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]
Calculus 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.
Sir Isaac Newton was an English mathematician, physicist, astronomer, alchemist, theologian, and author who was described in his time as a natural philosopher. He was a key figure in the Scientific Revolution and the Enlightenment that followed. His pioneering book Philosophiæ Naturalis Principia Mathematica, first published in 1687, consolidated many previous results and established classical mechanics. Newton also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for developing infinitesimal calculus.
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f′, and an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
Abraham de Moivre FRS 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 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 proof 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.
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics to astronomical objects, such as stars and planets, to produce ephemeris data.
James Gregory FRS was a Scottish mathematician and astronomer. His surname is sometimes spelt 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 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.
Method of Fluxions is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. 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.
In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context.
Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Infinitesimal calculus was developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other. An argument over priority led to the Leibniz–Newton calculus controversy which continued until the death of Leibniz in 1716. The development of calculus and its uses within the sciences have continued to the present day.
In mathematics, a differential equation is an equation that relates one or more unknown 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.
In the history of calculus, 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 disfavour 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 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.