Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz was a prominent German polymath and philosopher in the history of mathematics and the history of philosophy. His most notable accomplishment was conceiving the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments. Mathematical works have always favored Leibniz's notation as the conventional expression of calculus, while Newton's notation became unused, it was only in the 20th century that Leibniz's law of continuity and transcendental law of homogeneity found mathematical implementation. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator, he refined the binary number system, the foundation of all digital computers. In philosophy, Leibniz is most noted for his optimism, i.e. his conclusion that our universe is, in a restricted sense, the best possible one that God could have created, an idea, lampooned by others such as Voltaire.
Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism. The work of Leibniz anticipated modern logic and analytic philosophy, but his philosophy looks back to the scholastic tradition, in which conclusions are produced by applying reason to first principles or prior definitions rather than to empirical evidence. Leibniz made major contributions to physics and technology, anticipated notions that surfaced much in philosophy, probability theory, medicine, psychology and computer science, he wrote works on philosophy, law, theology and philology. Leibniz contributed to the field of library science. While serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would serve as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, in unpublished manuscripts, he wrote in several languages, but in Latin and German.
There is no complete gathering of the writings of Leibniz translated into English. Gottfried Leibniz was born on 1 July 1646, toward the end of the Thirty Years' War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his family journal: 21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, im Wassermann. In English: On Sunday 21 June 1646, my son Gottfried Wilhelm is born into the world a quarter before seven in the evening, in Aquarius. Leibniz was baptized on 3 July of that year at Leipzig, his father died when he was six years old, from that point on he was raised by his mother. Leibniz's father had been a Professor of Moral Philosophy at the University of Leipzig, the boy inherited his father's personal library, he was given free access to it from the age of seven. While Leibniz's schoolwork was confined to the study of a small canon of authorities, his father's library enabled him to study a wide variety of advanced philosophical and theological works—ones that he would not have otherwise been able to read until his college years.
Access to his father's library written in Latin led to his proficiency in the Latin language, which he achieved by the age of 12. He composed 300 hexameters of Latin verse, in a single morning, for a special event at school at the age of 13. In April 1661 he enrolled in his father's former university at age 14, completed his bachelor's degree in Philosophy in December 1662, he defended his Disputatio Metaphysica de Principio Individui, which addressed the principle of individuation, on 9 June 1663. Leibniz earned his master's degree in Philosophy on 7 February 1664, he published and defended a dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum, arguing for both a theoretical and a pedagogical relationship between philosophy and law, in December 1664. After one year of legal studies, he was awarded his bachelor's degree in Law on 28 September 1665, his dissertation was titled De conditionibus. In early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria, the first part of, his habilitation thesis in Philosophy, which he defended in March 1666.
His next goal was to earn his license and Doctorate in Law, which required three years of study. In 1666, the University of Leipzig turned down Leibniz's doctoral application and refused to grant him a Doctorate in Law, most due to his relative youth. Leibniz subsequently left Leipzig. Leibniz enrolled in the University of Altdorf and submitted a thesis, which he had been working on earlier in Leipzig; the title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure. Leibniz earned his license to practice law and his Doctorate in Law in November 1666, he next declined the offer of an academic appointment at Altdorf, saying that "my thoughts were turned in an different direction". As an adult, Leibniz often
Isaac Newton
Sir Isaac Newton was an English mathematician, astronomer and author, recognised as one of the most influential scientists of all time, a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton made seminal contributions to optics, shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. In Principia, Newton formulated the laws of motion and universal gravitation that formed the dominant scientific viewpoint until it was superseded by the theory of relativity. Newton used his mathematical description of gravity to prove Kepler's laws of planetary motion, account for tides, the trajectories of comets, the precession of the equinoxes and other phenomena, eradicating doubt about the Solar System's heliocentricity, he demonstrated that the motion of objects on Earth and celestial bodies could be accounted for by the same principles. Newton's inference that the Earth is an oblate spheroid was confirmed by the geodetic measurements of Maupertuis, La Condamine, others, convincing most European scientists of the superiority of Newtonian mechanics over earlier systems.
Newton built the first practical reflecting telescope and developed a sophisticated theory of colour based on the observation that a prism separates white light into the colours of the visible spectrum. His work on light was collected in his influential book Opticks, published in 1704, he formulated an empirical law of cooling, made the first theoretical calculation of the speed of sound, introduced the notion of a Newtonian fluid. In addition to his work on calculus, as a mathematician Newton contributed to the study of power series, generalised the binomial theorem to non-integer exponents, developed a method for approximating the roots of a function, classified most of the cubic plane curves. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, he was a devout but unorthodox Christian who rejected the doctrine of the Trinity. Unusually for a member of the Cambridge faculty of the day, he refused to take holy orders in the Church of England.
Beyond his work on the mathematical sciences, Newton dedicated much of his time to the study of alchemy and biblical chronology, but most of his work in those areas remained unpublished until long after his death. Politically and tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02, he was knighted by Queen Anne in 1705 and spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint, as well as president of the Royal Society. Isaac Newton was born on Christmas Day, 25 December 1642 "an hour or two after midnight", at Woolsthorpe Manor in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire, his father named Isaac Newton, had died three months before. Born prematurely, Newton was a small child; when Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Margery Ayscough.
Newton disliked his stepfather and maintained some enmity towards his mother for marrying him, as revealed by this entry in a list of sins committed up to the age of 19: "Threatening my father and mother Smith to burn them and the house over them." Newton's mother had three children from her second marriage. From the age of about twelve until he was seventeen, Newton was educated at The King's School, which taught Latin and Greek and imparted a significant foundation of mathematics, he was removed from school, returned to Woolsthorpe-by-Colsterworth by October 1659. His mother, widowed for the second time, attempted to make him an occupation he hated. Henry Stokes, master at The King's School, persuaded his mother to send him back to school. Motivated by a desire for revenge against a schoolyard bully, he became the top-ranked student, distinguishing himself by building sundials and models of windmills. In June 1661, he was admitted to Trinity College, Cambridge, on the recommendation of his uncle Rev William Ayscough, who had studied there.
He started as a subsizar—paying his way by performing valet's duties—until he was awarded a scholarship in 1664, guaranteeing him four more years until he could get his MA. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers such as Descartes, astronomers such as Galileo and Thomas Street, through whom he learned of Kepler's work, he set down in his notebook a series of "Quaestiones" about mechanical philosophy. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that became calculus. Soon after Newton had obtained his BA degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the subsequent two years saw the development of his theories on calculus and the law of gravitation. In April 1667, he returned in October was elected as a fellow of Trinity.
Fellows were required to become ordained priests, although this was no
Method of Fluxions
Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, published in 1736. Fluxion is Newton's term for a derivative, he 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 that of Leibniz, although Newton's dot notation for differentiation x ˙ for denoting derivatives with respect to time is still in current use throughout mechanics and circuit analysis. Newton's Method of Fluxions was formally published posthumously, but following Leibniz's publication of the calculus a bitter rivalry erupted between the two mathematicians over who had developed the calculus first and so Newton no longer hid his knowledge of fluxions.
For a period of time encompassing Newton's working life, the discipline of analysis was a subject of controversy in the mathematical community. Although analytic techniques provided solutions to long-standing problems, including problems of quadrature and the finding of tangents, the proofs of these solutions were not known to be reducible to the synthetic rules of Euclidean geometry. Instead, analysts were forced to invoke infinitesimal, or "infinitely small", quantities to justify their algebraic manipulations; some of Newton's mathematical contemporaries, such as Isaac Barrow, were skeptical of such techniques, which had no clear geometric interpretation. Although in his early work Newton used infinitesimals in his derivations without justifying them, he developed something akin to the modern definition of limits in order to justify his work. Method of Fluxions at the Internet Archive
John Herschel
Sir John Frederick William Herschel, 1st Baronet was an English polymath, astronomer, inventor, experimental photographer who invented the blueprint, did botanical work. Herschel originated the use of the Julian day system in astronomy, he named four moons of Uranus. He made many contributions to the science of photography, investigated colour blindness and the chemical power of ultraviolet rays. Herschel was born in Slough, the son of Mary Baldwin and William Herschel, he was the nephew of astronomer Caroline Herschel. He studied shortly at Eton College and St John's College, graduating as Senior Wrangler in 1813, it was during his time as an undergraduate that he became friends with the mathematicians Charles Babbage and George Peacock. He started working with his father, he took up astronomy in 1816, building a reflecting telescope with a mirror 18 inches in diameter, with a 20-foot focal length. Between 1821 and 1823 he re-examined, with the double stars catalogued by his father, he was one of the founders of the Royal Astronomical Society in 1820.
For his work with his father, he was presented with the Gold Medal of the Royal Astronomical Society in 1826, with the Lalande Medal of the French Academy of Sciences in 1825, while in 1821 the Royal Society bestowed upon him the Copley Medal for his mathematical contributions to their Transactions. Herschel was made a Knight of the Royal Guelphic Order in 1831. Herschel served as President of the Royal Astronomical Society three times: 1827–29, 1839–41 and 1847–49. Herschel's A preliminary discourse on the study of natural philosophy, published early in 1831 as part of Dionysius Lardner's Cabinet cyclopædia, set out methods of scientific investigation with an orderly relationship between observation and theorising, he described nature as being governed by laws which were difficult to discern or to state mathematically, the highest aim of natural philosophy was understanding these laws through inductive reasoning, finding a single unifying explanation for a phenomenon. This became an authoritative statement with wide influence on science at the University of Cambridge where it inspired the student Charles Darwin with "a burning zeal" to contribute to this work.
Herschel published a catalogue of his astronomical observations in 1864, as the General Catalogue of Nebulae and Clusters, a compilation of his own work and that of his father's, expanding on the senior Herschel's Catalogue of Nebulae. A further complementary volume was published posthumously, as the General Catalogue of 10,300 Multiple and Double Stars. Herschel considered astigmatism to be due to irregularity of the cornea and theorised that vision could be improved by the application of some animal jelly contained in a capsule of glass against the cornea, his views were published in an article entitled Light in 1828 and the Encyclopædia Metropolitana in 1845. Discoveries of Herschel include the galaxies NGC 7, NGC 10, NGC 25, NGC 28 Declining an offer from the Duke of Sussex that they travel to South Africa on a Navy ship and his wife paid £500 for passage on the S. S. Mountstuart Elphinstone, which departed from Portsmouth on 13 November 1833; the voyage to South Africa was made in order to catalogue the stars and other objects of the southern skies.
This was to be a completion as well as extension of the survey of the northern heavens undertaken by his father William Herschel. He arrived in Cape Town on 15 January 1834 and set up a private 21 ft telescope at Feldhausen at Claremont, a suburb of Cape Town. Amongst his other observations during this time was that of the return of Comet Halley. Herschel collaborated with Thomas Maclear, the Astronomer Royal at the Cape of Good Hope and the members of the two families became close friends. During this time, he witnessed the Great Eruption of Eta Carinae. In addition to his astronomical work, this voyage to a far corner of the British empire gave Herschel an escape from the pressures under which he found himself in London, where he was one of the most sought-after of all British men of science. While in southern Africa, he engaged in a broad variety of scientific pursuits free from a sense of strong obligations to a larger scientific community, it was, he recalled the happiest time in his life.
In an extraordinary departure from astronomy, Herschel combined his talents with those of his wife and between 1834 and 1838 they produced 131 botanical illustrations of fine quality, showing the Cape flora. Herschel used a camera lucida to obtain accurate outlines of the specimens and left the details to his wife. Though their portfolio had been intended as a personal record, despite the lack of floral dissections in the paintings, their accurate rendition makes them more valuable than many contemporary collections; some 112 of the 132 known flower studies were collected and published as Flora Herscheliana in 1996. As their home during their stay in the Cape, the Herschels had selected'Feldhausen', an old estate on the south-eastern side of Table Mountain. Here John set up his reflector to begin his survey of the southern skies. Herschel, at the same time, read widely. Intrigued by the ideas of gradual formation of landscapes set out in Charles Lyell's Principles of Geology, he wrote to Lyell on 20 February 1836 praising the book as a work that would bring "a complete revolution in subject, by alterin
William Whewell
Rev Dr William Whewell DD HFRSE was an English polymath, Anglican priest, philosopher and historian of science. He was Master of Cambridge. In his time as a student there, he achieved distinction in mathematics. What is most remarked about Whewell is the breadth of his endeavours. In a time of increasing specialisation, Whewell appears as a vestige of an earlier era when natural philosophers dabbled in a bit of everything, he researched ocean tides, published work in the disciplines of mechanics, geology and economics, while finding the time to compose poetry, author a Bridgewater Treatise, translate the works of Goethe, write sermons and theological tracts. In mathematics, Whewell introduced what is now called the Whewell equation, an equation defining the shape of a curve without reference to an arbitrarily chosen coordinate system. One of Whewell's greatest gifts to science was his wordsmithing, he corresponded with many in his field and helped them come up with new terms for their discoveries.
Whewell contributed the terms scientist, linguistics, catastrophism, uniformitarianism, astigmatism amongst others. Whewell died in Cambridge in 1866 as a result of a fall from his horse. Whewell was born in the son of John Whewell and his wife, Elizabeth Bennison, his father was a master carpenter, wished him to follow his trade, but William's success in mathematics at Lancaster and Heversham grammar schools won him an exhibition at Trinity College, Cambridge. In 1814 he was awarded the Chancellor's Gold Medal for poetry, he was Second Wrangler in 1816, President of the Cambridge Union Society in 1817, became fellow and tutor of his college, and, in 1841, succeeded Christopher Wordsworth as master. He was professor of mineralogy from 1828 to 1832 and Knightbridge Professor of Philosophy from 1838 to 1855. Whewell married, firstly, in Cordelia Marshall, daughter of John Marshall. In 1858 he married again, to Everina Frances, widow of Sir Gilbert Affleck, 5th Baronet who had died in 1865, he himself died in Cambridge in 1866 as a result of a fall from his horse..
A window dedicated to Lady Affleck, his second wife, was installed in her memory in the chancel of All Saints' Church and made by Morris & Co. His best-known works are two voluminous books which attempt to systematize the development of the sciences, History of the Inductive Sciences and The Philosophy of the Inductive Sciences, Founded Upon Their History. While the History traced how each branch of the sciences had evolved since antiquity, Whewell viewed the Philosophy as the "Moral" of the previous work as it sought to extract a universal theory of knowledge through history. In the latter, he attempted to follow Francis Bacon's plan for discovery, he examined ideas and by the "colligation of facts" endeavoured to unite these ideas with the facts and so construct science. Whewell analysed inductive reasoning into three steps: The selection of the idea, such as space, cause, or likeness. Upon these follow special methods of induction applicable to quantity: the method of curves, the method of means, the method of least squares and the method of residues, special methods depending on resemblance, such as the method of gradation and the method of natural classification.
In Philosophy of the Inductive Sciences Whewell was the first to use the term "consilience" to discuss the unification of knowledge between the different branches of learning. Here, as in his ethical doctrine, Whewell was moved by opposition to contemporary English empiricism. Following Immanuel Kant, he asserted against John Stuart Mill the a priori nature of necessary truth, by his rules for the construction of conceptions he dispensed with the inductive methods of Mill; as stated, one of Whewell's greatest gifts to science was his wordsmithing. He corresponded with many in his field and helped them come up with new terms for their discoveries. In fact, Whewell came up with the term scientist itself in 1833, it was first published in Whewell's anonymous 1834 review of Mary Somerville's On the Connexion of the Physical Sciences published in the Quarterly Review.. Whewell was prominent not only in scientific research and philosophy, but in university and college administration, his first work, An Elementary Treatise on Mechanics, cooperated with those of George Peacock and John Herschel in reforming the Cambridge method of mathematical teaching.
His work and publications helped influence the recognition of the moral and natural sciences as an integral part of the Cambridge curriculum. In general, however in years, he opposed reform: he defended the tutorial system, in a controversy with Connop Thirlwall, opposed the admission of Dissenters, he opposed the appointment of the University Commission, wrote two pamphle
Charles Babbage
Charles Babbage was an English polymath. A mathematician, philosopher and mechanical engineer, Babbage originated the concept of a digital programmable computer. Considered by some to be a "father of the computer", Babbage is credited with inventing the first mechanical computer that led to more complex electronic designs, though all the essential ideas of modern computers are to be found in Babbage's analytical engine, his varied work in other fields has led him to be described as "pre-eminent" among the many polymaths of his century. Parts of Babbage's incomplete mechanisms are on display in the Science Museum in London. In 1991, a functioning difference engine was constructed from Babbage's original plans. Built to tolerances achievable in the 19th century, the success of the finished engine indicated that Babbage's machine would have worked. Babbage's birthplace is disputed, but according to the Oxford Dictionary of National Biography he was most born at 44 Crosby Row, Walworth Road, England.
A blue plaque on the junction of Larcom Street and Walworth Road commemorates the event. His date of birth was given in his obituary in The Times as 26 December 1792; the parish register of St. Mary's, London, shows that Babbage was baptised on 6 January 1792, supporting a birth year of 1791. Babbage was one of four children of Betsy Plumleigh Teape, his father was a banking partner of William Praed in founding Praed's & Co. of Fleet Street, London, in 1801. In 1808, the Babbage family moved into the old Rowdens house in East Teignmouth. Around the age of eight, Babbage was sent to a country school in Alphington near Exeter to recover from a life-threatening fever. For a short time he attended King Edward VI Grammar School in Totnes, South Devon, but his health forced him back to private tutors for a time. Babbage joined the 30-student Holmwood Academy, in Baker Street, Middlesex, under the Reverend Stephen Freeman; the academy had a library. He studied with two more private tutors after leaving the academy.
The first was a clergyman near Cambridge. He was brought home, to study at the Totnes school: this was at age 16 or 17; the second was an Oxford tutor, under whom Babbage reached a level in Classics sufficient to be accepted by Cambridge. Babbage arrived at Trinity College, Cambridge, in October 1810, he was self-taught in some parts of contemporary mathematics. As a result, he was disappointed in the standard mathematical instruction available at the university. Babbage, John Herschel, George Peacock, several other friends formed the Analytical Society in 1812; as a student, Babbage was a member of other societies such as The Ghost Club, concerned with investigating supernatural phenomena, the Extractors Club, dedicated to liberating its members from the madhouse, should any be committed to one. In 1812 Babbage transferred to Cambridge, he did not graduate with honours. He instead received a degree without examination in 1814, he had defended a thesis, considered blasphemous in the preliminary public disputation.
Considering his reputation, Babbage made progress. He lectured to the Royal Institution on astronomy in 1815, was elected a Fellow of the Royal Society in 1816. After graduation, on the other hand, he applied for positions unsuccessfully, had little in the way of career. In 1816 he was a candidate for a teaching job at Haileybury College. In 1819, Babbage and Herschel visited Paris and the Society of Arcueil, meeting leading French mathematicians and physicists; that year Babbage applied to be professor at the University of Edinburgh, with the recommendation of Pierre Simon Laplace. With Herschel, Babbage worked on the electrodynamics of Arago's rotations, publishing in 1825, their explanations were only transitional, being broadened by Michael Faraday. The phenomena are now part of the theory of eddy currents, Babbage and Herschel missed some of the clues to unification of electromagnetic theory, staying close to Ampère's force law. Babbage purchased the actuarial tables of George Barrett, who died in 1821 leaving unpublished work, surveyed the field in 1826 in Comparative View of the Various Institutions for the Assurance of Lives.
This interest followed a project to set up an insurance company, prompted by Francis Baily and mooted in 1824, but not carried out. Babbage did calculate actuarial tables for that scheme, using Equitable Society mortality data from 1762 onwards. During this whole period Babbage depended awkwardly on his father's support, given his father's attitude to his early marriage, of 1814: he and Edward Ryan wedded the Whitmore sisters, he made a home in Marylebone in London, founded a large family. On his father's death in 1827, Babbage inherited a large estate. After his wife's death in the same year he spent time travelling. In Italy he met Leopold II, Grand Duke of Tuscany, foreshadowing a visit to Piedmont. In April 1828 he was in Rome, relying on Herschel to manage the difference engine project, when he heard that he had become professor at Cambridge, a positio
