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THE CLAIM OF LEIBNITZ



INVENTION OF THE DIFFERENTIAL CALCULUS.



' . ' The present publication is a revised and enlarged edition of an Essay
which appeared in German in 1858. (Leipsig and Kiel. Schwers'schc Buchhandlung.)



« VMBRIDI

VH1NI BD )IY n I l:ll > M 1.1 1 I .



THE CLAIM OF LEIBNITZ

TO THE

INVENTION

OF THE

DIFFERENTIAL CALCULUS,



BY

DR. H. SLOMAN.



TRANSLATED FROM THE GERMAN WITH CONSIDERABLE
ALTERATIONS AND NEW ADDENDA BY THE AUTHOR.



MACMILLAN AND CO.

AND 23, HENRIETTA STREET. COVENT GARDEN, LONDON.

1860.






SIR DAVID BR E W S T E R.

THE
BIOGRAPHER OF NEWTON,

AND THE

REV. J. EDLESTON,

IS KESPECTEULLY DEDICATED
BY

THE AUTHOR.



CHAPTER I.



BARROW AND THE METHOD OF TANGENTS.

From about the year 1650, the vigorous mathematical life, in
which England had never been deficient, is seen to receive there an
extraordinary impulse, and attain to such a degree of development,
that that country became the centre of all the mathematical activity
of the period, while in France, after the death of Descartes, there
are no important men to name in mathematics.*

* Perhaps even Descartes was much indebted to the English Harriot. For
not only does the upright Wallis, who would never knowingly have uttered an
untruth, affirm this with zealous warmth in many passages of his Tractatus Algebra
historicus et practicus, but it was also believed by contemporaries, and at the same
time countrymen of Descartes's, who are spoken of in Baillet's Vita Cartesii, and
by Roberval, qui s ' entretenant un jour avec Milord Cavendish, lui temoignant etre
inquiet, d'ou etait venu a Descartes I'idee, d'egaler tons les termes d'une equation
a zero, Milord Cavendish lui dit, qiCil ri ignorait cela que parcequ'il etait Frangais,
et lui offrit de lui montrer le livre auquel Descartes devait cette invention. En effet
il le mena chez lui, et lui montra Vendroit de Harriot, ou Von voit la inane chose;
sur quoi Roberval, transports de joie, s'ecria, "il Va vu, il Va vu!" et il le publia
de toute part. We quote this out of Montuclat. II., p. 144. When Colbert in 1666
was looking about him for men, out of whom to form an Academie des Sciences, he
found no geometers or astronomers in France, except the following: viz., Auzout,
Buot, Carcavi, Couplet, Frenicle, Niquet, Picart, Richer, Roberval and De la Voye —
none of them, with the exception of Roberval, who died soon after, persons of
any great eminence. It was on them and their immediate successors that Leibnitz
and Bernouilli, who were both their colleagues, pronounced the following judgment :
(See Gerhardt's edition of the Math. Works of Leibnitz, p. 814 : the earlier editions

B



2 BARBOW AND THE METHOD OF TANGENTS.

Two problems occupied at that time the attention of geometers,
namely the problem of Tangents and that of Quadratures, in which
Barrow and "Wallis, in England, had achieved the most advanced
positions.

The two problems had as yet no mutual connexion ; for the object
contemplated was measure in one of them, and direction in the other.
It will be readily understood, that Barrow's method of tangents cannot
be left unnoticed in an enquiry like ours; and so indeed a great
deal is said respecting him, by the most modern writers in France
and Germany — as Biot in the "Journal des Savants/' and Gerhardt
in his various writings — who have aroused in the present day a lively
interest in the question, was Leibnitz the discoverer of the Differential
Calculus, and to what extent?

AYe need not on this point speak at much length. Barrow says,*
Nulla est magnitude*) qua- non innumeris modi's intelligi producta jiossit,



of the Correspondence do not contain this passage) Verissimum est, quod de nonniillis
Academicis notas — ct sane qua a se habent plerumque sunt mediocria, ne dicam ridicula —
et si quid boni cdunt, dubitare non lied, quin ab aliisfurati sint.

* Compare p. 15 of his principal work, and the one which made the greatest
noise at that time, entitled, Lectiones Geometrical in quibus prcesertim generalia
curvarum symptomata declarantur. Of this work the date is not without importance :
it was puhlished in 1670, (and not in 1674, as Gerhardt says in his tract of 1848,
p. 15 — nor yet in 1672, as he supposes in his tract of 1855, p. 45). That Leihnitz
before his discovery of the Differential Calculus either in 1676 or in 1675 or in
1674, should not have read this work, (as Gerhardt affirms in the place quoted,)
is inconceivable. Books were not so abundant in those times. Indeed evidences
to the contrary are contained in the documents, which Gerhardt himself produces.
In App. 1, to Gerhardt's tract of 1848, p. 32, Leibnitz says expressly, that he had
seen from Barrow's Lectiones " cum prodirent" — what they contained. This proves
that Leibnitz possessed Barrow's book not long after its first appearance, 1670.

unit gives another document, (Tract of 1855, p. 129,) from which the same
conclusion may be drawn. This document is, as Gerhardt affirms, dated in Leibnitz'
hand-writing 1 Nov., 167.3, and therein we have again Leibnitz' own words: Plera-
que theoremata Geometries indivisibilium, qua apud VavdUerium, J'inccntium, Gre-
gorium, Barrovium, extant, etc.



I



BARROW AND THE METHOD OF TANGENTS. 3

per motus locales, per inter sectiones magnitudinum, per quantitate por-
tioneque determinatas ab assignatis locis distantias, per ductus magni-
tudinum in magnitudines, per applicationes magnitudinum ad magnitudines,
per aggregationem magnitudinum ordine certo dispositarum, per appo-
sitionem magnitudinum ad alias, vel subductionem ab aliis. Horum
modus primarius, et quern alii omnes quodammodo supponant oportet
est iste per motum localem. In spite of this idea, which involves his
peculiar mode of contemplating the subject, Barrow is entirely devoted
to the more important method of Cavalleri, who considers every figure
as composed of parts infinitely minute and numerous, and every curve of
an infinity of straight lines. So he says, for example, at p. 15 : curva
aliquis, vel e rectis (angulos efficientibus) composita, quae curvae quoque
nomen merito ferat; Archimedes enim e rectis compositas lineas [uti
figurarum circuit's inscriptarum perimetros) KafMirvXwv


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Online LibraryH SlomanThe claim of Leibnitz to the invention of the differential calculus → online text (page 1 of 13)