s. Most historians today date the manuscript much later, in 1701. This is the manuscript commonly referred to today as Cum Prodiisset. Dec 22 '15 at 7:51
• Note that what Leibniz refers to as Euclid book 5 definition 5 is numbered slightly differently at David Joyce's website that you linked. The edition Joyce uses lists the same definition as book 5 definition 4. Leibniz was using a different edition. At any rate this definition is what is commonly known today as the archimedean property. Dec 22 '15 at 8:51