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* Arithmetica Universalis* ("Universal Arithmetic") is a mathematics text by Isaac Newton. Written in Latin, it was edited and published by William Whiston, Newton's successor as Lucasian Professor of Mathematics at the University of Cambridge. The

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

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

**William Whiston** was an English theologian, historian, and mathematician, a leading figure in the popularisation of the ideas of Isaac Newton. He is now probably best known for helping to instigate the Longitude Act in 1714 and his important translations of the *Antiquities of the Jews* and other works by Josephus. He was a prominent exponent of Arianism and wrote *A New Theory of the Earth*.

Whiston's original edition was published in 1707. It was translated into English by Joseph Raphson, who published it in 1720 as the *Universal Arithmetick*. John Machin published a second Latin edition in 1722.

**Joseph Raphson** was an English mathematician known best for the Newton–Raphson method.

**John Machin**, a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.

None of these editions credits Newton as author; Newton was unhappy with the publication of the *Arithmetica*, and so refused to have his name appear. In fact, when Whiston's edition was published, Newton was so upset he considered purchasing all of the copies so he could destroy them.

The *Arithmetica* touches on algebraic notation, arithmetic, the relationship between geometry and algebra, and the solution of equations. Newton also applied Descartes' rule of signs to imaginary roots. He also offered, without proof, a rule to determine the number of imaginary roots of polynomial equations. Not for another 150 years would a rigorous proof to Newton's counting formula be found (by James Joseph Sylvester, published in 1865).

**Geometry** is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

**Algebra** is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

**René Descartes** was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. One of the most notable intellectual figures of the Dutch Golden Age, Descartes is also widely regarded as one of the founders of modern philosophy.

**Diophantus of Alexandria** was an Alexandrian Hellenistic mathematician, who was the author of a series of books called *Arithmetica*, many of which are now lost. His texts deal with solving algebraic equations. While reading Claude Gaspard Bachet de Méziriac's edition of Diophantus' *Arithmetica,* Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted in the margin without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as *adaequalitas* in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.

The area of study known as the **history of mathematics** is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, together with Ancient Egypt and Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.

**Number theory** is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers.

In mathematics, **Galois theory** provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. It has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated ; showing that there is no quintic formula; and showing which polygons are constructible.

* La Géométrie* was published in 1637 as an appendix to

** Muḥammad ibn Mūsā al-Khwārizmī**, Arabized as al-Khwarizmi with al- and formerly Latinized as

The * Disquisitiones Arithmeticae* is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having a revolutionary impact on the field of number theory as it not only turned the field truly rigorous and systematic but also paved the path for modern number theory. In this book Gauss brought together and reconciled results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange, and Legendre and added many profound and original results of his own.

* The Compendious Book on Calculation by Completion and Balancing*, also known as

The * Ars Magna* is an important Latin-language book on algebra written by Gerolamo Cardano. It was first published in 1545 under the title

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.

In mathematics, the **Newton inequalities** are named after Isaac Newton. Suppose *a*_{1}, *a*_{2}, ..., *a*_{n} are real numbers and let denote the *k*th elementary symmetric function in *a*_{1}, *a*_{2}, ..., *a*_{n}. Then the **elementary symmetric means**, given by

A timeline of key algebraic developments are as follows:

This is a timeline of pure and applied mathematics history.

**Richard Sault** was an English mathematician, editor and translator, one of The Athenian Society. On the strength of his *Second Spira* he is also now credited as a Christian Cartesian philosopher.

**Michael Stifel** or **Styfel** was a German monk, Protestant reformer and mathematician. He was an Augustinian who became an early supporter of Martin Luther. He was later appointed professor of mathematics at Jena University.

**John Radford Young** was an English mathematician, professor and author, who was almost entirely self-educated. He was born of humble parents in London. At an early age he became acquainted with Olinthus Gilbert Gregory, who perceived his mathematical ability and assisted him in his studies. In 1823, while working in a private establishment for the deaf, he published *An Elementary Treatise on Algebra* with a dedication to Gregory. This treatise was followed by a series of elementary works, in which, following in the steps of Robert Woodhouse, Young familiarized English students with continental methods of mathematical analysis.

*Summa de arithmetica, geometria, proportioni et proportionalita* is a book on mathematics written by Luca Pacioli and first published in 1494. It contains a comprehensive summary of Renaissance mathematics, including practical arithmetic, basic algebra, basic geometry and accounting, written in Italian for use as a textbook.

- The
*Arithmetica Universalis*from the Grace K. Babson Collection, including links to PDFs of English and Latin versions of the*Arithmetica* -
*Arithmetica Universalis*(1720), translated by Joseph Raphson - Centre College Library information on Newton's works

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Images, videos and audio are available under their respective licenses.