Lagrange's Algebraicization of Analysis and Wronski's Critique, in Bottazzini's The Higher Calculus and in Boyer's History of the Calculus

by Corry Shores
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From Bottazzini's The Higher Calculus:

In his Théorie des fonctions analytiques (1772), Lagrange attempted to

present the principles of the calculus in a systematic manner without making any reference to infinitesimals, evanescent quantities, differentials or limits. He instead stressed the need to reduce the calculus to simple algebraic manipulations of finite quantities. (Bottazzini 48ab)

He defines a function thus:

We call a function of one or more quantities every expression of the calculus in which these quantities enter in any way whatsoever, mixed or not with other quantitites that one takes as having given and invariable values, while the quantitities of the function can have all possible values. (Lagrange, 1797, p. 1; 1813, p. 15, qtd in Bottazzini 48bc)

Langrange's next step was to show that any given function can be expanded as a series. Bottazzini quotes Lagrange as writing:

"We therefore consider a function f (x) of any variable x. If in place of x we put x + i, i being any indeterminate quantity whatever, it becomes f (x + i) and, by the theory of series [Bottazzini's emphasis] we can expand it as a series of this form:

in which the quantities p, q, r, ... , the coefficients of the powers of i, will be new functions of x, derived from the primitive function x and independent of the [indeterminate] quantity i ... (1797, p.2; 1813, pp.21-2, qtd in Bottazzini 48d).

Because Lagrange is interested in the formation and calculation of these different functions, the new calculus for him is differential or fluxional (48d).

Lagrange critiqued early renditions of the calculus, arguing instead that "'the true metaphysics' of the calculus lies in the fact that the errors resulting from neglecting infinitesimals of higher degrees were 'corrected or compensated' by the procedures of the calculus themselves, when they were limited to infinitesimals of the same degree" (49a). Euler and D'Alembert failed to realize this, although they addressed the problem by claiming that infinitely small values equal zero, but the limits of the ratios of indefinite differences have quantitative value (49b). Lagrange critiques Newton's notion of the motion of quantities so to avoid the notion of infinitesimal value by saying

on the one hand, to introduce movement into a calculus that only has algebraic quantities as its object is to introduce a foreign idea which obligates one to think of these quantities as the lines traversed by a moving body. On the other hand, it must be granted that we do not even have a really clear idea of what the velocity of a point is at every instant when this velocity is variable (Lagrange 1797, pp. 3-4; 1813, p.17; qtd in Bottazzini 49bc).

Lagrange thinks that Landen and Arbogast make steps in the right direction, because their aim according to Lagrange was to deal with the functions arising when expanding any given function, and then to apply these "derived" functions in math and science. Lagrange claims that his approach is free of "every illicit supposition" and "all metaphysics," and that it bears "the rigor of the ancient demonstrations" [see Hegel's Science of Logic section 587] by basing it on his method of primitive functions and derivatives (49d). "Nevertheless," Bottazzini writes

the crucial point in his entire construction is his assertion that it is possible to expand any function in a series of ascending powers of an indeterminate increment i. (49d)

Because he wants to offer proof, Lagrange considers the

form of the series that must represent the expansion of every function f (x) when one substutites x + 1 in place of x, and which we have supposed must contain only integral and positive power of i. (Lagrange 1797, p.7; 1813, p. 22; qtd in Bottazzini 50a).

Lagrange wants to establish that this holds for all functions, so he shows that "when x and i remain indeterminate, the series cannot contain fractional or negative powers of i" (50b). But there is no way to prove such an expansion of a function, "there is only the fact that the series contains only positive powers of i," and it is only after the function is expanded that we learn what each terms means. So we begin with

where i is any indeterminate quantity whatever, P is a new function of x and i. We can determine this other function P in terms of x and i by moving fx to the other side of the equation, and dividing both sides by i so to obtain:

We then let i vanish so to separate P insofar as it is independent of i (which does not equal zero as it vanishes), leaving a differential value p. Lagrange then shows that

P = p + iQ.

Q can be further determined, and so on, hence:

But each time we substitute we continue to multiple i's with other i's

Lagrange says we may make i so small that any given term will be larger than those following it in the series.

He recognizes that this expansion does not hold for every possible function.

Lagrange then identifies the functions p, q, r, ... with f 'x, (f ''x/2), (f '''x/3!), ... , where f 'x, f ''x, f '''x, and so on are the successive derivatives of the function fx. [See the Taylor Series]

What is most notable about his technique is that it is purely algebraic.

Lagrange's theory was attacked a year later by Hoëne Wronski. In his Introduction à la philosophie de la mathématique et technie de l'algorithmie, he comes to conclude that all functions may take the form:

"where the

are any functions of x and the determination of the coefficients

of the series depend on the determinants that are today called 'Wronskian'" (54-55).

Wronski then vigorously critiques Lagrange in Réfutation de la théorie des fonctions analytiques de Lagrange. In it, Wronski notes that Lagrange's theory is based on two assumptions:

Wronski attacks three aspects of Lagrange's theory:

1) He wonders what the grounds are for the first formulation. There seem not to be any.

2) Lagrange claims that the second formulation is true because it is verified when i equals one, but Wronski wants to know what about when i does not equal 1?

3) Lagrange defines f '(x) as the coefficient of the second term in the expanded series in the bottom equation. But Wronski notes that the position of a function in a series does not by its place alone determine the meaning of that function; it seems Lagrange slips them together invalidly.

Bottazzini, Umberto. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Transl. Warren Van Egmond. Paris: Springer-Verlag, 1981.

----------------------------------------

from Boyer's History of the Calculus

Under the influence of Leibniz' differential method and Kant's transcendental philosophy, Hoëné Wronski objected against Lagrange's ban on the infinite in analysis.

Wronki rightly asks where Lagrange obtained the series

Wronski believed instead that mathematics should be based on what he considered the "suprime algorithmic law:"

"where the quantities

are any functions of the variable x. Being the supreme law of mathematics, the irrecusable truth of this law he held to be not mathematically derived, but given by transcendental philosophy" (Boyer 261c.d).

Wronski was correct to criticize Lagrange's theory for being limited to only expandable functions. But he otherwise held highly unconventional and controversial views on the calculus (262a).

Whereas Lagrange had attempted to give a formal logical justification of the subject, Wronski asserted that the differential calculus constituted a primitive algorithm governing the generation of quantities, rather than the laws of quantities already formed. (262ab).

The calculus' propositions, which express absolute truths, cannot be deduced within the sphere of mathematics. Wronski objected to construing calculus in terms of limits, ultimate ratios, vanishing quantities, and functions, because he thought it was fruitless to abandon the notion of the infinite (262b).

Although mathematicians continued to hold to the limit concept,

Wronski represents an extreme example of a view which we shall find recurring throughout the nineteenth century. In regarding the calculus as a means of explaining the growth of magnitudes, followers of this school of thought were to attempt to retain the concept of the infinitely small, not as an extensive quantity but as an intensive magnitude. Mathematics has excluded the fixed infinitely small because it has failed to establish the notion logically; but transcendental philosophy has sought to preserve primitive intuition in this respect by interpreting it as having an a priori metaphysical reality associated with the generation of magnitude. (262-263).

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.

Lagrange in the History of the Calculus

presentation of Edwards & Penney's work, by Corry Shores
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Joseph Louis Lagrange was skeptical of the infinitely small; for him, the accuracy of calculus results from a “compensation of errors.” And yet, he rejected the limit concept on account of its poor metaphysical grounding. The tangent was not a limit, for Lagrange, because after becoming the tangent, the secant could very well continue to the other side of the point and become a secant again. Nor did he accept the method of fluxions, because it made use of “the irrelevant notion of motion.” He rejected Euler’s presentation of dx and dy as 0, because Lagrange felt that we do not have a clear and precise notion of the ratio of two terms which become zero. As a result, Lagrange sought out a simple algebraic method that was free from these objections.

Lagrange found his means in the Taylor Series.

The series f (x + h) = f (x) + f ' (x)h + f '' (x)h^2/2! had been known at least from the time of Taylor, whose name it bears. In this series, the coefficients of the powers of h involve the ratios of differentials, or of fluxions. However, the series can be derived without reference to these notions. What would be more natural than to define differentials and fluxions in terms of the coefficients of such a series? This procedure would (only on the surface, as we know now) obviate the necessity of introducing either limits or infinitesimals into the work, and the calculus would thus be reduced to simple algebraic operations. (252c)

Taylor thought that such an approach would free calculus from "all metaphysics and of any theory of infinitely small or vanishing quantities" (Boyer quoting Lagrange 252d). Although his method was not completely satisfactory, it had the advantage that it did not make use of ideas from geometry, mechanics, or philosophy (253b).

And yet, Lagrange was later criticized for "giving up, in favor of mathematical formalism, the 'generative' concept which has frequently been felt to be the basis of the methods of fluxions and differentials" (253bc).

But Lagrange continued seeking a formalization of the notion of limit based on Euler's function concept.

Incidentally, in so doing he focused attention for almost the first time upon the quantity which is now the central conception in the calculus -- that of the derived function, or the derivative, or the differential coefficient. Lagrange, in this connection gave not only the name from which the word derivative was adopted, but also the notation f 'x, modifications of which are still conveniently used. (253d)

Newton did not interpret the ratio of infinitesimals as such a single number or quantity (derivative), for he considered it more as a ratio of increments or fluxions (254a).

Similarly, Leibniz did not consider the ratio of infinitesimals as a single number, but instead as a quotient of "inassignables" (254b).

Lagrange's method was first to properly make use of the notion of derivative as "merely a single coefficient of a term in an infinite series" that is also "completely divested of any idea of ratio, or limiting equality" (254c).

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.

Newton's Flux

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From the preface to Methods of Fluxions

The chief principle of the Method of Fluxions is that mathematical quantity, particularly extension, may be conceived as generated by continued local Motion and that all Quantities whatsoever can be conceived as being generated in this manner (xi). When these magnitudes are generated, they must be generated according to increases and decreases of comparative velocity whose relations are fixed and determinable.

Another principle of the text is that

Quantity is infinitely divisible, or that it may (mentally at least) so far continually diminish, as at last, before it is totally extinguished, to arrive at Quantities that may be call'd vanishing Quantitites, or which are infinitely little, and less than any assignable Quantity. Or it supposes that we amy form a Notion, not indeed of absolute, but of relative and comparative infinity.

(xi.d)

Newton's method, then, is unlike the "Method of Indivisibles" which takes there to be infinitely many little Quantities that actually exist. There are infinite orders and gradations of these indivisibles, "not relatively, but absolutely such" (xi-xii). The problems with this method arise if we do not distinguish absolute and relative Infinity. Absolute infinity cannot enter into our calculations, but relative infinitity can. Newton begins with finite Quantities, and diminishes them relatively and gradually to infinitely little Quantities. Thus he begins first with finite quantities rather than infinitesimal ones, which places his method in accordance with common Algebra and Geometry. The result being the "most curious Discoveries in Art and Nature" and "the sublimest Theories" (xii.b.c)

From Principia

Book I, Section I, Lemma I

Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.

Lemma II, integral calculus:

"If in any figure AacE, terminated by the right lines Aa. AE, and the curve acE, there be inscribed any number of parallelograms Ab, Be, Cd, etc., comprehended under equal bases AB, BC, CD, etc., and the sides, Bb, Cc, Dd, etc., parallel to one side Aa of the figure; and the parallelograms aKbl, bLcm, cMdn, etc.,  are completed. Then if the breadth of those parallelograms be supposed to be diminished, and their number to be augmented in infinitum; I say, that the ultimate ratios which the inscribed figure AKbLcMdD, the circumscribed figure AalbmcndoE, and curvilinear figure AabcdE, will have to one another, are ratios of equality."
Lemma XI, Scholium
Instead of using the method of indivisibles,
I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios ; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is performed as by the method of indivisibles ; and now those principles being demonstrated, we may use them with more safety. Therefore if hereafter I should happen to consider quantities as made up of particles, or should use little curve lines for right ones, I would not be understood to mean indivisibles, but evanescent divisible quantities : not the sums and ratios of determinate parts, but always the limits of sums and ratios.
Perhaps it may be objected, that there is no ultimate proportion, of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alleged, that a body arriving at a certain place, and there stopping has no ultimate velocity : because the velocity, before the body comes to the place, is not its ultimate velocity ; when it has arrived, is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities, not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and demonstrating any other thing that is likewise geometrical.
It may also be objected, that if the ultimate ratios of evanescent quantities are given, their ultimate magnitudes will be also given : and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables, in the 10th Book of his Elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, unless (till?) the quantities are diminished in infinitum. This thing will appear more evident in quantities infinitely great. If two quantities, whose difference is given, be augmented in infinitum, the ultimate ratio of these quantities will be given, to wit, the ratio of equality; but it does not from thence follow, that the ultimate or greatest quantities themselves, whose ratio that is, will be given. Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least (the least possible), or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be always diminished without end.
Book II, Section II, Lemma II:
"The moment of any genitum is equal to the moments of each of the generating sides drawn into the indices of the powers of those sides, and into their co-efficients continually. 
  
 
  
I call any quantity a genitum which is not made by addition or subduction of divers parts, but is generated or produced in arithmetic by the multiplication, division, or extraction of the root of any terms whatsoever: in geometry by the invention of contents and sides, or of the extremes and means of proportionals. Quantities of this kind are products, quotients, roots, rectangles, squares, cubes, square and cubic sides, and the like. These quantities I here consider as variable and indetermined, and increasing or decreasing, as it were, by a perpetual motion or flux; and I understand their momentaneous increments or decrements by the name of moments; so that the increments may be esteemed as added or affirmative moments; and the decrements as subducted or negative ones. But take care not to look upon finite particles as such. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive them as the just nascent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their first proportion, as nascent. It will be the same thing, if, instead of moments, we use either the velocities of the increments and decrements (which may also be called the motions, mutations, and fluxions of quantities), or any finite quantities proportional to those velocities. The co-efficient of any generating side is the quantity which arises by applying the genitum to that side.
  
 
  
Wherefore the sense of the Lemma is, that if the moments of any quantities A, B, C, &c., increasing or decreasing by a perpetual flux, or the velocities of the mutations which are proportional to them, be called a, b, c, &c., the moment or mutation of the generated rectangle AB will be aB + bA; the moment of the generated content ABC will be aBC + bAC + cAB; and the moments of the generated powers
will be
respectively, and in general, that the moment of any power 
 
Also, that the moment of the generated quantity
; 
the moment of the generated quantity
and the moment of the generated quantity
   
or
 
and so on.
CASE 1. Any rectangle, as AB, augmented by a perpetual flux, when, as yet, there wanted of the sides A and B half their moments

was

or

but but as soon as the sides A and B are augmented by the other half moments, the rectangle becomes 

into

or

From this rectangle subduct the former rectangle, and there will remain the excess aB + bA. Therefore with the whole increments a and b of the sides, the increment aB +bA of the rectangle is generated. Q.E.D. 
From: http://www.maths.tcd.ie/pub/HistMath/People/Newton/RouseBall/RB_Newton.html
The invention of the infinitesimal calculus was one of the great intellectual achievements of the seventeenth century. This method of analysis, expressed in the notation of fluxions and fluents, was used by Newton in or before 1666, but no account of it was published until 1693, though its general outline was known to his friends and pupils long anterior to that year, and no complete exposition of his methods was given before 1736.
The idea of a fluxion or differential coefficient, as treated at this time, is simple. When two quantities - e.g. the radius of a sphere and its volume - are so related that a change in one causes a change in the other, the one is said to be a function of the other. The ratio of the rates at which they change is termed the differential coefficient or fluxion of the one with regard to the other, and the process by which this ratio is determined is known as differentiation. Knowing the differential coefficient and one set of corresponding values of the two quantities, it is possible by summation to determine the relation between them, as Cavalieri and others had shewn; but often the process is difficult, if, however, we can reverse the process of differentiation we can obtain this result directly. This process of reversal is termed integration. It was at once seen that problems connected with the quadrature of curves, and the determination of volumes (which were soluble by summation, as had been shewn by the employment of indivisibles), were reducible to integration. In mechanics also, by integration, velocities could be deduced from known accelerations, and distances traversed from known velocities. In short, wherever things change according to known laws, here was a possible method of finding the relation between them. It is true that, when we try to express observed phenomena in the language of the calculus, we usually obtain an equation involving the variables, and their differential coefficients - and possibly the solution may be beyond our powers. Even so, the method is often fruitful, and its use marked a real advance in thought and power.
I proceed to describe somewhat fully Newton's methods as described by Colson.

From Walter William Rouse Ball: A Short Account of the History of Mathematics, Published by Macmillan, 1901:
The second part of this appendix to the Optics contains a description of Newton's method of fluxions. This is best considered in connection with Newton's manuscript on the same subject which was published by John Colson in 1736, and of which it is a summary.
The fluxional calculus is one form of the infinitesimal calculus expressed in a certain notation, just as the differential calculus is another aspect of the same calculus expressed in a different notation. Newton assumed that all geometrical magnitudes might be conceived as generated by continuous motion; thus a line may be considered as generated by the motion of a point, a surface by that of a line, a solid by that of a surface, a plane angle by the rotation of a line, and so on. The quantity thus generated was defined by him as the fluent or flowing quantity. The velocity of the moving magnitude was defined as the fluxion of the fluent. This seems to be the earliest definite recognition of the idea of a continuous function, though it had been foreshadowed in some of Napier's papers.
Newton's treatment of the subject is as follows. There are two kinds of problems. The object of the first is to find the fluxion of a given quantity, or more generally "the relation of the fluents being given, to find the relation of their fluxions.'' This is equivalent to differentiation. The object of the second or inverse method of fluxions is from the fluxion or some relations involving it to determine the fluent, or more generally "an equation being proposed exhibiting the relation of the fluxions of quantities, to find the relations of those quantities, or fluents, to one another.'' This is equivalent either to integration which Newton termed the method of quadrature, or to the solution of a differential equation which was called by Newton the inverse method of tangents. The methods for solving these problems are discussed at considerable length.
Newton then went on to apply these results to questions connected with the maxima and minima of quantities, the method of drawing tangents to curves, and the curvature of curves (namely, the determination of the centre of curvature, the radius of curvature, and the rate at which the radius of curvature increases). He next considered the quadrature of curves and the rectification of curves. In finding the maximum and minimum of functions of one variable we regard the change of sign of the difference between two consecutive values of the function as the true criterion; but his argument is that when a quantity increasing has attained its maximum it can have no further increment, or when decreasing it has attained its minimum it can have no further decrement; consequently the fluxion must be equal to nothing.
It has been remarked that neither Newton nor Leibnitz produced a calculus, that is, a classified collection of rules; and that the problems they discussed were treated from first principles. That, no doubt, is the usual sequence in the history of such discoveries, though the fact is frequently forgotten by subsequent writers. In this case I think the statement, so far as Newton's treatment of the differential or fluxional part of the calculus is concerned, is incorrect, as the foregoing account sufficiently shews.
If a flowing quantity or fluent were represented by x, Newton denoted its fluxion by , the fluxion of    or second fluxion of x by , and so on. Similarly the fluent of x was denoted by , or sometimes by x' or [x]. The infinitely small part by which a fluent such as x increased in a small interval of time measured by o was called the moment of the fluent; and its value was shewn to be o. Newton adds the important remark that thus we may in any problem neglect the terms multiplied by the second and higher powers of o, and we may always find an equation between the co-ordinates x, y of a point on a curve and their fluxions , . It is an application of this principle which constitutes one of the chief values of the calculus; for if we desire to find the effect produced by several causes on a system, then, if we can find the effect produced by each cause when acting alone in a very small time, the total effect produced in that time will be equal to the sum of the separate effects. I should here note the fact that Vince and other English writers in the eighteenth century used    to denote the increment of x and not the velocity with which it increased; that is   in their writings stands for what Newton would have expressed by o   and what Leibnitz would have written as dx.
I need not discuss in detail the manner in which Newton treated the problems above mentioned. I will only add that, in spite of the form of his definition, the introduction into geometry of the idea of time was evaded by supposing that some quantity (ex. gr. the abscissa of a point on a curve) increased equably; and the required results then depend on the rate at which other quantities (ex. gr. the ordinate or radius of curvature) increase relatively to the one so chosen. The fluent so chosen is what we now call the independent variable; its fluxion was termed the "principal fluxion''; and, of course, if it were denoted by x, then    was constant, and consequently    = 0.