120. Abraham de Moivre
Abraham de Moivre (26 May 1667 – 27 November 1754) was a French mathematician known for de Moivre's formula, one of those that link complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmond Halley, and James Stirling. Even though he faced religious persecution he remained a "steadfast Christian" throughout his life. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux.
De Moivre wrote a book on probability theory, The Doctrine of Chances, said to have been prized by gamblers. De Moivre first discovered Binet's formula, the closed-form expression for Fibonacci numbers linking the nth power of the golden ratio φ to the nth Fibonacci number. He also was the first to postulate the Central Limit Theorem, a cornerstone of probability theory.
Abraham de Moivre was born in Vitry-le-François in Champagne on May 26, 1667. His father, Daniel de Moivre, was a surgeon who believed in the value of education. Though Abraham de Moivre's parents were Protestant, he first attended Christian Brothers' Catholic school in Vitry, which was unusually tolerant given religious tensions in France at the time. When he was eleven, his parents sent him to the Protestant Academy at Sedan, where he spent four years studying Greek under Jacques du Rondel. The Protestant Academy of Sedan had been founded in 1579 at the initiative of Françoise de Bourbon, the widow of Henri-Robert de la Marck.
In 1682 the Protestant Academy at Sedan was suppressed, and de Moivre enrolled to study logic at Saumur for two years. Although mathematics was not part of his course work, de Moivre read several works on mathematics on his own including Elémens des mathématiques by Jean Prestet and a short treatise on games of chance, De Ratiociniis in Ludo Aleae, by Christiaan Huygens. In 1684, de Moivre moved to Paris to study physics, and for the first time had formal mathematics training with private lessons from Jacques Ozanam.
Religious persecution in France became severe when King Louis XIV issued the Edict of Fontainebleau in 1685, which revoked the Edict of Nantes, that had given substantial rights to French Protestants. It forbade Protestant worship and required that all children be baptized by Catholic priests. De Moivre was sent to the Prieure de Saint-Martin, a school that the authorities sent Protestant children to for indoctrination into Catholicism.
It is unclear when de Moivre left the Prieure de Saint-Martin and moved to England, since the records of the Prieure de Saint-Martin indicate that he left the school in 1688, but de Moivre and his brother presented themselves as Huguenots admitted to the Savoy Church in London on August 28, 1687.
By the time he arrived in London, de Moivre was a competent mathematician with a good knowledge of many of the standard texts. To make a living, de Moivre became a private tutor of mathematics, visiting his pupils or teaching in the coffee houses of London. De Moivre continued his studies of mathematics after visiting the Earl of Devonshire and seeing Newton's recent book, Principia Mathematica. Looking through the book, he realized that it was far deeper than the books that he had studied previously, and he became determined to read and understand it. However, as he was required to take extended walks around London to travel between his students, de Moivre had little time for study, so he tore pages from the book and carried them around in his pocket to read between lessons.
According to a possibly apocryphal story, Newton, in the later years of his life, used to refer people posing mathematical questions to him to de Moivre, saying, "He knows all these things better than I do."
By 1692, de Moivre became friends with Edmond Halley and soon after with Isaac Newton himself. In 1695, Halley communicated de Moivre's first mathematics paper, which arose from his study of fluxions in the Principia Mathematica, to the Royal Society. This paper was published in the Philosophical Transactions that same year. Shortly after publishing this paper, de Moivre also generalized Newton's noteworthy binomial theorem into the multinomial theorem. The Royal Society became apprised of this method in 1697, and it made de Moivre a member two months later.
After de Moivre had been accepted, Halley encouraged him to turn his attention to astronomy. In 1705, de Moivre discovered, intuitively, that "the centripetal force of any planet is directly related to its distance from the centre of the forces and reciprocally related to the product of the diameter of the evolute and the cube of the perpendicular on the tangent." In other words, if a planet, M, follows an elliptical orbit around a focus F and has a point P where PM is tangent to the curve and FPM is a right angle so that FP is the perpendicular to the tangent, then the centripetal force at point P is proportional to FM/(R*(FP)3) where R is the radius of the curvature at M. The mathematician Johann Bernoulli proved this formula in 1710.
Despite these successes, de Moivre was unable to obtain an appointment to a chair of mathematics at any university, which would have released him from his dependence on time-consuming tutoring that burdened him more than it did most other mathematicians of the time. At least a part of the reason was a bias against his French origins.
In November 1697 he was elected a Fellow of the Royal Society and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus.
Throughout his life de Moivre remained poor. It is reported that he was a regular customer of Slaughter's Coffee House, St. Martin's Lane at Cranbourn Street, where he earned a little money from playing chess.
De Moivre continued studying the fields of probability and mathematics until his death in 1754 and several additional papers were published after his death. As he grew older, he became increasingly lethargic and needed longer sleeping hours. A common, though disputable, claim is that he noted he was sleeping an extra 15 minutes each night and correctly calculated the date of his death as the day when the sleep time reached 24 hours, November 27, 1754. He died in London and his body was buried at St Martin-in-the-Fields, although his body was later moved.
119. John Napier
John Napier, Napier also spelled Neper (born 1550, Merchiston Castle, near Edinburgh, Scot.—died April 4, 1617, Merchiston Castle) Scottish mathematician and theological writer who originated the concept of logarithms as a mathematical device to aid in calculations.
At the age of 13, Napier entered the University of St. Andrews, but his stay appears to have been short, and he left without taking a degree.
Little is known of Napier’s early life, but it is thought that he traveled abroad, as was then the custom of the sons of the Scottish landed gentry. He was certainly back home in 1571, and he stayed either at Merchiston or at Gartness for the rest of his life. He married the following year. A few years after his wife’s death in 1579, he married again.
Theology and inventions
Napier’s life was spent amid bitter religious dissensions. A passionate and uncompromising Protestant, in his dealings with the Church of Rome he sought no quarter and gave none. It was well known that James VI of Scotland hoped to succeed Elizabeth I to the English throne, and it was suspected that he had sought the help of the Catholic Philip II of Spain to achieve this end. Panic stricken at the peril that seemed to be impending, the general assembly of the Scottish Church, a body with which Napier was closely associated, begged James to deal effectively with the Roman Catholics, and on three occasions Napier was a member of a committee appointed to make representations to the King concerning the welfare of the church and to urge him to see that “justice be done against the enemies of God’s Church.”
In January 1594, Napier addressed to the King a letter that forms the dedication of his Plaine Discovery of the Whole Revelation of Saint John, a work that, while it professed to be of a strictly scholarly character, was calculated to influence contemporary events. In it he declared:
Let it be your Majesty’s continuall study to reforme the universall enormities of your country, and first to begin at your Majesty’s owne house, familie and court, and purge the same of all suspicion of Papists and Atheists and Newtrals, whereof this Revelation forthtelleth that the number shall greatly increase in these latter daies.
The work occupies a prominent place in Scottish ecclesiastical history.
Following the publication of this work, Napier seems to have occupied himself with the invention of secret instruments of war, for in a manuscript collection now at Lambeth Palace, London, there is a document bearing his signature, enumerating various inventions “designed by the Grace of God, and the worke of expert craftsmen” for the defense of his country. These inventions included two kinds of burning mirrors, a piece of artillery, and a metal chariot from which shot could be discharged through small holes.
Contribution to mathematics
Napier devoted most of his leisure to the study of mathematics, particularly to devising methods of facilitating computation, and it is with the greatest of these, logarithms, that his name is associated. He began working on logarithms probably as early as 1594, gradually elaborating his computational system whereby roots, products, and quotients could be quickly determined from tables showing powers of a fixed number used as a base.
His contributions to this powerful mathematical invention are contained in two treatises: Mirifici Logarithmorum Canonis Descriptio (Description of the Marvelous Canon of Logarithms), which was published in 1614, and Mirifici Logarithmorum Canonis Constructio (Construction of the Marvelous Canon of Logarithms), which was published two years after his death. In the former, he outlined the steps that had led to his invention.
Logarithms were meant to simplify calculations, especially multiplication, such as those needed in astronomy. Napier discovered that the basis for this computation was a relationship between an arithmetical progression—a sequence of numbers in which each number is obtained, following a geometric progression, from the one immediately preceding it by multiplying by a constant factor, which may be greater than unity (e.g., the sequence 2, 4, 8, 16 . . . ) or less than unity (e.g., 8, 4, 2, 1, 1/2 . . . ).
In the Descriptio, besides giving an account of the nature of logarithms, Napier confined himself to an account of the use to which they might be put. He promised to explain the method of their construction in a later work. This was the Constructio, which claims attention because of the systematic use in its pages of the decimal point to separate the fractional from the integral part of a number. Decimal fractions had already been introduced by the Flemish mathematician Simon Stevin in 1586, but his notation was unwieldy. The use of a point as the separator occurs frequently in the Constructio. Joost Bürgi, the Swiss mathematician, between 1603 and 1611 independently invented a system of logarithms, which he published in 1620. But Napier worked on logarithms earlier than Bürgi and has the priority due to his prior date of publication in 1614.
Although Napier’s invention of logarithms overshadows all his other mathematical work, he made other mathematical contributions. In 1617 he published his Rabdologiae, seu Numerationis per Virgulas Libri Duo (Study of Divining Rods, or Two Books of Numbering by Means of Rods, 1667); in this he described ingenious methods of multiplying and dividing of small rods known as Napier’s bones, a device that was the forerunner of the slide rule. He also made important contributions to spherical trigonometry, particularly by reducing the number of equations used to express trigonometrical relationships from 10 to 2 general statements. He is also credited with certain trigonometrical relations—Napier’s analogies—but it seems likely that the English mathematician Henry Briggs had a share in these.
The solution M#217 is correct. Excellent, bobbym!
The solutions M#216 and M#217 are correct. Splendid, Relentless!
M#218. A 20 meter deep well with diameter 7m is dug and the earth from digging is evenly spread out to for a platform 22 m by 14 m. Find the height of the platform.
118. Claudius Ptolemy
The Greek astronomer, astrologer, and geographer Claudius Ptolemy (ca. 100-ca. 170) established the system of mathematical astronomy that remained standard in Christian and Moslem countries until the 16th century.
Ptolemy is known to have made astronomical observations at Alexandria in Egypt between 127 and 141, and he probably lived on into the reign of Marcus Aurelius (161-180). Beyond the fact that his On the Faculty of Judgment indicates his adherence to Stoic doctrine, nothing more of his biography is available.
The earliest and most influential of Ptolemy's major writings is the Almagest. In 13 books it establishes the kinematic models (purely mathematical and nonphysical) used to explain solar, lunar, and planetary motion and determines the parameters which quantify these models and permit the computation of longitudes and latitudes; of the times, durations, and magnitudes of lunar and solar eclipses; and of the times of heliacal risings and settings. Ptolemy also provides a catalog of 1, 022 fixed stars, giving for each its longitude and latitude according to an ecliptic coordinate system.
Ptolemy's is a geocentric system, though the earth is the actual center only of the sphere of the fixed stars and of the "crank mechanism" of the moon; the orbits of all the other planets are slightly eccentric. Ptolemy thus hypothesizes a mathematical system which cannot be made to agree with the rules of Aristotelian physics, which require that the center of the earth be the center of all celestial circular motions.
In solar astronomy Ptolemy accepts and confirms the eccentric model and its parameters established by Hipparchus. For the moon Ptolemy made enormous improvements in Hipparchus's model, though he was unable to surmount all the difficulties of lunar motion evident even to ancient astronomers. Ptolemy discerned two more inequalities and proposed a complicated model to account for them. The effect of the Ptolemaic lunar model is to draw the moon close enough to the earth at quadratures to produce what should be a visible increase in apparent diameter; the increase, however, was not visible. The Ptolemaic models for the planets generally account for the two inequalities in planetary motion and are represented by combinations of circular motions: eccentrics and epicycles. Such a combination of eccentric and epicyclic models represents Ptolemy's principal original contribution in the Almagest.
This brief text was inscribed on a stele erected at Canobus near Alexandria in Egypt in 146 or 147. It contains the parameters of Ptolemy's solar, lunar, and planetary models as given in the Almagest but modified in some instances. There is also a section on the harmony of the spheres. The epoch of the Canobic Inscription is the first year of Augustus, or 30 B.C.
In the two books of Planetary Hypotheses, an important cosmological work, Ptolemy "corrects" some of the parameters of the Almagest and suggests an improved model to explain planetary latitude. In the section of the first book preserved only in Arabic, he proposes absolute dimensions for the celestial spheres (maximum and minimum distances of the planets, their apparent and actual diameters, and their volumes). The second book, preserved only in Arabic, describes a physical actualization of the mathematical models of the planets in the Almagest. Here the conflict with Aristotelian physics becomes unavoidable (Ptolemy uses Aristotelian terminology but makes no attempt to reconcile his views of the causes of the inequalities of planetary motion with Aristotle's), and it was in attempting to remove the discrepancies that the "School of Maragha" and also Ibn al-Shatir in the 13th and 14th centuries devised new planetary models that largely anticipate Copernicus's.
This work originally contained two books, but only the second has survived. It is a calendar of the parapegma type, giving for each day of the Egyptian year the time of heliacal rising or setting of certain fixed stars. The views of Eudoxus, Hipparchus, Philip of Opus, Callippus, Euctemon, and others regarding the meteorological phenomena associated with these risings and settings are quoted. This makes the Phases useful to the historian of early Greek astronomy, though it is certainly the least important of Ptolemy's astronomical works.
Consisting of four books, the Apotelesmatica is Ptolemy's contribution to astrological theory. He attempts in the first book to place astrology on a sound scientific basis. Astrology for Ptolemy is less exact than astronomy is, as the former deals with objects influenced by many other factors besides the positions of the planets at a particular point in time, whereas the latter describes the unswerving motions of the eternal stars themselves. In the second book, general astrology affecting whole states, societies, and regions is described; this general astrology is largely derived from Mesopotamian astral omina. The final two books are devoted to genethlialogy, the science of predicting the events in the life of a native from the horoscope cast for the moment of his birth. The Apotelesmatica was long the main handbook for astrologers.
In the eight books of the Geography, Ptolemy sets forth mathematical solutions to the problems of representing the spherical surface of the earth on a plane surface (a map), but the work is largely devoted to a list of localities with their coordinates. This list is arranged by regions, with the river and mountain systems and the ethnography of each region also usually described. He begins at the West in book 2 (his prime meridian ran through the "Fortunate Islands, " apparently the Canaries) and proceeds eastward to India, the Malay Peninsula, and China in book 7.
Despite his brilliant mathematical theory of map making, Ptolemy had not the requisite material to construct the accurate picture of the world that he desired. Aside from the fact that, following Marinus in this as in much else, he underestimated the size of the earth, concluding that the distance from the Canaries to China is about 180° instead of about 130°, he was seriously hampered by the lack of all the gnomon observations that are necessary to establish the latitudes of the places he lists. For longitudes he could not utilize astronomical observations because no systematic exploitation of this method of determining longitudinal differences had been organized. He was compelled to rely on travelers' estimates of distances, which varied widely in their reliability and were most uncertain guides. His efforts, however, provided western Europe, Byzantium, and Islam with their most detailed conception of the inhabited world.
Harmonics and Optics
These, the last two works in the surviving corpus of Ptolemy's writings, investigate two other fields included in antiquity in the general field of mathematics. The Harmonics in three books became one of the standard works on the mathematical theory of music in late antiquity and throughout the Byzantine period. The Optics in five books discussed the geometry of vision, especially mirror reflection and refraction. The Optics survives only in a Latin translation prepared by Eugenius, Admiral of Sicily, toward the end of the 12th century, from an Arabic version in which the first book and the end of the fifth were lost. The doubts surrounding its authenticity as a work of Ptolemy seem to have been overcome by recent scholarship.
Ptolemy's brilliance as a mathematician, his exactitude, and his masterful presentation seemed to his successors to have exhausted the possibilities of mathematical astronomy and geography. To a large extent they were right. Without better instrumentation only minor adjustments in the Ptolemaic parameters or models could be made. The major "improvements" in the models—those of the School of Maragha—are designed primarily to satisfy philosophy, not astronomy; the lunar theory was the only exception. Most of the deviations from Ptolemaic methods in medieval astronomy are due to the admixture of non-Greek material and the continued use of pre-Ptolemaic elements. The Geography was never seriously challenged before the 15th century.
The authority of the astronomical and geographical works carries over to the astrological treatise and, to a lesser extent, to the Harmonics and Optics. The Apotelesmatica was always recognized as one of the works most clearly defending the scientific basis of astrology in general, and of genethlialogy in particular. But Neoplatonism as developed by the pagans of Harran provided a more extended theory of the relationship of the celestial spheres to the sublunar world, and this theory was popularized in Islam in the 9th century. The Harmonics ceased to be popular as Greek music ceased to follow the classical modes, and the Optics was rendered obsolete by Moslem scientists. Ptolemy's fame and influence, then, rest primarily on the Almagest, his most original work, justly subtitled The Greatest.