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.
117. Joseph Henry, (born December 17, 1797, Albany, New York, U.S. - died May 13, 1878, Washington, D.C.) one of the first great American scientists after Benjamin Franklin. He aided and discovered several important principles of electricity, including self-induction, a phenomenon of primary importance in electronic circuitry.
While working with electromagnets at the Albany Academy (New York) in 1829, he made important design improvements. By insulating the wire instead of the iron core, he was able to wrap a large number of turns of wire around the core and thus greatly increase the power of the magnet. He made an electromagnet for Yale College that could support 2,063 pounds, a world record at the time.
Henry also searched for electromagnetic induction—the process of converting magnetism into electricity - and in 1831 he started building a large electromagnet for that purpose. Because the room at the Albany Academy in which he wanted to build his experiment was not available, he had to postpone his work until June 1832, when he learned that British physicist Michael Faraday had already discovered induction the previous year. However, when he resumed his experiments, he was the first to notice the principle of self-induction.
In 1831 Henry built and successfully operated, over a distance of 2.4 km (1.5 miles), a telegraph of his own design. He became professor of natural philosophy at the College of New Jersey (later Princeton University) in 1832. Continuing his researches, he discovered the laws upon which the transformer is based. He also found that currents could be induced at a distance and in one case magnetized a needle by using a lightning flash 13 km (8 miles) away. That experiment was apparently the first use of radio waves across a distance. He aided Samuel F.B. Morse in the development of the telegraph by giving him 8 km (5 miles) of copper wire and writing a letter to Congress in 1842 encouraging it to support an 80-km (50-mile) test line. By using a thermogalvanometer, a heat-detection device, he showed that sunspots radiate less heat than the general solar surface.
In 1846 Henry became the first secretary of the Smithsonian Institution, Washington, D.C., where he organized and supported a corps of volunteer weather observers. The success of the Smithsonian meteorological work led to the creation of the U.S. Weather Bureau (later Service). One of Lincoln’s chief technical advisers during the U.S. Civil War, he was a primary organizer of the National Academy of Sciences and its second president. In 1893 his name was given to the standard electrical unit of inductive resistance, the henry.
116. Jean Charles Athanase Peltier
Jean Charles Athanase Peltier (22 February 1785 – 27 October 1845) was a French physicist. He was originally a watch dealer, but at 30 years old took up experiments and observations in the physics.
Peltier was the author of numerous papers in different departments of physics, but his name is specially associated with the thermal effects at junctions in a voltaic circuit. He introduced the Peltier effect. Peltier also introduced the concept of electrostatic induction (1840), based on the modification of the distribution of electric charge in a material under the influence of a second object closest to it and its own electrical charge. This effect has been very important in the recent development of non-polluting cooling mechanisms.
Peltier initially trained as a watchmaker and was up to his 30s working as a watch dealer. Peltier worked with Abraham Louis Breguet in Paris. Later, he worked with various experiments on electrodynamics and noticed that in an electronic element when current flows through, a temperature difference or temperature difference is generated at a current flow. In 1836 he published his work and in 1838 his findings were confirmed by Emil Lenz. Furthermore, Peltier dealt with topics from the atmospheric electricity and meteorology. In 1840, he published a work on the causes and formation of hurricanes.
Peltier's papers, which are numerous, are devoted in great part to atmospheric electricity, waterspouts, cyanometry and polarization of sky-light, the temperature of water in the spheroidal state, and the boiling-point at great elevations. There are also a few devoted to curious points of natural history. But his name will always be associated with the thermal effects at junctions in a voltaic circuit, a discovery of importance quite comparable with those of Seebeck and Cumming.
Peltier discovered the calorific effect of electric current passing through the junction of two different metals. This is now called the Peltier effect (or Peltier–Seebeck effect). By switching the direction of current, either heating or cooling may be achieved. Junctions always come in pairs, as the two different metals are joined at two points. Thus heat will be moved from one junction to the other.
The Peltier effect is the presence of heating or cooling at an electrified junction of two different conductors (1834).His great experimental discovery was the heating or cooling of the junctions in a heterogeneous circuit of metals according to the direction in which an electric current is made to pass round the circuit. This reversible effect is proportional directly to the strength of the current, not to its square, as is the irreversible generation of heat due to resistance in all parts of the circuit. It is found that, if a current pass from an external source through a circuit of two metals, it cools one junction and heats the other. It cools the junction if it be in the same direction as the thermoelectric current which would be caused by directly heating that junction. In other words, the passage of a current from an external source produces in the junctions of the circuit a distribution of temperature which leads to the weakening of the current by the superposition of a thermo-electric current running in the opposite direction.
When electromotive current is made to flow through a electronic junction between two conductors (A and B), heat is removed at the junction. To make a typical pump, multiple junctions are created between two plates. One side heats and the other side cools. A dissipation device is attached to the hot side to maintain cooling effect on the cold side. Typically, the use of the Peltier effect as a heat pump device involves multiple junctions in series, through which a current is driven. Some of the junctions lose heat due to the Peltier effect, while others gain heat. Thermoelectric pumps exploit this phenomenon, as do thermoelectric cooling Peltier modules found in refrigerators.