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John Wallis (3 December 1616 – 8 November 1703 ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the symbol ∞ to represent the concept of infinity. He similarly used 1/∞ for an infinitesimal. John Wallis is often known to be a modern-day Newton and one of the greatest intellectuals of the early renaissance of mathematics.

John Wallis was born in Ashford, Kent, the third of five children of Reverend John Wallis and Joanna Chapman, where his father was a rector. He was initially educated at a school in Ashford but moved to James Movat's school in Tenterden in 1625 following an outbreak of plague. Wallis was first exposed to mathematics in 1631, at Martin Holbeach's school in Felsted; he enjoyed maths, but his study was erratic, since "mathematics, at that time with us, were scarce looked on as academical studies, but rather mechanical" (Scriba 1970). At the school in Felsted, Wallis learned how to speak and write Latin. By this time, he also was proficient in other languages such French, Greek, and Hebrew.

As it was intended he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge. While there, he kept an act on the doctrine of the circulation of the blood; that was said to have been the first occasion in Europe on which this theory was publicly maintained in a disputation. His interests, however, centred on mathematics. He received his Bachelor of Arts degree in 1637 and a Master's in 1640, afterwards entering the priesthood. From 1643 to 1649, he served as a nonvoting scribe at the Westminster Assembly. He was elected to a fellowship at Queens' College, Cambridge in 1644, from which he had to resign following his marriage.

Throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to Holbeach at Felsted School. He rendered them great practical assistance in deciphering Royalist dispatches. The quality of cryptography at that time was mixed; despite the individual successes of mathematicians such as François Viète, the principles underlying cipher design and analysis were very poorly understood. Most ciphers were ad hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. Wallis realised that the latter were far more secure – even describing them as "unbreakable", though he was not confident enough in this assertion to encourage revealing cryptographic algorithms. He was also concerned about the use of ciphers by foreign powers, refusing, for example, Gottfried Leibniz's request of 1697 to teach Hanoverian students about cryptography.

Returning to London – he had been made chaplain at St Gabriel Fenchurch in 1643 – Wallis joined the group of scientists that was later to evolve into the Royal Society. He was finally able to indulge his mathematical interests, mastering William Oughtred's Clavis Mathematicae in a few weeks in 1647. He soon began to write his own treatises, dealing with a wide range of topics, which he continued for the rest of his life.

Wallis joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I, by which he incurred the lasting hostility of the Independents. In spite of their opposition he was appointed in 1649 to the Savilian Chair of Geometry at Oxford University, where he lived until his death on 28 October 1703 (O.S.). Wallis always wanted to enter the church since he was little. In 1650, Wallis was ordained as a minister. After, he spent two years with Sir Richard Darley and Lady Vere as a private chaplain. In 1661, he was one of twelve Presbyterian representatives at the Savoy Conference.

Besides his mathematical works he wrote on theology, logic, English grammar and philosophy, and he was involved in devising a system for teaching a deaf boy to speak at Littlecote House. William Holder had earlier taught a deaf man, Alexander Popham, to speak "plainly and distinctly, and with a good and graceful tone". Wallis later claimed credit for this, leading Holder to accuse Wallis of "rifling his Neighbours, and adorning himself with their spoyls".

Contributions with Newton
John Wallis was also considered an editor of Isaac Newton’s mathematical work. Wallis was always open to the idea of making algebraic methods in print form, however Isaac Newton was not. Newton believed that algebra was not “worthy” enough to be published in any kind of form. Wallis eventually convinced Newton to the idea, and Newton agreed to publish some of his algebraic methods in Wallis’s published books “Algebra”, and “Latin Opera”. Working with Newton, Wallis learned a lot from it which helped him come up with more advanced ideas later in the future. = Michael Maestlin =

Michael Maestlin (also Mästlin, Möstlin, or Moestlin) (30 September 1550, Göppingen &#x2013; 20 October 1631, Tübingen) was a German astronomer and mathematician, known for being the mentor of Johannes Kepler. He was a student of Petrus Apianus and was known as the teacher who most influenced Kepler. Maestlin was considered to be one of the most significant astronomers between the time of Copernicus and Kepler.

Career
Maestlin studied theology, mathematics, and astronomy/astrology at the University of Tübingen—the Tübinger Stift. (Tübingen was part of the Duchy of Württemberg.) He graduated as magister in 1571 and became in 1576 a Lutheran deacon in Backnang, continuing his studies there.

In 1580 he became a professor of mathematics, first at the University of Heidelberg, then at the University of Tübingen, where he taught for 47 years from 1583. In 1582 Maestlin wrote a popular introduction to astronomy.

Among his students was Johannes Kepler (1571-1630). Although he primarily taught the traditional geocentric Ptolemaic view of the solar system, Maestlin was also one of the first to accept and teach the heliocentric Copernican view. Maestlin corresponded with Kepler frequently and played a sizable part in his adoption of the Copernican system. Galileo Galilei's adoption of heliocentrism was also attributed to Maestlin.

The first known calculation of the (inverse) golden ratio as a decimal of "about 0.6180340" was written in 1597 by Maestlin in a letter to Kepler.

Michael Maestlin was one of the very few astronomers that fully adopted the Copernican hypothesis, that proposed that the Earth was a planet and that it moved around the sun. Maestlin reacted to the thought of distant stars spinning around a fixed earth every 24 hours and taught everything that he could about Copernicus to Kepler.

Michael Maestlin also published a treatise and gave a short piece on the nova of 1573. This nova was called the Nova of Cassiopeia. It fascinated and intrigued lots of people, including astronomer Tycho Brache. However, it was said that Maestlin’s treatise was practically similar to Tycho’s treatise, De Stella Nova, which was published about three months later.

In 1580, Maestlin observed a comet where he paid close attention to it and began to gather up some ideas on how it formed. Nine years later in 1589, Maestlin came up with conclusions and informed his friend and astrologer, Helisaeus Roselin, who said that the moon was located in front of the Great Comet of 1577, about the appearance of the comet and the reasoning behind his conclusions.

In 1582, Maestlin wrote a popular introduction to astronomy by publishing the first edition of his astronomy textbook, that eventually consisted of six editions, called Epitome Astronomiae. Using Ptolemy’s work, including Ptolemy’s famous geocentric model, Maestlin used this work to help himself publish this textbook that contained a description of astronomy.

The preface in the book Narratio Prima was also written by Maestlin. This preface was an introduction to the work of Copernicus. Additionally, Maestlin made many contributions to several tables and diagrams in one of Kepler’s books, adding a treatise together with an introduction of his own. A discussion of the great sphere and the lunar sphere, as well as more discussion and conclusions to his descriptions of the Copernican planetary theory was also added by Maestlin in Kepler’s book.

= William Oughtred =

William Oughtred ( 5 March 1574 – 30 June 1660 ) was an English mathematician and Anglican clergyman. After John Napier invented logarithms and Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, Oughtred was the first to use two such scales sliding by one another to perform direct multiplication and division. He is credited with inventing the slide rule in about 1622. He also introduced the "×" symbol for multiplication and the abbreviations "sin" and "cos" for the sine and cosine functions. He is considered as one of the world’s greatest mathematics teachers. Oughtred was born at Eton in Buckinghamshire (now part of Berkshire), and educated at Eton College and King's College, Cambridge of which he became fellow. Being admitted to holy orders, he left the University of Cambridge about 1603, for a living at Shalford in Surrey; he was presented in 1610 to the rectory of Albury, near Guildford in Surrey, where he settled. He was rector of Albury for fifty years. His father was a writing-master at Eton College and taught him as well. Oughtred had a passion for mathematics, as he would stay up most nights and learn, while others were sleeping.

Books
Other works were a treatise on navigation entitled Circles of Proportion, in 1632, and a book on trigonometry and dialling, and his Opuscula Mathematica, published posthumously in 1676. He invented a universal equinoctial ring dial of two rings. These three books are important in the development of basic and elementary mathematics.

Clavis Mathematicae
The first edition of the Clavis Mathematicae was published in 1631 that consisted of 20 chapters and 88 pages that included algebra and several fundamentals of mathematics. Some changes were then added by Oughtred to the first edition, and a second and third edition were made in 1647 and 1648, having no preface and reducing the book by one chapter.

This book opens up with a discussion of the Hindu-Arabic notation of decimal fractions and later in the book, introduces multiplication and division sign abbreviations of decimal fractions. He also discusses two types of ways to do long division and introduces the “~” symbol, in terms of mathematics, expressing the difference between two variables.

Circle Proportion and the Horizontal Instrument
In the Circles of Proportion and the Horizontal Instrument, Oughtred introduces the abbreviations for trigonometric functions. This book was originally in manuscript before it eventually became published. Also, the slide rule is discussed, an invention that was made by Oughtred which provided a mechanical method of finding logarithmic results.

It is mentioned in this book that John Napier was the first person to ever use to the decimal point and comma, however German Pitiscus was actually the first to do so.

Trigonometria
Trigonometria contains about 36 pages of writing. In this book, the abbreviations for the trigonometric functions are explained in further detail consisting of mathematical tables.

Sundials
At the age of 23, Oughtred invented the double horizontal sundial. He invented the double horizontal sundial, now named Oughtred-type after him. A short description The description and use of the double Horizontall Dyall (16 pages) was added to a 1653 edition (in English translation) of the pioneer book on recreational mathematics, Récréations Mathématiques (1624) by Hendrik van Etten, a pseudonym of Jean Leurechon. The translation itself is no longer attributed to Oughtred, but (probably) to Francis Malthus.

He also invented the Universal equinoctial ring dial.