De analysi per aequationes numero terminorum infinitas

Last updated

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.



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 ] , [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]


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]

See also

Related Research Articles

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

Isaac Newton Influential English physicist and mathematician

Sir Isaac Newton was an English mathematician, physicist, astronomer, theologian, and author who is widely recognised as one of the greatest mathematicians and most influential scientists of all time and as a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, established classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

Numerical analysis Field of mathematics

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. The growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, and subtle numerical analysis is required to implement these detailed models of the world. For example, ordinary differential equations appear in celestial mechanics ; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Mathematical analysis Branch of mathematics

Analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

Abraham de Moivre French mathematician

Abraham de Moivre was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

Isaac Barrow English Christian theologian, and mathematician

Isaac Barrow was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was the first to calculate the tangents of the kappa curve. He is also notable for being the inaugural holder of the prestigious Lucasian Professorship of Mathematics, a post later held by his student, Isaac Newton.

James Gregory (mathematician) Scottish mathematician and astronomer

James Gregory FRS was a Scottish mathematician and astronomer. His surname is sometimes spelled as Gregorie, the original Scottish spelling. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.

Roger Cotes

Roger Cotes FRS was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the Principia, before publication. He also invented the quadrature formulas known as Newton–Cotes formulas, and made a geometric argument that can be interpreted as a logarithmic version of Euler's formula. He was the first Plumian Professor at Cambridge University from 1707 until his death.

<i>Method of Fluxions</i> Book by Isaac Newton

Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxion is Newton's term for a derivative. He originally developed the method at Woolsthorpe Manor during the closing of Cambridge during the Great Plague of London from 1665 to 1667, but did not choose to make his findings known. Gottfried Leibniz developed his form of calculus independently around 1673, 7 years after Newton had developed the basis for differential calculus, as seen in surviving documents like “the method of fluxions and fluents..." from 1666. Leibniz however published his discovery of differential calculus in 1684, nine years before Newton formally published his fluxion notation form of calculus in part during 1693. The calculus notation in use today is mostly that of Leibniz, although Newton's dot notation for differentiation for denoting derivatives with respect to time is still in current use throughout mechanics and circuit analysis.

Johann Bernoulli Swiss mathematician

Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Leonhard Euler in the pupil's youth.

In mathematics education, precalculus or college algebra is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Wilhelm Leibniz independently developed the theory of infinitesimal calculus in the later 17th century. By the end of the 17th century, both Leibniz and Newton claimed that the other had stolen his work, and the Leibniz–Newton calculus controversy continued until the death of Leibniz in 1716.

Harlan J. Brothers American mathematician

Harlan J. Brothers is an inventor, composer, mathematician, and educator based in Branford, Connecticut.

Differential equation Mathematical equation involving derivatives of an unknown function

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Iriññāttappiḷḷi Mādhavan Nampūtiri known as Mādhava of Sangamagrāma was an Indian mathematician and astronomer from the town believed to be present-day Aloor, Irinjalakuda in Thrissur District, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".

Leibniz–Newton calculus controversy

The calculus controversy was an argument between the mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus. The question was a major intellectual controversy, which began simmering in 1699 and broke out in full force in 1711. Leibniz had published his work first, but Newton's supporters accused Leibniz of plagiarizing Newton's unpublished ideas. Leibniz died in disfavor in 1716 after his patron, the Elector Georg Ludwig of Hanover, became King George I of Great Britain in 1714. The modern consensus is that the two men developed their ideas independently.

The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among the most prolific and successful mathematicians in the history of the field. His seminal work had a profound impact in numerous areas of mathematics and he is widely credited for introducing and popularizing modern notation and terminology.

A timeline of calculus and mathematical analysis.


A fluxion is the instantaneous rate of change, or gradient, of a fluent at a given point. Fluxions were introduced by Isaac Newton to describe his form of a time derivative. Newton introduced the concept in 1665 and detailed them in his mathematical treatise, Method of Fluxions. Fluxions and fluents made up Newton's early calculus.


  1. The Mathematical Association of America .org 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. 1 2 3 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. 1 2 3 4 Britannica Educational The Britannica Guide to Analysis and Calculus. – 288 pages The Rosen Publishing Group, 1 July 2010. 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 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 Matematico) "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 Washington & Jefferson College Retrieved 8 February 2012