1911 Encyclopædia Britannica/Infinitesimal Calculus/History 2
| II. History (§13-22) | Infinitesimal Calculus II. History (§23-31) |
III. Outlines of the Infinitesimal Calculus (§32-39) |
23. It is difficult to account for the fragmentary manner of publication of the Fluxional Calculus and for the long delays which took place. At the time (1671) when Newton composed the Methodus fluxionum he contemplated bringing out an edition of Gerhard Kinckhuysen’s Retarded Publication of the method of Fluxions. treatise on algebra and prefixing his tract to this treatise. In the same year his “Theory of Light and Colours” was published in the Philosophical Transactions, and the opposition which it excited led to the abandonment of the project with regard to fluxions. In 1680 Collins sought the assistance of the Royal Society for the publication of the tract, and this was granted in 1682. Yet it remained unpublished. The reason is unknown; but it is known that about 1679, 1680, Newton took up again the studies in natural philosophy which he had intermitted for several years, and that in 1684 he wrote the tract De motu which was in some sense a first draft of the Principia, and it may be conjectured that the fluxions were held over until the Principia should be finished. There is also reason to think that Newton had become dissatisfied with the arguments about infinitesimals on which his calculus was based. In the preface to the De quadratura curvarum (1704), in which he describes this tract as something which he once wrote (“olim scripsi”) he says that there is no necessity to introduce into the method of fluxions any argument about infinitely small quantities; and in the Principia (1687) he adopted instead of the method of fluxions a new method, that of “Prime and Ultimate Ratios.” By the aid of this method it is possible, as Newton knew, and as was afterwards seen by others, to found the calculus of fluxions on an irreproachable method of limits. For the purpose of explaining his discoveries in dynamics and astronomy Newton used the method of limits only, without the notation of fluxions, and he presented all his results and demonstrations in a geometrical form. There is no doubt that he arrived at most of his theorems in the first instance by using the method of fluxions. Further evidence of Newton’s dissatisfaction with arguments about infinitely small quantities is furnished by his tract Methodus differentialis, published in 1711 by William Jones, in which he laid the foundations of the “Calculus of Finite Differences.”
24. Leibnitz, unlike Newton, was practically a self-taught
mathematician. He seems to have been first attracted to
mathematics as a means of symbolical expression, and
on the occasion of his first visit to London, early in
1673, he learnt about the doctrine of infinite series
Leibnitz’s course
of discovery.
which James Gregory, Nicolaus Mercator, Lord
Brouncker and others, besides Newton, had used in their investigations.
It appears that he did not on this occasion become
acquainted with Collins, or see Newton’s Analysis per aequationes,
but he purchased Barrow’s Lectiones. On returning to
Paris he made the acquaintance of Huygens, who recommended
him to read Descartes’ Géométrie. He also read Pascal’s Lettres
de Dettonville, Gregory of St Vincent’s Opus geometricum,
Cavalieri’s Indivisibles and the Synopsis geometrica of Honoré
Fabri, a book which is practically a commentary on Cavalieri;
it would never have had any importance but for the influence
which it had on Leibnitz’s thinking at this critical period. In
August of this year (1673) he was at work upon the problem of
tangents, and he appears to have made out the nature of the
solution—the method involved in Barrow’s differential triangle—for
himself by the aid of a diagram drawn by Pascal in a demonstration
of the formula for the area of a spherical surface. He
saw that the problem of the relation between the differences
of neighbouring ordinates and the ordinates themselves was the
important problem, and then that the solution of this problem
was to be effected by quadratures. Unlike Newton, who arrived
at differentiation and tangents through integration and areas,
Leibnitz proceeded from tangents to quadratures. When he
turned his attention to quadratures and indivisibles, and
realized the nature of the process of finding areas by summing
“infinitesimal” rectangles, he proposed to replace the rectangles
by triangles having a common vertex, and obtained by this
method the result which we write
In 1674 he sent an account of his method, called “transmutation,” along with this result to Huygens, and early in 1675 he sent it to Henry Oldenburg, secretary of the Royal Society, with inquiries as to Newton’s discoveries in regard to quadratures. In October of 1675 he had begun to devise a symbolical notation for quadratures, starting from Cavalieri’s indivisibles. At first he proposed to use the word omnia as an abbreviation for Cavalieri’s “sum of all the lines,” thus writing omnia y for that which we write “∫ydx,” but within a day or two he wrote “∫y.” He regarded the symbol “∫” as representing an operation which raises the dimensions of the subject of operation—a line becoming an area by the operation—and he devised his symbol “d ” to represent the inverse operation, by which the dimensions are diminished. He observed that, whereas “∫” represents “sum,” “d ” represents “difference.” His notation appears to have been practically settled before the end of 1675, for in November he wrote ∫ydy = 12y 2, just as we do now.
25. In July of 1676 Leibnitz received an answer to his inquiry in regard to Newton’s methods in a letter written by Newton to Oldenburg. In this letter Newton gave a general statement of the binomial theorem and many results relating to series. He stated that by means of such Correspondence of Newton and Leibnitz. series he could find areas and lengths of curves, centres of gravity and volumes and surfaces of solids, but, as this would take too long to describe, he would illustrate it by examples. He gave no proofs. Leibnitz replied in August, stating some results which he had obtained, and which, as it seemed, could not be obtained easily by the method of series, and he asked for further information. Newton replied in a long letter to Oldenburg of the 24th of October 1676. In this letter he gave a much fuller account of his binomial theorem and indicated a method of proof. Further he gave a number of results relating to quadratures; they were afterwards printed in the tract De quadratura curvarum. He gave many other results relating to the computation of natural logarithms and other calculations in which series could be used. He gave a general statement, similar to that in the letter to Collins, as to the kind of problems relating to tangents, maxima and minima, &c., which he could solve by his method, but he concealed his formulation of the calculus in an anagram of transposed letters. The solution of the anagram was given eleven years later in the Principia in the words we have quoted from Wallis’s Algebra. In neither of the letters to Oldenburg does the characteristic notation of the fluxional calculus occur, and the words “fluxion” and “fluent” occur only in anagrams of transposed letters. The letter of October 1676 was not despatched until May 1677, and Leibnitz answered it in June of that year. In October 1676 Leibnitz was in London, where he made the acquaintance of Collins and read the Analysis per aequationes, and it seems to have been supposed afterwards that he then read Newton’s letter of October 1676, but he left London before Oldenburg received this letter. In his answer of June 1677 Leibnitz gave Newton a candid account of his differential calculus, nearly in the form in which he afterwards published it, and explained how he used it for quadratures and inverse problems of tangents. Newton never replied.
26. In the Acta eruditorum of 1684 Leibnitz published a short memoir entitled Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus. In this memoir the differential dx of a variable x, Leibnitz’s Differential Calculus. considered as the abscissa of a point of a curve, is said to be an arbitrary quantity, and the differential dy of a related variable y, considered as the ordinate of the point, is defined as a quantity which has to dx the ratio of the ordinate to the subtangent, and rules are given for operating with differentials. These are the rules for forming the differential of a constant, a sum (or difference), a product, a quotient, a power (or root). They are equivalent to our rules (i.)-(iv.) of § 11 and the particular result
The rule for a function of a function is not stated explicitly but is illustrated by examples in which new variables are introduced, in much the same way as in Newton’s Methodus fluxionum. In connexion with the problem of maxima and minima, it is noted that the differential of y is positive or negative according as y increases or decreases when x increases, and the discrimination of maxima from minima depends upon the sign of ddy, the differential of dy. In connexion with the problem of tangents the differentials are said to be proportional to the momentary increments of the abscissa and ordinate. A tangent is defined as a line joining two “infinitely” near points of a curve, and the “infinitely” small distances (e.g., the distance between the feet of the ordinates of such points) are said to be expressible by means of the differentials (e.g., dx). The method is illustrated by a few examples, and one example is given of its application to “inverse problems of tangents.” Barrow’s inversion-theorem and its application to quadratures are not mentioned. No proofs are given, but it is stated that they can be obtained easily by any one versed in such matters. The new methods in regard to differentiation which were contained in this memoir were the use of the second differential for the discrimination of maxima and minima, and the introduction of new variables for the purpose of differentiating complicated expressions. A greater novelty was the use of a letter (d), not as a symbol for a number or magnitude, but as a symbol of operation. None of these novelties account for the far-reaching effect which this memoir has had upon the development of mathematical analysis. This effect was a consequence of the simplicity and directness with which the rules of differentiation were stated. Whatever indistinctness might be felt to attach to the symbols, the processes for solving problems of tangents and of maxima and minima were reduced once for all to a definite routine.
27. This memoir was followed in 1686 by a second, entitled
De Geometria recondita et analysi indivisibilium atque infinitorum,
in which Leibnitz described the method of using his
new differential calculus for the problem of quadratures.
This was the first publication of the notation ∫ydx.
Development of
the Calculus.
The new method was called calculus summatorius.
The brothers Jacob (James) and Johann (John) Bernoulli were
able by 1690 to begin to make substantial contributions to
the development of the new calculus, and Leibnitz adopted
their word “integral” in 1695, they at the same time adopting
his symbol “∫.” In 1696 the marquis de l’Hospital published
the first treatise on the differential calculus with the title Analyse
des infiniment petits pour l’intelligence des lignes courbes. The
few references to fluxions in Newton’s Principia (1687) must
have been quite unintelligible to the mathematicians of the time,
and the publication of the fluxional notation and calculus by
Wallis in 1693 was too late to be effective. Fluxions had been
supplanted before they were introduced.
The differential calculus and the integral calculus were rapidly developed in the writings of Leibnitz and the Bernoullis. Leibnitz (1695) was the first to differentiate a logarithm and an exponential, and John Bernoulli was the first to recognize the property possessed by an exponential (ax) of becoming infinitely great in comparison with any power (xn) when x is increased indefinitely. Roger Cotes (1722) was the first to differentiate a trigonometrical function. A great development of infinitesimal methods took place through the founding in 1696–1697 of the “Calculus of Variations” by the brothers Bernoulli.
28. The famous dispute as to the priority of Newton and Leibnitz in the invention of the calculus began in 1699 through the publication by Nicolas Fatio de Duillier of a tract in which he stated that Newton was not only the first, but by many years the first inventor, and insinuated Dispute concerning Priority. that Leibnitz had stolen it. Leibnitz in his reply (Acta Eruditorum, 1700) cited Newton’s letters and the testimony which Newton had rendered to him in the Principia as proofs of his independent authorship of the method. Leibnitz was especially hurt at what he understood to be an endorsement of Duillier’s attack by the Royal Society, but it was explained to him that the apparent approval was an accident. The dispute was ended for a time. On the publication of Newton’s tract De quadratura curvarum, an anonymous review of it, written, as has since been proved, by Leibnitz, appeared in the Acta Eruditorum, 1705. The anonymous reviewer said: “Instead of the Leibnitzian differences Newton uses and always has used fluxions . . . just as Honoré Fabri in his Synopsis Geometrica substituted steps of movements for the method of Cavalieri.” This passage, when it became known in England, was understood not merely as belittling Newton by comparing him with the obscure Fabri, but also as implying that he had stolen his calculus of fluxions from Leibnitz. Great indignation was aroused; and John Keill took occasion, in a memoir on central forces which was printed in the Philosophical Transactions for 1708, to affirm that Newton was without doubt the first inventor of the calculus, and that Leibnitz had merely changed the name and mode of notation. The memoir was published in 1710. Leibnitz wrote in 1711 to the secretary of the Royal Society (Hans Sloane) requiring Keill to retract his accusation. Leibnitz’s letter was read at a meeting of the Royal Society, of which Newton was then president, and Newton made to the society a statement of the course of his invention of the fluxional calculus with the dates of particular discoveries. Keill was requested by the society “to draw up an account of the matter under dispute and set it in a just light.” In his report Keill referred to Newton’s letters of 1676, and said that Newton had there given so many indications of his method that it could have been understood by a person of ordinary intelligence. Leibnitz wrote to Sloane asking the society to stop these unjust attacks of Keill, asserting that in the review in the Acta Eruditorum no one had been injured but each had received his due, submitting the matter to the equity of the Royal Society, and stating that he was persuaded that Newton himself would do him justice. A committee was appointed by the society to examine the documents and furnish a report. Their report, presented in April 1712, concluded as follows:
“The differential method is one and the same with the method of fluxions, excepting the name and mode of notation; Mr Leibnitz calling those quantities differences which Mr Newton calls moments or fluxions, and marking them with the letter d, a mark not used by Mr Newton. And therefore we take the proper question to be, not who invented this or that method, but who was the first inventor of the method; and we believe that those who have reputed Mr Leibnitz the first inventor, knew little or nothing of his correspondence with Mr Collins and Mr Oldenburg long before; nor of Mr Newton’s having that method above fifteen years before Mr. Leibnitz began to publish it in the Acta Eruditorum of Leipzig. For which reasons we reckon Mr Newton the first inventor, and are of opinion that Mr Keill, in asserting the same, has been no ways injurious to Mr Leibnitz.”
The report with the letters and other documents was printed (1712) under the title Commercium Epistolicum D. Johannis Collins et aliorum de analysi promota, jussu Societatis Regiae in lucem editum, not at first for publication. An account of the contents of the Commercium Epistolicum was printed in the Philosophical Transactions for 1715. A second edition of the Commercium Epistolicum was published in 1722. The dispute was continued for many years after the death of Leibnitz in 1716. To translate the words of Moritz Cantor, it “redounded to the discredit of all concerned.”
29. One lamentable consequence of the dispute was a severance of British methods from continental ones. In Great Britain it became a point of honour to use fluxions and other Newtonian methods, while on the continent the notation of Leibnitz was universally adopted. This British and Continental Schools of Mathematics. severance did not at first prevent a great advance in mathematics in Great Britain. So long as attention was directed to problems in which there is but one independent variable (the time, or the abscissa of a point of a curve), and all the other variables depend upon this one, the fluxional notation could be used as well as the differential and integral notation, though perhaps not quite so easily. Up to about the middle of the 18th century important discoveries continued to be made by the use of the method of fluxions. It was the introduction of partial differentiation by Leonhard Euler (1734) and Alexis Claude Clairaut (1739), and the developments which followed upon the systematic use of partial differential coefficients, which led to Great Britain being left behind; and it was not until after the reintroduction of continental methods into England by Sir John Herschel, George Peacock and Charles Babbage in 1815 that British mathematics began to flourish again. The exclusion of continental mathematics from Great Britain was not accompanied by any exclusion of British mathematics from the continent. The discoveries of Brook Taylor and Colin Maclaurin were absorbed into the rapidly growing continental analysis, and the more precise conceptions reached through a critical scrutiny of the true nature of Newton’s fluxions and moments stimulated a like scrutiny of the basis of the method of differentials.
30. This method had met with opposition from the first.
Christiaan Huygens, whose opinion carried more weight than
that of any other scientific man of the day, declared
that the employment of differentials was unnecessary,
and that Leibnitz’s second differential was meaningless
Opposition to
the calculus.
(1691). A Dutch physician named Bernhard Nieuwentijt
attacked the method on account of the use of quantities
which are at one stage of the process treated as somethings and
at a later stage as nothings, and he was especially severe in
commenting upon the second and higher differentials (1694, 1695).
Other attacks were made by Michel Rolle (1701), but they
were directed rather against matters of detail than against the
general principles. The fact is that, although Leibnitz in his
answers to Nieuwentijt (1695), and to Rolle (1702), indicated
that the processes of the calculus could be justified by the
methods of the ancient geometry, he never expressed himself
very clearly on the subject of differentials, and he conveyed,
probably without intending it, the impression that the calculus
leads to correct results by compensation of errors. In England
the method of fluxions had to face similar attacks. George
Berkeley, bishop and philosopher, wrote in 1734 a tract entitled
The Analyst; or a Discourse addressed to an Infidel Mathematician,
in which he proposed to destroy the presumption that the
The “Analyst” controversy.
opinions of mathematicians in matters of faith are
likely to be more trustworthy than those of divines,
by contending that in the much vaunted fluxional
calculus there are mysteries which are accepted
unquestioningly by the mathematicians, but are incapable of
logical demonstration. Berkeley’s criticism was levelled against
all infinitesimals, that is to say, all quantities vaguely conceived
as in some intermediate state between nullity and finiteness,
as he took Newton’s moments to be conceived. The tract
occasioned a controversy which had the important consequence
of making it plain that all arguments about infinitesimals must
be given up, and the calculus must be founded on the method of
limits. During the controversy Benjamin Robins gave an
exceedingly clear explanation of Newton’s theories of fluxions
and of prime and ultimate ratios regarded as theories of limits.
In this explanation he pointed out that Newton’s moment
(Leibnitz’s “differential”) is to be regarded as so much of the
actual difference between two neighbouring values of a variable
as is needful for the formation of the fluxion (or differential
coefficient) (see G. A. Gibson, “The Analyst Controversy,”
Proc. Math. Soc., Edinburgh, xvii., 1899). Colin Maclaurin
published in 1742 a Treatise of Fluxions, in which he reduced
the whole theory to a theory of limits, and demonstrated it by
the method of Archimedes. This notion was gradually transferred
to the continental mathematicians. Leonhard Euler
in his Institutiones Calculi differentialis (1755) was reduced to the
position of one who asserts that all differentials are zero, but,
as the product of zero and any finite quantity is zero, the ratio
of two zeros can be a finite quantity which it is the business
of the calculus to determine. Jean le Rond d’Alembert in the
Encyclopédie méthodique (1755, 2nd ed. 1784) declared that
differentials were unnecessary, and that Leibnitz’s calculus was
a calculus of mutually compensating errors, while Newton’s
method was entirely rigorous. D’Alembert’s opinion of Leibnitz’s
calculus was expressed also by Lazare N. M. Carnot in his
Réflexions sur la métaphysique du calcul infinitésimal (1799)
and by Joseph Louis de la Grange (generally called Lagrange)
in writings from 1760 onwards. Lagrange proposed in his
Théorie des fonctions analytiques (1797) to found the whole of the
calculus on the theory of series. It was not until 1823 that a
treatise on the differential calculus founded upon the method
of limits was published. The treatise was the Résumé des leçons
Cauchy’s method
of limits.
. . . sur le calcul infinitésimal of Augustin Louis Cauchy.
Since that time it has been understood that the use of the
phrase “infinitely small” in any mathematical argument
is a figurative mode of expression pointing to a
limiting process. In the opinion of many eminent
mathematicians such modes of expression are
confusing to students, but in treatises on the
calculus the traditional modes of expression are still largely
adopted.
31. Defective modes of expression did not hinder constructive
work. It was the great merit of Leibnitz’s symbolism that
a mathematician who used it knew what was to be
done in order to formulate any problem analytically,
even though he might not be absolutely clear as to the
Arithmetical basis
of modern analysis.
proper interpretation of the symbols, or able to render
a satisfactory account of them. While new and varied
results were promptly obtained by using them, a long time elapsed
before the theory of them was placed on a sound basis. Even
after Cauchy had formulated his theory much remained to be
done, both in the rapidly growing department of complex
variables, and in the regions opened up by the theory of expansions
in trigonometric series. In both directions it was seen
that rigorous demonstration demanded greater precision in
regard to fundamental notions, and the requirement of precision
led to a gradual shifting of the basis of analysis from geometrical
intuition to arithmetical law. A sketch of the outcome of this
movement—the “arithmetization of analysis,” as it has been
called—will be found in Function. Its general tendency has
been to show that many theories and processes, at first accepted
as of general validity, are liable to exceptions, and much of the
work of the analysts of the latter half of the 19th century was
directed to discovering the most general conditions in which
particular processes, frequently but not universally applicable,
can be used without scruple.