Marx's Mathematical Manuscripts 1881

Appendix IV. John Landen's ‘Residual Analysis

Written: August, 1881;
Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;
First published: in Russian translation, in Pod znamenem marksizma, 1933.

Notice of Marx’s intention to acquaint himself with the works of John Landen in the British Museum is evident at several places in the mathematical manuscripts of Marx (see p.33).

Marx saw in Landen a possible precursor of Lagrange, attempting to ‘rebuild on strictly algebraic lines the foundation of differential calculus’(p.113), and he proposed that the Landen method should be compared to the method Marx categorised as ‘algebraic differentiation’, but he himself doubted that Landen really understood the essential difference between this method and any other. To convince himself of the truth of this proposal Marx wanted to study in the Museum Landen’s Residual Analysis.

In the sources available to him Marx could find two earlier opinions of this book: in Hind’s textbook (p.128, 2nd ed.) and in Lacroix’s long ‘Treatise’ (Vol.I, pp.239-240) - which are in fact almost identical since Hind had essentially translated into English the appropriate passage from Lacroix. In Hind we read: ‘The notion of establishing this kind of calculus [that is, differential calculus] upon principles purely algebraical, seems however to have originated with Mr John Landen, a celebrated English mathematician who flourished about the middle of the 18th century. In what is termed his Residual Analysis, the first object is to exhibit the algebraical development of the difference of the same functions of quantities x and x’ divided by the difference of the quantities themselves, or the development of the expression (f(x’) - f(x)/(x’ - x), and afterwards to find what is called the special value of the result when x’ is made = x and when therefore all trace of the divisor x’ - x has disappeared.’ (And in Lacroix, ‘... and when this quotient [(f(x’) - f(x))/(x’ - x)] is obtained in order not to conserve any trace of the divisor x’ - x, one sets x’ = x, since the final goal of the calculation is to arrive at a special value of the above ratio.’)

Marx apparently did not succeed in his intention to study Landen’s book in the British Museum. An analysis of the contents of the book, however, completely confirms Marx’s expressed opinion, which he himself considered ‘highly probable’.

The complete title of the Landen book is ‘The Residual Analysis, a new branch of the algebraic art, of very extensive use, both in pure mathematics and natural philosophy. Book I, By John Landen. London. Printed for the author, and sold by L.Haws, W.Clarke and R.Collins, at the Red Lion in Paternoster Row, 1764.’

The preface begin with the words:

‘Having some time ago stumbled across a new and easy method of investigating the binomial theorem with the help of a purely algebraic process, I turned to see whether the means used to investigate this theorem might be of service with other theorems, and I soon found that a certain type of calculation founded on this method may be used in many researches. I call this special method Residual Analysis, since in all problems where it is used the basic tools which we employ to obtain the desired result are those quantities and algebraic expression which mathematicians call residuals.’

Later the author criticises the fluxions calculus of Newton and the differentials of Leibnitz as based on the introduction into mathematics of undefined new ‘principles’. Those applied in the calculus of fluxions of Newton he considers the explanation of the significant new terms introduced into the theory, such as the not really existent but nonetheless apparent (as self-evident) concepts, imaginary motion and graphically continuous flow, which do not belong in any mathematics of clear and distinct ideas but do continue to speak for example of such things as the speed of time, the velocity of velocity and so on as unnecessary in the proof (and therefore on the other hand serve as the means of definition of several exact mathematical concepts). In the analysis of Leibnitz he considers undefined the introduction, under cover of new ‘principles’ of infinitely small quantities and the quantity infinitely smaller than any infinitely small quantity, the suppression of which (when it is not a matter of accepted approximate results) is: ‘a very unsatisfactory (if not erroneous) method to rid us of such quantities’ (p.IV). Landen believed that mathematics had no need of such alien principles and that his Residual Analysis ‘does not require any principles other than those accepted since antiquity in algebra and geometry’, ‘no less (if not more) in use, than the calculus of fluxions or differential calculus’ (p.IV).

The starting-point of residual analysis is in the formula

(ar - br)/(a - b) = ar-1 + ar-2b + ... + br-1 (1)

(where r is a positive whole number) with the help of which and the formulae* derived from it

(vm/r - wm/r)/(v - w) = (vm/r - 1)⋅((1 + w/v + w/v]² + ... + w/v]m-1)/(1 + w/v]m/r + w/v]2m/r + ... + w/v](r-1)m/r)) (2)

(v-m/r - w-m/r)/(v - w) = - v-1⋅w-m/r⋅((1 + w/v + w/v]² + ... + w/v]m-1)/(1 + w/v]m/r + w/v]2m/r + ... + w/v](r-1)m/r)) (3)

(where m and r are positive whole numbers), Landen obtains the derivative of the power function xp for whole and fractional (positive or negative) values of p as a ‘special value’ of the ratio

(xp - x1p)/(x - x1)

at x = x1. In other words, he predefines the ratio (xp - x1p)/(x - x1) at x = x1 as that which fulfils the equality of formulae (1), (2) and (3).

The ‘special value’ of the ratio (y - y1)/(x - x1), where y = f(x), y1 = f(x1), at x = x1, Landen designates [x - y].

He obtains the transition to the irrational powers in his examples, beginning with the determination of the ‘special value’ of the ratio (v4/3 - w4/3)/(v - w) at v = w (the derivative of v4/3 with respect to v) by two different means, one employing formula (2) with m = 4 and r = 3, the other by the same formula, but ‘since 4/3 = 1.333...’ using the pairs (m = 13,333, r = 10,000), (m = 133,333, r = 100,000), and so on. Landen saves himself from the difficulties attending this infinite process by remarking that the ‘final value’ of

(1 + 1 + 1 + 1 + ... (13,333 times))/(1 + 1 + 1 + 1 + ... (10,000 times))

is obviously equal to 4/3, the quantity from which [the number] 1.333 ... is derived (p.7).

After this he makes the transition to the case where m/r = sqrt{2} = 1.4142 ... , treating it by means of the second method, that is, as he himself notes, ‘approximately’, but such that it can in any case be made more ‘closely approximate’, he again concludes that the ‘final value’ of

(1 + 1 + 1 + 1 + ... (14,142 ... times))/(1 + 1 + 1 + 1 + ... (10,000 times))

‘is equal to sqrt{2}, the value from which [the number] 1.4142 etc. is derived (by the taking of the root).’ (p.8)

It is not surprising that Landen cannot construct his Residual Analysis without employing in one form or another the concept of limit. However, in practice he speaks of the limit from the viewpoint of Newton, treating the limit as the ‘final value’ (as the end) of an infinite (that is, without having any end) sequence. Naturally he did not in fact use this definition, but he approached by this means an approximate evaluation of the point and of the convergence (or divergence) of the process of their sequential values, which prompted the concrete contents of the question to him.

Like other mathematicians of his time, Landen considers it possible to employ freely divergent series in formally structured expressions of infinite series if the former only play an intermittent role in the construction. If a series had to express the value of some sort of quantity which was subject to calculation, then in order for it to be used it had to converge. Landen did not consider it necessary to explain precisely what he had in mind for ‘converget’ or ‘divergent’ series but instead, having expanded (by means of some sort of formative arrangement) the function into a series, he usually points out the radius of convergence of the derived series and introduces methods by which to ‘improve’ the convergence (to replace the series with another which converges ‘more rapidly’ to the same limit). Landen thus, among the number of ‘principles’ ‘already accepted since Antiquity in algebra and geometry’, obviously includes some concepts of the passage to a limit, with which he deals in practice (when speaking of an approximate calculation, for example). But he had no general concept of ‘convergence’ or ‘limit’. Nor did he have methods for calculating limits (or proving their non-existence) which included a wide variety of classes of functions. Landen therefore looked for a definition of the derivative (the ‘special value’) which would contain within itself its own algorithm.

Just like Newton, he spoke in terms of the function of x as an analogue of the concept of real numbers. In detail, just as any real number can be regarded as the (finite or infinite) sum of powers to the base 10, of which each one is denoted by the figures 0, 1, 2 ... 9, so any function of x, according to Newton, ought to be represented as the (finite or infinite) sum of powers of the base x, with each denoted by numbers (coefficients) - that is, as a power series. (A series was considered ‘representing’ a certain function given in terms of a finite ‘algebraic’ expression if the series is obtained by formal manipulation from the given function. So, for instance, the series 1 + x + x² + ... + xn + ... was considered to ‘represent’ the function 1/(1 - x) since it can be obtained by the division of 1 by 1 - x by means of the division of the polynomial.) The task of finding the derivative of the function f(x) could be represented as equivalent to the analogous task for the power xp and to the task, once knowing the derivatives of the elements (or factors), of finding the derivative of the sum. Just these problems Landen solved first of all in his Residual Analysis. The extension of these methods into functions of several variables and into partial derivatives of various orders, accompanied by a host of technical difficulties, Landen dealt with by means of occasionally very clever formal calculations.

In this it is usually implicitly assumed that the power series corresponding to the function is single-valued, that is, if two power series are to represent one and the same function of x, then the coefficients for each of the powers on them must be equal (hence the widespread use of the so-called ‘method of undefined coefficients’).

As an example illustrating Landen’s use of these methods we present his proposed (with several more precise definitions in use even today) demonstration of the binomial theorem of Newton for the general case of a binomial raised to a real exponent. Since Marx devoted special attention to this theorem of Newton, Primarily with respect to the theorems of Taylor and MacLaurin (see for example pp.109,116), Landen’s proof may provide interest in this connection.


(a + x)p = A1 + A2x + A3x² + ... , (1)

where p is any real number and A1, A2 ... are undefined coefficients assumed to be independent of x. Letting x = 0 on both sides of the equation yields AI = ap. The differentiation of the complete equation (1) with respect to x (Landen, of course, did not speak of the derivative with respect to x but of the corresponding ‘special value’ which he had for Axr where A is independent of x and r is real) becomes

p(a + x)p-1 = A2 + 2A3x + 3A4x² + ... (2)

Multiplying equation (1) by p and equation (2) by (a + x), we obtain

p(a + x)p = pA1 + pA2x + pA3x² + ... , (1´)

p(a + x)p = aA2 + ((2aA3)/A²)}x + ((3aA4)/(2A3)/}x² + ... , (2´)

from which, recalling the assumed single valuation of the expansion of the expression p(a + x)p into a series of powers of x, we have

aA2 = pA1, implies A2 = (p/a)A1 = pap-1 ,

2aA3 + A2 = pA2 , implies A3 = ((p - 1)/2a)A2 = (p(p - 1)/2)ap-2 ,

3aA4 + 2A3 = pA3 , implies A4 = ((p-2)/3a)A3

= ((p(p - 1) (p - 2))/(2⋅3))ap - 3 ,

.. .. .. ....

and therefore

(a + x)p = ap + (p/1)ap - 1 + ((p(p - 1))/(1⋅2))ap - 2x² + ((p(p - 1) (p - 2))/(1⋅2⋅3))ap - 3x³ + ... ,

which is the binomial theorem of Newton.

Although the residual analysis of John Landen did not become an everyday working instrument among mathematicians - Landen’s notation was cumbersome and he (perhaps therefore) did not reach the theorems of Taylor and MacLauring - it does not follow that Landen’s work was generally without influence in the development of mathematics. Landen himself writes (p.45) that several of his theorems from the Residual Analysis have ‘struck the attention of Mr De Moivre, Mr Stirling, and other eminent mathematicians’. In his Traité (Vol 1, p.240) Lacroix agrees that he employs the Landen method as an ‘imitation a l’algèbre’ for the proof of the binomial theorem and the expansion of exponential and logarithmic functions into a series. Lacroix’s textbook enjoyed a widespread popularity among mathematicians.

However, Lacroix’s notice was drawn to Landen through the influence of Lagrange, whose Théorie des fonctions analytique Lacroix made the basis for his Traité. In the introduction of this book, speaking of the difficulties remaining in the fundamental concepts of analysis according to Newton, Lagrange writes: ‘In order to avoid these difficulties, a skillful English geometer having made an important discovery in analysis, proposed to replace the method of fluxions, which until then all English geometricians used consistently, with another method, purely analytical and analagous to the method of differentials, but in which, instead of employing differences of variable quantities which are infinitely small or equal to zero, one uses at first the different values of these quantities which are then set equal, after having made, by division, the factor disappear which this equality sets equal to zero. By this means one truly avoids the infinitely small and vanishing quantities; but the results and the application of this calculus are embarrassing and inconvenient, and one must admit that this means of rendering the principles of calculus more rigorous at the same time sacrifices its principal advantages, simplicity of method and ease of operation.’ (In addition to the Residual Analysis Lagrange also cites ‘the discourse on the same subject published ... in 1758. See Oeuvres des Lagrange, Vol. IX, Paris, 1881, p.18).

The last comment of Lagrange is obviously related to the fact that Landen uses an extremely awkward notation and did not obtain the differential and the operations with the differential symbols of calculus.

Separate from Lagrange, Lacroix concludes that the method of Landen ‘reduces essentially to the method of limits’ (Traité, p.XVII).


* In order to show (2) using (1) it is sufficient to note that vm/r - wm/r)/(v - w) = (vm - wm)/(v - w) :(vm - wm)/(vm/r - wm/r) = (vm - wm)/(v - w) : ((vm/r)r - (wm/r)r)/(vm/r - wm/r) . Formula (3) follows easily from Formula (2) vm/r - wm/r)/(v - w) = ((vw)m/r(vm/r - wm/r))/((vw)m/r(v - w)m/r) = - (wm/r - vm/r)/((vw)m/r(w - v))