De analysi per aequationes numero terminorum infinitas (or On analysis by infinite series,[1] On Analysis by Equations with an infinite number of terms,[2] or On the Analysis by means of equations of an infinite number of terms)[3] is a mathematical work by Isaac Newton.

## Creation

Composed in 1669,[4] during the mid-part of that year probably,[5] from ideas Newton had acquired during the period 1665–1666.[4] Newton wrote

And whatever the common Analysis performs by Means of Equations of a finite number of Terms (provided that can be done) this new method can always perform the same by means of infinite Equations. So that I have not made any Question of giving this the name of Analysis likewise. For the Reasonings in this are no less certain than in the other, nor the Equations less exact; albeit we Mortals whose reasoning Powers are confined within narrow Limits, can neither express, nor so conceive the Terms of these Equations as to know exactly from thence the Quantities we want. To conclude, we may justly reckon that to belong to the Analytic Art, by the help of which the Areas and Lengths, etc. of Curves may be exactly and geometrically determined. Newton[4]

The explication was written to remedy apparent weaknesses in the logarithmic series[6] [infinite series for ${\displaystyle \log(1+x)}$] ,[7] that had become republished due to Nicolaus Mercator,[6][8] or through the encouragement of Isaac Barrow in 1669, to ascertain the knowing of the prior authorship of a general method of infinite series. The writing was circulated amongst scholars as a manuscript in 1669,[6][9] including John Collins a mathematics intelligencer[10] for a group of British and continental mathematicians. His relationship with Newton in the capacity of informant proved instrumental in securing Newton recognition and contact with John Wallis at the Royal Society.[11][12] Both Cambridge University Press and Royal Society rejected the treatise from publication,[6] being instead published in London in 1711[13] by William Jones,[14] and again in 1744,[15] as Methodus fluxionum et serierum infinitarum cum eisudem applicatione ad curvarum geometriam[16] in Opuscula mathematica, philosophica et philologica by Marcum-Michaelem Bousquet at that time edited by Johann Castillioneus.[17]

## Content

The exponential series, i.e., tending toward infinity, was discovered by Newton and is contained within the Analysis. The treatise contains also the sine series and cosine series and arc series, the logarithmic series and the binomial series.[18]

## References

1. ^ The Mathematical Association of America .org Archived 1 July 2013 at the Wayback Machine Retrieved 3 February 2012 & newtonproject Retrieved 6 February 2012
2. ^ Nicholls State University Thibodaux, Louisiana .edu heck teaching 573 Retrieved 3 February 2012
3. ^ I. Grattan-Guinness 2005 – Landmark writings in Western mathematics 1640–1940 – 1022 pages (Google eBook) Elsevier, 20 May 2005 Retrieved 27 January 2012 ISBN 0-444-50871-6
4. ^ a b c Carl B. Boyer, Uta C. Merzbach A History of Mathematics. – 640 pages John Wiley and Sons, 11 November 2010. 2011. Retrieved 27 January 2012. ISBN 0-470-63056-6
5. ^ Endre Süli, David Francis Mayers 2003 – An introduction to numerical analysis – 433 pages Cambridge University Press, 28 Aug 2003 Retrieved 27 January 2012 ISBN 0-521-00794-1
6. ^ a b c d Britannica EducationalThe Britannica Guide to Analysis and Calculus. – 288 pages The Rosen Publishing Group, 1 July 2010. 2010. ISBN 9781615302208. Retrieved 27 January 2012. ISBN 1-61530-220-4
7. ^ B.B.Blank reviewing The Calculus Wars: Newton, Leibniz and the greatest mathematical clash of all time by J.S.Bardi pdf Retrieved 8 February 2012
8. ^ Babson College archives-and-collections Archived 22 January 2018 at the Wayback Machine Retrieved 8 February 2012
9. ^ King's College London © 2010 – 2012 King's College London Retrieved 27 January 2012
10. ^ Birch, History of Royal Society, et al. (Richard S. Westfall ed.) Rice University galileo.edu Retrieved 8 February 2012
11. ^ D.Harper – index Retrieved 8 February 2012
12. ^ Niccolò Guicciardini & University of Bergamo – Isaac Newton on mathematical certainty and method, Issue 4 – 422 pages ISBN 0-262-01317-7 Transformations: Studies in the History of Science and Technology MIT Press, 30 Oct 2009 & John Wallis as editor of Newton's mathematical work The Royal Society 2012 Retrieved 8 February 2012
13. ^ Anders Hald 2003 – A history of probability and statistics and their applications before 1750 – 586 pages Volume 501 of Wiley series in probability and statistics Wiley-IEEE, 2003 Retrieved 27 January 2012 ISBN 0-471-47129-1
14. ^ Alexander Gelbukh, Eduardo F. Morales – MICAI 2008: advances in artificial intelligence : 7th Mexican International Conference on Artificial Intelligence, Atizapán de Zaragoza, Mexico, 27–31 October 2008 : proceedings (Google eBook) – 1034 pages Volume 5317 of Lecture Notes in Artificial Intelligence Springer, 2008 Retrieved 27 January 2012 ISBN 3-540-88635-4
15. ^ Nicolas Bourbaki (Henri Cartan, Claude Chevalley, Jean Dieudonné, André Weil et al) – Functions of a real variable: elementary theory – 338 pages Springer, 2004 Retrieved 27 January 2012
16. ^ Department of Mathematics (Dipartimento di Matematica) "Ulisse Dini" html Retrieved 27 January 2012
17. ^ ISAACI NEWTONI – Opuscula [ apud Marcum-Michaelem Bousquet & socios, 1744 ] Retrieved 2012-01-27 originally from Ghent University digitalized on 26 October 2007
18. ^ M. Woltermann Archived 5 August 2012 at archive.today Washington & Jefferson College[1] Archived 17 April 2018 at the Wayback Machine Retrieved 8 February 2012