Chapter X: Act 1890: , the effect of which is explained in the article Insanity. Any (6)

From the similarity of the triangles P´TM´, PTM we have

y´:A - E = y:A,

where A denotes the subtangent TM. The point P´ being near the curve,
we may substitute in the equation of the curve x - E for x and (yA -
yE)/A for y. The equation of the curve is approximately satisfied. If
it is taken to be satisfied exactly, the result is an equation of the
form [phi](x, y, A, E) = 0, the left-hand member of which is divisible
by E. Omitting the factor E, and putting E = 0 in the remaining
factor, we have an equation which gives A. In this problem of tangents
also Fermat found the required result by a process equivalent to
differentiation.

Fermat gave several examples of the application of his method; among them was one in which he showed that he could differentiate very complicated irrational functions. For such functions his method was to begin by obtaining a rational equation. In rationalizing equations Fermat, in other writings, used the device of introducing new variables, but he did not use this device to simplify the process of differentiation. Some of his results were published by Pierre Hérigone in his _Supplementum cursus mathematici_ (1642). His communication to Descartes was not published in full until after his death (Fermat, _Opera varia_, 1679). Methods similar to Fermat's were devised by René de Sluse (1652) for tangents, and by Johannes Hudde (1658) for maxima and minima. Other methods for the solution of the problem of tangents were devised by Roberval and Torricelli, and published almost simultaneously in 1644. These methods were founded upon the composition of motions, the theory of which had been taught by Galileo (1638), and, less completely, by Roberval (1636). Roberval and Torricelli could construct the tangents of many curves, but they did not arrive at Fermat's artifice. This artifice is that which we have noted in § 10 as the fundamental artifice of the infinitesimal calculus.

Barrow's Differential Triangle.

17. Among the comparatively few mathematicians who before 1665 could perform differentiations was Isaac Barrow. In his book entitled _Lectiones opticae et geometricae_, written apparently in 1663, 1664, and published in 1669, 1670, he gave a method of tangents like that of Roberval and Torricelli, compounding two velocities in the directions of the axes of x and y to obtain a resultant along the tangent to a curve. In an appendix to this book he gave another method which differs from Fermat's in the introduction of a differential equivalent to our dy as well as dx. Two neighbouring ordinates PM and QN of a curve (fig. 7) are regarded as containing an indefinitely small (_indefinite parvum_) arc, and PR is drawn parallel to the axis of x. The tangent PT at P is regarded as identical with the secant PQ, and the position of the tangent is determined by the similarity of the triangles PTM, PQR. The increments QR, PR of the ordinate and abscissa are denoted by a and e; and the ratio of a to e is determined by substituting x + e for x and y + a for y in the equation of the curve, rejecting all terms which are of order higher than the first in a and e, and omitting the terms which do not contain a or e. This process is equivalent to differentiation. Barrow appears to have invented it himself, but to have put it into his book at Newton's request. The triangle PQR is sometimes called "Barrow's differential triangle."

Barrow's Inversion-theorem.

The reciprocal relation between differentiation and integration (§ 6)
was first observed explicitly by Barrow in the book cited above. If
the quadrature of a curve y = f(x) is known, so that the area up to
the ordinate x is given by F(x), the curve y = F(x) can be drawn, and
Barrow showed that the subtangent of this curve is measured by the
ratio of its ordinate to the ordinate of the original curve. The curve
y = F(x) is often called the "quadratrix" of the original curve; and
the result has been called "Barrow's inversion-theorem." He did not
use it as we do for the determination of quadratures, or indefinite
integrals, but for the solution of problems of the kind which were
then called "inverse problems of tangents." In these problems it was
sought to determine a curve from some property of its tangent, e.g.
the property that the subtangent is proportional to the square of the
abscissa. Such problems are now classed under "differential
equations." When Barrow wrote, quadratures were familiar and
differentiation unfamiliar, just as hyperbolas were trusted while
logarithms were strange. The functional notation was not invented till
long afterwards (see FUNCTION), and the want of it is felt in reading
all the mathematics of the 17th century.

Nature of the discovery called the Infinitesimal Calculus.

18. The great secret which afterwards came to be called the "infinitesimal calculus" was almost discovered by Fermat, and still more nearly by Barrow. Barrow went farther than Fermat in the theory of differentiation, though not in the practice, for he compared two increments; he went farther in the theory of integration, for he obtained the inversion-theorem. The great discovery seems to consist partly in the recognition of the fact that differentiation, known to be a useful process, could always be performed, at least for the functions then known, and partly in the recognition of the fact that the inversion-theorem could be applied to problems of quadrature. By these steps the problem of tangents could be solved once for all, and the operation of integration, as we call it, could be rendered systematic. A further step was necessary in order that the discovery, once made, should become accessible to mathematicians in general; and this step was the introduction of a suitable notation. The definite abandonment of the old tentative methods of integration in favour of the method in which this operation is regarded as the inverse of differentiation was especially the work of Isaac Newton; the precise formulation of simple rules for the process of differentiation in each special case, and the introduction of the notation which has proved to be the best, were especially the work of Gottfried Wilhelm Leibnitz. This statement remains true although Newton invented a systematic notation, and practised differentiation by rules equivalent to those of Leibnitz, before Leibnitz had begun to work upon the subject, and Leibnitz effected integrations by the method of recognizing differential coefficients before he had had any opportunity of becoming acquainted with Newton's methods.

Newton's investigations.

19. Newton was Barrow's pupil, and he knew to start with in 1664 all that Barrow knew, and that was practically all that was known about the subject at that time. His original thinking on the subject dates from the year of the great plague (1665-1666), and it issued in the invention of the "Calculus of Fluxions," the principles and methods of which were developed by him in three tracts entitled _De analysi per aequationes numero terminorum infinitas, Methodus fluxionum et serierum infinitarum, and De quadratura curvarum_. None of these was published until long after they were written. The _Analysis per aequationes_ was composed in 1666, but not printed until 1711, when it was published by William Jones. The _Methodus fluxionum_ was composed in 1671 but not printed till 1736, nine years after Newton's death, when an English translation was published by John Colson. In Horsley's edition of Newton's works it bears the title _Geometria analytica_. The _Quadratura_ appears to have been composed in 1676, but was first printed in 1704 as an appendix to Newton's _Opticks_.

Newton's method of Series.

20. The tract _De Analysi per aequationes ..._ was sent by Newton to
Barrow, who sent it to John Collins with a request that it might be
made known. One way of making it known would have been to print it in
the _Philosophical Transactions_ of the Royal Society, but this course
was not adopted. Collins made a copy of the tract and sent it to Lord
Brouncker, but neither of them brought it before the Royal Society.
The tract contains a general proof of Barrow's inversion-theorem which
is the same in principle as that in § 6 above. In this proof and
elsewhere in the tract a notation is introduced for the momentary
increment (_momentum_) of the abscissa or area of a curve; this
"moment" is evidently meant to represent a moment of time, the
abscissa representing time, and it is effectively the same as our
differential element--the thing that Fermat had denoted by E, and
Barrow by e, in the case of the abscissa. Newton denoted the moment of
the abscissa by o, that of the area z by ov. He used the letter v for
the ordinate y, thus suggesting that his curve is a velocity-time
graph such as Galileo had used. Newton gave the formula for the area
of a curve v = x^m (m ± -1) in the form z = x^(m+1)/(m + 1). In the
proof he transformed this formula to the form z^n = c^n x^p, where n
and p are positive integers, substituted x + o for x and z + ov for z,
and expanded by the binomial theorem for a positive integral exponent,
thus obtaining the relation

z^n + nz^(n-1) ov + ... = c^n (x_p + px^(p-1)o + ...),

from which he deduced the relation

nz_(n-1)v = c^n px^(p-1)

by omitting the equal terms z^n and c^n·x^p and dividing the remaining
terms by o, tacitly putting o = 0 after division. This relation is the
same as v = x^m. Newton pointed out that, conversely, from the
relation v = x^m the relation z = x^(m+1) / (m + 1) follows. He
applied his formula to the quadrature of curves whose ordinates can be
expressed as the sum of a finite number of terms of the form ax^m; and
gave examples of its application to curves in which the ordinate is
expressed by an infinite series, using for this purpose the binomial
theorem for negative and fractional exponents, that is to say, the
expansion of (1 + x)^n in an infinite series of powers of x. This
theorem he had discovered; but he did not in this tract state it in a
general form or give any proof of it. He pointed out, however, how it
may be used for the solution of equations by means of infinite series.
He observed also that all questions concerning lengths of curves,
volumes enclosed by surfaces, and centres of gravity, can be
formulated as problems of quadratures, and can thus be solved either
in finite terms or by means of infinite series. In the _Quadratura_
(1676) the method of integration which is founded upon the
inversion-theorem was carried out systematically. Among other results
there given is the quadrature of curves expressed by equations of the
form y = x^n·(a + bx^m)^p; this has passed into text-books under the
title "integration of binomial differentials" (see § 49). Newton
announced the result in letters to Collins and Oldenburg of 1676.

Newton's method of Fluxions.

21. In the _Methodus fluxionum_ (1671) Newton introduced his
characteristic notation. He regarded variable quantities as generated
by the motion of a point, or line, or plane, and called the generated
quantity a "fluent" and its rate of generation a "fluxion." The
fluxion of a fluent x is represented by x, and its moment, or
"infinitely" small increment accruing in an "infinitely" short time,
is represented by [.x]o. The problems of the calculus are stated to be
(i.) to find the velocity at any time when the distance traversed is
given; (ii.) to find the distance traversed when the velocity is
given. The first of these leads to differentiation. In any rational
equation containing x and y the expressions x + [.x]o and y +[.y]o are
to be substituted for x and y, the resulting equation is to be divided
by o, and afterwards o is to be omitted. In the case of irrational
functions, or rational functions which are not integral, new variables
are introduced in such a way as to make the equations contain rational
integral terms only. Thus Newton's rules of differentiation would be
in our notation the rules (i.), (ii.), (v.) of § 11, together with the
particular result which we write

dx^m
---- = mx^(m-1), (m integral).
dx

a result which Newton obtained by expanding (x = [.x]o)^m by the
binomial theorem. The second problem is the problem of integration,
and Newton's method for solving it was the method of series founded
upon the particular result which we write
_
/ x^(m+1)
| x^m dx = -------.
_/ m + 1

Newton added applications of his methods to maxima and minima,
tangents and curvature. In a letter to Collins of date 1672 Newton
stated that he had certain methods, and he described certain results
which he had found by using them. These methods and results are those
which are to be found in the _Methodus fluxionum_; but the letter
makes no mention of fluxions and fluents or of the characteristic
notation. The rule for tangents is said in the letter to be analogous
to de Sluse's, but to be applicable to equations that contain
irrational terms.

Publication of the Fluxional Notation.

22. Newton gave the fluxional notation also in the tract De
_Quadratura curvarum_ (1676), and he there added to it notation for
the higher differential coefficients and for indefinite integrals, as
we call them. Just as x, y, z, ... are fluents of which [.x], [.y],
[.z], ... are the fluxions, so [.x], [.y], [.z], ... can be treated as
fluents of which the fluxions may be denoted by [:x], [:y], [:z],...
In like manner the fluxions of these may be denoted by [:x], [:y],
[:z], ... and so on. Again x, y, z, ... may be regarded as fluxions of
which the fluents may be denoted by ['x], ['y], ['z], ... and these
again as fluxions of other quantities denoted by ["x], ["y], ["z], ...
and so on. No use was made of the notation ['x], ["x], ... in the
course of the tract. The first publication of the fluxional notation
was made by Wallis in the second edition of his _Algebra_ (1693) in
the form of extracts from communications made to him by Newton in
1692. In this account of the method the symbols 0, [.x], [:x], ...
occur, but not the symbols ['x], ["x], .... Wallis's treatise also
contains Newton's formulation of the problems of the calculus in the
words _Data aequatione fluentes quotcumque quantitates involvente
fluxiones invenire et vice versa_ ("an equation containing any number
of fluent quantities being given, to find their fluxions and vice
versa"). In the _Philosophiae naturalis principia mathematica_ (1687),
commonly called the "Principia," the words "fluxion" and "moment"
occur in a lemma in the second book; but the notation which is
characteristic of the calculus of fluxions is nowhere used.

Retarded Publication of the method of Fluxions.

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 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 diferentialis_, published in 1711 by William Jones, in which he laid the foundations of the "Calculus of Finite Differences."

Leibnitz's course of discovery.

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 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

1 1 1 1
--- [pi] = 1 - --- + --- - --- + ...
4 3 5 7

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 "[int] ydx," but within a day or two he wrote "[int] y". He regarded the symbol "[int]" 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 "[int]" represents "sum," "d" represents "difference." His notation appears to have been practically settled before the end of 1675, for in November he wrote [int] y dy = ½y², just as we do now.

Correspondence of Newton and Leibnitz.

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 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.

Leibnitz's Differential Calculus.

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, 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

d(x^m) = mx^(m-1) dx.

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.

Development of the Calculus.

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 [int] ydx. 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 "[int]." 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 (a^x) of becoming infinitely great in comparison with any power (x^n) 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.

Dispute concerning Priority.

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 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."

British and Continental Schools of Mathematics.

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 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.

Oppositions to the calculus.

The "Analyst" controversy.

Cauchy's method of limits.

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 (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 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 ... 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.

Arithmetical basis of modern analysis.

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 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.

III. _Outlines of the Infinitesimal Calculus._

32. The general notions of functionality, limits and continuity are explained in the article FUNCTION. Illustrations of the more immediate ways in which these notions present themselves in the development of the differential and integral calculus will be useful in what follows.

Geometrical limits.

Tangents.

33. Let y be given as a function of x, or, more generally, let x and y
be given as functions of a variable t. The first of these cases is
included in the second by putting x = t. If certain conditions are
satisfied the aggregate of the points determined by the functional
relations form a curve. The first condition is that the aggregate of
the values of t to which values of x and y correspond must be
continuous, or, in other words, that these values must consist of all
real numbers, or of all those real numbers which lie between assigned
extreme numbers. When this condition is satisfied the points are
"ordered," and their order is determined by the order of the numbers
t, supposed to be arranged in order of increasing or decreasing
magnitude; also there are two senses of description of the curve,
according as t is taken to increase or to diminish. The second
condition is that the aggregate of the points which are determined by
the functional relations must be "continuous." This condition means
that, if any point P determined by a value of t is taken, and any
distance [delta], however small, is chosen, it is possible to find two
points Q, Q´ of the aggregate which are such that (i.) P is between Q
and Q´, (ii.) if R, R´ are any points between Q and Q´ the distance
RR´ is less than [delta]. The meaning of the word "between" in this
statement is fixed by the ordering of the points. Sometimes additional
conditions are imposed upon the functional relations before they are
regarded as defining a curve. An aggregate of points which satisfies
the two conditions stated above is sometimes called a "Jordan curve."
It by no means follows that every curve of this kind has a tangent. In
order that the curve may have a tangent at P it is necessary that, if
any angle [alpha], however small, is specified, a distance [delta] can
be found such that when P is between Q and Q´, and PQ and PQ´ are less
than [delta], the angle RPR´ is less than [alpha] for all pairs of
points R, R´ which are between P and Q, or between P and Q´ (fig. 8).
When this condition is satisfied y is a function of x which has a
differential coefficient. The only way of finding out whether this
condition is satisfied or not is to attempt to form the differential
coefficient. If the quotient of differences [Delta]y/[Delta]x has a
limit when [Delta]x tends to zero, y is a differentiable function of
x, and the limit in question is the differential coefficient. The
derived function, or differential coefficient, of a function [f](x) is
always defined by the formula

d[f](x) [f](x + h) - [f](x)
[f]´(x) = ------- = lim. -------------------.
dx h=0 h

Rules for the formation of differential coefficients in particular
cases have been given in § 11 above. The definition of a differential
coefficient, and the rules of differentiation are quite independent of
any geometrical interpretation, such as that concerning tangents to a
curve, and the tangent to a curve is properly defined by means of the
differential coefficient of a function, not the differential
coefficient by means of the tangent.

Progressive and Regressive Differential Coefficients.

It may happen that the limit employed in defining the differential
coefficient has one value when h approaches zero through positive
values, and a different value when h approaches zero through negative
values. The two limits are then called the "progressive" and
"regressive" differential coefficients. In applications to dynamics,
when x denotes a coordinate and t the time, dx/dt denotes a velocity.
If the velocity is changed suddenly the progressive differential
coefficient measures the velocity just after the change, and the
regressive differential coefficient measures the velocity just before
the change. Variable velocities are properly defined by means of
differential coefficients.

Areas.

Lengths of Curves.

All geometrical limits may be specified in terms similar to those
employed in specifying the tangent to a curve; in difficult cases they
must be so specified. Geometrical intuition may fail to answer the
question of the existence or non-existence of the appropriate limits.
In the last resort the definitions of many quantities of geometrical
import must be analytical, not geometrical. As illustrations of this
statement we may take the definitions of the areas and lengths of
curves. We may not assume that every curve has an area or a length. To
find out whether a curve has an area or not, we must ascertain whether
the limit expressed by [f]ydx exists. When the limit exists the curve
has an area. The definition of the integral is quite independent of
any geometrical interpretation. The length of a curve again is defined
by means of a limiting process. Let P, Q be two points of a curve, and
R1, R2, ... R_(n-1) a set of intermediate points of the curve,
supposed to be described in the sense in which Q comes after P. The
points R are supposed to be reached successively in the order of the
suffixes when the curve is described in this sense. We form a sum of
lengths of chords

PR1 + R1R2 + ... + R_(n-1)Q.

If this sum has a limit when the number of the points R is increased
indefinitely and the lengths of all the chords are diminished
indefinitely, this limit is the length of the arc PQ. The limit is the
same whatever law may be adopted for inserting the intermediate points
R and diminishing the lengths of the chords. It appears from this
statement that the differential element of the arc of a curve is the
length of the chord joining two neighbouring points. In accordance
with the fundamental artifice for forming differentials (§§ 9, 10),
the differential element of arc ds may be expressed by the formula

ds = [root] {(dx)² + (dy)²},

of which the right-hand member is really the measure of the distance
between two neighbouring points on the tangent. The square root must
be taken to be positive. We may describe this differential element as
being so much of the actual arc between two neighbouring points as
need be retained for the purpose of forming the integral expression
for an arc. This is a description, not a definition, because the
length of the short arc itself is only definable by means of the
integral expression. Similar considerations to those used in defining
the areas of plane figures and the lengths of plane curves are
applicable to the formation of expressions for differential elements
of volume or of the areas of curved surfaces.

Constants of Integration.

34. In regard to differential coefficients it is an important theorem
that, if the derived function [f]´(x) vanishes at all points of an
interval, the function [f](x) is constant in the interval. It follows
that, if two functions have the same derived function they can only
differ by a constant. Conversely, indefinite integrals are
indeterminate to the extent of an additive constant.

Higher Differential Coefficients.

35. The differential coefficient dy/dx, or the derived function
[f]´(x), is itself a function of x, and its differential coefficient
is denoted by [f]´´(x) or d²y/dx². In the second of these notations
d/dx is regarded as the symbol of an operation, that of
differentiation with respect to x, and the index 2 means that the
operation is repeated. In like manner we may express the results of n
successive differentiations by [f]^(n)(x) or by d^n·y/dx^n. When the
second differential coefficient exists, or the first is
differentiable, we have the relation

[f](x + h) - 2[f](x) + [f](x - h)
[f]´´(x) = lim. --------------------------------- (i.)
h=0 h²

The limit expressed by the right-hand member of this equation may
exist in cases in which [f]´(x) does not exist or is not
differentiable. The result that, when the limit here expressed can be
shown to vanish at all points of an interval, then [f](x) must be a
linear function of x in the interval, is important.

The relation (i.) is a particular case of the more general relation

[f]^(n)(x) = lim.(h=0) h^-n [[f](x + nh) -n[f] {(x + (n - 1)h}

n(n - 1)
+ -------- [f]{x + (n - 2)h} - ... +(-1)^n [f](x)]. (ii.)
2!

As in the case of relation (i.) the limit expressed by the right-hand
member may exist although some or all of the derived functions
[f]´(x), [f]´´(x), ... [f]^(n-1)(x) do not exist.

Corresponding to the rule iii. of § 11 we have the rule for forming
the nth differential coefficient of a product in the form

d^n(uv) d^n v du d^(n-1)v n(n - 1) d²u d^(n-2)v d^n u
------- = u ----- + n -- -------- + -------- ---- -------- + ... + ----- v,
dx^n dx^n dx dx^(n-1) 1.2 dx² dx^(n-2) dx^n

where the coefficients are those of the expansion of (1 + x)^n in
powers of x (n being a positive integer). The rule is due to Leibnitz,
(1695).

_Differentials of higher orders_ may be introduced in the same way as
the differential of the first order. In general when y = [f](x), the
nth differential d^n·y is defined by the equation

d^n·y = [f]^n(x)(dx)^n,

in which dx is the (arbitrary) differential of x.

Symbols of operation.

When d/dx is regarded as a single symbol of operation the symbol [f]
... dx represents the inverse operation. If the former is denoted by
D, the latter may be denoted by D^-1. D^n means that the operation D
is to be performed n times in succession; D^-n that the operation of
forming the indefinite integral is to be performed n times in
succession. Leibnitz's course of thought (§ 24) naturally led him to
inquire after an interpretation of D^n. where n is not an integer. For
an account of the researches to which this inquiry gave rise,
reference may be made to the article by A. Voss in _Ency. d. math.
Wiss._ Bd. ii. A, 2 (Leipzig, 1889). The matter is referred to as
"fractional" or "generalized" differentiation.

Theorem of Intermediate Value.

36. After the formation of differential coefficients the most
important theorem of the differential calculus is the _theorem of
intermediate value_ ("theorem of mean value," "theorem of finite
increments," "Rolle's theorem," are other names for it). This theorem
may be explained as follows: Let A, B be two points of a curve y =
[f](x) (fig. 9). Then there is a point P between A and B at which the
tangent is parallel to the secant AB. This theorem is expressed
analytically in the statement that if [f]´(x) is continuous between a
and b, there is a value x1 of x between a and b which has the property
expressed by the equation

[f](b) - [f](a)
--------------- = [f]´(x1). (i.)
b - a

The value x1 can be expressed in the form a + [theta](b - a) where
[theta] is a number between 0 and 1.

A slightly more general theorem was given by Cauchy (1823) to the
effect that, if [f]´(x) and F´(x) are continuous between x = a and x =
b, then there is a number [theta] between 0 and 1 which has the
property expressed by the equation

F(b) - F(a) F´{a + [theta](b - a)}
--------------- = ------------------------.
[f](b) - [f](a) [f]´{a + [theta](b - a)}

The theorem expressed by the relation (i.) was first noted by Rolle
(1690) for the case where [f](x) is a rational integral function which
vanishes when x = a and also when x = b. The general theorem was given
by Lagrange (1797). Its fundamental importance was first recognized by
Cauchy (1823). It may be observed here that the theorem of integral
calculus expressed by the equation
_
/ b
F(b) - F(a) = | F´(x) dx
_/ a

follows at once from the definition of an integral and the theorem of
intermediate value.

The theorem of intermediate value may be generalized in the statement
that, if [f](x) and all its differential coefficients up to the nth
inclusive are continuous in the interval between x = a and x = b, then
there is a number [theta] between 0 and 1 which has the property
expressed by the equation

(b - a)² (b - a)^(n-1)
[f](b) = [f](a) + (b - a)[f]´(a) + -------- [f]´´ (a) + ... + ------------- [f]^(n-1)(a)
2! (n - 1)!

(b - a)^n
+ --------- [f]^(n) {a + [theta](b - a)}. (i.)
n!

Taylor's Theorem.

37. This theorem provides a means for computing the values of a
function at points near to an assigned point when the value of the
function and its differential coefficients at the assigned point are
known. The function is expressed by a terminated series, and, when the
remainder tends to zero as n increases, it may be transformed into an
infinite series. The theorem was first given by Brook Taylor in his
_Methodus Incrementorum_ (1717) as a corollary to a theorem concerning
finite differences. Taylor gave the expression for [f](x + z) in terms
of [f](x), [f]´(x), ... as an infinite series proceeding by powers of
z. His notation was that appropriate to the method of fluxions which
he used. This rule for expressing a function as an infinite series is
known as Taylor's theorem. The relation (i.), in which the remainder
after n terms is put in evidence, was first obtained by Lagrange
(1797). Another form of the remainder was given by Cauchy (1823) viz.,

(b - a)^n
--------- (1 - [theta])^(n-1) [f]^n {a + [theta](b - a)}.
(n - 1)!

The conditions of validity of Taylor's expansion in an infinite series
have been investigated very completely by A. Pringsheim (_Math. Ann._
Bd. xliv., 1894). It is not sufficient that the function and all its
differential coefficients should be finite at x = a; there must be a
_neighbourhood_ of a within which Cauchy's form of the remainder tends
to zero as n increases (cf. FUNCTION).

An example of the necessity of this condition is afforded by the
function f(x) which is given by the equation

__ n = [oo]
1 \ (-1)^n 1
[f](x) = ------ + ) ------ ------------ (i.)
1 + x² /__ n = 1 n! 1 + 3^(2n)x²

The sum of the series

x²
[f](0) + x[f]´(0) + -- [f]´´(0) + ... (ii.)
2!

is the same as that of the series

e^-1 - x² e^-3² + x^4 e^(-3^4) - ...