Newton and Leibniz: the
birth of calculus
works in the section
- Gottfried Wilhelm von
Leibniz, Nova methodus pro maximis et minimis, itemque
tangentibus, quae nec fractas nec irrationales quantitates moratur,
et singulare pro illis calculi genus, in Acta eruditorum,
- Isaac Newton,
Philosophiae naturalis Principia mathematica, editio secunda,
Cantabrigiae, [Cambridge University], 1713 [first edition
- Isaac Newton,
Methodus fluxionum et serierum infinitarum cum eisudem
applicatione ad curvarum geometriam, in Opuscola mathematica
philosophica et philologica, tomus primus, Lausannae et Genevae,
apud Marcum Michaelem Bousquet, 1744.
In October 1684
Leibniz published on the Acta
eruditorum his Nova methodus pro maximis et minimis,
itemque tangentibus, quae nec fractas nec irrationales quantitates
moratur, et singulare pro illis calculi genus.
This is traditionally considered to be the birth of infinitesimal calculus.
The title could be translated as "New method for maxima and minima, and for tangents, that is not hindered by fractional or irrational quantities,
and a singular kind of calculus for the above
mentioned". Reference to the work of Fermat is evident.
In the short report, Leibniz directly introduced the differentiation rules.
He manages to overcome the limits of the previous methods by
separating the difficulties deriving from the complexity of the equation
that was until then considered in its totality.
Almost twenty years before the publication of the
Nova Methodus of Leibniz, in 1665-1666, Newton
had already elaborated his calculus. The fundamental elements, with the
systematic use of developments in series, find their first version in the
De analysi per aequationes numero terminorum
1669 but published only in 1711.
The typical formulation of the problem in terms of finding the relations
(meaning the velocity of increasing) of given
"fluents or flowing quantities"
(meaning variables) appears in the following two works: the
Methodus fluxionum et serierum infinitarum and the De
quadratura curvarum, written respectively in 1671 and in 1676
but published themselves only later. The first publication of the results that
Newton had obtained takes place only in 1687, sometime later the
of Leibniz, with the
Philosophiae naturalis Principia mathematica.
At the opening of the first book, some lemmas illustrate the
fundaments of calculus in the form of "the prime and ultimate ratios for
evanescent quantities" and in the second book we find the algorithms
In these last ones Newton recognizes in an annotation, the fundaments
of his method as well as Leibniz's, method that the two scientists had
communicated to each other through their correspondence
of the previous ten years. In the third edition of the
Principia the reference disappears. This is the sign of the well known
argument between the two regarding the priority of the invention of calculus
that broke out at the end of the century and that divided the mathematicians
of the time.
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