[1]See Mrs. De Morgan'sMemoir of Augustus De Morgan, London, 1882, p 61.[2]In the first edition this reference was to page 11.[3]In the first edition this read "at page 438," the work then appearing in a single volume.[4]"Just as it would surely have been better not to have considered it (i.e., the trinity) as a mystery, and with Cl. Kleckermann to have investigated by the aid of philosophy according to the teaching of true logic what it might be, before they determined what it was; just so would it have been better to withdraw zealously and industriously into the deepest caverns and darkest recesses of metaphysical speculations and suppositions in order to establish their opinion beyond danger from the weapons of their adversaries.... Indeed that great man so explains and demonstrates this dogma (although to theologians the word has not much charm) from the immovable foundations of philosophy, that with but few changes and additions a mind sincerely devoted to truth can desire nothing more."[5]Mrs. Wititterly, inNicholas Nickleby.—A. De M.[6]The brackets mean that the paragraph is substantially from some one of theAthenæum Supplements.—S. E. De M.[7]"It is annoying that this ingenious naturalist who has already given us more useful works and has still others in preparation, uses for this odious task, a pen dipped in gall and wormwood. It is true that many of his remarks have some foundation, and that to each error that he points out he at the same time adds its correction. But he is not always just and never fails to insult. After all, what does his book prove except that a forty-fifth part of a very useful review is not free from mistakes? Must we confuse him with those superficial writers whose liberty of body does not permit them to restrain their fruitfulness, that crowd of savants of the highest rank whose writings have adorned and still adorn theTransactions? Has he forgotten that the names of the Boyles, Newtons, Halleys, De Moivres, Hans Sloanes, etc. have been seen frequently? and that still are found those of the Wards, Bradleys, Grahams, Ellicots, Watsons, and of an author whom Mr. Hill prefers to all others, I mean Mr. Hill himself?"[8]"Let no free man be seized or imprisoned or in any way harmed except by trial of his peers."[9]"The master can rob, wreck and punish his slave according to his pleasure save only that he may not maim him."[10]An Irish antiquary informs me that Virgil is mentioned in annals at A.D. 784, as "Verghil, i.e., the geometer, Abbot of Achadhbo [and Bishop of Saltzburg] died in Germany in the thirteenth year of his bishoprick." No allusion is made to his opinions; but it seems he was, by tradition, a mathematician. The Abbot of Aghabo (Queen's County) was canonized by Gregory IX, in 1233. The story of the second, or scapegoat, Virgil would be much damaged by the character given to the real bishop, if there were anything in it to dilapidate.—A. De M.[11]"He performed many acts befitting the Papal dignity, and likewise many excellent (to be sure!) works."[12]"After having been on the throne during ten years of pestilence."[13]The work is theQuestiones Joannis Buridani super X libros Aristotelis ad Nicomachum, curante Egidio Delfo... Parisiis, 1489, folio. It also appeared at Paris in editions of 1499, 1513, and 1518, and at Oxford in 1637.[14]Jean Buridan was born at Béthune about 1298, and died at Paris about 1358. He was professor of philosophy at the University of Paris and several times held the office of Rector. As a philosopher he was classed among the nominalists.[15]So in the original.[16]Baruch Spinoza, or Benedict de Spinoza as he later called himself, the pantheistic philosopher, excommunicated from the Jewish faith for heresy, was born at Amsterdam in 1632 and died there in 1677.[17]Michael Scott, or Scot, was born about 1190, probably in Fifeshire, Scotland, and died about 1291. He was one of the best known savants of the court of Emperor Frederick II, and wrote upon astrology, alchemy, and the occult sciences. He was looked upon as a great magician and is mentioned among the wizards in Dante'sInferno."That other, round the loinsSo slender of his shape, was Michael Scot,Practised in every slight of magic wile."Inferno, XX.Boccaccio also speaks of him: "It is not long since there was in this city (Florence) a great master in necromancy, who was called Michele Scotto, because he was a Scot."Decameron, Dec. Giorno.Scott's mention of him in Canto Second of hisLay of the Last Minstrel, is well known:"In these fair climes, it was my lotTo meet the wondrous Michael Scott;A wizard of such dreaded fame,That when, in Salamanca's cave,Him listed his magic wand to wave,The bells would ring in Notre Dame!"Sir Walter's notes upon him are of interest.[18]These were some of the forgeries which Michel Chasles (1793-1880) was duped into buying. They purported to be a correspondence between Pascal and Newton and to show that the former had anticipated some of the discoveries of the great English physicist and mathematician. That they were forgeries was shown by Sir David Brewster in 1855.[19]"Let the serpent also break from its appointed path."[20]Guglielmo Brutus Icilius Timoleon Libri-Carucci della Sommaja, born at Florence in 1803; died at Fiesole in 1869. HisHistoire des Sciences Mathématiquesappeared at Paris in 1838, the entire first edition of volume I, save some half dozen that he had carried home, being burned on the day that the printing was completed. He was a great collector of early printed works on mathematics, and was accused of having stolen large numbers of them from other libraries. This accusation took him to London, where he bitterly attacked his accusers. There were two auction sales of his library, and a number of his books found their way into De Morgan's collection.[21]Philo of Gadara lived in the second century B.C. He was a pupil of Sporus, who worked on the problem of the two mean proportionals.[22]In hisHistoire des Mathématiques, the first edition of which appeared in 1758. Jean Etienne Montucla was born at Lyons in 1725 and died at Versailles in 1799. He was therefore only thirty-three years old when his great work appeared. The second edition, with additions by D'Alembert, appeared in 1799-1802. He also wrote a work on the quadrature of the circle,Histoire des recherches sur la Quadrature du Cercle, which appeared in 1754.[23]Eutocius of Ascalon was born in 480 A.D. He wrote commentaries on the first four books of the conics of Apollonius of Perga (247-222 B.C.). He also wrote on the Sphere and Cylinder and the Quadrature of the Circle, and on the two books on Equilibrium of Archimedes (287-212 B.C.)[24]Edward Cocker was born in 1631 and died between 1671 and 1677. His famous arithmetic appeared in 1677 and went through many editions. It was written in a style that appealed to teachers, and was so popular that the expression "According to Cocker" became a household phrase. Early in the nineteenth century there was a similar saying in America, "According to Daboll," whose arithmetic had some points of analogy to that of Cocker. Each had a well-known prototype in the ancient saying, "He reckons like Nicomachus of Gerasa."[25]So in the original, for Barrême. François Barrême was to France what Cocker was to England. He was born at Lyons in 1640, and died at Paris in 1703. He published several arithmetics, dedicating them to his patron, Colbert. One of the best known of his works isL'arithmétique, ou le livre facile pour apprendre l'arithmétique soi-mème, 1677. The French wordbarêmeorbarrême, a ready-reckoner, is derived from his name.[26]Born at Rome, about 480 A.D.; died at Pavia, 524. Gibbon speaks of him as "the last of the Romans whom Cato or Tully could have acknowledged for their countryman." His works on arithmetic, music, and geometry were classics in the medieval schools.[27]Johannes Campanus, of Novarra, was chaplain to Pope Urban IV (1261-1264). He was one of the early medieval translators of Euclid from the Arabic into Latin, and the first printed edition of theElements(Venice, 1482) was from his translation. In this work he probably depended not a little upon at least two or three earlier scholars. He also wroteDe computo ecclesiastico Calendarium, andDe quadratura circuli.[28]Archimedes gave 3-1/7, and 3-10/71 as the limits of the ratio of the circumference to the diameter of a circle.[29]Friedrich W. A. Murhard was born at Cassel in 1779 and died there in 1853. HisBibliotheca Mathematica, Leipsic, 1797-1805, is ill arranged and inaccurate, but it is still a helpful bibliography. De Morgan speaks somewhere of his indebtedness to it.[30]Abraham Gotthelf Kästner was born at Leipsic in 1719, and died at Göttingen in 1800. He was professor of mathematics and physics at Göttingen. HisGeschichte der Mathematik(1796-1800) was a work of considerable merit. In the text of theBudget of Paradoxesthe name appears throughout as Kastner instead of Kästner.[31]Lucas Gauricus, or Luca Gaurico, born at Giffoni, near Naples, in 1476; died at Rome in 1558. He was an astrologer and mathematician, and was professor of mathematics at Ferrara in 1531. In 1545 he became bishop of Cività Ducale.[32]John Couch Adams was born at Lidcot, Cornwall, in 1819, and died in 1892. He and Leverrier predicted the discovery of Neptune from the perturbations in Uranus.[33]Urbain-Jean-Joseph Leverrier was born at Saint-Lô, Manche, in 1811, and died at Paris in 1877. It was his data respecting the perturbations of Uranus that were used by Adams and himself in locating Neptune.[34]Joseph-Juste Scaliger, the celebrated philologist, was born at Agen in 1540, and died at Leyden in 1609. HisCyclometrica elementa, to which De Morgan refers, appeared at Leyden in 1594.[35]The title is:In hoc libra contenta.... Introductio i geometriā.... Liber de quadratura circuli. Liber de cubicatione sphere. Perspectiva introductio. Carolus Bovillus, or Charles Bouvelles (Boüelles, Bouilles, Bouvel), was born at Saucourt, Picardy, about 1470, and died at Noyon about 1533. He was canon and professor of theology at Noyon. HisIntroductiocontains considerable work on star polygons, a favorite study in the Middle Ages and early Renaissance. His workQue hoc volumine continētur. Liber de intellectu. Liber de sensu, etc., appeared at Paris in 1509-10.[36]Nicolaus Cusanus, Nicolaus Chrypffs or Krebs, was born at Kues on the Mosel in 1401, and died at Todi, Umbria, August 11, 1464. He held positions of honor in the church, including the bishopric of Brescia. He was made a cardinal in 1448. He wrote several works on mathematics, hisOpuscula variaappearing about 1490, probably at Strasburg, but published without date or place. HisOperaappeared at Paris in 1511 and again in 1514, and at Basel in 1565.[37]Henry Stephens (born at Paris about 1528, died at Lyons in 1598) was one of the most successful printers of his day. He was known asTypographus Parisiensis, and to his press we owe some of the best works of the period.[38]Jacobus Faber Stapulensis (Jacques le Fèvre d'Estaples) was born at Estaples, near Amiens, in 1455, and died at Nérac in 1536. He was a priest, vicar of the bishop of Meaux, lecturer on philosophy at the Collège Lemoine in Paris, and tutor to Charles, son of Francois I. He wrote on philosophy, theology, and mathematics.[39]Claude-François Milliet de Challes was born at Chambéry in 1621, and died at Turin in 1678. He editedEuclidis Elementorum libri octoin 1660, and published aCursus seu mundus mathematicus, which included a short history of mathematics, in 1674. He also wrote on mathematical geography.[40]This date should be 1503, if he refers to the first edition. It is well known that this is the first encyclopedia worthy the name to appear in print. It was written by Gregorius Reisch (born at Balingen, and died at Freiburg in 1487), prior of the cloister at Freiburg and confessor to Maximilian I. The first edition appeared at Freiburg in 1503, and it passed through many editions in the sixteenth and seventeenth centuries. The title of the 1504 edition reads:Aepitoma omnis phylosophiae. alias Margarita phylosophica tractans de omni genere scibili: Cum additionibus: Quae in alijs non habentur.[41]This is theIntroductio in arithmeticam Divi S. Boetii.... Epitome rerum geometricarum ex geometrica introductio C. Bovilli. De quadratura circuli demonstratio ex Campano, that appeared without date about 1507.[42]Born at Liverpool in 1805, and died there about 1872. He was a merchant, and in 1865 he published, at Liverpool, a work entitledThe Quadrature of the Circle, or the True Ratio between the Diameter and Circumference geometrically and mathematically demonstrated. In this he gives the ratio as exactly 3⅛.[43]"That it would be impossible to tell him exactly, since no one had yet been able to find precisely the ratio of the circumference to the diameter."[44]This is the Paris edition: "Parisiis: ex officina Ascensiana anno Christi ... MDXIIII," as appears by the colophon of the second volume to which De Morgan refers.[45]Regiomontanus, or Johann Müller of Königsberg (Regiomontanus), was born at Königsberg in Franconia, June 5, 1436, and died at Rome July 6, 1476. He studied at Vienna under the great astronomer Peuerbach, and was his most famous pupil. He wrote numerous works, chiefly on astronomy. He is also known by the names Ioannes de Monte Regio, de Regiomonte, Ioannes Germanus de Regiomonte, etc.[46]Henry Cornelius Agrippa was born at Cologne in 1486 and died either at Lyons in 1534 or at Grenoble in 1535. He was professor of theology at Cologne and also at Turin. After the publication of hisDe Occulta Philosophiahe was imprisoned for sorcery. Both works appeared at Antwerp in 1530, and each passed through a large number of editions. A French translation appeared in Paris in 1582, and an English one in London in 1651.[47]Nicolaus Remegius was born in Lorraine in 1554, and died at Nancy in 1600. He was a jurist and historian, and held the office of procurator general to the Duke of Lorraine.[48]This was at the storming of the city by the British on May 4, 1799. From his having been born in India, all this appealed strongly to the interests of De Morgan.[49]Orontius Finaeus, or Oronce Finé, was born at Briançon in 1494 and died at Paris, October 6, 1555. He was imprisoned by François I for refusing to recognize the concordat (1517). He was made professor of mathematics in the Collège Royal (later called the Collège de France) in 1532. He wrote extensively on astronomy and geometry, but was by no means a great scholar. He was a pretentious man, and his works went through several editions. HisProtomathesisappeared at Paris in 1530-32. The work referred to by De Morgan is theQuadratura circuli tandem inventa & clarissime demonstrata... Lutetiae Parisiorum, 1544, fol. In the 1556 edition of hisDe rebus mathematicis, hactenus desideratis, Libri IIII, published at Paris, the subtitle is:Quibus inter cætera, Circuli quadratura Centum modis, & suprà, per eundem Orontium recenter excogitatis, demonstratus, so that he kept up his efforts until his death.[50]Johannes Buteo (Boteo, Butéon, Bateon) was born in Dauphiné c. 1485-1489, and died in a cloister in 1560 or 1564. Some writers give Charpey as the place and 1492 as the date of his birth, and state that he died at Canar in 1572. He belonged to the order of St. Anthony, and wrote chiefly on geometry, exposing the pretenses of Finaeus. HisOpera geometricaappeared at Lyons in 1554, and hisLogisticaandDe quadratura circuli libri duoat Lyons in 1559.[51]This is the great French algebraist, François Viète (Vieta), who was born at Fontenay-le-Comte in 1540, and died at Paris, December 13, 1603. His well-knownIsagoge in artem analyticamappeared at Tours in 1591. HisOpera mathematicawas edited by Van Schooten in 1646.[52]This is theDe Rebus mathematicis hactenus desideratis, Libri IIII, that appeared in Paris in 1556. For the title page see Smith, D. E.,Rara Arithmetica, Boston, 1908, p. 280.[53]The title is correct except for a colon afterAstronomicum. Nicolaus Raimarus Ursus was born in Henstede or Hattstede, in Dithmarschen, and died at Prague in 1599 or 1600. He was a pupil of Tycho Brahe. He also wroteDe astronomis hypothesibus(1597) andArithmetica analytica vulgo Cosa oder Algebra(1601).[54]Born at Dôle, Franche-Comté, about 1550, died in Holland about 1600. The work to which reference is made is theQuadrature du cercle, ou manière de trouver un quarré égal au cercle donné, which appeared at Delft in 1584. Duchesne had the courage of his convictions, not only on circle-squaring but on religion as well, for he was obliged to leave France because of his conversion to Calvinism. De Morgan's statement that his real name is Van der Eycke is curious, since he was French born. The Dutch may have translated his name when he became professor at Delft, but we might equally well say, that his real name was Quercetanus or à Quercu.[55]This was the father of Adriaan Metius (1571-1635). He was a mathematician and military engineer, and suggested the ratio 355/113 forπ, a ratio afterwards published by his son. The ratio, then new to Europe, had long been known and used in China, having been found by Tsu Ch'ung-chih (428-499 A.D.).[56]This was Jost Bürgi, or Justus Byrgius, the Swiss mathematician of whom Kepler wrote in 1627: "Apices logistici Justo Byrgio multis annis ante editionem Neperianam viam præiverunt ad hos ipsissimos logarithmos." He constructed a table of antilogarithms (Arithmetische und geometrische Progress-Tabulen), but it was not published until after Napier's work appeared.[57]Ludolphus Van Ceulen, born at Hildesheim, and died at Leyden in 1610. It was he who first carried the computation ofπto 35 decimal places.[58]Jens Jenssen Dodt, van Flensburg, a Dutch historian, who died in 1847.[59]I do not know this edition. There was one "Antverpiae apud Petrum Bellerum sub scuto Burgundiae," 4to, in 1591.[60]Archytas of Tarentum (430-365 B.C.) who wrote on proportions, irrationals, and the duplication of the cube.[61]The Circle Speaks."At first a circle I was called,And was a curve around aboutLike lofty orbit of the sunOr rainbow arch among the clouds.A noble figure then was I—And lacking nothing but a start,And lacking nothing but an end.But now unlovely do I seemPolluted by some angles new.This thing Archytas hath not doneNor noble sire of IcarusNor son of thine, Iapetus.What accident or god can thenHave quadrated mine area?"The Author Replies."By deepest mouth of TuriaAnd lake of limpid clearness, liesA happy state not far removedFrom old Saguntus; farther yetA little way from Sucro town.In this place doth a poet dwell,Who oft the stars will closely scan,And always for himself doth claimWhat is denied to wiser men;—An old man musing here and thereAnd oft forgetful of himself,Not knowing how to rightly placeThe compasses, nor draw a line,As he doth of himself relate.This craftsman fine, in sooth it isHath quadrated thine area."[62]Pietro Bongo, or Petrus Bungus, was born at Bergamo, and died there in 1601. His work on the Mystery of Numbers is one of the most exhaustive and erudite ones of the mystic writers. The first edition appeared at Bergamo in 1583-84; the second, at Bergamo in 1584-85; the third, at Venice in 1585; the fourth, at Bergamo in 1590; and the fifth, which De Morgan calls the second, in 1591. Other editions, before the Paris edition to which he refers, appeared in 1599 and 1614; and the colophon of the Paris edition is dated 1617. See the editor'sRara Arithmetica, pp. 380-383.[63]William Warburton (1698-1779), Bishop of Gloucester, whose works got him into numerous literary quarrels, being the subject of frequent satire.[64]Thomas Galloway (1796-1851), who was professor of mathematics at Sandhurst for a time, and was later the actuary of the Amicable Life Assurance Company of London. In the latter capacity he naturally came to be associated with De Morgan.[65]Giordano Bruno was born near Naples about 1550. He left the Dominican order to take up Calvinism, and among his publications wasL'expulsion de la bête triomphante. He taught philosophy at Paris and Wittenberg, and some of his works were published in England in 1583-86. Whether or not he was roasted alive "for the maintenance and defence of the holy Church," as De Morgan states, depends upon one's religious point of view. At any rate, he was roasted as a heretic.[66]Referring to part of hisDiscours de la méthode, Leyden, 1637.[67]Bartholomew Legate, who was born in Essex about 1575. He denied the divinity of Christ and was the last heretic burned at Smithfield.[68]Edward Wightman, born probably in Staffordshire. He was anti-Trinitarian, and claimed to be the Messiah. He was the last man burned for heresy in England.[69]Gaspar Schopp, born at Neumarck in 1576, died at Padua in 1649; grammarian, philologist, and satirist.[70]Konrad Ritterhusius, born at Brunswick in 1560; died at Altdorf in 1613. He was a jurist of some power.[71]Johann Jakob Brucker, born at Augsburg in 1696, died there in 1770. He wrote on the history of philosophy (1731-36, and 1742-44).[72]Daniel Georg Morhof, born at Wismar in 1639, died at Lübeck in 1691. He was rector of the University of Kiel, and professor of eloquence, poetry, and history.[73]In theHistoire des Sciences Mathématiques, vol. IV, note X, pp. 416-435 of the 1841 edition.[74]Colenso (1814-1883), missionary bishop of Natal, was one of the leaders of his day in the field of higher biblical criticism. De Morgan must have admired his mathematical works, which were not without merit.[75]Samuel Roffey Maitland, born at London in 1792; died at Gloucester in 1866. He was an excellent linguist and a critical student of the Bible. He became librarian at Lambeth in 1838.[76]Archbishop Howley (1766-1848) was a thorough Tory. He was one of the opponents of the Roman Catholic Relief bill, the Reform bill, and the Jewish Civil Disabilities Relief bill.[77]We have, in America at least, almost forgotten the great stir made by Edward B. Pusey (1800-1882) in the great Oxford movement in the middle of the nineteenth century. He was professor of Hebrew at Oxford, and canon of Christ Church.[78]That is, hisMagia universalis naturae et artis sive recondita naturalium et artificialium rerum scientia, Würzburg, 1657, 4to, with editions at Bamberg in 1671, and at Frankfort in 1677. Gaspard Schott (Königshofen 1608, Würzburg 1666) was a physicist and mathematician, devoting most of his attention to the curiosities of his sciences. His type of mind must have appealed to De Morgan.[79]Salicetti Quadratura circuli nova, perspicua, expedita, veraque tum naturalis, tum geometrica, etc., 1608.—Consideratio nova in opusculum Archimedis de circuli dimensione, etc., 1609.[80]Melchior Adam, who died at Heidelberg in 1622, wrote a collection of biographies which was published at Heidelberg and Frankfort from 1615 to 1620.[81]Born at Baden in 1524; died at Basel in 1583. The Erastians were related to the Zwinglians, and opposed all power of excommunication and the infliction of penalties by a church.[82]See Acts xii. 20.[83]Theodore de Bèse, a French theologian; born at Vezelay, in Burgundy, in 1519; died at Geneva, in 1605.[84]Dr. Robert Lee (1804-1868) had some celebrity in De Morgan's time through his attempt to introduce music and written prayers into the service of the Scotch Presbyterian church.[85]Born at Veringen, Hohenzollern, in 1512; died at Röteln in 1564.[86]Born at Kinnairdie, Bannfshire, in 1661; died at London in 1708. HisAstronomiae Physicae et Geometriae Elementa, Oxford, 1702, was an influential work.[87]The title was carelessly copied by De Morgan, not an unusual thing in his case. The original reads: A Plaine Discovery, of the whole Revelation of S. Iohn: set downe in two treatises ... set foorth by John Napier L. of Marchiston ... whereunto are annexed, certaine Oracles of Sibylla ... London ... 1611.[88]I have not seen the first edition, but it seems to have appeared in Edinburgh, in 1593, with a second edition there in 1594. The 1611 edition was the third.[89]It seems rather certain that Napier felt his theological work of greater importance than that in logarithms. He was born at Merchiston, near (now a part of) Edinburgh, in 1550, and died there in 1617, three years after the appearance of hisMirifici logarithmorum canonis descriptio.[90]Followed, in the third edition, from which he quotes, by a comma.[91]There was an edition published at Stettin in 1633. An English translation by P. F. Mottelay appeared at London in 1893. Gilbert (1540-1603) was physician to Queen Elizabeth and President of the College of Physicians at London. HisDe Magnetewas the first noteworthy treatise on physics printed in England. He treated of the earth as a spherical magnet and suggested the variation and declination of the needle as a means of finding latitude at sea.[92]The title says "ab authoris fratre collectum," although it was edited by J. Gruterus.[93]Porta was born at Naples in 1550 and died there in 1615. He studied the subject of lenses and the theory of sight, did some work in hydraulics and agriculture, and was well known as an astrologer. HisMagiae naturalis libri XXwas published at Naples in 1589. The above title should readcurvilineorum.[94]Cataldi was born in 1548 and died at Bologna in 1626. He was professor of mathematics at Perugia, Florence, and Bologna, and is known in mathematics chiefly for his work in continued fractions. He was one of the scholarly men of his day.[95]Georg Joachim Rheticus was born at Feldkirch in 1514 and died at Caschau, Hungary, in 1576. He was one of the most prominent pupils of Copernicus, hisNarratio de libris revolutionum Copernici(Dantzig, 1540) having done much to make the theory of his master known.[96]Henry Briggs, who did so much to make logarithms known, and who used the base 10, was born at Warley Wood, in Yorkshire, in 1560, and died at Oxford in 1630. He was Savilian professor of mathematics at Oxford, and his grave may still be seen there.[97]He lived at "Reggio nella Emilia" in the 16th and 17th centuries. HisRegola e modo facilissimo di quadrare il cerchiowas published at Reggio in 1609.[98]Christoph Klau (Clavius) was born at Bamberg in 1537, and died at Rome in 1612. He was a Jesuit priest and taught mathematics in the Jesuit College at Rome. He wrote a number of works on mathematics, including excellent text-books on arithmetic and algebra.[99]Christopher Gruenberger, or Grienberger, was born at Halle in Tyrol in 1561, and died at Rome in 1636. He was, like Clavius, a Jesuit and a mathematician, and he wrote a little upon the subject of projections. HisProspectiva nova coelestisappeared at Rome in 1612.[100]The name should, of course, be Lansbergii in the genitive, and is so in the original title. Philippus Lansbergius was born at Ghent in 1560, and died at Middelburg in 1632. He was a Protestant theologian, and was also a physician and astronomer. He was a well-known supporter of Galileo and Copernicus. HisCommentationes in motum terrae diurnum et annuumappeared at Middelburg in 1630 and did much to help the new theory.[101]I have never seen the work. It is rare.[102]The African explorer, born in Somersetshire in 1827, died at Bath in 1864. He was the first European to cross Central Africa from north to south. He investigated the sources of the Nile.[103]Prester (Presbyter, priest) John, the legendary Christian king whose realm, in the Middle Ages, was placed both in Asia and in Africa, is first mentioned in the chronicles of Otto of Freisingen in the 12th century. In the 14th century his kingdom was supposed to be Abyssinia.[104]"It is a profane and barbarous nation, dirty and slovenly, who eat their meat half raw and drink mare's milk, and who use table-cloths and napkins only to wipe their hands and mouths."[105]"The great Prester John, who is the fourth in rank, is emperor of Ethiopia and of the Abyssinians, and boasts of his descent from the race of David, as having descended from the Queen of Sheba, Queen of Ethiopia. She, having gone to Jerusalem to see the wisdom of Solomon, about the year of the world 2952, returned pregnant with a son whom they called Moylech, from whom they claim descent in a direct line. And so he glories in being the most ancient monarch in the world, saying that his empire has endured for more than three thousand years, which no other empire is able to assert. He also puts into his titles the following: 'We, the sovereign in my realms, uniquely beloved of God, pillar of the faith, sprung from the race of Judah, etc.' The boundaries of this empire touch the Red Sea and the mountains of Azuma on the east, and on the western side it is bordered by the River Nile which separates it from Nubia. To the north lies Egypt, and to the south the kingdoms of Congo and Mozambique. It extends forty degrees in length, or one thousand twenty-five leagues, from Congo or Mozambique on the south to Egypt on the north; and in width it reaches from the Nile on the west to the mountains of Azuma on the east, seven hundred twenty-five leagues, or twenty-nine degrees. This empire contains thirty large provinces, namely Medra, Gaga, Alchy, Cedalon, Mantro, Finazam, Barnaquez, Ambiam, Fungy, Angoté, Cigremaon, Gorga, Cafatez, Zastanla, Zeth, Barly, Belangana, Tygra, Gorgany, Barganaza, d'Ancut, Dargaly, Ambiacatina, Caracogly, Amara, Maon (sic), Guegiera, Bally, Dobora, and Macheda. All of these provinces are situated directly under the equinoctial line between the tropics of Capricorn and Cancer; but they are two hundred fifty leagues nearer our tropic than the other. The name of Prester John signifies Great Lord, and is not Priest [Presbyter] as many think. He has always been a Christian, but often schismatic. At the present time he is a Catholic and recognizes the Pope as sovereign pontiff. I met one of his bishops in Jerusalem, and often conversed with him through the medium of our guide. He was of grave and serious bearing, pleasant of speech, but wonderfully subtle in everything he said. He took great delight in what I had to relate concerning our beautiful ceremonies and the dignity of our prelates in their pontifical vestments. As to other matters I will only say that the Ethiopian is joyous and merry, not at all like the Tartar in the matter of filth, nor like the wretched Arab. They are refined and subtle, trusting no one, wonderfully suspicious, and very devout. They are not at all black as is commonly supposed, by which I refer to those who do not live under the equator or too near to it, for these are Moors as we shall see."With respect to this translation it should be said that the original forms of the proper names have been preserved, although they are not those found in modern works. It should also be stated that the meaning of Prester is not the one that was generally accepted by scholars at the time the work was written, nor is it the one accepted to-day. There seems to be no doubt that the word is derived from Presbyter as stated in note103on page71, since the above-mentioned chronicles of Otto, bishop of Freisingen about the middle of the twelfth century, states this fact clearly. Otto received his information from the bishop of Gabala (the Syrian Jibal) who told him the story of John,rex et sacerdos, or Presbyter John as he liked to be called. He goes on to say "Should it be asked why, with all this power and splendor, he calls himself merely 'presbyter,' this is because of his humility, and because it was not fitting for one whose server was a primate and king, whose butler an archbishop and king, whose chamberlain a bishop and king, whose master of the horse an archimandrite and king, whose chief cook an abbot and king, to be called by such titles as these."[106]Thomas Fienus (Fyens) was born at Antwerp in 1567 and died in 1631. He was professor of medicine at Louvain. Besides the editions mentioned below, hisDe cometis anni 1618appeared at Leipsic in 1656. He also wrote aDisputatio an coelum moveatur et terra quiescat, which appeared at Antwerp in 1619, and again at Leipsic in 1656.[107]Libertus Fromondus (1587-c 1653), a Belgian theologian, dean of the College Church at Harcourt, and professor at Louvain. The name also appears as Froidmont and Froimont.[108]L. Fromondi ... meteorologicorum libri sex.... Cui accessit T. Fieni et L. Fromondi dissertationes de cometa anni 1618....This is from the 1670 edition. The 1619 edition was published at Antwerp. TheMeteorologicorum libri VI, appeared at Antwerp in 1627. He also wroteAnti-Aristarchus sive orbis terrae immobilis liber unicus(Antwerp, 1631);Labyrrinthus sive de compositione continui liber unus, Philosophis, Mathematicis, Theologis utilis et jucundus(Antwerp, 1631) andVesta sive Anti-Aristarchi vindex adversus Jac. Lansbergium (Philippi filium) et copernicanos(Antwerp, 1634).[109]Snell was born at Leyden in 1591, and died there in 1626. He studied under Tycho Brahe and Kepler, and is known for Snell's law of the refraction of light. He was the first to determine the size of the earth by measuring the arc of a meridian with any fair degree of accuracy. The title should read:Willebrordi Snellii R. F. Cyclometricus, de circuli dimensione secundum Logistarum abacos, et ad Mechanicem accuratissima....[110]Bacon was born at York House, London, in 1561, and died near Highgate, London, in 1626. HisNovum Organum Scientiarum or New Method of employing the reasoning faculties in the pursuits of Truthappeared at London in 1620. He had previously published a work entitledOf the Proficience and Advancement of Learning, divine and humane(London, 1605), which again appeared in 1621. HisDe augmentis scientiarum Libri IXappeared at Paris in 1624, and hisHistoria naturalis et experimentalis de ventisat Leyden in 1638. He was successively solicitor general, attorney general, lord chancellor (1619), Baron Verulam and Viscount St. Albans. He was deprived of office and was imprisoned in the Tower of London in 1621, but was later pardoned.[111]The Greek form,Organon, is sometimes used.[112]James Spedding (1808-1881), fellow of Cambridge, who devoted his life to his edition of Bacon.[113]R. Leslie Ellis (1817-1859), editor of theCambridge Mathematical Journal. He also wrote on Roman aqueducts, on Boole's Laws of Thought, and on the formation of a Chinese dictionary.[114]Douglas Derion Heath (1811-1897), a classical and mathematical scholar.[115]There have been numerous editions of Bacon's complete works, including the following: Frankfort, 1665; London, 1730, 1740, 1764, 1765, 1778, 1803, 1807, 1818, 1819, 1824, 1825-36, 1857-74, 1877. The edition to which De Morgan refers is that of 1857-74, 14 vols., of which five were apparently out at the time he wrote. There were also French editions in 1800 and 1835.[116]So in the original for Tycho Brahe.[117]In general these men acted before Baron wrote, or at any rate, before he wrote theNovum Organum, but the statement must not be taken too literally. The dates are as follows: Copernicus, 1473-1543; Tycho Brahe, 1546-1601; Gilbert, 1540-1603; Kepler, 1571-1630; Galileo, 1564-1642; Harvey, 1578-1657. For example, Harvey'sExercitatio Anatomica de Motu Cordis et Sanguinisdid not appear until 1628, and hisExercitationes de Generationeuntil 1651.[118]Robert Hooke (1635-1703) studied under Robert Boyle at Oxford. He was "Curator of Experiments" to the Royal Society and its secretary, and was professor of geometry at Gresham College, London. It is true that he was "very little of a mathematician" although he wrote on the motion of the earth (1674), on helioscopes and other instruments (1675), on the rotation of Jupiter (1666), and on barometers and sails.[119]The son of the Sir William mentioned below. He was born in 1792 and died in 1871. He wrote a treatise on light (1831) and one on astronomy (1836), and established an observatory at the Cape of Good Hope where he made observations during 1834-1838, publishing them in 1847. On his return to England he was knighted, and in 1848 was made president of the Royal Society. The title of the work to which reference is made is:A preliminary discourse on the Study of Natural Philosophy. It appeared at London in 1831.[120]Sir William was horn at Hanover in 1738 and died at Slough, near Windsor in 1822. He discovered the planet Uranus and six satellites, besides two satellites of Saturn. He was knighted by George III.[121]This was the work of 1836. He also published a work entitledOutlines of Astronomyin 1849.[122]While Newton does not tell the story, he refers in thePrincipia(1714 edition, p. 293) to the accident caused by his cat.[123]Marino Ghetaldi (1566-1627), whosePromotus Archimedesappeared at Rome in 1603,Nonnullae propositiones de parabolaat Rome in 1603. andApollonius redivivusat Venice in 1607. He was a nobleman and was ambassador from Venice to Rome.[124]Simon Stevin (born at Bruges, 1548; died at the Hague, 1620). He was an engineer and a soldier, and hisLa Disme(1585) was the first separate treatise on the decimal fraction. The contribution referred to above is probably that on the center of gravity of three bodies (1586).[125]Habakuk Guldin (1577-1643), who took the name Paul on his conversion to Catholicism. He became a Jesuit, and was professor of mathematics at Vienna and later at Gratz. In hisCentrobaryca seu de centro gravitatis trium specierum quantitatis continuae(1635), of the edition of 1641, appears the Pappus rule for the volume of a solid formed by the revolution of a plane figure about an axis, often spoken of as Guldin's Theorem.[126]Edward Wright was born at Graveston, Norfolkshire, in 1560, and died at London in 1615. He was a fellow of Caius College, Cambridge, and in his work entitledThe correction of certain errors in Navigation(1599) he gives the principle of Mercator's projection. He translated thePortuum investigandorum ratioof Stevin in 1599.[127]De Morgan never wrote a more suggestive sentence. Its message is not for his generation alone.[128]The eminent French physicist, Jean Baptiste Biot (1779-1862), professor in the Collège de France. His workSur les observatoires météorologiquesappeared in 1855.[129]George Biddell Airy (1801-1892), professor of astronomy and physics at Cambridge, and afterwards director of the Observatory at Greenwich.[130]De Morgan would have rejoiced in the rôle played by Intuition in the mathematics of to-day, notably among the followers of Professor Klein.[131]Colburn was the best known of the calculating boys produced in America. He was born at Cabot, Vermont, in 1804, and died at Norwich, Vermont, in 1840. Having shown remarkable skill in numbers as early as 1810, he was taken to London in 1812, whence he toured through Great Britain and to Paris. The Earl of Bristol placed him in Westminster School (1816-1819). On his return to America he became a preacher, and later a teacher of languages.[132]The history of calculating boys is interesting. Mathieu le Coc (about 1664), a boy of Lorraine, could extract cube roots at sight at the age of eight. Tom Fuller, a Virginian slave of the eighteenth century, although illiterate, gave the number of seconds in 7 years 17 days 12 hours after only a minute and a half of thought. Jedediah Buxton, an Englishman of the eighteenth century, was studied by the Royal Society because of his remarkable powers. Ampère, the physicist, made long calculations with pebbles at the age of four. Gauss, one of the few infant prodigies to become an adult prodigy, corrected his father's payroll at the age of three. One of the most remarkable of the French calculating boys was Henri Mondeux. He was investigated by Arago, Sturm, Cauchy, and Liouville, for the Académie des Sciences, and a report was written by Cauchy. His specialty was the solution of algebraic problems mentally. He seems to have calculated squares and cubes by a binomial formula of his own invention. He died in obscurity, but was the subject of aBiographieby Jacoby (1846). George P. Bidder, the Scotch engineer (1806-1878), was exhibited as an arithmetical prodigy at the age of ten, and did not attend school until he was twelve. Of the recent cases two deserve special mention, Inaudi and Diamandi. Jacques Inaudi (born in 1867) was investigated for the Académie in 1892 by a commission including Poincaré, Charcot, and Binet. (See theRevue des Deux Mondes, June 15, 1892, and the laboratory bulletins of the Sorbonne). He has frequently exhibited his remarkable powers in America. Périclès Diamandi was investigated by the same commission in 1893. See Alfred Binet,Psychologie des Grands Calculateurs et Joueurs d'Echecs, Paris, 1894.[133]John Flamsteed's (1646-1719) "old white house" was the first Greenwich observatory. He was the Astronomer Royal and first head of this observatory.[134]It seems a pity that De Morgan should not have lived to lash those of our time who are demanding only the immediately practical in mathematics. His satire would have been worth the reading against those who seek to stifle the science they pretend to foster.[135]Ismael Bouillaud, or Boulliau, was born in 1605 and died at Paris in 1694. He was well known as an astronomer, mathematician, and jurist. He lived with De Thou at Paris, and accompanied him to Holland. He traveled extensively, and was versed in the astronomical work of the Persians and Arabs. It was in hisAstronomia philolaica, opus novum(Paris, 1645) that he attacked Kepler's laws. His tables were shown to be erroneous by the fact that the solar eclipse did not take place as predicted by him in 1645.[136]As it did, until 1892, when Airy had reached the ripe age of ninety-one.[137]Didaci a Stunica ... In Job commentariaappeared at Toledo in 1584.[138]"The false Pythagorean doctrine, absolutely opposed to the Holy Scriptures, concerning the mobility of the earth and the immobility of the sun."[139]Paolo Antonio Foscarini (1580-1616), who taught theology and philosophy at Naples and Messina, was one of the first to champion the theories of Copernicus. This was in hisLettera sopra l'opinione de' Pittagorici e del Copernico, della mobilità della Terra e stabilità del Sole, e il nuovo pittagorico sistema del mondo, 4to, Naples, 1615. The condemnation of the Congregation was published in the following spring, and in the year of Foscarini's death at the early age of thirty-six.[140]"To be wholly prohibited and condemned," because "it seeks to show that the aforesaid doctrine is consonant with truth and is not opposed to the Holy Scriptures."[141]"As repugnant to the Holy Scriptures and to its true and Catholic interpretation (which in a Christian man cannot be tolerated in the least), he does not hesitate to treat (of his subject) 'by hypothesis', but he even adds 'as most true'!"[142]"To the places in which he discusses not by hypothesis but by making assertions concerning the position and motion of the earth."[143]"Copernicus.If by chance there shall be vain talkers who, although ignorant of all mathematics, yet taking it upon themselves to sit in judgment upon the subject on account of a certain passage of Scripture badly distorted for their purposes, shall have dared to criticize and censure this teaching of mine, I pay no attention to them, even to the extent of despising their judgment as rash. For it is not unknown that Lactantius, a writer of prominence in other lines although but little versed in mathematics, spoke very childishly about the form of the earth when he ridiculed those who declared that it was spherical. Hence it should not seem strange to the learned if some shall look upon us in the same way. Mathematics is written for mathematicians, to whom these labors of ours will seem, if I mistake not, to add something even to the republic of the Church....Emend.Here strike out everything from 'if by chance' to the words 'these labors of ours,' and adapt it thus: 'But these labors of ours.'"[144]"Copernicus.However if we consider the matter more carefully it will be seen that the investigation is not yet completed, and therefore ought by no means to be condemned.Emend.However, if we consider the matter more carefully it is of no consequence whether we regard the earth as existing in the center of the universe or outside of the center, so far as the solution of the phenomena of celestial movements is concerned."[145]"The whole of this chapter may be cut out, since it avowedly treats of the earth's motion, while it refutes the reasons of the ancients proving its immobility. Nevertheless, since it seems to speak problematically, in order that it may satisfy the learned and keep intact the sequence and unity of the book let it be emended as below."[146]"Copernicus.Therefore why do we still hesitate to concede to it motion which is by nature consistent with its form, the more so because the whole universe is moving, whose end is not and cannot be known, and not confess that there is in the sky an appearance of daily revolution, while on the earth there is the truth of it? And in like manner these things are as if Virgil's Æneas should say, 'We are borne from the harbor' ...Emend.Hence I cannot concede motion to this form, the more so because the universe would fall, whose end is not and cannot be known, and what appears in the heavens is just as if ..."[147]"Copernicus. I also add that it would seem very absurd that motion should be ascribed to that which contains and locates, and not rather to that which is contained and located, that is the earth.Emend.I also add that it is not more difficult to ascribe motion to the contained and located, which is the earth, than to that which contains it."[148]"Copernicus.You see, therefore, that from all these things the motion of the earth is more probable than its immobility, especially in the daily revolution which is as it were a particular property of it.Emend.Omit from 'You see' to the end of the chapter."[149]"Copernicus.Therefore, since there is nothing to hinder the motion of the earth, it seems to me that we should consider whether it has several motions, to the end that it may be looked upon as one of the moving stars.Emend.Therefore, since I have assumed that the earth moves, it seems to me that we should consider whether it has several motions."[150]"Copernicus.We are not ashamed to acknowledge ... that this is preferably verified in the motion of the earth.Emend.We are not ashamed to assume ... that this is consequently verified in the motion."[151]"Copernicus.So divine is surely this work of the Best and Greatest.Emend.Strike out these last words."[152]This should be Cap. 11, lib. i, p. 10.[153]"Copernicus.Demonstration of the threefold motion of the earth.Emend.On the hypothesis of the threefold motion of the earth and its demonstration."[154]This should be Cap. 20, lib. iv, p. 122.[155]"Copernicus.Concerning the size of these three stars, the sun, the moon and the earth.Emend.Strike out the words 'these three stars,' because the earth is not a star as Copernicus would make it."[156]He seems to speak problematically in order to satisfy the learned.[157]One of the Church Fathers, born about 250 A.D., and died about 330, probably at Trèves. He wroteDivinarum Institutionum Libri VII.and other controversial and didactic works against the learning and philosophy of the Greeks.[158]Giovanni Battista Riccioli (1598-1671) taught philosophy and theology at Parma and Bologna, and was later professor of astronomy. HisAlmagestum novumappeared in 1651, and hisArgomento fisico-matematico contro il moto diurno della terrain 1668.[159]He was a native of Arlington, Sussex, and a pensioner of Christ's College, Cambridge. In 1603 he became a master of arts at Oxford.[160]Straying, i.e., from the right way.[161]"Private subjects may, in the presence of danger, defend themselves or their families against a monarch as against any malefactor, if the monarch assaults them like a bandit or a ravisher, and provided they are unable to summon the usual protection and cannot in any way escape the danger."[162]Daniel Neal (1678-1743), an independent minister, wrote aHistory of the Puritansthat appeared in 1732. The account may be found in the New York edition of 1843-44, vol. I, p. 271.[163]Anthony Wood (1632-1695), whoseHistoria et Antiquitates Universitatis Oxoniensis(1674) andAthenae Oxoniensis(1691) are among the classics on Oxford.[164]Part of the title, not here quoted, shows the nature of the work more clearly: "liber unicus, in quo decretum S. Congregationis S. R. E. Cardinal. an. 1616, adversus Pythagorico-Copernicanos editum defenditur."[165]This was John Elliot Drinkwater Bethune (1801-1851), the statesman who did so much for legislative and educational reform in India. His father, John Drinkwater Bethune, wrote a history of the siege of Gibraltar.[166]The article referred to is about thirty years old; since it appeared another has been given (Dubl. Rev., Sept. 1865) which is of much greater depth. In it will also be found the Roman view of Bishop Virgil (ante, p. 32).—A. De M.[167]Jean Baptiste Morin (1583-1656), in his younger days physician to the Bishop of Boulogne and the Duke of Luxemburg, became in 1630 professor of mathematics at the Collège Royale. His chief contribution to the problem of the determination of longitude is hisLongitudinum terrestrium et coelestium nova et hactenus optata scientia(1634). He also wrote against Copernicus in hisFamosi problematis de telluris motu vel quiete hactenus optata solutio(1631), and against Lansberg in hisResponsio pro telluris quiete(1634).
[1]See Mrs. De Morgan'sMemoir of Augustus De Morgan, London, 1882, p 61.
[2]In the first edition this reference was to page 11.
[3]In the first edition this read "at page 438," the work then appearing in a single volume.
[4]"Just as it would surely have been better not to have considered it (i.e., the trinity) as a mystery, and with Cl. Kleckermann to have investigated by the aid of philosophy according to the teaching of true logic what it might be, before they determined what it was; just so would it have been better to withdraw zealously and industriously into the deepest caverns and darkest recesses of metaphysical speculations and suppositions in order to establish their opinion beyond danger from the weapons of their adversaries.... Indeed that great man so explains and demonstrates this dogma (although to theologians the word has not much charm) from the immovable foundations of philosophy, that with but few changes and additions a mind sincerely devoted to truth can desire nothing more."
[5]Mrs. Wititterly, inNicholas Nickleby.—A. De M.
[6]The brackets mean that the paragraph is substantially from some one of theAthenæum Supplements.—S. E. De M.
[7]"It is annoying that this ingenious naturalist who has already given us more useful works and has still others in preparation, uses for this odious task, a pen dipped in gall and wormwood. It is true that many of his remarks have some foundation, and that to each error that he points out he at the same time adds its correction. But he is not always just and never fails to insult. After all, what does his book prove except that a forty-fifth part of a very useful review is not free from mistakes? Must we confuse him with those superficial writers whose liberty of body does not permit them to restrain their fruitfulness, that crowd of savants of the highest rank whose writings have adorned and still adorn theTransactions? Has he forgotten that the names of the Boyles, Newtons, Halleys, De Moivres, Hans Sloanes, etc. have been seen frequently? and that still are found those of the Wards, Bradleys, Grahams, Ellicots, Watsons, and of an author whom Mr. Hill prefers to all others, I mean Mr. Hill himself?"
[8]"Let no free man be seized or imprisoned or in any way harmed except by trial of his peers."
[9]"The master can rob, wreck and punish his slave according to his pleasure save only that he may not maim him."
[10]An Irish antiquary informs me that Virgil is mentioned in annals at A.D. 784, as "Verghil, i.e., the geometer, Abbot of Achadhbo [and Bishop of Saltzburg] died in Germany in the thirteenth year of his bishoprick." No allusion is made to his opinions; but it seems he was, by tradition, a mathematician. The Abbot of Aghabo (Queen's County) was canonized by Gregory IX, in 1233. The story of the second, or scapegoat, Virgil would be much damaged by the character given to the real bishop, if there were anything in it to dilapidate.—A. De M.
[11]"He performed many acts befitting the Papal dignity, and likewise many excellent (to be sure!) works."
[12]"After having been on the throne during ten years of pestilence."
[13]The work is theQuestiones Joannis Buridani super X libros Aristotelis ad Nicomachum, curante Egidio Delfo... Parisiis, 1489, folio. It also appeared at Paris in editions of 1499, 1513, and 1518, and at Oxford in 1637.
[14]Jean Buridan was born at Béthune about 1298, and died at Paris about 1358. He was professor of philosophy at the University of Paris and several times held the office of Rector. As a philosopher he was classed among the nominalists.
[15]So in the original.
[16]Baruch Spinoza, or Benedict de Spinoza as he later called himself, the pantheistic philosopher, excommunicated from the Jewish faith for heresy, was born at Amsterdam in 1632 and died there in 1677.
[17]Michael Scott, or Scot, was born about 1190, probably in Fifeshire, Scotland, and died about 1291. He was one of the best known savants of the court of Emperor Frederick II, and wrote upon astrology, alchemy, and the occult sciences. He was looked upon as a great magician and is mentioned among the wizards in Dante'sInferno.
"That other, round the loinsSo slender of his shape, was Michael Scot,Practised in every slight of magic wile."Inferno, XX.
"That other, round the loinsSo slender of his shape, was Michael Scot,Practised in every slight of magic wile."Inferno, XX.
"That other, round the loins
So slender of his shape, was Michael Scot,
Practised in every slight of magic wile."Inferno, XX.
Boccaccio also speaks of him: "It is not long since there was in this city (Florence) a great master in necromancy, who was called Michele Scotto, because he was a Scot."Decameron, Dec. Giorno.
Scott's mention of him in Canto Second of hisLay of the Last Minstrel, is well known:
"In these fair climes, it was my lotTo meet the wondrous Michael Scott;A wizard of such dreaded fame,That when, in Salamanca's cave,Him listed his magic wand to wave,The bells would ring in Notre Dame!"
"In these fair climes, it was my lotTo meet the wondrous Michael Scott;A wizard of such dreaded fame,That when, in Salamanca's cave,Him listed his magic wand to wave,The bells would ring in Notre Dame!"
"In these fair climes, it was my lot
To meet the wondrous Michael Scott;
A wizard of such dreaded fame,
That when, in Salamanca's cave,
Him listed his magic wand to wave,
The bells would ring in Notre Dame!"
Sir Walter's notes upon him are of interest.
[18]These were some of the forgeries which Michel Chasles (1793-1880) was duped into buying. They purported to be a correspondence between Pascal and Newton and to show that the former had anticipated some of the discoveries of the great English physicist and mathematician. That they were forgeries was shown by Sir David Brewster in 1855.
[19]"Let the serpent also break from its appointed path."
[20]Guglielmo Brutus Icilius Timoleon Libri-Carucci della Sommaja, born at Florence in 1803; died at Fiesole in 1869. HisHistoire des Sciences Mathématiquesappeared at Paris in 1838, the entire first edition of volume I, save some half dozen that he had carried home, being burned on the day that the printing was completed. He was a great collector of early printed works on mathematics, and was accused of having stolen large numbers of them from other libraries. This accusation took him to London, where he bitterly attacked his accusers. There were two auction sales of his library, and a number of his books found their way into De Morgan's collection.
[21]Philo of Gadara lived in the second century B.C. He was a pupil of Sporus, who worked on the problem of the two mean proportionals.
[22]In hisHistoire des Mathématiques, the first edition of which appeared in 1758. Jean Etienne Montucla was born at Lyons in 1725 and died at Versailles in 1799. He was therefore only thirty-three years old when his great work appeared. The second edition, with additions by D'Alembert, appeared in 1799-1802. He also wrote a work on the quadrature of the circle,Histoire des recherches sur la Quadrature du Cercle, which appeared in 1754.
[23]Eutocius of Ascalon was born in 480 A.D. He wrote commentaries on the first four books of the conics of Apollonius of Perga (247-222 B.C.). He also wrote on the Sphere and Cylinder and the Quadrature of the Circle, and on the two books on Equilibrium of Archimedes (287-212 B.C.)
[24]Edward Cocker was born in 1631 and died between 1671 and 1677. His famous arithmetic appeared in 1677 and went through many editions. It was written in a style that appealed to teachers, and was so popular that the expression "According to Cocker" became a household phrase. Early in the nineteenth century there was a similar saying in America, "According to Daboll," whose arithmetic had some points of analogy to that of Cocker. Each had a well-known prototype in the ancient saying, "He reckons like Nicomachus of Gerasa."
[25]So in the original, for Barrême. François Barrême was to France what Cocker was to England. He was born at Lyons in 1640, and died at Paris in 1703. He published several arithmetics, dedicating them to his patron, Colbert. One of the best known of his works isL'arithmétique, ou le livre facile pour apprendre l'arithmétique soi-mème, 1677. The French wordbarêmeorbarrême, a ready-reckoner, is derived from his name.
[26]Born at Rome, about 480 A.D.; died at Pavia, 524. Gibbon speaks of him as "the last of the Romans whom Cato or Tully could have acknowledged for their countryman." His works on arithmetic, music, and geometry were classics in the medieval schools.
[27]Johannes Campanus, of Novarra, was chaplain to Pope Urban IV (1261-1264). He was one of the early medieval translators of Euclid from the Arabic into Latin, and the first printed edition of theElements(Venice, 1482) was from his translation. In this work he probably depended not a little upon at least two or three earlier scholars. He also wroteDe computo ecclesiastico Calendarium, andDe quadratura circuli.
[28]Archimedes gave 3-1/7, and 3-10/71 as the limits of the ratio of the circumference to the diameter of a circle.
[29]Friedrich W. A. Murhard was born at Cassel in 1779 and died there in 1853. HisBibliotheca Mathematica, Leipsic, 1797-1805, is ill arranged and inaccurate, but it is still a helpful bibliography. De Morgan speaks somewhere of his indebtedness to it.
[30]Abraham Gotthelf Kästner was born at Leipsic in 1719, and died at Göttingen in 1800. He was professor of mathematics and physics at Göttingen. HisGeschichte der Mathematik(1796-1800) was a work of considerable merit. In the text of theBudget of Paradoxesthe name appears throughout as Kastner instead of Kästner.
[31]Lucas Gauricus, or Luca Gaurico, born at Giffoni, near Naples, in 1476; died at Rome in 1558. He was an astrologer and mathematician, and was professor of mathematics at Ferrara in 1531. In 1545 he became bishop of Cività Ducale.
[32]John Couch Adams was born at Lidcot, Cornwall, in 1819, and died in 1892. He and Leverrier predicted the discovery of Neptune from the perturbations in Uranus.
[33]Urbain-Jean-Joseph Leverrier was born at Saint-Lô, Manche, in 1811, and died at Paris in 1877. It was his data respecting the perturbations of Uranus that were used by Adams and himself in locating Neptune.
[34]Joseph-Juste Scaliger, the celebrated philologist, was born at Agen in 1540, and died at Leyden in 1609. HisCyclometrica elementa, to which De Morgan refers, appeared at Leyden in 1594.
[35]The title is:In hoc libra contenta.... Introductio i geometriā.... Liber de quadratura circuli. Liber de cubicatione sphere. Perspectiva introductio. Carolus Bovillus, or Charles Bouvelles (Boüelles, Bouilles, Bouvel), was born at Saucourt, Picardy, about 1470, and died at Noyon about 1533. He was canon and professor of theology at Noyon. HisIntroductiocontains considerable work on star polygons, a favorite study in the Middle Ages and early Renaissance. His workQue hoc volumine continētur. Liber de intellectu. Liber de sensu, etc., appeared at Paris in 1509-10.
[36]Nicolaus Cusanus, Nicolaus Chrypffs or Krebs, was born at Kues on the Mosel in 1401, and died at Todi, Umbria, August 11, 1464. He held positions of honor in the church, including the bishopric of Brescia. He was made a cardinal in 1448. He wrote several works on mathematics, hisOpuscula variaappearing about 1490, probably at Strasburg, but published without date or place. HisOperaappeared at Paris in 1511 and again in 1514, and at Basel in 1565.
[37]Henry Stephens (born at Paris about 1528, died at Lyons in 1598) was one of the most successful printers of his day. He was known asTypographus Parisiensis, and to his press we owe some of the best works of the period.
[38]Jacobus Faber Stapulensis (Jacques le Fèvre d'Estaples) was born at Estaples, near Amiens, in 1455, and died at Nérac in 1536. He was a priest, vicar of the bishop of Meaux, lecturer on philosophy at the Collège Lemoine in Paris, and tutor to Charles, son of Francois I. He wrote on philosophy, theology, and mathematics.
[39]Claude-François Milliet de Challes was born at Chambéry in 1621, and died at Turin in 1678. He editedEuclidis Elementorum libri octoin 1660, and published aCursus seu mundus mathematicus, which included a short history of mathematics, in 1674. He also wrote on mathematical geography.
[40]This date should be 1503, if he refers to the first edition. It is well known that this is the first encyclopedia worthy the name to appear in print. It was written by Gregorius Reisch (born at Balingen, and died at Freiburg in 1487), prior of the cloister at Freiburg and confessor to Maximilian I. The first edition appeared at Freiburg in 1503, and it passed through many editions in the sixteenth and seventeenth centuries. The title of the 1504 edition reads:Aepitoma omnis phylosophiae. alias Margarita phylosophica tractans de omni genere scibili: Cum additionibus: Quae in alijs non habentur.
[41]This is theIntroductio in arithmeticam Divi S. Boetii.... Epitome rerum geometricarum ex geometrica introductio C. Bovilli. De quadratura circuli demonstratio ex Campano, that appeared without date about 1507.
[42]Born at Liverpool in 1805, and died there about 1872. He was a merchant, and in 1865 he published, at Liverpool, a work entitledThe Quadrature of the Circle, or the True Ratio between the Diameter and Circumference geometrically and mathematically demonstrated. In this he gives the ratio as exactly 3⅛.
[43]"That it would be impossible to tell him exactly, since no one had yet been able to find precisely the ratio of the circumference to the diameter."
[44]This is the Paris edition: "Parisiis: ex officina Ascensiana anno Christi ... MDXIIII," as appears by the colophon of the second volume to which De Morgan refers.
[45]Regiomontanus, or Johann Müller of Königsberg (Regiomontanus), was born at Königsberg in Franconia, June 5, 1436, and died at Rome July 6, 1476. He studied at Vienna under the great astronomer Peuerbach, and was his most famous pupil. He wrote numerous works, chiefly on astronomy. He is also known by the names Ioannes de Monte Regio, de Regiomonte, Ioannes Germanus de Regiomonte, etc.
[46]Henry Cornelius Agrippa was born at Cologne in 1486 and died either at Lyons in 1534 or at Grenoble in 1535. He was professor of theology at Cologne and also at Turin. After the publication of hisDe Occulta Philosophiahe was imprisoned for sorcery. Both works appeared at Antwerp in 1530, and each passed through a large number of editions. A French translation appeared in Paris in 1582, and an English one in London in 1651.
[47]Nicolaus Remegius was born in Lorraine in 1554, and died at Nancy in 1600. He was a jurist and historian, and held the office of procurator general to the Duke of Lorraine.
[48]This was at the storming of the city by the British on May 4, 1799. From his having been born in India, all this appealed strongly to the interests of De Morgan.
[49]Orontius Finaeus, or Oronce Finé, was born at Briançon in 1494 and died at Paris, October 6, 1555. He was imprisoned by François I for refusing to recognize the concordat (1517). He was made professor of mathematics in the Collège Royal (later called the Collège de France) in 1532. He wrote extensively on astronomy and geometry, but was by no means a great scholar. He was a pretentious man, and his works went through several editions. HisProtomathesisappeared at Paris in 1530-32. The work referred to by De Morgan is theQuadratura circuli tandem inventa & clarissime demonstrata... Lutetiae Parisiorum, 1544, fol. In the 1556 edition of hisDe rebus mathematicis, hactenus desideratis, Libri IIII, published at Paris, the subtitle is:Quibus inter cætera, Circuli quadratura Centum modis, & suprà, per eundem Orontium recenter excogitatis, demonstratus, so that he kept up his efforts until his death.
[50]Johannes Buteo (Boteo, Butéon, Bateon) was born in Dauphiné c. 1485-1489, and died in a cloister in 1560 or 1564. Some writers give Charpey as the place and 1492 as the date of his birth, and state that he died at Canar in 1572. He belonged to the order of St. Anthony, and wrote chiefly on geometry, exposing the pretenses of Finaeus. HisOpera geometricaappeared at Lyons in 1554, and hisLogisticaandDe quadratura circuli libri duoat Lyons in 1559.
[51]This is the great French algebraist, François Viète (Vieta), who was born at Fontenay-le-Comte in 1540, and died at Paris, December 13, 1603. His well-knownIsagoge in artem analyticamappeared at Tours in 1591. HisOpera mathematicawas edited by Van Schooten in 1646.
[52]This is theDe Rebus mathematicis hactenus desideratis, Libri IIII, that appeared in Paris in 1556. For the title page see Smith, D. E.,Rara Arithmetica, Boston, 1908, p. 280.
[53]The title is correct except for a colon afterAstronomicum. Nicolaus Raimarus Ursus was born in Henstede or Hattstede, in Dithmarschen, and died at Prague in 1599 or 1600. He was a pupil of Tycho Brahe. He also wroteDe astronomis hypothesibus(1597) andArithmetica analytica vulgo Cosa oder Algebra(1601).
[54]Born at Dôle, Franche-Comté, about 1550, died in Holland about 1600. The work to which reference is made is theQuadrature du cercle, ou manière de trouver un quarré égal au cercle donné, which appeared at Delft in 1584. Duchesne had the courage of his convictions, not only on circle-squaring but on religion as well, for he was obliged to leave France because of his conversion to Calvinism. De Morgan's statement that his real name is Van der Eycke is curious, since he was French born. The Dutch may have translated his name when he became professor at Delft, but we might equally well say, that his real name was Quercetanus or à Quercu.
[55]This was the father of Adriaan Metius (1571-1635). He was a mathematician and military engineer, and suggested the ratio 355/113 forπ, a ratio afterwards published by his son. The ratio, then new to Europe, had long been known and used in China, having been found by Tsu Ch'ung-chih (428-499 A.D.).
[56]This was Jost Bürgi, or Justus Byrgius, the Swiss mathematician of whom Kepler wrote in 1627: "Apices logistici Justo Byrgio multis annis ante editionem Neperianam viam præiverunt ad hos ipsissimos logarithmos." He constructed a table of antilogarithms (Arithmetische und geometrische Progress-Tabulen), but it was not published until after Napier's work appeared.
[57]Ludolphus Van Ceulen, born at Hildesheim, and died at Leyden in 1610. It was he who first carried the computation ofπto 35 decimal places.
[58]Jens Jenssen Dodt, van Flensburg, a Dutch historian, who died in 1847.
[59]I do not know this edition. There was one "Antverpiae apud Petrum Bellerum sub scuto Burgundiae," 4to, in 1591.
[60]Archytas of Tarentum (430-365 B.C.) who wrote on proportions, irrationals, and the duplication of the cube.
[61]
The Circle Speaks."At first a circle I was called,And was a curve around aboutLike lofty orbit of the sunOr rainbow arch among the clouds.A noble figure then was I—And lacking nothing but a start,And lacking nothing but an end.But now unlovely do I seemPolluted by some angles new.This thing Archytas hath not doneNor noble sire of IcarusNor son of thine, Iapetus.What accident or god can thenHave quadrated mine area?"The Author Replies."By deepest mouth of TuriaAnd lake of limpid clearness, liesA happy state not far removedFrom old Saguntus; farther yetA little way from Sucro town.In this place doth a poet dwell,Who oft the stars will closely scan,And always for himself doth claimWhat is denied to wiser men;—An old man musing here and thereAnd oft forgetful of himself,Not knowing how to rightly placeThe compasses, nor draw a line,As he doth of himself relate.This craftsman fine, in sooth it isHath quadrated thine area."
The Circle Speaks."At first a circle I was called,And was a curve around aboutLike lofty orbit of the sunOr rainbow arch among the clouds.A noble figure then was I—And lacking nothing but a start,And lacking nothing but an end.But now unlovely do I seemPolluted by some angles new.This thing Archytas hath not doneNor noble sire of IcarusNor son of thine, Iapetus.What accident or god can thenHave quadrated mine area?"
The Circle Speaks.
"At first a circle I was called,
And was a curve around about
Like lofty orbit of the sun
Or rainbow arch among the clouds.
A noble figure then was I—
And lacking nothing but a start,
And lacking nothing but an end.
But now unlovely do I seem
Polluted by some angles new.
This thing Archytas hath not done
Nor noble sire of Icarus
Nor son of thine, Iapetus.
What accident or god can then
Have quadrated mine area?"
The Author Replies."By deepest mouth of TuriaAnd lake of limpid clearness, liesA happy state not far removedFrom old Saguntus; farther yetA little way from Sucro town.In this place doth a poet dwell,Who oft the stars will closely scan,And always for himself doth claimWhat is denied to wiser men;—An old man musing here and thereAnd oft forgetful of himself,Not knowing how to rightly placeThe compasses, nor draw a line,As he doth of himself relate.This craftsman fine, in sooth it isHath quadrated thine area."
The Author Replies.
"By deepest mouth of Turia
And lake of limpid clearness, lies
A happy state not far removed
From old Saguntus; farther yet
A little way from Sucro town.
In this place doth a poet dwell,
Who oft the stars will closely scan,
And always for himself doth claim
What is denied to wiser men;—
An old man musing here and there
And oft forgetful of himself,
Not knowing how to rightly place
The compasses, nor draw a line,
As he doth of himself relate.
This craftsman fine, in sooth it is
Hath quadrated thine area."
[62]Pietro Bongo, or Petrus Bungus, was born at Bergamo, and died there in 1601. His work on the Mystery of Numbers is one of the most exhaustive and erudite ones of the mystic writers. The first edition appeared at Bergamo in 1583-84; the second, at Bergamo in 1584-85; the third, at Venice in 1585; the fourth, at Bergamo in 1590; and the fifth, which De Morgan calls the second, in 1591. Other editions, before the Paris edition to which he refers, appeared in 1599 and 1614; and the colophon of the Paris edition is dated 1617. See the editor'sRara Arithmetica, pp. 380-383.
[63]William Warburton (1698-1779), Bishop of Gloucester, whose works got him into numerous literary quarrels, being the subject of frequent satire.
[64]Thomas Galloway (1796-1851), who was professor of mathematics at Sandhurst for a time, and was later the actuary of the Amicable Life Assurance Company of London. In the latter capacity he naturally came to be associated with De Morgan.
[65]Giordano Bruno was born near Naples about 1550. He left the Dominican order to take up Calvinism, and among his publications wasL'expulsion de la bête triomphante. He taught philosophy at Paris and Wittenberg, and some of his works were published in England in 1583-86. Whether or not he was roasted alive "for the maintenance and defence of the holy Church," as De Morgan states, depends upon one's religious point of view. At any rate, he was roasted as a heretic.
[66]Referring to part of hisDiscours de la méthode, Leyden, 1637.
[67]Bartholomew Legate, who was born in Essex about 1575. He denied the divinity of Christ and was the last heretic burned at Smithfield.
[68]Edward Wightman, born probably in Staffordshire. He was anti-Trinitarian, and claimed to be the Messiah. He was the last man burned for heresy in England.
[69]Gaspar Schopp, born at Neumarck in 1576, died at Padua in 1649; grammarian, philologist, and satirist.
[70]Konrad Ritterhusius, born at Brunswick in 1560; died at Altdorf in 1613. He was a jurist of some power.
[71]Johann Jakob Brucker, born at Augsburg in 1696, died there in 1770. He wrote on the history of philosophy (1731-36, and 1742-44).
[72]Daniel Georg Morhof, born at Wismar in 1639, died at Lübeck in 1691. He was rector of the University of Kiel, and professor of eloquence, poetry, and history.
[73]In theHistoire des Sciences Mathématiques, vol. IV, note X, pp. 416-435 of the 1841 edition.
[74]Colenso (1814-1883), missionary bishop of Natal, was one of the leaders of his day in the field of higher biblical criticism. De Morgan must have admired his mathematical works, which were not without merit.
[75]Samuel Roffey Maitland, born at London in 1792; died at Gloucester in 1866. He was an excellent linguist and a critical student of the Bible. He became librarian at Lambeth in 1838.
[76]Archbishop Howley (1766-1848) was a thorough Tory. He was one of the opponents of the Roman Catholic Relief bill, the Reform bill, and the Jewish Civil Disabilities Relief bill.
[77]We have, in America at least, almost forgotten the great stir made by Edward B. Pusey (1800-1882) in the great Oxford movement in the middle of the nineteenth century. He was professor of Hebrew at Oxford, and canon of Christ Church.
[78]That is, hisMagia universalis naturae et artis sive recondita naturalium et artificialium rerum scientia, Würzburg, 1657, 4to, with editions at Bamberg in 1671, and at Frankfort in 1677. Gaspard Schott (Königshofen 1608, Würzburg 1666) was a physicist and mathematician, devoting most of his attention to the curiosities of his sciences. His type of mind must have appealed to De Morgan.
[79]Salicetti Quadratura circuli nova, perspicua, expedita, veraque tum naturalis, tum geometrica, etc., 1608.—Consideratio nova in opusculum Archimedis de circuli dimensione, etc., 1609.
[80]Melchior Adam, who died at Heidelberg in 1622, wrote a collection of biographies which was published at Heidelberg and Frankfort from 1615 to 1620.
[81]Born at Baden in 1524; died at Basel in 1583. The Erastians were related to the Zwinglians, and opposed all power of excommunication and the infliction of penalties by a church.
[82]See Acts xii. 20.
[83]Theodore de Bèse, a French theologian; born at Vezelay, in Burgundy, in 1519; died at Geneva, in 1605.
[84]Dr. Robert Lee (1804-1868) had some celebrity in De Morgan's time through his attempt to introduce music and written prayers into the service of the Scotch Presbyterian church.
[85]Born at Veringen, Hohenzollern, in 1512; died at Röteln in 1564.
[86]Born at Kinnairdie, Bannfshire, in 1661; died at London in 1708. HisAstronomiae Physicae et Geometriae Elementa, Oxford, 1702, was an influential work.
[87]The title was carelessly copied by De Morgan, not an unusual thing in his case. The original reads: A Plaine Discovery, of the whole Revelation of S. Iohn: set downe in two treatises ... set foorth by John Napier L. of Marchiston ... whereunto are annexed, certaine Oracles of Sibylla ... London ... 1611.
[88]I have not seen the first edition, but it seems to have appeared in Edinburgh, in 1593, with a second edition there in 1594. The 1611 edition was the third.
[89]It seems rather certain that Napier felt his theological work of greater importance than that in logarithms. He was born at Merchiston, near (now a part of) Edinburgh, in 1550, and died there in 1617, three years after the appearance of hisMirifici logarithmorum canonis descriptio.
[90]Followed, in the third edition, from which he quotes, by a comma.
[91]There was an edition published at Stettin in 1633. An English translation by P. F. Mottelay appeared at London in 1893. Gilbert (1540-1603) was physician to Queen Elizabeth and President of the College of Physicians at London. HisDe Magnetewas the first noteworthy treatise on physics printed in England. He treated of the earth as a spherical magnet and suggested the variation and declination of the needle as a means of finding latitude at sea.
[92]The title says "ab authoris fratre collectum," although it was edited by J. Gruterus.
[93]Porta was born at Naples in 1550 and died there in 1615. He studied the subject of lenses and the theory of sight, did some work in hydraulics and agriculture, and was well known as an astrologer. HisMagiae naturalis libri XXwas published at Naples in 1589. The above title should readcurvilineorum.
[94]Cataldi was born in 1548 and died at Bologna in 1626. He was professor of mathematics at Perugia, Florence, and Bologna, and is known in mathematics chiefly for his work in continued fractions. He was one of the scholarly men of his day.
[95]Georg Joachim Rheticus was born at Feldkirch in 1514 and died at Caschau, Hungary, in 1576. He was one of the most prominent pupils of Copernicus, hisNarratio de libris revolutionum Copernici(Dantzig, 1540) having done much to make the theory of his master known.
[96]Henry Briggs, who did so much to make logarithms known, and who used the base 10, was born at Warley Wood, in Yorkshire, in 1560, and died at Oxford in 1630. He was Savilian professor of mathematics at Oxford, and his grave may still be seen there.
[97]He lived at "Reggio nella Emilia" in the 16th and 17th centuries. HisRegola e modo facilissimo di quadrare il cerchiowas published at Reggio in 1609.
[98]Christoph Klau (Clavius) was born at Bamberg in 1537, and died at Rome in 1612. He was a Jesuit priest and taught mathematics in the Jesuit College at Rome. He wrote a number of works on mathematics, including excellent text-books on arithmetic and algebra.
[99]Christopher Gruenberger, or Grienberger, was born at Halle in Tyrol in 1561, and died at Rome in 1636. He was, like Clavius, a Jesuit and a mathematician, and he wrote a little upon the subject of projections. HisProspectiva nova coelestisappeared at Rome in 1612.
[100]The name should, of course, be Lansbergii in the genitive, and is so in the original title. Philippus Lansbergius was born at Ghent in 1560, and died at Middelburg in 1632. He was a Protestant theologian, and was also a physician and astronomer. He was a well-known supporter of Galileo and Copernicus. HisCommentationes in motum terrae diurnum et annuumappeared at Middelburg in 1630 and did much to help the new theory.
[101]I have never seen the work. It is rare.
[102]The African explorer, born in Somersetshire in 1827, died at Bath in 1864. He was the first European to cross Central Africa from north to south. He investigated the sources of the Nile.
[103]Prester (Presbyter, priest) John, the legendary Christian king whose realm, in the Middle Ages, was placed both in Asia and in Africa, is first mentioned in the chronicles of Otto of Freisingen in the 12th century. In the 14th century his kingdom was supposed to be Abyssinia.
[104]"It is a profane and barbarous nation, dirty and slovenly, who eat their meat half raw and drink mare's milk, and who use table-cloths and napkins only to wipe their hands and mouths."
[105]"The great Prester John, who is the fourth in rank, is emperor of Ethiopia and of the Abyssinians, and boasts of his descent from the race of David, as having descended from the Queen of Sheba, Queen of Ethiopia. She, having gone to Jerusalem to see the wisdom of Solomon, about the year of the world 2952, returned pregnant with a son whom they called Moylech, from whom they claim descent in a direct line. And so he glories in being the most ancient monarch in the world, saying that his empire has endured for more than three thousand years, which no other empire is able to assert. He also puts into his titles the following: 'We, the sovereign in my realms, uniquely beloved of God, pillar of the faith, sprung from the race of Judah, etc.' The boundaries of this empire touch the Red Sea and the mountains of Azuma on the east, and on the western side it is bordered by the River Nile which separates it from Nubia. To the north lies Egypt, and to the south the kingdoms of Congo and Mozambique. It extends forty degrees in length, or one thousand twenty-five leagues, from Congo or Mozambique on the south to Egypt on the north; and in width it reaches from the Nile on the west to the mountains of Azuma on the east, seven hundred twenty-five leagues, or twenty-nine degrees. This empire contains thirty large provinces, namely Medra, Gaga, Alchy, Cedalon, Mantro, Finazam, Barnaquez, Ambiam, Fungy, Angoté, Cigremaon, Gorga, Cafatez, Zastanla, Zeth, Barly, Belangana, Tygra, Gorgany, Barganaza, d'Ancut, Dargaly, Ambiacatina, Caracogly, Amara, Maon (sic), Guegiera, Bally, Dobora, and Macheda. All of these provinces are situated directly under the equinoctial line between the tropics of Capricorn and Cancer; but they are two hundred fifty leagues nearer our tropic than the other. The name of Prester John signifies Great Lord, and is not Priest [Presbyter] as many think. He has always been a Christian, but often schismatic. At the present time he is a Catholic and recognizes the Pope as sovereign pontiff. I met one of his bishops in Jerusalem, and often conversed with him through the medium of our guide. He was of grave and serious bearing, pleasant of speech, but wonderfully subtle in everything he said. He took great delight in what I had to relate concerning our beautiful ceremonies and the dignity of our prelates in their pontifical vestments. As to other matters I will only say that the Ethiopian is joyous and merry, not at all like the Tartar in the matter of filth, nor like the wretched Arab. They are refined and subtle, trusting no one, wonderfully suspicious, and very devout. They are not at all black as is commonly supposed, by which I refer to those who do not live under the equator or too near to it, for these are Moors as we shall see."
With respect to this translation it should be said that the original forms of the proper names have been preserved, although they are not those found in modern works. It should also be stated that the meaning of Prester is not the one that was generally accepted by scholars at the time the work was written, nor is it the one accepted to-day. There seems to be no doubt that the word is derived from Presbyter as stated in note103on page71, since the above-mentioned chronicles of Otto, bishop of Freisingen about the middle of the twelfth century, states this fact clearly. Otto received his information from the bishop of Gabala (the Syrian Jibal) who told him the story of John,rex et sacerdos, or Presbyter John as he liked to be called. He goes on to say "Should it be asked why, with all this power and splendor, he calls himself merely 'presbyter,' this is because of his humility, and because it was not fitting for one whose server was a primate and king, whose butler an archbishop and king, whose chamberlain a bishop and king, whose master of the horse an archimandrite and king, whose chief cook an abbot and king, to be called by such titles as these."
[106]Thomas Fienus (Fyens) was born at Antwerp in 1567 and died in 1631. He was professor of medicine at Louvain. Besides the editions mentioned below, hisDe cometis anni 1618appeared at Leipsic in 1656. He also wrote aDisputatio an coelum moveatur et terra quiescat, which appeared at Antwerp in 1619, and again at Leipsic in 1656.
[107]Libertus Fromondus (1587-c 1653), a Belgian theologian, dean of the College Church at Harcourt, and professor at Louvain. The name also appears as Froidmont and Froimont.
[108]L. Fromondi ... meteorologicorum libri sex.... Cui accessit T. Fieni et L. Fromondi dissertationes de cometa anni 1618....This is from the 1670 edition. The 1619 edition was published at Antwerp. TheMeteorologicorum libri VI, appeared at Antwerp in 1627. He also wroteAnti-Aristarchus sive orbis terrae immobilis liber unicus(Antwerp, 1631);Labyrrinthus sive de compositione continui liber unus, Philosophis, Mathematicis, Theologis utilis et jucundus(Antwerp, 1631) andVesta sive Anti-Aristarchi vindex adversus Jac. Lansbergium (Philippi filium) et copernicanos(Antwerp, 1634).
[109]Snell was born at Leyden in 1591, and died there in 1626. He studied under Tycho Brahe and Kepler, and is known for Snell's law of the refraction of light. He was the first to determine the size of the earth by measuring the arc of a meridian with any fair degree of accuracy. The title should read:Willebrordi Snellii R. F. Cyclometricus, de circuli dimensione secundum Logistarum abacos, et ad Mechanicem accuratissima....
[110]Bacon was born at York House, London, in 1561, and died near Highgate, London, in 1626. HisNovum Organum Scientiarum or New Method of employing the reasoning faculties in the pursuits of Truthappeared at London in 1620. He had previously published a work entitledOf the Proficience and Advancement of Learning, divine and humane(London, 1605), which again appeared in 1621. HisDe augmentis scientiarum Libri IXappeared at Paris in 1624, and hisHistoria naturalis et experimentalis de ventisat Leyden in 1638. He was successively solicitor general, attorney general, lord chancellor (1619), Baron Verulam and Viscount St. Albans. He was deprived of office and was imprisoned in the Tower of London in 1621, but was later pardoned.
[111]The Greek form,Organon, is sometimes used.
[112]James Spedding (1808-1881), fellow of Cambridge, who devoted his life to his edition of Bacon.
[113]R. Leslie Ellis (1817-1859), editor of theCambridge Mathematical Journal. He also wrote on Roman aqueducts, on Boole's Laws of Thought, and on the formation of a Chinese dictionary.
[114]Douglas Derion Heath (1811-1897), a classical and mathematical scholar.
[115]There have been numerous editions of Bacon's complete works, including the following: Frankfort, 1665; London, 1730, 1740, 1764, 1765, 1778, 1803, 1807, 1818, 1819, 1824, 1825-36, 1857-74, 1877. The edition to which De Morgan refers is that of 1857-74, 14 vols., of which five were apparently out at the time he wrote. There were also French editions in 1800 and 1835.
[116]So in the original for Tycho Brahe.
[117]In general these men acted before Baron wrote, or at any rate, before he wrote theNovum Organum, but the statement must not be taken too literally. The dates are as follows: Copernicus, 1473-1543; Tycho Brahe, 1546-1601; Gilbert, 1540-1603; Kepler, 1571-1630; Galileo, 1564-1642; Harvey, 1578-1657. For example, Harvey'sExercitatio Anatomica de Motu Cordis et Sanguinisdid not appear until 1628, and hisExercitationes de Generationeuntil 1651.
[118]Robert Hooke (1635-1703) studied under Robert Boyle at Oxford. He was "Curator of Experiments" to the Royal Society and its secretary, and was professor of geometry at Gresham College, London. It is true that he was "very little of a mathematician" although he wrote on the motion of the earth (1674), on helioscopes and other instruments (1675), on the rotation of Jupiter (1666), and on barometers and sails.
[119]The son of the Sir William mentioned below. He was born in 1792 and died in 1871. He wrote a treatise on light (1831) and one on astronomy (1836), and established an observatory at the Cape of Good Hope where he made observations during 1834-1838, publishing them in 1847. On his return to England he was knighted, and in 1848 was made president of the Royal Society. The title of the work to which reference is made is:A preliminary discourse on the Study of Natural Philosophy. It appeared at London in 1831.
[120]Sir William was horn at Hanover in 1738 and died at Slough, near Windsor in 1822. He discovered the planet Uranus and six satellites, besides two satellites of Saturn. He was knighted by George III.
[121]This was the work of 1836. He also published a work entitledOutlines of Astronomyin 1849.
[122]While Newton does not tell the story, he refers in thePrincipia(1714 edition, p. 293) to the accident caused by his cat.
[123]Marino Ghetaldi (1566-1627), whosePromotus Archimedesappeared at Rome in 1603,Nonnullae propositiones de parabolaat Rome in 1603. andApollonius redivivusat Venice in 1607. He was a nobleman and was ambassador from Venice to Rome.
[124]Simon Stevin (born at Bruges, 1548; died at the Hague, 1620). He was an engineer and a soldier, and hisLa Disme(1585) was the first separate treatise on the decimal fraction. The contribution referred to above is probably that on the center of gravity of three bodies (1586).
[125]Habakuk Guldin (1577-1643), who took the name Paul on his conversion to Catholicism. He became a Jesuit, and was professor of mathematics at Vienna and later at Gratz. In hisCentrobaryca seu de centro gravitatis trium specierum quantitatis continuae(1635), of the edition of 1641, appears the Pappus rule for the volume of a solid formed by the revolution of a plane figure about an axis, often spoken of as Guldin's Theorem.
[126]Edward Wright was born at Graveston, Norfolkshire, in 1560, and died at London in 1615. He was a fellow of Caius College, Cambridge, and in his work entitledThe correction of certain errors in Navigation(1599) he gives the principle of Mercator's projection. He translated thePortuum investigandorum ratioof Stevin in 1599.
[127]De Morgan never wrote a more suggestive sentence. Its message is not for his generation alone.
[128]The eminent French physicist, Jean Baptiste Biot (1779-1862), professor in the Collège de France. His workSur les observatoires météorologiquesappeared in 1855.
[129]George Biddell Airy (1801-1892), professor of astronomy and physics at Cambridge, and afterwards director of the Observatory at Greenwich.
[130]De Morgan would have rejoiced in the rôle played by Intuition in the mathematics of to-day, notably among the followers of Professor Klein.
[131]Colburn was the best known of the calculating boys produced in America. He was born at Cabot, Vermont, in 1804, and died at Norwich, Vermont, in 1840. Having shown remarkable skill in numbers as early as 1810, he was taken to London in 1812, whence he toured through Great Britain and to Paris. The Earl of Bristol placed him in Westminster School (1816-1819). On his return to America he became a preacher, and later a teacher of languages.
[132]The history of calculating boys is interesting. Mathieu le Coc (about 1664), a boy of Lorraine, could extract cube roots at sight at the age of eight. Tom Fuller, a Virginian slave of the eighteenth century, although illiterate, gave the number of seconds in 7 years 17 days 12 hours after only a minute and a half of thought. Jedediah Buxton, an Englishman of the eighteenth century, was studied by the Royal Society because of his remarkable powers. Ampère, the physicist, made long calculations with pebbles at the age of four. Gauss, one of the few infant prodigies to become an adult prodigy, corrected his father's payroll at the age of three. One of the most remarkable of the French calculating boys was Henri Mondeux. He was investigated by Arago, Sturm, Cauchy, and Liouville, for the Académie des Sciences, and a report was written by Cauchy. His specialty was the solution of algebraic problems mentally. He seems to have calculated squares and cubes by a binomial formula of his own invention. He died in obscurity, but was the subject of aBiographieby Jacoby (1846). George P. Bidder, the Scotch engineer (1806-1878), was exhibited as an arithmetical prodigy at the age of ten, and did not attend school until he was twelve. Of the recent cases two deserve special mention, Inaudi and Diamandi. Jacques Inaudi (born in 1867) was investigated for the Académie in 1892 by a commission including Poincaré, Charcot, and Binet. (See theRevue des Deux Mondes, June 15, 1892, and the laboratory bulletins of the Sorbonne). He has frequently exhibited his remarkable powers in America. Périclès Diamandi was investigated by the same commission in 1893. See Alfred Binet,Psychologie des Grands Calculateurs et Joueurs d'Echecs, Paris, 1894.
[133]John Flamsteed's (1646-1719) "old white house" was the first Greenwich observatory. He was the Astronomer Royal and first head of this observatory.
[134]It seems a pity that De Morgan should not have lived to lash those of our time who are demanding only the immediately practical in mathematics. His satire would have been worth the reading against those who seek to stifle the science they pretend to foster.
[135]Ismael Bouillaud, or Boulliau, was born in 1605 and died at Paris in 1694. He was well known as an astronomer, mathematician, and jurist. He lived with De Thou at Paris, and accompanied him to Holland. He traveled extensively, and was versed in the astronomical work of the Persians and Arabs. It was in hisAstronomia philolaica, opus novum(Paris, 1645) that he attacked Kepler's laws. His tables were shown to be erroneous by the fact that the solar eclipse did not take place as predicted by him in 1645.
[136]As it did, until 1892, when Airy had reached the ripe age of ninety-one.
[137]Didaci a Stunica ... In Job commentariaappeared at Toledo in 1584.
[138]"The false Pythagorean doctrine, absolutely opposed to the Holy Scriptures, concerning the mobility of the earth and the immobility of the sun."
[139]Paolo Antonio Foscarini (1580-1616), who taught theology and philosophy at Naples and Messina, was one of the first to champion the theories of Copernicus. This was in hisLettera sopra l'opinione de' Pittagorici e del Copernico, della mobilità della Terra e stabilità del Sole, e il nuovo pittagorico sistema del mondo, 4to, Naples, 1615. The condemnation of the Congregation was published in the following spring, and in the year of Foscarini's death at the early age of thirty-six.
[140]"To be wholly prohibited and condemned," because "it seeks to show that the aforesaid doctrine is consonant with truth and is not opposed to the Holy Scriptures."
[141]"As repugnant to the Holy Scriptures and to its true and Catholic interpretation (which in a Christian man cannot be tolerated in the least), he does not hesitate to treat (of his subject) 'by hypothesis', but he even adds 'as most true'!"
[142]"To the places in which he discusses not by hypothesis but by making assertions concerning the position and motion of the earth."
[143]"Copernicus.If by chance there shall be vain talkers who, although ignorant of all mathematics, yet taking it upon themselves to sit in judgment upon the subject on account of a certain passage of Scripture badly distorted for their purposes, shall have dared to criticize and censure this teaching of mine, I pay no attention to them, even to the extent of despising their judgment as rash. For it is not unknown that Lactantius, a writer of prominence in other lines although but little versed in mathematics, spoke very childishly about the form of the earth when he ridiculed those who declared that it was spherical. Hence it should not seem strange to the learned if some shall look upon us in the same way. Mathematics is written for mathematicians, to whom these labors of ours will seem, if I mistake not, to add something even to the republic of the Church....Emend.Here strike out everything from 'if by chance' to the words 'these labors of ours,' and adapt it thus: 'But these labors of ours.'"
[144]"Copernicus.However if we consider the matter more carefully it will be seen that the investigation is not yet completed, and therefore ought by no means to be condemned.Emend.However, if we consider the matter more carefully it is of no consequence whether we regard the earth as existing in the center of the universe or outside of the center, so far as the solution of the phenomena of celestial movements is concerned."
[145]"The whole of this chapter may be cut out, since it avowedly treats of the earth's motion, while it refutes the reasons of the ancients proving its immobility. Nevertheless, since it seems to speak problematically, in order that it may satisfy the learned and keep intact the sequence and unity of the book let it be emended as below."
[146]"Copernicus.Therefore why do we still hesitate to concede to it motion which is by nature consistent with its form, the more so because the whole universe is moving, whose end is not and cannot be known, and not confess that there is in the sky an appearance of daily revolution, while on the earth there is the truth of it? And in like manner these things are as if Virgil's Æneas should say, 'We are borne from the harbor' ...Emend.Hence I cannot concede motion to this form, the more so because the universe would fall, whose end is not and cannot be known, and what appears in the heavens is just as if ..."
[147]"Copernicus. I also add that it would seem very absurd that motion should be ascribed to that which contains and locates, and not rather to that which is contained and located, that is the earth.Emend.I also add that it is not more difficult to ascribe motion to the contained and located, which is the earth, than to that which contains it."
[148]"Copernicus.You see, therefore, that from all these things the motion of the earth is more probable than its immobility, especially in the daily revolution which is as it were a particular property of it.Emend.Omit from 'You see' to the end of the chapter."
[149]"Copernicus.Therefore, since there is nothing to hinder the motion of the earth, it seems to me that we should consider whether it has several motions, to the end that it may be looked upon as one of the moving stars.Emend.Therefore, since I have assumed that the earth moves, it seems to me that we should consider whether it has several motions."
[150]"Copernicus.We are not ashamed to acknowledge ... that this is preferably verified in the motion of the earth.Emend.We are not ashamed to assume ... that this is consequently verified in the motion."
[151]"Copernicus.So divine is surely this work of the Best and Greatest.Emend.Strike out these last words."
[152]This should be Cap. 11, lib. i, p. 10.
[153]"Copernicus.Demonstration of the threefold motion of the earth.Emend.On the hypothesis of the threefold motion of the earth and its demonstration."
[154]This should be Cap. 20, lib. iv, p. 122.
[155]"Copernicus.Concerning the size of these three stars, the sun, the moon and the earth.Emend.Strike out the words 'these three stars,' because the earth is not a star as Copernicus would make it."
[156]He seems to speak problematically in order to satisfy the learned.
[157]One of the Church Fathers, born about 250 A.D., and died about 330, probably at Trèves. He wroteDivinarum Institutionum Libri VII.and other controversial and didactic works against the learning and philosophy of the Greeks.
[158]Giovanni Battista Riccioli (1598-1671) taught philosophy and theology at Parma and Bologna, and was later professor of astronomy. HisAlmagestum novumappeared in 1651, and hisArgomento fisico-matematico contro il moto diurno della terrain 1668.
[159]He was a native of Arlington, Sussex, and a pensioner of Christ's College, Cambridge. In 1603 he became a master of arts at Oxford.
[160]Straying, i.e., from the right way.
[161]"Private subjects may, in the presence of danger, defend themselves or their families against a monarch as against any malefactor, if the monarch assaults them like a bandit or a ravisher, and provided they are unable to summon the usual protection and cannot in any way escape the danger."
[162]Daniel Neal (1678-1743), an independent minister, wrote aHistory of the Puritansthat appeared in 1732. The account may be found in the New York edition of 1843-44, vol. I, p. 271.
[163]Anthony Wood (1632-1695), whoseHistoria et Antiquitates Universitatis Oxoniensis(1674) andAthenae Oxoniensis(1691) are among the classics on Oxford.
[164]Part of the title, not here quoted, shows the nature of the work more clearly: "liber unicus, in quo decretum S. Congregationis S. R. E. Cardinal. an. 1616, adversus Pythagorico-Copernicanos editum defenditur."
[165]This was John Elliot Drinkwater Bethune (1801-1851), the statesman who did so much for legislative and educational reform in India. His father, John Drinkwater Bethune, wrote a history of the siege of Gibraltar.
[166]The article referred to is about thirty years old; since it appeared another has been given (Dubl. Rev., Sept. 1865) which is of much greater depth. In it will also be found the Roman view of Bishop Virgil (ante, p. 32).—A. De M.
[167]Jean Baptiste Morin (1583-1656), in his younger days physician to the Bishop of Boulogne and the Duke of Luxemburg, became in 1630 professor of mathematics at the Collège Royale. His chief contribution to the problem of the determination of longitude is hisLongitudinum terrestrium et coelestium nova et hactenus optata scientia(1634). He also wrote against Copernicus in hisFamosi problematis de telluris motu vel quiete hactenus optata solutio(1631), and against Lansberg in hisResponsio pro telluris quiete(1634).