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

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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.

The Taylor Series and Taylor Polynomials in Edwards & Penney

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We previously examined infinite series with constant terms whose sums (when convergent) were numbers. However, series have more practical usage when the series have variable terms.

We noted before that regarding this equation:

we found:

if we were to write r = x for the ratio in a geometric series, then Theorem 1 gives us as the infinite series representation

of the function f (x) = 1/(1-x). In other words, for each fixed number x with |x| <>1/(1-x). The nth partial sum

of the geometric series in the equation above this one above is now an nth-degree polynomial that approximates the function f (x) = 1/(1 - x). The convergence of the infinite series for

should then be accurate if n is sufficiently large. The figure

Shows the graphs of 1/(1 - x) and the three approximations

It appears that the approximations are more accurate when n is larger and when x is closer to zero.

Polynomial Approximations:

Perhaps we want to calculate or approximate a specific value

of a given function f. We would only need to find a polynomial P (x) whose graph is close to the graph of f on some interval containing

Because if so, we could then use the value

as an approximation of the actual value for

Then as soon as we know how to find such an approximated polynomial P (x), we next would need to know how accurately

approximates the desired value

The simplest example of polynomial approximation is the linear approximation

which we obtained by writing

in the linear approximation formula

The graph for the first-degree polynomial

is the line tangent to the curve y = f (x) at the point (a, f (a)):

This first-degree polynomial agrees with f and with its first derivative at x = a. That is,

Example 2: We suppose that f (x) = ln x and that a = 0. [The ln x means 'the power which e must be raised to get x.'] Then f (1) = 0 [because when x is at point 1, y is at zero] and f ' (1) = 1 [because was determined as the value whose differential is its own value, so when e^x is made just e when x = 1, then we obtain the value for the slope that when multiplied to e obtains e again, which is 1/1]; so,

[because the polynomial takes the a value to be 1]. Hence we expect that

for x near 1 [because we are substituting the polynomial for the function producing e.] With x = 1.1, we find that

Thus the error in this polynomial approximation for ln x is about 5%.

To better approximate ln x near x = 1, we will find a second-degree polynomial

that not only has the same value and the same first derivative as does f at x = 1, but also has the same second derivative there:

In order to satisfy these conditions, we must have:

When we solve these equations we get:

so

With x = 1.1, we see that

which is accurate to three decimal places because

The graph of

is a parabola through (1,0) with the same value, slope, and curvature there as y = ln x:

The tangent line and the parabola used in the computations of this above example illustrate one general approach to polynomial approximation. To approximate the function f (x) near x = a , we look for an nth-degree polynomial

such that its value at a and the values of its first n derivatives at a agree with the corresponding values of f. That is, we require that

We can use these n + 1 conditions to evaluate the values of the n + 1 coefficients

The algebra involved is much simpler, however, if we begin with

expressed as an nth-degree polynomial in powers of x - a rather than in powers of x:

The substituting x = a in the above equation yields

[because the x's cancel to zero, leaving only the first coefficient] by the first condition of the vertical series of equations.

Next, substituting x = a into

yields

so,

[by dividing out the 2 from the left side]. We continue the process to find

In general, the constant term in the kth derivative

because it is the kth derivative of the kth-degree term

(Recall that

denotes the factorial of the positive integer k, read "k factorial"). So when we substitute x = a into

we find that

and thus that

for k = 1, 2, 3, . . . , n.

The above equation also holds for k = 0 if we use the universal convention that 0! = 1 and agree that the zeroth derivative

of the function g is just g itself. With such conventions, our computations establish the following theorem.

Theorem: The nth-Degree Taylor Polynomial:

Suppose that the first n derivatives of the function f (x) exist at x = a. Let

be the nth-degree polynomial

Then the values of

and its first n derivatives agree, at x = a, with the values of f and its first n derivatives there. That is, the equations below hold:

The polynomial in the above equation

is called the nth-degree Taylor polynomial of the function f at the point x = a. We take note that

is a polynomial in powers of x - a rather than in powers of x. To use

effectively for the approximation of f (x) near a, we must be able to compute the value f (a) and the values of its derivatives f ' (a), f '' (a), and so on, all the way to

The line

is simply the line tangent to the curve y = f (x) at the point (a, f (a)). Thus y = f (x) and

have the same slope at this point. We recall from before that the second derivative measures the way the curve y = f (x) is bending as it passes through (a, f (a)). Thus, we call f '' (a) the "concavity" of y = f (x) at (a, f (a)). Then, because

it follows that

has the same value, the same slope, and the same concavity at (a, f (a)) as does y = f (x). In addition,

and f (x) will also have the same rate of change of concavity at (a, f (a)). Such observations suggest that the larger n is, the more closely the nth-degree Taylor polynomial will approximate f (x) for x near a.

Example 3: Find the nth-degree Taylor polynomial of f (x) = 1

The pattern is clear:

hence

so the equation

gives

With n = 2 we obtain the quadratic polynomial:

which is the same as from our previous example. With the third-degree Taylor polynomial

we can go a step further in approximating

The value

is accurate to four decimal places (rounded). In the figure below

we see that the higher the degree and the closer x is to 1, the more accurate the approximation

appears to be.

In the common case a = 0, the nth-degree Taylor polynomial in equation

reduces to

Example 4: Find the nth-degree Taylor polynomial for

This is the easiest of all Taylor polynomials to compute, because

for all

Thus the equation

yields

[by using the nth-Degree Taylor Polynomial theorem:

]. The first few Taylor polynomials of the natural exponential function at a = 0 are, therefore,

The figure below shows the graphs of

The table below shows how these polynomial approximate

for x = 0.1 and for x = 0.5.

At least for these two values of x, the closer x is to a = 0, the more rapidly

appears to approach f (x) as n increases.

Taylor's Formula:

The closeness with which the polynomial

approximates the function f (x) is measure by the difference

for which

This difference

is called the nth-degree remainder for f (x) at x = a. It is the error made if the value f (x) is replaced with the approximation

The theorem that lets us estimate the error, or remainder,

is called Taylor's formula, after Brook Taylor (1685-1731), a follower of Newton who introduced Taylor polynomials in an article published in 1715. The particular expression for

that we give next is called the Lagrange form for the remainder because it first appeared in 1797 in a book by the French mathematician Joseph Louis Lagrange (1936-1813).

Theorem 2: Taylor's Formula:

Suppose that the (n + 1)th derivative of the function f exists on an interval containing the points a and b. Then

for some number z between a and b.

If we replace b with x in the above equation, we get the nth-degree Taylor formula with remainder at x = a:

where z is some number between a and x. Thus the nth-degree remainder term is

[because it is the last term in the series].

Example 3 continued: Estimate the accuracy of the approximation

we substitute x = 1 into the formula

for the kth derivative of f (x) = ln x and get

Hence the third-degree Taylor formula with remainder at x = 1 is

with z between a = 1 and x. With x = 1.1 this gives

where 1 <>z = 1 gives the largest possible magnitude

[more clearly: (0.1)^4/4 = 0.000025] of the remainder term. It follows that

o.095308 <>

so we can conclude that ln(1.1) = 0.0953 to four-place accuracy.

Taylor Series:

If the function f has derivatives of all orders, then we can write Taylor's formula [Suppose that the (n + 1)th derivative of the function f exists on an interval containing the points a and b. Then

for some number z between a and b.] with any degree n that we please. Ordinarily, the exact value of z in the Taylor remainder term in equation

is unknown. Nevertheless, we can sometimes use this equation to show that the remainder approaches zero as

for some particular fixed value of x. Then the equation

gives:

[because

]; that is,

The infinite series:

is called the Taylor Series of the function f at x = a. Its partial sums are the successive Taylor polynomials of f at x = a.

We can write the Taylor series of a function f without knowing that it converges. But if the limit in equation

can be established, then it follows as in equation

that the Taylor Series in equation

actually converges to f (x). If so, then we can approximate the value of f (x) sufficiently accurately by calculating the value of a Taylor polynomial of f of sufficiently high degree.

Example 5: In example 4 we noted that if

for all integers

Hence the Taylor formula

at a = 0 gives

for some z between 0 and x. If x and hence z are negative then

if both are positive. Thus the remainder term

satisfies the inequalities

Therefore, the fact that

for all x implies that

for all x. This means that the Taylor series for

converges to

for all x, and we may write

The series in this above equation is the most famous and most important of all Taylor series. With x = 1, the above equation yields a numerical series

for the number e itself. The 10th and 20th partial sums of this series give the approximations

both of which are accurate to the number of decimal places shown.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.702a-709c.