Notes

See Mrs. De Morgan's Memoir of Augustus De Morgan, London, 1882, p 61.

In the first edition this reference was to page 11.

In the first edition this read "at page 438," the work then appearing in a single volume.

"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."

Mrs. Wititterly, in Nicholas Nickleby.—A. De M.

The brackets mean that the paragraph is substantially from some one of the Athenæum Supplements.—S. E. De M.

"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 the Transactions? 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?"

"Let no free man be seized or imprisoned or in any way harmed except by trial of his peers."

"The master can rob, wreck and punish his slave according to his pleasure save only that he may not maim him."

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.

"He performed many acts befitting the Papal dignity, and likewise many excellent (to be sure!) works."

"After having been on the throne during ten years of pestilence."

The work is the Questiones 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.

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.

So in the original.

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.

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's Inferno.

"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 his Lay of the Last Minstrel, is well known:

"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.

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.

"Let the serpent also break from its appointed path."

Guglielmo Brutus Icilius Timoleon Libri-Carucci della Sommaja, born at Florence in 1803; died at Fiesole in 1869. His Histoire des Sciences Mathématiques appeared 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.

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.

In his Histoire 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.

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.)

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."

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 is L'arithmétique, ou le livre facile pour apprendre l'arithmétique soi-mème, 1677. The French word barême or barrême, a ready-reckoner, is derived from his name.

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.

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 the Elements (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 wrote De computo ecclesiastico Calendarium, and De quadratura circuli.

Archimedes gave 3-1/7, and 3-10/71 as the limits of the ratio of the circumference to the diameter of a circle.

Friedrich W. A. Murhard was born at Cassel in 1779 and died there in 1853. His Bibliotheca 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.

Abraham Gotthelf Kästner was born at Leipsic in 1719, and died at Göttingen in 1800. He was professor of mathematics and physics at Göttingen. His Geschichte der Mathematik (1796-1800) was a work of considerable merit. In the text of the Budget of Paradoxes the name appears throughout as Kastner instead of Kästner.

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.

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.

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.

Joseph-Juste Scaliger, the celebrated philologist, was born at Agen in 1540, and died at Leyden in 1609. His Cyclometrica elementa, to which De Morgan refers, appeared at Leyden in 1594.

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. His Introductio contains considerable work on star polygons, a favorite study in the Middle Ages and early Renaissance. His work Que hoc volumine continētur. Liber de intellectu. Liber de sensu, etc., appeared at Paris in 1509-10.

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, his Opuscula varia appearing about 1490, probably at Strasburg, but published without date or place. His Opera appeared at Paris in 1511 and again in 1514, and at Basel in 1565.

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 as Typographus Parisiensis, and to his press we owe some of the best works of the period.

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.

Claude-François Milliet de Challes was born at Chambéry in 1621, and died at Turin in 1678. He edited Euclidis Elementorum libri octo in 1660, and published a Cursus seu mundus mathematicus, which included a short history of mathematics, in 1674. He also wrote on mathematical geography.

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.

This is the Introductio 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.

Born at Liverpool in 1805, and died there about 1872. He was a merchant, and in 1865 he published, at Liverpool, a work entitled The 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.

"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."

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.

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.

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 his De Occulta Philosophia he 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.

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.

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.

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. His Protomathesis appeared at Paris in 1530-32. The work referred to by De Morgan is the Quadratura circuli tandem inventa & clarissime demonstrata ... Lutetiae Parisiorum, 1544, fol. In the 1556 edition of his De 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.

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. His Opera geometrica appeared at Lyons in 1554, and his Logistica and De quadratura circuli libri duo at Lyons in 1559.

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-known Isagoge in artem analyticam appeared at Tours in 1591. His Opera mathematica was edited by Van Schooten in 1646.

This is the De 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.

The title is correct except for a colon after Astronomicum. 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 wrote De astronomis hypothesibus (1597) and Arithmetica analytica vulgo Cosa oder Algebra (1601).

Born at Dôle, Franche-Comté, about 1550, died in Holland about 1600. The work to which reference is made is the Quadrature 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.

This was the father of Adriaan Metius (1571-1635). He was a mathematician and military engineer, and suggested the ratio 355/113 for π, 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.).

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.

Ludolphus Van Ceulen, born at Hildesheim, and died at Leyden in 1610. It was he who first carried the computation of π to 35 decimal places.

Jens Jenssen Dodt, van Flensburg, a Dutch historian, who died in 1847.

I do not know this edition. There was one "Antverpiae apud Petrum Bellerum sub scuto Burgundiae," 4to, in 1591.

Archytas of Tarentum (430-365 B.C.) who wrote on proportions, irrationals, and the duplication of the cube.

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 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."

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's Rara Arithmetica, pp. 380-383.

William Warburton (1698-1779), Bishop of Gloucester, whose works got him into numerous literary quarrels, being the subject of frequent satire.

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.

Giordano Bruno was born near Naples about 1550. He left the Dominican order to take up Calvinism, and among his publications was L'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.

Referring to part of his Discours de la méthode, Leyden, 1637.

Bartholomew Legate, who was born in Essex about 1575. He denied the divinity of Christ and was the last heretic burned at Smithfield.

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.

Gaspar Schopp, born at Neumarck in 1576, died at Padua in 1649; grammarian, philologist, and satirist.

Konrad Ritterhusius, born at Brunswick in 1560; died at Altdorf in 1613. He was a jurist of some power.

Johann Jakob Brucker, born at Augsburg in 1696, died there in 1770. He wrote on the history of philosophy (1731-36, and 1742-44).

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.

In the Histoire des Sciences Mathématiques, vol. IV, note X, pp. 416-435 of the 1841 edition.

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.

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.

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.

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.

That is, his Magia 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.

Salicetti Quadratura circuli nova, perspicua, expedita, veraque tum naturalis, tum geometrica, etc., 1608.—Consideratio nova in opusculum Archimedis de circuli dimensione, etc., 1609.

Melchior Adam, who died at Heidelberg in 1622, wrote a collection of biographies which was published at Heidelberg and Frankfort from 1615 to 1620.

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.

See Acts xii. 20.

Theodore de Bèse, a French theologian; born at Vezelay, in Burgundy, in 1519; died at Geneva, in 1605.

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.

Born at Veringen, Hohenzollern, in 1512; died at Röteln in 1564.

Born at Kinnairdie, Bannfshire, in 1661; died at London in 1708. His Astronomiae Physicae et Geometriae Elementa, Oxford, 1702, was an influential work.

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.

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.

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 his Mirifici logarithmorum canonis descriptio.

Followed, in the third edition, from which he quotes, by a comma.

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. His De Magnete was 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.

The title says "ab authoris fratre collectum," although it was edited by J. Gruterus.

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. His Magiae naturalis libri XX was published at Naples in 1589. The above title should read curvilineorum.

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.

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, his Narratio de libris revolutionum Copernici (Dantzig, 1540) having done much to make the theory of his master known.

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.

He lived at "Reggio nella Emilia" in the 16th and 17th centuries. His Regola e modo facilissimo di quadrare il cerchio was published at Reggio in 1609.

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.

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. His Prospectiva nova coelestis appeared at Rome in 1612.

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. His Commentationes in motum terrae diurnum et annuum appeared at Middelburg in 1630 and did much to help the new theory.

I have never seen the work. It is rare.

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.

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.

"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."

"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 note on page , 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."

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, his De cometis anni 1618 appeared at Leipsic in 1656. He also wrote a Disputatio an coelum moveatur et terra quiescat, which appeared at Antwerp in 1619, and again at Leipsic in 1656.

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.

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. The Meteorologicorum libri VI, appeared at Antwerp in 1627. He also wrote Anti-Aristarchus sive orbis terrae immobilis liber unicus (Antwerp, 1631); Labyrrinthus sive de compositione continui liber unus, Philosophis, Mathematicis, Theologis utilis et jucundus (Antwerp, 1631) and Vesta sive Anti-Aristarchi vindex adversus Jac. Lansbergium (Philippi filium) et copernicanos (Antwerp, 1634).

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....

Bacon was born at York House, London, in 1561, and died near Highgate, London, in 1626. His Novum Organum Scientiarum or New Method of employing the reasoning faculties in the pursuits of Truth appeared at London in 1620. He had previously published a work entitled Of the Proficience and Advancement of Learning, divine and humane (London, 1605), which again appeared in 1621. His De augmentis scientiarum Libri IX appeared at Paris in 1624, and his Historia naturalis et experimentalis de ventis at 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.

The Greek form, Organon, is sometimes used.

James Spedding (1808-1881), fellow of Cambridge, who devoted his life to his edition of Bacon.

R. Leslie Ellis (1817-1859), editor of the Cambridge Mathematical Journal. He also wrote on Roman aqueducts, on Boole's Laws of Thought, and on the formation of a Chinese dictionary.

Douglas Derion Heath (1811-1897), a classical and mathematical scholar.

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.

So in the original for Tycho Brahe.

In general these men acted before Baron wrote, or at any rate, before he wrote the Novum 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's Exercitatio Anatomica de Motu Cordis et Sanguinis did not appear until 1628, and his Exercitationes de Generatione until 1651.

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.

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.

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.

This was the work of 1836. He also published a work entitled Outlines of Astronomy in 1849.

While Newton does not tell the story, he refers in the Principia (1714 edition, p. 293) to the accident caused by his cat.

Marino Ghetaldi (1566-1627), whose Promotus Archimedes appeared at Rome in 1603, Nonnullae propositiones de parabola at Rome in 1603. and Apollonius redivivus at Venice in 1607. He was a nobleman and was ambassador from Venice to Rome.

Simon Stevin (born at Bruges, 1548; died at the Hague, 1620). He was an engineer and a soldier, and his La 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).

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 his Centrobaryca 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.

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 entitled The correction of certain errors in Navigation (1599) he gives the principle of Mercator's projection. He translated the Portuum investigandorum ratio of Stevin in 1599.

De Morgan never wrote a more suggestive sentence. Its message is not for his generation alone.

The eminent French physicist, Jean Baptiste Biot (1779-1862), professor in the Collège de France. His work Sur les observatoires météorologiques appeared in 1855.

George Biddell Airy (1801-1892), professor of astronomy and physics at Cambridge, and afterwards director of the Observatory at Greenwich.

De Morgan would have rejoiced in the rôle played by Intuition in the mathematics of to-day, notably among the followers of Professor Klein.

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.

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 a Biographie by 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 the Revue 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.

John Flamsteed's (1646-1719) "old white house" was the first Greenwich observatory. He was the Astronomer Royal and first head of this observatory.

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.

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 his Astronomia 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.

As it did, until 1892, when Airy had reached the ripe age of ninety-one.

Didaci a Stunica ... In Job commentaria appeared at Toledo in 1584.

"The false Pythagorean doctrine, absolutely opposed to the Holy Scriptures, concerning the mobility of the earth and the immobility of the sun."

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 his Lettera 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.

"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."

"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'!"

"To the places in which he discusses not by hypothesis but by making assertions concerning the position and motion of the earth."

"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.'"

"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."

"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."

"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 ..."

"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."

"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."

"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."

"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."

"Copernicus. So divine is surely this work of the Best and Greatest. Emend. Strike out these last words."

This should be Cap. 11, lib. i, p. 10.

"Copernicus. Demonstration of the threefold motion of the earth. Emend. On the hypothesis of the threefold motion of the earth and its demonstration."

This should be Cap. 20, lib. iv, p. 122.

"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."

He seems to speak problematically in order to satisfy the learned.

One of the Church Fathers, born about 250 A.D., and died about 330, probably at Trèves. He wrote Divinarum Institutionum Libri VII. and other controversial and didactic works against the learning and philosophy of the Greeks.

Giovanni Battista Riccioli (1598-1671) taught philosophy and theology at Parma and Bologna, and was later professor of astronomy. His Almagestum novum appeared in 1651, and his Argomento fisico-matematico contro il moto diurno della terra in 1668.

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.

Straying, i.e., from the right way.

"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."

Daniel Neal (1678-1743), an independent minister, wrote a History of the Puritans that appeared in 1732. The account may be found in the New York edition of 1843-44, vol. I, p. 271.

Anthony Wood (1632-1695), whose Historia et Antiquitates Universitatis Oxoniensis (1674) and Athenae Oxoniensis (1691) are among the classics on Oxford.

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."

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.

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.

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 his Longitudinum terrestrium et coelestium nova et hactenus optata scientia (1634). He also wrote against Copernicus in his Famosi problematis de telluris motu vel quiete hactenus optata solutio (1631), and against Lansberg in his Responsio pro telluris quiete (1634).

The work appeared at Leyden in 1626, at Amsterdam in 1634, at Copenhagen in 1640 and again at Leyden in 1650. The title of the 1640 edition is Arithmeticae Libri II et Geometriae Libri VI. The work on which it is based is the Arithmeticae et Geometriae Practica, which appeared in 1611.

The father's name was Adriaan, and Lalande says that it was Montucla who first made the mistake of calling him Peter, thinking that the initials P. M. stood for Petrus Metius, when in reality they stood for piae memoriae! The ratio 355/113 was known in China hundreds of years before his time. See note , page .

Adrian Metius (1571-1635) was professor of medicine at the University of Franeker. His work was, however, in the domain of astronomy, and in this domain he published several treatises.

The first edition was entitled: The Discovery of a World in the Moone. Or, a Discourse Tending to prove that 'tis probable there may be another habitable World in that Planet. 1638, 8vo. The fourth edition appeared in 1684. John Wilkins (1614-1672) was Warden of Wadham College, Oxford; master of Trinity, Cambridge; and, later, Bishop of Chester. He was influential in founding the Royal Society.

The first edition was entitled: C. Hugenii Κοσμοθεωρος, sive de Terris coelestibus, earumque ornatu, conjecturae, The Hague, 1698, 4to. There were several editions. It was also translated into French (1718), and there was another English edition (1722). Huyghens (1629-1695) was one of the best mathematical physicists of his time.

It is hardly necessary to say that science has made enormous advance in the chemistry of the universe since these words were written.

William Whewell (1794-1866) is best known through his History of the Inductive Sciences (1837) and Philosophy of the Inductive Sciences (1840).

Thomas Chalmers (1780-1847), the celebrated Scotch preacher. These discourses were delivered while he was minister in a large parish in the poorest part of Glasgow, and in them he attempted to bring science into harmony with the Bible. He was afterwards professor of moral philosophy at St. Andrew's (1823-28), and professor of theology at Edinburgh (1828). He became the leader of a schism from the Scotch Presbyterian Church,—the Free Church.

That is, in Robert Watt's (1774-1819) Bibliotheca Britannica (posthumous, 1824). Nor is it given in the Dictionary of National Biography.

The late Greek satirist and poet, c. 120-c. 200 A.D.

François Rabelais (c. 1490-1553) the humorist who created Pantagruel (1533) and Gargantua (1532). His work as a physician and as editor of the works of Galen and Hippocrates is less popularly known.

Francis Godwin (1562-1633) bishop of Llandaff and Hereford. Besides some valuable historical works he wrote The Man in the Moone, or a Discourse of a voyage thither by Domingo Gonsales, the Speed Messenger of London, 1638.

Bernard Le Bovier de Fontenelle (1657-1757), historian, critic, mathematician, Secretary of the Académie des Sciences, and member of the Académie Française. His Entretien sur la pluralité des mondes appeared at Paris in 1686.

Athanasius Kircher (1602-1680), Jesuit, professor of mathematics and philosophy, and later of Hebrew and Syriac, at Wurzburg; still later professor of mathematics and Hebrew at Rome. He wrote several works on physics. His collection of mathematical instruments and other antiquities became the basis of the Kircherian Museum at Rome.

"Both belief and non-belief are dangerous. Hippolitus died because his stepmother was believed. Troy fell because Cassandra was not believed. Therefore the truth should be investigated long before foolish opinion can properly judge." (Prove = probe?).

Jacobus Grandamicus (Jacques Grandami) was born at Nantes in 1588 and died at Paris in 1672. He was professor of theology and philosophy in the Jesuit colleges at Rennes, Tours, Rouen, and other places. He wrote several works on astronomy.

"And I, if I be lifted up from the earth, will draw all men unto me." John xii. 32.

Andrea Argoli (1568-1657) wrote a number of works on astronomy, and computed ephemerides from 1621 to 1700.

So in the original edition of the Budget. It is Johannem Pellum in the original title. John Pell (1610 or 1611-1685) studied at Cambridge and Oxford, and was professor of mathematics at Amsterdam (1643-46) and Breda (1646-52). He left many manuscripts but published little. His name attaches by accident to an interesting equation recently studied with care by Dr. E. E. Whitford (New York, 1912).

Christianus Longomontanus (Christen Longberg or Lumborg) was born in 1569 at Longberg, Jutland, and died in 1647 at Copenhagen. He was an assistant of Tycho Brahe and accepted the diurnal while denying the orbital motion of the earth. His Cyclometria e lunulis reciproce demonstrata appeared in 1612 under the name of Christen Severin, the latter being his family name. He wrote several other works on the quadrature problem, and some treatises on astronomy.

The names are really pretty well known. Giles Persone de Roberval was born at Roberval near Beauvais in 1602, and died at Paris in 1675. He was professor of philosophy at the Collège Gervais at Paris, and later at the Collège Royal. He claimed to have discovered the theory of indivisibles before Cavalieri, and his work is set forth in his Traité des indivisibles which appeared posthumously in 1693.

Hobbes (1588-1679), the political and social philosopher, lived a good part of his time (1610-41) in France where he was tutor to several young noblemen, including the Cavendishes. His Leviathan (1651) is said to have influenced Spinoza, Leibnitz, and Rousseau. His Quadratura circuli, cubatio sphaerae, duplicatio cubi ... (London, 1669), Rosetum geometricum ... (London, 1671), and Lux Mathematica, censura doctrinae Wallisianae contra Rosetum Hobbesii (London, 1674) are entirely forgotten to-day. (See a further note, infra.)

Pierre de Carcavi, a native of Lyons, died at Paris in 1684. He was a member of parliament, royal librarian, and member of the Académie des Sciences. His attempt to prove the impossibility of the quadrature appeared in 1645. He was a frequent correspondent of Descartes.

Cavendish (1591-1654) was Sir (not Lord) Charles. He was, like De Morgan himself, a bibliophile in the domain of mathematics. His life was one of struggle, his term as member of parliament under Charles I being followed by gallant service in the royal army. After the war he sought refuge on the continent where he met most of the mathematicians of his day. He left a number of manuscripts on mathematics, which his widow promptly disposed of for waste paper. If De Morgan's manuscripts had been so treated we should not have had his revision of his Budget of Paradoxes.

Marin Mersenne (1588-1648), a minorite, living in the cloisters at Nevers and Paris, was one of the greatest Franciscan scholars. He edited Euclid, Apollonius, Archimedes, Theodosius, and Menelaus (Paris, 1626), translated the Mechanics of Galileo into French (1634), wrote Harmonicorum Libri XII (1636), and Cogitata physico-mathematica (1644), and taught theology and philosophy at Nevers.

Johann Adolph Tasse (Tassius) was born in 1585 and died at Hamburg in 1654. He was professor of mathematics in the Gymnasium at Hamburg, and wrote numerous works on astronomy, chronology, statics, and elementary mathematics.

Johann Ludwig, Baron von Wolzogen, seems to have been one of the early unitarians, called Fratres Polonorum because they took refuge in Poland. Some of his works appear in the Bibliotheca Fratrum Polonorum (Amsterdam, 1656). I find no one by the name who was contributing to mathematics at this time.

Descartes is too well known to need mention in this connection.

Bonaventura Cavalieri (1598-1647) was a Jesuit, a pupil of Galileo, and professor of mathematics at Bologna. His greatest work, Geometria indivisibilibus continuorum nova quadam ratione promota, in which he makes a noteworthy step towards the calculus, appeared in 1635.

Jacob (Jacques) Golius was born at the Hague in 1596 and died at Leyden in 1667. His travels in Morocco and Asia Minor (1622-1629) gave him such knowledge of Arabic that he became professor of that language at Leyden. After Snell's death he became professor of mathematics there. He translated Arabic works on mathematics and astronomy into Latin.

It would be interesting to follow up these rumors, beginning perhaps with the tomb of Archimedes. The Ludolph van Ceulen story is very likely a myth. The one about Fagnano may be such. The Bernoulli tomb does have the spiral, however (such as it is), as any one may see in the cloisters at Basel to-day.

Collins (1625-1683) was secretary of the Royal Society, and was "a kind of register of all new improvements in mathematics." His office brought him into correspondence with all of the English scientists, and he was influential in the publication of various important works, including Branker's translation of the algebra by Rhonius, with notes by Pell, which was the first work to contain the present English-American symbol of division. He also helped in the publication of editions of Archimedes and Apollonius, of Kersey's Algebra, and of the works of Wallis. His profession was that of accountant and civil engineer, and he wrote three unimportant works on mathematics (one published posthumously, and the others in 1652 and 1658).

Heinrich Christian Schumacher (1780-1850) was professor of astronomy at Copenhagen and director of the observatory at Altona. His translation of Carnot's Géométrie de position (1807) brought him into personal relations with Gauss, and the friendship was helpful to Schumacher. He was a member of many learned societies and had a large circle of acquaintances. He published numerous monographs and works on astronomy.

Gassendi (1592-1655) might well have been included by De Morgan in the group, since he knew and was a friend of most of the important mathematicians of his day. Like Mersenne, he was a minorite, but he was a friend of Galileo and Kepler, and wrote a work under the title Institutio astronomica, juxta hypotheses Copernici, Tychonis-Brahaei et Ptolemaei (1645). He taught philosophy at Aix, and was later professor of mathematics at the College Royal at Paris.

Burnet is the Bishop Gilbert Burnet (1643-1715) who was so strongly anti-Romanistic that he left England during the reign of James II and joined the ranks of the Prince of Orange. William made him bishop of Salisbury.

There is some substantial basis for De Morgan's doubts as to the connection of that mirandula of his age, Sir Kenelm Digby (1603-1665), with the famous poudre de sympathie. It is true that he was just the one to prepare such a powder. A dilletante in everything,—learning, war, diplomacy, religion, letters, and science—he was the one to exploit a fraud of this nature. He was an astrologer, an alchemist, and a fabricator of tales, and well did Henry Stubbes characterize him as "the very Pliny of our age for lying." He first speaks of the powder in a lecture given at Montpellier in 1658, and in the same year he published the address at Paris under the title: Discours fait en une célèbre assemblée par le chevalier Digby .... touchant la guérison de playes par la poudre de sympathie. The London edition referred to by De Morgan also came out in 1658, and several editions followed it in England, France and Germany. But Nathaniel Highmore in his History of Generation (1651) referred to the concoction as "Talbot's Powder" some years before Digby took it up. The basis seems to have been vitriol, and it was claimed that it would heal a wound by simply being applied to a bandage taken from it.

This work by Thomas Birch (1705-1766) came out in 1756-57. Birch was a voluminous writer on English history. He was a friend of Dr. Johnson and of Walpole, and he wrote a life of Robert Boyle.

We know so much about John Evelyn (1620-1706) through the diary which he began at the age of eleven, that we forget his works on navigation and architecture.

I suppose this was the seventh Earl of Shrewsbury (1553-1616).

This is interesting in view of the modern aseptic practice of surgery and the antiseptic treatment of wounds inaugurated by the late Lord Lister.

Perhaps De Morgan had not heard the bon mot of Dr. Holmes: "I firmly believe that if the whole materia medica could be sunk to the bottom of the sea, it would be all the better for mankind and all the worse for the fishes."

The full title is worth giving, because it shows the mathematical interests of Hobbes, and the nature of the six dialogues: Examinatio et emendatio mathematicae hodiernae qualis explicatur in libris Johannis Wallisii geometriae professoris Saviliani in Academia Oxoniensi: distributa in sex dialogos (1. De mathematicae origine ...; 2. De principiis traditis ab Euclide; 3. De demonstratione operationum arithmeticarum ...; 4. De rationibus; 5. De angula contactus, de sectionibus coni, et arithmetica infinitorum; 6. Dimensio circuli tribus methodis demonstrata ... item cycloidis verae descriptio et proprietates aliquot.) Londini, 1660 (not 1666). For a full discussion of the controversy over the circle, see George Croom Robertson's biography of Hobbes in the eleventh edition of the Encyclopaedia Britannica.

This is his Animadversions upon Mr. Hobbes' late book De principiis et ratiocinatione geometrarum, 1666, or his Hobbianae quadraturae circuli, cubationis sphaerae et duplicationis cubi confutatio, also of 1669.

This is the work of 1669 referred to above.

Gregoire de St. Vincent (1584-1667) published his Opus geometricum quadraturae circuli et sectionum coni at Antwerp in 1647.

This appears in J. Scaligeri cyclometrica elementa duo, Lugduni Batav., 1594.

Adriaen van Roomen (1561-1615) gave the value of π to sixteen decimal places in his Ideae mathematicae pars prima (1593), and wrote his In Archimedis circuli dimensionem expositio & analysis in 1597.

Kästner. See note on page .

Bentley (1662-1742) might have done it, for as the head of Trinity College, Cambridge, and a follower of Newton, he knew some mathematics. Erasmus (1466-1536) lived a little too early to attempt it, although his brilliant satire might have been used to good advantage against those who did try.

"In grammar, to give the winds to the ships and to give the ships to the winds mean the same thing. But in geometry it is one thing to assume the circle BCD not greater than thirty-six segments BCDF, and another (to assume) the thirty-six segments BCDF not greater than the circle. The one assumption is true, the other false."

The Greek scholar (1559-1614) who edited a Greek and Latin edition of Aristotle in 1590.

Jacques Auguste de Thou (1553-1617), the historian and statesman.

"To value Scaliger higher even when wrong, than the multitude when right."

"I would rather err with Scaliger than be right with Clavius."

"The perimeter of the dodecagon to be inscribed in a circle is greater than the perimeter of the circle. And the more sides a polygon to be inscribed in a circle successively has, so much the greater will the perimeter of the polygon be than the perimeter of the circle."

De Morgan took, perhaps, the more delight in speaking thus of Sir William Hamilton (1788-1856) because of a spirited controversy that they had in 1847 over the theory of logic. Possibly, too, Sir William's low opinion of mathematics had its influence.

Edwards (1699-1757) wrote The canons of criticism (1747) in which he gave a scathing burlesque on Warburton's Shakespeare. It went through six editions.

Antoine Teissier (born in 1632) published his Eloges des hommes savants, tirés de l'histoire de M. de Thou in 1683.

"He boasted without reason of having found the quadrature of the circle. The glory of this admirable discovery was reserved for Joseph Scaliger, as Scévole de St. Marthe has written."

Natural and political observations mentioned in the following Index, and made upon the Bills of Mortality.... With reference to the government, religion, trade, growth, ayre, and diseases of the said city. London, 1662, 4to. The book went through several editions.

Ne sutor ultra crepidam, "Let the cobbler stick to his last," as we now say.

The author (1632-1695) of the Historia et Antiquitates Universitatis Oxoniensis (1674). See note , page .

The mathematical guild owes Samuel Pepys (1633-1703) for something besides his famous diary (1659-1669). Not only was he president of the Royal Society (1684), but he was interested in establishing Sir William Boreman's mathematical school at Greenwich.

John Graunt (1620-1674) was a draper by trade, and was a member of the Common Council of London until he lost office by turning Romanist. Although a shopkeeper, he was elected to the Royal Society on the special recommendation of Charles II. Petty edited the fifth edition of his work, adding much to its size and value, and this may be the basis of Burnet's account of the authorship.

Petty (1623-1687) was a mathematician and economist, and a friend of Pell and Sir Charles Cavendish. His survey of Ireland, made for Cromwell, was one of the first to be made on a large scale in a scientific manner. He was one of the founders of the Royal Society.

The story probably arose from Graunt's recent conversion to the Roman Catholic faith.

He was born in 1627 and died in 1704. He published a series of ephemerides, beginning in 1659. He was imprisoned in 1679, at the time of the "Popish Plot," and again for treason in 1690. His important astrological works are the Animal Cornatum, or the Horn'd Beast (1654) and The Nativity of the late King Charls (1659).

Isaac D'Israeli (1766-1848), in his Curiosities of Literature (1791), speaking of Lilly, says: "I shall observe of this egregious astronomer, that there is in this work, so much artless narrative, and at the same time so much palpable imposture, that it is difficult to know when he is speaking what he really believes to be the truth." He goes on to say that Lilly relates that "those adepts whose characters he has drawn were the lowest miscreants of the town. Most of them had taken the air in the pillory, and others had conjured themselves up to the gallows. This seems a true statement of facts."

It is difficult to estimate William Lilly (1602-1681) fairly. His Merlini Anglici ephemeris, issued annually from 1642 to 1681, brought him a great deal of money. Sir George Wharton (1617-1681) also published an almanac annually from 1641 to 1666. He tried to expose John Booker (1603-1677) by a work entitled Mercurio-Coelicio-Mastix; or, an Anti-caveat to all such, as have (heretofore) had the misfortune to be Cheated and Deluded by that Grand and Traiterous Impostor of this Rebellious Age, John Booker, 1644. Booker was "licenser of mathematical [astrological] publications," and as such he had quarrels with Lilly, Wharton, and others.

See note on page .

This is the Ars Signorum, vulgo character universalis et lingua philosophica, that appeared at London in 1661, 8vo. George Dalgarno anticipated modern methods in the teaching of the deaf and dumb.

See note on page .

If the hyperbola is referred to the asymptotes as axes, the area between two ordinates (x = a, x = b) is the difference of the logarithms of a and b to the base e. E.g., in the case of the hyperbola xy = 1, the area between x = a and x = 1 is log a.

"On ne peut lui refuser la justice de remarquer que personne avant lui ne s'est porté dans cette recherche avec autant de génie, & même, si nous en exceptons son objet principal, avec autant de succès." Quadrature du Cercle, p. 66.

The title proceeds: Seu duae mediae proportionales inter extremas datas per circulum et per infinitas hyperbolas, vel ellipses et per quamlibet exhibitae.... René Francois, Baron de Sluse (1622-1685) was canon and chancellor of Liège, and a member of the Royal Society. He also published a work on tangents (1672). The word mesolabium is from the Greek μεσολάβιον or μεσόλαβον, an instrument invented by Eratosthenes for finding two mean proportionals.

The full title has some interest: Vera circuli et hyperbolae quadratura cui accedit geometriae pars universalis inserviens quantitatum curvarum transmutationi et mensurae. Authore Jacobo Gregorio Abredonensi Scoto ... Patavii, 1667. That is, James Gregory (1638-1675) of Aberdeen (he was really born near but not in the city), a good Scot, was publishing his work down in Padua. The reason was that he had been studying in Italy, and that this was a product of his youth. He had already (1663) published his Optica promota, and it is not remarkable that his brilliancy brought him a wide circle of friends on the continent and the offer of a pension from Louis XIV. He became professor of mathematics at St Andrews and later at Edinburgh, and invented the first successful reflecting telescope. The distinctive feature of his Vera quadratura is his use of an infinite converging series, a plan that Archimedes used with the parabola.

Jean de Beaulieu wrote several works on mathematics, including La lumière de l'arithmétique (n.d.), La lumière des mathématiques (1673), Nouvelle invention d'arithmétique (1677), and some mathematical tables.

A just estimate. There were several works published by Gérard Desargues (1593-1661), of which the greatest was the Brouillon Proiect (Paris, 1639). There is an excellent edition of the Œuvres de Desargues by M. Poudra, Paris, 1864.

"A certain M. de Beaugrand, a mathematician, very badly treated by Descartes, and, as it appears, rightly so."

This is a very old approximation for π. One of the latest pretended geometric proofs resulting in this value appeared in New York in 1910, entitled Quadrimetry (privately printed).

"Copernicus, a German, made himself no less illustrious by his learned writings; and we might say of him that he stood alone and unique in the strength of his problems, if his excessive presumption had not led him to set forth in this science a proposition so absurd that it is contrary to faith and reason, namely that the circumference of a circle is fixed and immovable while the center is movable: on which geometrical principle he has declared in his astrological treatise that the sun is fixed and the earth is in motion."

So in the original.

Franciscus Maurolycus (1494-1575) was really the best mathematician produced by Sicily for a long period. He made Latin translations of Theodosius, Menelaus, Euclid, Apollonius, and Archimedes, and wrote on cosmography and other mathematical subjects.

"Nicolaus Copernicus is also tolerated who asserted that the sun is fixed and that the earth whirls about it; and he rather deserves a whip or a lash than a reproof."

"Algebra is the curious science of scholars, and particularly for a general of an army, or a captain, in order quickly to draw up an army in battle array and to number the musketeers and pikemen who compose it, without the figures of arithmetic. This science has five special figures of this kind: P means plus in commerce and pikemen in the army; M means minus, and musketeer in the art of war;... R signifies root in the measurement of a cube, and rank in the army; Q means square (French quarè, as then spelled) in both cases; C means cube in mensuration, and cavalry in arranging batallions and squadrons. As for the operations of this science, they are as follows: to add a plus and a plus, the sum will be plus; to add minus with plus, take the less from the greater and the remainder will be the sum required or the number to be found. I say this only in passing, for the benefit of those who are wholly ignorant of it."

He refers to the Joannis de Beaugrand ... Geostatice, seu de vario pondere gravium secundum varia a terrae (centro) intervalla dissertatio mathematica, Paris, 1636. Pascal relates that de Beaugrand sent all of Roberval's theorems on the cycloid and Fermat's on maxima and minima to Galileo in 1638, pretending that they were his own.

More (1614-1687) was a theologian, a fellow of Christ College, Cambridge, and a Christian Platonist.

Matthew Hale (1609-1676) the famous jurist, wrote a number of tracts on scientific, moral, and religious subjects. These were collected and published in 1805.

They might have been attributed to many a worse man than Dr. Hales (1677-1761), who was a member of the Royal Society and of the Paris Academy, and whose scheme for the ventilation of prisons reduced the mortality at the Savoy prison from one hundred to only four a year. The book to which reference is made is Vegetable Staticks or an Account of some statical experiments on the sap in Vegetables, 1727.

Pleas of the Crown; or a Methodical Summary of the Principal Matters relating to the subject, 1678.

Thomae Streete Astronomia Carolina, a new theory of the celestial motions, 1661. It also appeared at Nuremberg in 1705, and at London in 1710 and 1716 (Halley's editions). He wrote other works on astronomy.

This was the Sir Thomas Street (1626-1696) who passed sentence of death on a Roman Catholic priest for saying mass. The priest was reprieved by the king, but in the light of the present day one would think the justice more in need of pardon. He took part in the trial of the Rye House Conspirators in 1683.

Edmund Halley (1656-1742), who succeeded Wallis (1703) as Savilian professor of mathematics at Oxford, and Flamsteed (1720) as head of the Greenwich observatory. It is of interest to note that he was instrumental in getting Newton's Principia printed.

Shepherd (born in 1760) was one of the most famous lawyers of his day. He was knighted in 1814 and became Attorney General in 1817.

This was William Hone (1780-1842), a book publisher, who wrote satires against the government, and who was tried three times because of his parodies on the catechism, creed, and litany (illustrated by Cruikshank). He was acquitted on all of the charges.

Valentinus was a Benedictine monk and was still living at Erfurt in 1413. His Currus triumphalis antimonii appeared in 1624. Synesius was Bishop of Ptolemaide, who died about 430. His works were printed at Paris in 1605. Theodor Kirckring (1640-1693) was a fellow-student of Spinoza's. Besides the commentary on Valentine he left several works on anatomy. His commentary appeared at Amsterdam in 1671. There were several editions of the Chariot.

The chief difficulty with this curious "monk-bane" etymology is its absurdity. The real origin of the word has given etymologists a good deal of trouble.

Robert Boyle (1627-1691), son of "the Great Earl" (of Cork). Perhaps his best-known discovery is the law concerning the volume of gases.

The real name of Eirenaeus Philalethes (born in 1622) is unknown. It may have been Childe. He claimed to have discovered the philosopher's stone in 1645. His tract in this work is The Secret of the Immortal Liquor Alkahest or Ignis-Aqua. See note , infra.

Johann Baptist van Helmont, Herr von Merode, Royenborg etc. (1577-1644). His chemical discoveries appeared in his Ortus medicinae (1648), which went through many editions.

De Morgan should have written up Francis Anthony (1550-1623), whose Panacea aurea sive tractatus duo de auro potabili (Hamburg, 1619) described a panacea that he gave for every ill. He was repeatedly imprisoned for practicing medicine without a license from the Royal College of Physicians.

Bernardus Trevisanus (1406-1490), who traveled even through Barbary, Egypt, Palestine, and Persia in search of the philosopher's stone. He wrote several works on alchemy,—De Chemica (1567), De Chemico Miraculo (1583), Traité de la nature de l'oeuf des philosophes (1659), etc., all published long after his death.

George Ripley (1415-1490) was an Augustinian monk, later a chamberlain of Innocent VIII, and still later a Carmelite monk. His Liber de mercuris philosophico and other tracts first appeared in Opuscula quaedam chymica (Frankfort, 1614).

Besides the Opus majus, and other of the better known works of this celebrated Franciscan (1214-1294), there are numerous tracts on alchemy that appeared in the Thesaurus chymicus (Frankfort, 1603).

George Starkey (1606-1665 or 1666) has special interest for American readers. He seems to have been born in the Bermudas and to have obtained the bachelor's degree in England. He then went to America and in 1646 obtained the master's degree at Harvard, apparently under the name of Stirk. He met Eirenaeus Philalethes (see note above) in America and learned alchemy from him. Returning to England, he sold quack medicines there, and died in 1666 from the plague after dissecting a patient who had died of the disease. Among his works was the Liquor Alcahest, or a Discourse of that Immortal Dissolvent of Paracelsus and Helmont, which appeared (1675) some nine years after his death.

Platt (1552-1611) was the son of a London brewer. Although he left a manuscript on alchemy, and wrote a book entitled Delights for Ladies to adorne their Persons (1607), he was knighted for some serious work on the chemistry of agriculture, fertilizing, brewing, and the preserving of foods, published in The Jewell House of Art and Nature (1594).

"Those who wish to call a man a liar and deceiver speak of him a writer of almanacs; but those who (would call him) a scoundrel and an imposter (speak of him as) a chemist."

"Trust your barque to the winds but not your body to a chemist; any breeze is safer than the faith of a chemist."

Probably the Jesuit, Père Claude François Menestrier (1631-1705), a well known historian.

The author was Christopher Nesse (1621-1705), a belligerent Calvinist, who wrote many controversial works and succeeded in getting excommunicated four times. One of his most virulent works was A Protestant Antidote against the Poison of Popery.

John Case (c. 1660-1700) was a famous astrologer and physician. He succeeded to Lilly's practice in London. In a darkened room, wherein he kept an array of mystical apparatus, he pretended to show the credulous the ghosts of their departed relatives. Besides his astrological works he wrote one serious treatise, the Compendium Anatomicum nova methodo institutum (1695), in which he defends Harvey's theories of embryology.

Marcelis (1636-after 1714) was a soap maker of Amsterdam. It is to be hoped that he made better soap than values of π.

John Craig (died in 1731) was a Scotchman, but most of his life was spent at Cambridge reading and writing on mathematics. He endeavored to introduce the Leibnitz differential calculus into England. His mathematical works include the Methodus Figurarum ... Quadraturas determinandi (1685), Tractatus ... de Figurarum Curvilinearum Quadraturis et locis Geometricis (1693), and De Calculo Fluentium libri duo (1718).

As is well known, this subject owes much to the Bernoullis. Craig's works on the calculus brought him into controversy with them. He also wrote on other subjects in which they were interested, as in his memoir On the Curve of the quickest descent (1700), On the Solid of least resistance (1700), and the Solution of Bernoulli's problem on Curves (1704).

This is Samuel Lee (1783-1852), the young prodigy in languages. He was apprenticed to a carpenter at twelve and learned Greek while working at the trade. Before he was twenty-five he knew Hebrew, Chaldee, Syriac, Samaritan, Persian, and Hindustani. He later became Regius professor of Hebrew at Cambridge.

"Where the devil, Master Ludovico, did you pick up such a collection?"

Lord William Brounker (c. 1620-1684), the first president of the Royal Society, is best known in mathematics for his contributions to continued fractions.

Horace Walpole (1717-1797) published his Catalogue of the Royal and Noble Authors of England in 1758. Since his time a number of worthy names in the domain of science in general and of mathematics in particular might be added from the peerage of England.

It was written by Charles Hayes (1678-1760), a mathematician and scholar of no mean attainments. He travelled extensively, and was deputy governor of the Royal African Company. His Treatise on Fluxions (London, 1704) was the first work in English to explain Newton's calculus. He wrote a work entitled The Moon (1723) to prove that our satellite shines by its own as well as by reflected light. His Chronographia Asiatica & Aegyptica (1758) gives the results of his travels.

Publick in the original.

Whiston (1667-1752) succeeded Newton as Lucasian professor of mathematics at Cambridge. In 1710 he turned Arian and was expelled from the university. His work on Primitive Christianity appeared the following year. He wrote many works on astronomy and religion.

Ditton (1675-1715) was, on Newton's recommendation, made Head of the mathematical school at Christ's Hospital, London. He wrote a work on fluxions (1706). His idea for finding longitude at sea was to place stations in the Atlantic to fire off bombs at regular intervals, the time between the sound and the flash giving the distance. He also corresponded with Huyghens concerning the use of chronometers for the purpose.

This was John Arbuthnot (c. 1658-1735), the mathematician, physician and wit. He was intimate with Pope and Swift, and was Royal physician to Queen Anne. Besides various satires he published a translation of Huyghens's work on probabilities (1692) and a well-known treatise on ancient coins, weights, and measures (1727).

Greene (1678-1730) was a very eccentric individual and was generally ridiculed by his contemporaries. In his will he directed that his body be dissected and his skeleton hung in the library of King's College, Cambridge. Unfortunately for his fame, this wish was never carried out.

This was the historian, Robert Sanderson (1660-1741), who spent most of his life at Cambridge.

I presume this was William Jones (1675-1749) the friend of Newton and Halley, vice-president of the Royal Society, in whose Synopsis Palmariorum Matheseos (1706) the symbol π is first used for the circle ratio.

This was the Geometrica solidorum, sive materiae, seu de varia compositione, progressione, rationeque velocitatum, Cambridge, 1712. The work was parodied in A Taste of Philosophical Fanaticism ... by a gentleman of the University of Gratz.

The antiquary and scientist (1690-1754), president of the Royal Society, member of the Académie, friend of Newton, and authority on numismatics.

She was Catherine Barton, Newton's step-niece. She married John Conduitt, master of the mint, who collected materials for a life of Newton.

A propos of Mrs. Conduitt's life of her illustrious uncle, Sir George Greenhill tells a very good story on Poincaré, the well-known French mathematician. At an address given by the latter at the International Congress of Mathematicians held in Rome in 1908 he spoke of the story of Newton and the apple as a mere fable. After the address Sir George asked him why he had done so, saying that the story was first published by Voltaire, who had heard it from Newton's niece, Mrs. Conduitt. Poincaré looked blank and said, "Newton, et la nièce de Newton, et Voltaire,—non! je ne vous comprends pas!" He had thought Sir George meant Professor Volterra of Rome, whose name in French is Voltaire, and who could not possibly have known a niece of Newton without bridging a century or so.

This was the Edmund Turnor (1755-1829) who wrote the Collections for the Town and Soke of Grantham, containing authentic Memoirs of Sir Isaac Newton, from Lord Portsmouth's Manuscripts, London, 1806.

It may be recalled to mind that Sir David (1781-1868) wrote a life of Newton (1855).

"They are in the country. We rejoice."

"I am here, chatterbox, suck!"

"I have been graduated! I decline!"

Giovanni Castiglioni (Castillon, Castiglione), was born at Castiglione, in Tuscany, in 1708, and died at Berlin in 1791. He was professor of mathematics at Utrecht and at Berlin. He wrote on De Moivre's equations (1762), Cardan's rule (1783), and Euclid's treatment of parallels (1788-89).

This was the Isaaci Newtoni, equitis aurati, opuscula mathematica, philosophica et philologica, Lausannae & Genevae, 1744.

At London, 4to.

"All the English attribute it to Newton."

Stephen Peter Rigaud (1774-1839), Savilian professor of geometry at Oxford (1810-27) and later professor of astronomy and head of the Radcliffe Observatory. He wrote An historical Essay on first publication of Sir Isaac Newton's Principia, Oxford, 1838, and a two-volume work entitled Correspondence of Scientific Men of the 17th Century, 1841.

It is no longer considered by scholars as the work of Newton.

J. Edleston, the author of the Correspondence of Sir Isaac Newton and Professor Cotes, London, 1850.

Palmer (1601-1647) was Master of Queen's College, Cambridge, a Puritan but not a separatist. His work, The Characters of a believing Christian, in Paradoxes and seeming contradictions, appeared in 1645.

Grosart (1827-1899) was a Presbyterian clergyman. He was a great bibliophile, and issued numerous reprints of rare books.

This was the year after Palmer's death. The title was, The Remaines of ... Francis Lord Verulam....; being Essays and severall Letters to severall great personages, and other pieces of various and high concernment not heretofore published, London, 1648, 4to.

Shaw (1694-1763) was physician extraordinary to George II. He wrote on chemistry and medicine, and his edition of the Philosophical Works of Francis Bacon appeared at London in 1733.

John Locke (1632-1704), the philosopher. This particular work appeared in 1695. There was an edition in 1834 (vol. 25 of the Sacred Classics) and one in 1836 (vol. 2 of the Christian Library).

I use the word Socinian because it was so much used in Locke's time: it is used in our own day by the small fry, the unlearned clergy and their immediate followers, as a term of reproach for all Unitarians. I suspect they have a kind of liking for the word; it sounds like so sinful. The learned clergy and the higher laity know better: they know that the bulk of the modern Unitarians go farther than Socinus, and are not correctly named as his followers. The Unitarians themselves neither desire nor deserve a name which puts them one point nearer to orthodoxy than they put themselves. That point is the doctrine that direct prayer to Jesus Christ is lawful and desirable: this Socinus held, and the modern Unitarians do not hold. Socinus, in treating the subject in his own Institutio, an imperfect catechism which he left, lays much more stress on John xiv. 13 than on xv. 16 and xvi. 23. He is not disinclined to think that Patrem should be in the first citation, where some put it; but he says that to ask the Father in the name of the Son is nothing but praying to the Son in prayer to the Father. He labors the point with obvious wish to secure a conclusive sanction. In the Racovian Catechism, of which Faustus Socinus probably drew the first sketch, a clearer light is arrived at. The translation says: "But wherein consists the divine honor due to Christ? In adoration likewise and invocation. For we ought at all times to adore Christ, and may in our necessities address our prayers to him as often as we please; and there are many reasons to induce us to do this freely." There are some who like accuracy, even in aspersion—A. De M.

Socinus, or Fausto Paolo Sozzini (1539-1604), was an antitrinitarian who believed in prayer and homage to Christ. Leaving Italy after his views became known, he repaired to Basel, but his opinions were too extreme even for the Calvinists. He then tried Transylvania, attempting to convert to his views the antitrinitarian Bishop Dávid. The only result of his efforts was the imprisonment of Dávid and his own flight to Poland, in which country he spent the rest of his life (1579-1604). His complete works appeared first at Amsterdam in 1668, in the Bibliotheca Fratres Polonorum. The Racovian Catechism (1605) appeared after his death, but it seems to have been planned by him.

"As much of faith as is necessary to salvation is contained in this article, Jesus is the Christ."

Edwards (1637-1716) was a Cambridge fellow, strongly Calvinistic. He published many theological works, attacking the Arminians and Socinians. Locke and Whiston were special objects of attack.

Sir I. Newton's views on points of Trinitarian Doctrine; his Articles of Faith, and the General Coincidence of his Opinions with those of J. Locke; a Selection of Authorities, with Observations, London, 1856.

A Confession of the Faith, Bristol, 1752, 8vo.

This was really very strange, because Laud (1573-1644), while he was Archbishop of Canterbury, forced a good deal of High Church ritual on the Puritan clergy, and even wished to compel the use of a prayer book in Scotland. It was this intolerance that led to his impeachment and execution.

The name is Jonchère. He was a man of some merit, proposing (1718) an important canal in Burgundy, and publishing a work on the Découverte des longitudes estimées généralement impossible à trouver, 1734 (or 1735).

Locke invented a kind of an instrument for finding longitude, and it is described in the appendix, but I can find nothing about the man. There was published some years later (London, 1751) another work of his, A new Problem to discover the longitude at sea.

Baxter, concerning whom I know merely that he was a schoolmaster, starts with the assumption of this value, and deduces from it some fourteen properties relating to the circle.

John, who died in 1780, was a well-known character in his way. He was a bookseller on Fleet Street, and his shop was a general rendezvous for the literary men of his time. He wrote the Memoirs of the Life and Writings of Mr. William Whiston (1749, with another edition in 1753). He was one of the first to issue regular catalogues of books with prices affixed.

The name appears both as Hulls and as Hull. He was born in Gloucestershire in 1699. In 1754 he published The Art of Measuring made Easy by the help of a new Sliding Scale.

Thomas Newcomen (1663-1729) invented the first practical steam engine about 1710. It was of about five and a half horse power, and was used for pumping water from coal mines. Savery had described such an engine in 1702, but Newcomen improved upon it and made it practical.

The well-known benefactor of art (1787-1863).

The tract was again reprinted in 1860.

Hulls made his experiment on the Avon, at Evesham, in 1737, having patented his machine in 1736. He had a Newcomen engine connected with six paddles. This was placed in the front of a small tow boat. The experiment was a failure.

William Symington (1763-1831). In 1786 he constructed a working model of a steam road carriage. The machinery was applied to a small boat in 1788, and with such success as to be tried on a larger boat in 1789. The machinery was clumsy, however, and in 1801 he took out a new patent for the style of engine still used on paddle wheel steamers. This engine was successfully used in 1802, on the Charlotte Dundas. Fulton (1765-1815) was on board, and so impressed Robert Livingston with the idea that the latter furnished the money to build the Clermont (1807), the beginning of successful river navigation.

Louis Bertrand Castel (1688-1757), most of whose life was spent in trying to perfect his Clavecin oculaire, an instrument on the order of the harpsichord, intended to produce melodies and harmonies of color. He also wrote L'Optique des couleurs (1740) and Sur le fond de la Musique (1754).

Dr. Robinson (1680-1754) was professor of physic at Trinity College, Dublin, and three times president of King and Queen's College of Physicians. In his Treatise on the Animal Economy (1732-3, with a third edition in 1738) he anticipated the discoveries of Lavoisier and Priestley on the nature of oxygen.

There was another edition, published at London in 1747, 8vo.

The author seems to have shot his only bolt in this work. I can find nothing about him.

Quod Deus sit, mundusque ab ipso creatus fuerit in tempore, ejusque providentia gubernetur. Selecta aliquot theoremata adversos atheos, etc., Paris, 1635, 4to.

The British Museum Catalogue mentions a copy of 1740, but this is possibly a misprint.

This was Johann II (1710-1790), son of Johann I, who succeeded his father as professor of mathematics at Basel.

Samuel Koenig (1712-1757), who studied under Johann Bernoulli I. He became professor of mathematics at Franeker (1747) and professor of philosophy at the Hague (1749).

"In accordance with the hypotheses laid down in this memoir it is so evident that t must = 34, y = 1, and z = 1, that there is no need of proof or authority for it to be recognized by every one."

"I subscribe to the judgment of Mr. Bernoulli as a result of these hypotheses."

"It clearly appears from my present analysis and demonstration that they have already recognized and perfectly agreed to the fact that the quadrature of the circle is mathematically demonstrated."

Dr. Knight (died in 1772) made some worthy contributions to the literature of the mariner's compass. As De Morgan states, he was librarian of the British Museum.

Sir Anthony Panizzi (1797-1879) fled from Italy under sentence of death (1822). He became assistant (1831) and chief (1856) librarian of the British Museum, and was knighted in 1869. He began the catalogue of printed books of the Museum.

Wright (1711-1786) was a physicist. He was offered the professorship of mathematics at the Imperial Academy of St. Petersburg but declined to accept it. This work is devoted chiefly to the theory of the Milky Way, the via lactea as he calls it after the manner of the older writers.

Troughton (1753-1835) was one of the world's greatest instrument makers. He was apprenticed to his brother John, and the two succeeded (1770) Wright and Cole in Fleet Street. Airy called his method of graduating circles the greatest improvement ever made in instrument making. He constructed (1800) the first modern transit circle, and his instruments were used in many of the chief observatories of the world.

William Simms (1793-1860) was taken into partnership by Troughton (1826) after the death of the latter's brother. The firm manufactured some well-known instruments.

This was George Horne (1730-1792), fellow of Magdalen College, Oxford, vice-Chancellor of the University (1776), Dean of Canterbury (1781), and Bishop of Norwich (1790). He was a great satirist, but most of his pamphlets against men like Adam Smith, Swedenborg, and Hume, were anonymous, as in the case of this one against Newton. He was so liberal in his attitude towards the Methodists that he would not have John Wesley forbidden to preach in his diocese. He was twenty-one when this tract appeared.

Martin (1704-1782) was by no means "old Benjamin Martin" when Horne wrote this pamphlet in 1749. In fact he was then only forty-five. He was a physicist and a well-known writer on scientific instruments. He also wrote Philosophia Britannica or a new and comprehensive system of the Newtonian Philosophy (1759).

Jean Théophile Desaguliers, or Des Aguliers (1683-1744) was the son of a Protestant who left France after the revocation of the Edict of Nantes. He became professor of physics at Oxford, and afterwards gave lectures in London. Later he became chaplain to the Prince of Wales. He published several works on physics.

Charles Hutton (1737-1823), professor of mathematics at Woolwich (1772-1807). His Mathematical Tables (1785) and Mathematical and Philosophical Dictionary (1795-1796) are well known.

James Epps (1773-1839) contributed a number of memoirs on the use and corrections of instruments. He was assistant secretary of the Astronomical Society.

John Hutchinson (1674-1737) was one of the first to try to reconcile the new science of geology with Genesis. He denied the Newtonian hypothesis as dangerous to religion, and because it necessitated a vacuum. He was a mystic in his interpretation of the Scriptures, and created a sect that went under the name of Hutchinsonians.

John Rowning, a Lincolnshire rector, died in 1771. He wrote on physics, and published a memoir on A machine for finding the roots of equations universally (1770).

It is always difficult to sanction this spelling of the name of this Jesuit father who is so often mentioned in the analytic treatment of conics. He was born in Ragusa in 1711, and the original spelling was Ruđer Josip Bošković. When he went to live in Italy, as professor of mathematics at Rome (1740) and at Pavia, the name was spelled Ruggiero Giuseppe Boscovich, although Boscovicci would seem to a foreigner more natural. His astronomical work was notable, and in his De maculis solaribus (1736) there is the first determination of the equator of a planet by observing the motion of spots on its surface. Boscovich came near having some contact with America, for he was delegated to observe in California the transit of Venus in 1755, being prevented by the dissolution of his order just at that time. He died in 1787, at Milan.

James Granger (1723-1776) who wrote the Biographical History of England, London, 1769. His collection of prints was remarkable, numbering some fourteen thousand.

He was curator of experiments for the Royal Society. He wrote a large number of books and monographs on physics. He died about 1713.

Lee seems to have made no impression on biographers.

This work appeared at London in 1852.

Of course this is no longer true. The most scholarly work to-day is that of Rudio, Archimedes, Huygens, Lambert, Legendre, vier Abhandlungen über die Kreismessung ... mit einer Uebersicht über die Geschichte des Problems von der Quadratur des Zirkels, von den ältesten Zeiten bis auf unsere Tage, Leipsic, 1892.

Joseph Jérome le François de Lalande (1732-1807), professor of astronomy in the Collège de France (1753) and director of the Paris Observatory (1761). His writings on astronomy and his Bibliographie astronomique, avec l'histoire de l'astronomie depuis 1781 jusqu'en 1802 (Paris, 1803) are well known.

De Morgan refers to his Histoire de l'Astronomie au 18e siècle, which appeared in 1827, five years after Delambre's death. Jean Baptiste Joseph Delambre (1749-1822) was a pupil of and a collaborator with Lalande, following his master as professor of astronomy in the Collège de France. His work on the measurements for the metric system is well known, and his four histories of astronomy, ancienne (1817), au moyen âge (1819), moderne (1821), and au 18e siècle (posthumous, 1827) are highly esteemed.

Jean-Joseph Rive (1730-1792), a priest who left his cure under grave charges, and a quarrelsome character. His attack on Montucla was a case of the pot calling the kettle black; for while he was a brilliant writer he was a careless bibliographer.

Isaac Barrow (1630-1677) was quite as well known as a theologian as he was from his Lucasian professorship of mathematics at Cambridge.

"Besides we can see by this that Barrow was a poor philosopher; for he believed in the immortality of the soul and in a Divinity other than universal nature."

The Récréations mathématiques et physiques (Paris, 1694) of Jacques Ozanam (1640-1717) is a work that is still highly esteemed. Among various other works he wrote a Dictionnaire mathématique ou Idée générale des mathématiques (1690) that was not without merit. The Récréations went through numerous editions (Paris, 1694, 1696, 1741, 1750, 1770, 1778, and the Montucla edition of 1790; London, 1708, the Montucla-Hutton edition of 1803 and the Riddle edition of 1840; Dublin, 1790).

Hendryk van Etten, the nom de plume of Jean Leurechon (1591-1670), rector of the Jesuit college at Bar, and professor of philosophy and mathematics. He wrote on astronomy (1619) and horology (1616), and is known for his Selecta Propositiones in tota sparsim mathematica pulcherrime propositae in solemni festo SS. Ignatii et Francesci Xaverii, 1622. The book to which De Morgan refers is his Récréation mathématicque, composée de plusieurs problèmes plaisants et facetieux, Lyons, 1627, with an edition at Pont-à-Mousson, 1629. There were English editions published at London in 1633, 1653, and 1674, and Dutch editions in 1662 and 1672.

I do not understand how De Morgan happened to miss owning the work by Claude Gaspar Bachet de Meziriac (1581-1638), Problèmes plaisans et délectables, which appeared at Lyons in 1612, 8vo, with a second edition in 1624. There was a fifth edition published at Paris in 1884.

His title page closes with "Paris, Chez Ch. Ant. Jombert.... M DCC LIV."

This was Charles-Antoine Jombert (1712-1784), a printer and bookseller with some taste for painting and architecture. He wrote several works and edited a number of early treatises.

The late Professor Newcomb made the matter plain even to the non-mathematical mind, when he said that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope."

Antinewtonianismi pars prima, in qua Newtoni de coloribus systema ex propriis principiis geometrice evertitur, et nova de coloribus theoria luculentissimis experimentis demonstrantur.... Naples, 1754; pars secunda, Naples, 1756.

Celestino Cominale (1722-1785) was professor of medicine at the University of Naples.

The work appeared in the years from 1844 to 1849.

There was a Vienna edition in 1758, 4to, and another in 1759, 4to. This edition is described on the title page as Editio Veneta prima ipso auctore praesente, et corrigente.

The first edition was entitled De solis ac lunae defectibus libri V. P. Rogerii Josephi Boscovich ... cum ejusdem auctoris adnotationibus, London, 1760. It also appeared in Venice in 1761, and in French translation by the Abbé de Baruel in 1779, and was a work of considerable influence.

Paulian (1722-1802) was professor of physics at the Jesuit college at Avignon. He wrote several works, the most popular of which, the Dictionnaire de physique (Avignon, 1761), went through nine editions by 1789.

This is correct.

Probably referring to the fact that Hill (1795-1879), who had done so much for postal reform, was secretary to the postmaster general (1846), and his name was a synonym for the post office directory.

Richard Lovett (1692-1780) was a good deal of a charlatan. He claimed to have studied electrical phenomena, and in 1758 advertised that he could effect marvelous cures, especially of sore throat, by means of electricity. Before publishing the works mentioned by De Morgan he had issued others of similar character, including The Subtile Medium proved (London, 1756) and The Reviewers Reviewed (London, 1760).

Jean Sylvain Bailly (1736-1793), member of the Académie française and of the Académie des sciences, first deputy elected to represent Paris in the Etats-généraux (1789), president of the first National Assembly, and mayor of Paris (1789-1791). For his vigor as mayor in keeping the peace, and for his manly defence of the Queen, he was guillotined. He was an astronomer of ability, but is best known for his histories of the science.

These were the Histoire de l'Astronomie ancienne (1775), Histoire de l'Astronomie moderne (1778-1783), Histoire de l'Astronomie indienne et orientale (1787), and Lettres sur l'origine des peuples de l'Asie (1775).

"The sick old man of Ferney, V., a boy of a hundred years." Voltaire was born in 1694, and hence was eighty-three at this time.

In Palmézeaux's Vie de Bailly, in Bailly's Ouvrage Posthume (1810), M. de Sales is quoted as saying that the Lettres sur l'Atlantide were sent to Voltaire and that the latter did not approve of the theory set forth.

The British Museum catalogue gives two editions, 1781 and 1782.

A mystic and a spiritualist. His chief work was the one mentioned here.

Jacob Behmen, or Böhme (1575-1624), known as "the German theosophist," was founder of the sect of Boehmists, a cult allied to the Swedenborgians. He was given to the study of alchemy, and brought the vocabulary of the science into his mystic writings. His sect was revived in England in the eighteenth century through the efforts of William Law. Saint-Martin translated into French two of his Latin works under the titles L'Aurore naissante, ou la Racine de la philosophie (1800), and Les trois principes de l'essence divine (1802). The originals had appeared nearly two hundred years earlier,—Aurora in 1612, and De tribus principiis in 1619.

"Unknown."

"Skeptical."

"Man, man, man."

"Men, men, men."

It is interesting to read De Morgan's argument against Saint-Martin's authorship of this work. It is attributed to Saint-Martin both by the Biographie Universelle and by the British Museum Catalogue, and De Morgan says by "various catalogues and biographies."

"To explain things by man and not man by things. On Errors and Truth, by a Ph.... Inc...."

"If we would preserve ourselves from all illusions, and above all from the allurements of pride, by which man is so often seduced, we should never take man, but always God, for our term of comparison."

"And here is found already an explanation of the numbers four and nine which caused some perplexity in the work cited above. Man is lost in passing from four to nine."

Williams also took part in the preparation of some tables for the government to assist in the determination of longitude. He had published a work two years before the one here cited, on the same subject,—An entire new work and method to discover the variation of the Earth's Diameters, London, 1786.

This is Gabriel Mouton (1618-1694), a vicar at Lyons, who suggested as a basis for a natural system of measures the mille, a minute of a degree of the meridian. This appeared in his Observationes diametrorum solis et lunae apparentium, meridianarumque aliquot altitudinum cum tabula declinationum solis.... Lyons, 1670.

Jacques Cassini (1677-1756), one of the celebrated Cassini family of astronomers. After the death of his father he became director of the observatory at Paris. The basis for a metric unit was set forth by him in his Traité de la grandeur et de la figure de la terre, Paris, 1720. He was a prolific writer on astronomy.

Alexis Jean Pierre Paucton (1732-1798). He was, for a time, professor of mathematics at Strassburg, but later (1796) held office in Paris. His leading contribution to metrology was his Métrologie ou Traité des mesures, Paris, 1780.

He was an obscure writer, born at Deptford.

He was also a writer of no scientific merit, his chief contributions being religious tracts. One of his productions, however, went through many editions, even being translated into French; Three dialogues between a Minister and one of his Parishioners; on the true principles of Religion and salvation for sinners by Jesus Christ. The twentieth edition appeared at Cambridge in 1786.

This was the Reflections on the Revolution in France, and on the proceedings in certain societies in London relative to that event (London, 1790) by Edmund Burke (1729-1797). Eleven editions of the work appeared the first year.

Paine (1736-1809) was born in Norfolkshire, of Quaker parents. He went to America at the beginning of the Revolution and published, in January 1776, a violent pamphlet entitled Common Sense. He was a private soldier under Washington, and on his return to England after the war he published The Rights of Man. He was indicted for treason and was outlawed to France. He was elected to represent Calais at the French convention, but his plea for moderation led him perilously near the guillotine. His Age of Reason (1794) was dedicated to Washington. He returned to America in 1802 and remained there until his death.

Part I appeared in 1791 and was so popular that eight editions appeared in that year. It was followed in 1792 by Part II, of which nine editions appeared in that year. Both parts were immediately republished in Paris, and there have been several subsequent editions.

Mary Wollstonecraft (1759-1797) was only thirty-three when this work came out. She had already published An historical and moral View of the Origin and Progress of the French Revolution (1790), and Original Stories from Real Life (1791). She went to Paris in 1792 and remained during the Reign of Terror.

Samuel Parr (1747-1827) was for a time head assistant at Harrow (1767-1771), afterwards headmaster in other schools. At the time this book was written he was vicar of Hatton, where he took private pupils (1785-1798) to the strictly limited number of seven. He was a violent Whig and a caustic writer.

On Mary Wollstonecraft's return from France she married (1797) William Godwin (1756-1836). He had started as a strong Calvinistic Nonconformist minister, but had become what would now be called an anarchist, at least by conservatives. He had written an Inquiry concerning Political Justice (1793) and a novel entitled Caleb Williams, or Things as they are (1794), both of which were of a nature to attract his future wife.

This child was a daughter. She became Shelley's wife, and Godwin's influence on Shelley was very marked.

This was John Nichols (1745-1826), the publisher and antiquary. He edited the Gentleman's Magazine (1792-1826) and his works include the Literary Anecdotes of the Eighteenth Century (1812-1815), to which De Morgan here refers.

William Bellenden, a Scotch professor at the University of Paris, who died about 1633. His textbooks are now forgotten, but Parr edited an edition of his works in 1787. The Latin preface, Praefatio ad Bellendum de Statu, was addressed to Burke, North, and Fox, and was a satire on their political opponents.

As we have seen, he had been head-master before he began taking "his handful of private pupils."

The story has evidently got mixed up in the telling, for Tom Sheridan (1721-1788), the great actor, was old enough to have been Dr. Parr's father. It was his son, Richard Brinsley Sheridan (1751-1816), the dramatist and politician, who was the pupil of Parr. He wrote The Rivals (1775) and The School for Scandal (1777) soon after Parr left Harrow.

Horner (1785-1864) was a geologist and social reformer. He was very influential in improving the conditions of child labor.

William Cobbett (1762-1835), the journalist, was a character not without interest to Americans. Born in Surrey, he went to America at the age of thirty and remained there eight years. Most of this time he was occupied as a bookseller in Philadelphia, and while thus engaged he was fined for libel against the celebrated Dr. Rush. On his return to England he edited the Weekly Political Register (1802-1835), a popular journal among the working classes. He was fined and imprisoned for two years because of his attack (1810) on military flogging, and was also (1831) prosecuted for sedition. He further showed his paradox nature by his History of the Protestant Reformation (1824-1827), an attack on the prevailing Protestant opinion. He also wrote a Life of Andrew Jackson (1834). After repeated attempts he succeeded in entering parliament, a result of the Reform Bill.

Robinson (1735-1790) was a Baptist minister who wrote several theological works and a number of hymns. His work at Cambridge so offended the students that they at one time broke up the services.

This work had passed through twelve editions by 1823.

Dyer (1755-1841), the poet and reformer, edited Robinson's Ecclesiastical Researches (1790). He was a life-long friend of Charles Lamb, and in their boyhood they were schoolmates at Christ's Hospital. His Complaints of the Poor People of England (1793) made him a worthy companion of the paradoxers above mentioned.

These were John Thelwall (1764-1834) whose Politics for the People or Hogswash (1794) took its title from the fact that Burke called the people the "swinish multitude." The book resulted in sending the author to the Tower for sedition. In 1798 he gave up politics and started a school of elocution which became very famous. Thomas Hardy (1752-1832), who kept a bootmaker's shop in Piccadilly, was a fellow prisoner with Thelwall, being arrested for high treason. He was founder (1792) of The London Corresponding Society, a kind of clearing house for radical associations throughout the country. Horne Tooke was really John Horne (1736-1812), he having taken the name of his friend William Tooke in 1782. He was a radical of the radicals, and organized a number of reform societies. Among these was the Constitutional Society that voted money (1775) to assist the American revolutionists, appointing him to give the contribution to Franklin. For this he was imprisoned for a year. With his fellow rebels in the Tower in 1794, however, he was acquitted. As a philologist he is known for his early advocacy of the study of Anglo-Saxon and Gothic, and his Diversions of Purley (1786) is still known to readers.

This was the admiral, Adam Viscount Duncan (1731-1804), who defeated the Dutch off Camperdown in 1797.

He was created Duke of Clarence and St. Andrews in 1789 and was Admiral of the Fleet escorting Louis XVIII on his return to France in 1814. He became Lord High Admiral in 1827, and reigned as William IV from 1830 to 1837.

This was Charles Abbott (1762-1832) first Lord Tenterden. He succeeded Lord Ellenborough as Chief Justice (1818) and was raised to the peerage in 1827. He was a strong Tory and opposed the Catholic Relief Bill, the Reform Bill, and the abolition of the death penalty for forgery.

Edward Law (1750-1818), first Baron Ellenborough. He was chief counsel for Warren Hastings, and his famous speech in defense of his client is well known. He became Chief Justice and was raised to the peerage in 1802. He opposed all efforts to modernize the criminal code, insisting upon the reactionary principle of new death penalties.

Edmund Law (1703-1787), Bishop of Carlisle (1768), was a good deal more liberal than his son. His Considerations on the Propriety of requiring subscription to the Articles of Faith (1774) was published anonymously. In it he asserts that not even the clergy should be required to subscribe to the thirty-nine articles.

Joe Miller (1684-1738), the famous Drury Lane comedian, was so illiterate that he could not have written the Joe Miller's Jests, or the Wit's Vade-Mecum that appeared the year after his death. It was often reprinted and probably contained more or less of Miller's own jokes.

The sixth duke (1766-1839) was much interested in parliamentary reform. He was a member of the Society of Friends of the People. He was for fourteen years a member of parliament (1788-1802) and was later Lord Lieutenant of Ireland (1806-1807). He afterwards gave up politics and became interested in agricultural matters.

George Jeffreys (c. 1648-1689), the favorite of James II, who was active in prosecuting the Rye House conspirators. He was raised to the peerage in 1684 and held the famous "bloody assize" in the following year, being made Lord Chancellor as a result. He was imprisoned in the Tower by William III and died there.

The Every Day Book, forming a Complete History of the Year, Months, and Seasons, and a perpetual Key to the Almanack, 1826-1827.

The first and second editions appeared in 1820. Two others followed in 1821.

The three trials of W. H., for publishing three parodies; viz the late John Wilkes' Catechism, the Political Litany, and the Sinecurists Creed; on three ex-officio informations, at Guildhall, London, ... Dec. 18, 19, & 20, 1817,... London, 1818.

The Political Litany appeared in 1817.

That is, Castlereagh's.

The well-known caricaturist (1792-1878), then only twenty-nine years old.

Robert Stewart (1769-1822) was second Marquis of Londonderry and Viscount Castlereagh. As Chief Secretary for Ireland he was largely instrumental in bringing about the union of Ireland and Great Britain. He was at the head of the war department during most of the Napoleonic wars, and was to a great extent responsible for the European coalition against the Emperor. He suicided in 1822.

John Murray (1778-1843), the well-known London publisher. He refused to finish the publication of Don Juan, after the first five cantos, because of his Tory principles.

Only the first two cantos appeared in 1819.

Proclus (412-485), one of the greatest of the neo-Platonists, studied at Alexandria and taught philosophy at Athens. He left commentaries on Plato and on part of Euclid's Elements.

Thomas Taylor (1758-1835), called "the Platonist," had a liking for mathematics, and was probably led by his interest in number mysticism to a study of neo-Platonism. He translated a number of works from the Latin and Greek, and wrote two works on theoretical arithmetic (1816, 1823).

There was an earlier edition, 1788-89.

Georgius Gemistus, or Georgius Pletho (Plethon), lived in the fourteenth and fifteenth centuries. He was a native of Constantinople, but spent most of his time in Greece. He devoted much time to the propagation of the Platonic philosophy, but also wrote on divinity, geography, and history.

Hannah More (1745-1833), was, in her younger days, a friend of Burke, Reynolds, Dr. Johnson, and Garrick. At this time she wrote a number of poems and aspired to become a dramatist. Her Percy (1777), with a prologue and epilogue by Garrick, had a long run at Covent Garden. Somewhat later she came to believe that the playhouse was a grave public evil, and refused to attend the revival of her own play with Mrs. Siddons in the leading part. After 1789 she and her sisters devoted themselves to starting schools for poor children, teaching them religion and housework, but leaving them illiterate.

These were issued at the rate of three each month,—a story, a ballad, and a Sunday tract. They were collected and published in one volume in 1795. It is said that two million copies were sold the first year. There were also editions in 1798, 1819, 1827, and 1836-37.

That is, Dr. Johnson (1709-1784). The Rambler was published in 1750-1752, and was an imitation of Addison's Spectator.

Dr. Moore, referred to below.

Dr. John Moore (1729-1802), physician and novelist, is now best known for his Journal during a Residence in France from the beginning of August to the middle of December, 1792, a work quoted frequently by Carlyle in his French Revolution.

Sir John Moore (1761-1809), Lieutenant General in the Napoleonic wars. He was killed in the battle of Corunna. The poem by Charles Wolfe (1791-1823), The Burial of Sir John Moore (1817), is well known.

Referring to the novels of Thomas Love Peacock (1785-1866), who succeeded James Mill as chief examiner of the East India Company, and was in turn succeeded by John Stuart Mill.

Frances Burney, Madame d'Arblay (1752-1840), married General d'Arblay, a French officer and companion of Lafayette, in 1793. She was only twenty-five when she acquired fame by her Evelina, or a Young Lady's Entrance into the World. Her Letters and Diaries appeared posthumously (1842-45).

Henry Peter, Baron Brougham and Vaux (1778-1868), well known in politics, science, and letters. He was one of the founders of the Edinburgh Review, became Lord Chancellor in 1830, and took part with men like William Frend, De Morgan's father-in-law, in the establishing of London University. He was also one of the founders of the Society for the Diffusion of Useful Knowledge. He was always friendly to De Morgan, who entered the faculty of London University, whose work on geometry was published by the Society mentioned, and who was offered the degree of doctor of laws by the University of Edinburgh while Lord Brougham was Lord Rector. The Edinburgh honor was refused by De Morgan who said he "did not feel like an LL.D."

Maria Edgeworth (1767-1849).

Sydney Owenson (c. 1783-1859) married Sir Thomas Morgan, a well-known surgeon, in 1812. Her Irish stories were very popular with the patriots but were attacked by the Quarterly Review. The Wild Irish Girl (1806) went through seven editions in two years.

1775-1817.

1771-1832.

The famous preacher (1732-1808). He was the first chairman of the Religious Tract Society. He is also known as one of the earliest advocates of vaccination, in his Cow-pock Inoculation vindicated and recommended from matters of fact, 1806.

Sir Rowland Hill (1795-1879), the father of penny postage.

Beilby Porteus (1731-1808), Bishop of Chester (1776) and Bishop of London (1787). He encouraged the Sunday-school movement and the dissemination of Hannah More's tracts. He was an active opponent of slavery, but also of Catholic emancipation.

Henrietta Maria Bowdler (1754-1830), generally known as Mrs. Harriet Bowdler. She was the author of many religious tracts and poems. Her Poems and Essays (1786) were often reprinted. The story goes that on the appearance of her Sermons on the Doctrines and duties of Christianity (published anonymously), Bishop Porteus offered the author a living under the impression that it was written by a man.

William Frend (1757-1841), whose daughter Sophia Elizabeth became De Morgan's wife (1837), was at one time a clergyman of the Established Church, but was converted to Unitarianism (1787). He came under De Morgan's definition of a true paradoxer, carrying on a zealous warfare for what he thought right. As a result of his Address to the Inhabitants of Cambridge (1787), and his efforts to have abrogated the requirement that candidates for the M.A. must subscribe to the thirty-nine articles, he was deprived of his tutorship in 1788. A little later he was banished (see De Morgan's statement in the text) from Cambridge because of his denunciation of the abuses of the Church and his condemnation of the liturgy. His eccentricity is seen in his declining to use negative quantities in the operations of algebra. He finally became an actuary at London and was prominent in radical associations. He was a mathematician of ability, having been second wrangler and having nearly attained the first place, and he was also an excellent scholar in Latin, Greek, and Hebrew.

George Peacock (1791-1858), Fellow of Trinity College, Cambridge, Lowndean professor of astronomy, and Dean of Ely Cathedral (1839). His tomb may be seen at Ely where he spent the latter part of his life. He was one of the group that introduced the modern continental notation of the calculus into England, replacing the cumbersome notation of Newton, passing from "the dotage of fluxions to the deism of the calculus."

Robert Simson (1687-1768); professor of mathematics at Glasgow. His restoration of Apollonius (1749) and his translation and restoration of Euclid (1756, and 1776—posthumous) are well known.

Francis Maseres (1731-1824), a prominent lawyer. His mathematical works had some merit.

These appeared annually from 1804 to 1822.

Henry Gunning (1768-1854) was senior esquire bedell of Cambridge. The Reminiscences appeared in two volumes in 1854.

John Singleton Copley, Baron Lyndhurst (1772-1863), the son of John Singleton Copley the portrait painter, was born in Boston. He was educated at Trinity College, Cambridge, and became a lawyer. He was made Lord Chancellor in 1827.

Sir William Rough (c. 1772-1838), a lawyer and poet, became Chief Justice of Ceylon in 1836. He was knighted in 1837.

Herbert Marsh, afterwards Bishop of Peterborough, a relation of my father.—S. E. De M.

He was born in 1757 and died in 1839. On the trial of Frend he publicly protested against testifying against a personal confidant, and was excused. He was one of the first of the English clergy to study modern higher criticism of the Bible, and amid much opposition he wrote numerous works on the subject. He was professor of theology at Cambridge (1707), Bishop of Llandaff (1816), and Bishop of Peterborough.

George Butler (1774-1853), Headmaster of Harrow (1805-1829), Chancellor of Peterborough (1836), and Dean of Peterborough (1842).

James Tate (1771-1843), Headmaster of Richmond School (1796-1833) and Canon of St. Paul's Cathedral (1833). He left several works on the classics.

Francis Place (1771-1854), at first a journeyman breeches maker, and later a master tailor. He was a hundred years ahead of his time as a strike leader, but was not so successful as an agitator as he was as a tailor, since his shop in Charing Cross made him wealthy. He was a well-known radical, and it was largely due to his efforts that the law against the combinations of workmen was repealed in 1824. His chief work was The Principles of Population (1822).

Speed (1552-1629) was a tailor until Grevil (Greville) made him independent of his trade. He was not only an historian of some merit, but a skilful cartographer. His maps of the counties were collected in the Theatre of the Empire of Great Britaine, 1611. About this same time he also published Genealogies recorded in Sacred Scripture, a work that had passed through thirty-two editions by 1640.

The history of Great Britaine under the conquests of ye Romans, Saxons, Danes, and Normans.... London, 1611, folio. The second edition appeared in 1623; the third, to which De Morgan here refers, posthumously in 1632; and the fourth in 1650.

William Nicolson (1655-1727) became Bishop of Carlisle in 1702, and Bishop of Derry in 1718. His chief work was the Historical Library (1696-1724), in the form of a collection of documents and chronicles. It was reprinted in 1736 and in 1776.

Sir Fulk Grevil, or Fulke Greville (1554-1628), was a favorite of Queen Elizabeth, Chancellor of the Exchequer under James I, a patron of literature, and a friend of Sir Philip Sidney.

See note on page .

See note on page .

See note on page .

Edward Waring (1736-1796) was Lucasian professor of mathematics at Cambridge. He published several works on analysis and curves. The work referred to was the Miscellanea Analytica de aequationibus algebraicis et curvarum proprietatibus, Cambridge, 1762.

A Dissertation on the use of the Negative Sign in Algebra...; to which is added, Machin's Quadrature of the Circle, London, 1758.

The paper was probably one on complex numbers, or possibly one on quaternions, in which direction as well as absolute value is involved.

De Morgan quotes from one of the Latin editions. Descartes wrote in French, the title of his first edition being: Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences, plus la dioptrique, les météores et la géométrie qui sont des essais de cette méthode, Leyden, 1637, 4to.

"I have observed that algebra indeed, as it is usually taught, is so restricted by definite rules and formulas of calculation, that it seems rather a confused kind of an art, by the practice of which the mind is in a certain manner disturbed and obscured, than a science by which it is cultivated and made acute."

It appeared in 93 volumes, from 1758 to 1851.

The principles of the doctrine of life-annuities; explained in a familiar manner ... with a variety of new tables ..., London, 1783.

I suppose the one who wrote Conjectures on the physical causes of Earthquakes and Volcanoes, Dublin, 1820.

Scriptores Logarithmici; or, a Collection of several curious tracts on the nature and construction of Logarithms ... together with same tracts on the Binomial Theorem ..., 6 vols., London, 1791-1807.

Charles Babbage (1792-1871), whose work on the calculating machine is well known. Maseres was, it is true, ninety-two at this time, but Babbage was thirty-one instead of twenty-nine. He had already translated Lacroix's Treatise on the differential and integral calculus (1816), in collaboration with Herschel and Peacock. He was Lucasian professor of mathematics at Cambridge from 1828 to 1839.

The great and new Art of weighing Vanity, or a discovery of the ignorance of the great and new artist in his pseudo-philosophical writings. The "great and new artist" was Sinclair.

George Sinclair, probably a native of East Lothian, who died in 1696. He was professor of philosophy and mathematics at Glasgow, and was one of the first to use the barometer in measuring altitudes. The work to which De Morgan refers is his Hydrostaticks (1672). He was a firm believer in evil spirits, his work on the subject going through four editions: Satan's Invisible World Discovered; or, a choice collection of modern relations, proving evidently against the Saducees and Athiests of this present age, that there are Devils, Spirits, Witches, and Apparitions, Edinburgh, 1685.

This was probably William Sanders, Regent of St. Leonard's College, whose Theses philosophicae appeared in 1674, and whose Elementa geometriae came out a dozen years later.

Ars nova et magna gravitatis et levitatis; sive dialogorum philosophicorum libri sex de aeris vera ac reali gravitate, Rotterdam, 1669, 4to.

Volume I, Nos. 1 and 2, appeared in 1803.

His daughter, Mrs. De Morgan, says in her Memoir of her husband: "My father had been second wrangler in a year in which the two highest were close together, and was, as his son-in-law afterwards described him, an exceedingly clear thinker. It is possible, as Mr. De Morgan said, that this mental clearness and directness may have caused his mathematical heresy, the rejection of the use of negative quantities in algebraical operations; and it is probable that he thus deprived himself of an instrument of work, the use of which might have led him to greater eminence in the higher branches." Memoir of Augustus De Morgan, London, 1882, p. 19.

"If it is not true it is a good invention." A well-known Italian proverb.

See page , note .

He was born at Paris in 1713, and died there in 1765.

Recherches sur les courbes à double courbure, Paris, 1731. Clairaut was then only eighteen, and was in the same year made a member of the Académie des sciences. His Elémens de géométrie appeared in 1741. Meantime he had taken part in the measurement of a degree in Lapland (1736-1737). His Traité de la figure de la terre was published in 1741. The Academy of St. Petersburg awarded him a prize for his Théorie de la lune (1750). His various works on comets are well known, particularly his Théorie du mouvement des comètes (1760) in which he applied the "problem of three bodies" to Halley's comet as retarded by Jupiter and Saturn.

Joseph Privat, Abbé de Molières (1677-1742), was a priest of the Congregation of the Oratorium. In 1723 he became a professor in the Collège de France. He was well known as an astronomer and a mathematician, and wrote in defense of Descartes's theory of vortices (1728, 1729). He also contributed to the methods of finding prime numbers (1705).

"Deserves not only to be printed, but to be admired as a marvel of imagination, of understanding, and of ability."

Blaise Pascal (1623-1662), the well-known French philosopher and mathematician. He lived for some time with the Port Royalists, and defended them against the Jesuits in his Provincial Letters. Among his works are the following: Essai pour les coniques (1640); Recit de la grande expérience de l'équilibre des liqueurs (1648), describing his experiment in finding altitudes by barometric readings; Histoire de la roulette (1658); Traité du triangle arithmétique (1665); Aleae geometria (1654).

This proposition shows that if a hexagon is inscribed in a conic (in particular a circle) and the opposite sides are produced to meet, the three points determined by their intersections will be in the same straight line.

Jacques Curabelle, Examen des Œuvres du Sr. Desargues, Paris, 1644. He also published without date a work entitled: Foiblesse pitoyable du Sr. G. Desargues employée contre l'examen fait de ses œuvres.

See page , note .

Until "this great proposition called Pascal's should see the light."

The story is that his father, Etienne Pascal, did not wish him to study geometry until he was thoroughly grounded in Latin and Greek. Having heard the nature of the subject, however, he began at the age of twelve to construct figures by himself, drawing them on the floor with a piece of charcoal. When his father discovered what he was doing he was attempting to demonstrate that the sum of the angles of a triangle equals two right angles. The story is given by his sister, Mme. Perier.

Sir John Wilson (1741-1793) was knighted in 1786 and became Commissioner of the Great Seal in 1792. He was a lawyer and jurist of recognized merit. He stated his theorem without proof, the first demonstration having been given by Lagrange in the Memoirs of the Berlin Academy for 1771,—Demonstration d'un théorème nouveau concernant les nombres premiers. Euler also gave a proof in his Miscellanea Analytica (1773). Fermat's works should be consulted in connection with the early history of this theorem.

He wrote, in 1760, a tract in defense of Waring, a point of whose algebra had been assailed by a Dr. Powell. Waring wrote another tract of the same date.—A. De M.

William Samuel Powell (1717-1775) was at this time a fellow of St. John's College, Cambridge. In 1765 he became Vice Chancellor of the University. Waring was a Magdalene man, and while candidate for the Lucasian professorship he circulated privately his Miscellanea Analytica. Powell attacked this in his Observations on the First Chapter of a Book called Miscellanea (1760). This attack was probably in the interest of another candidate, a man of his own college (St. John's), William Ludlam.

William Paley (1743-1805) was afterwards a tutor at Christ's College, Cambridge. He never contributed anything to mathematics, but his Evidences of Christianity (1794) was long considered somewhat of a classic. He also wrote Principles of Morality and Politics (1785), and Natural Theology (1802).

Edward, first Baron Thurlow (1731-1806) is known to Americans because of his strong support of the Royal prerogative during the Revolution. He was a favorite of George III, and became Lord Chancellor in 1778.

George Wilson Meadley (1774-1818) published his Memoirs of ... Paley in 1809. He also published Memoirs of Algernon Sidney in 1813. He was a merchant and banker, and had traveled extensively in Europe and the East. He was a convert to unitarianism, to which sect Paley had a strong leaning.

Watson (1737-1816) was a strange kind of man for a bishopric. He was professor of chemistry at Cambridge (1764) at the age of twenty-seven. It was his experiments that led to the invention of the black-bulb thermometer. He is said to have saved the government £100,000 a year by his advice on the manufacture of gunpowder. Even after he became professor of divinity at Cambridge (1771) he published four volumes of Chemical Essays (vol. I, 1781). He became Bishop of Llandaff in 1782.

James Adair (died in 1798) was counsel for the defense in the trial of the publishers of the Letters of Junius (1771). As King's Serjeant he assisted in prosecuting Hardy and Horne Tooke.

Morgan (1750-1833) was actuary of the Equitable Assurance Society of London (1774-1830), and it was to his great abilities that the success of that company was due at a time when other corporations of similar kind were meeting with disaster. The Royal Society awarded him a medal (1783) for a paper on Probability of Survivorship. He wrote several important works on insurance and finance.

Dr. Price (1723-1791) was a non-conformist minister and a writer on ethics, economics, politics, and insurance. He was a defender of the American Revolution and a personal friend of Franklin. In 1778 Congress invited him to America to assist in the financial administration of the new republic, but he declined. His famous sermon on the French Revolution is said to have inspired Burke's Reflections on the Revolution in France.

Elizabeth Gurney (1780-1845), a Quaker, who married Joseph Fry (1800), a London merchant. She was the prime mover in the Association for the Improvement of the Female Prisoners in Newgate, founded in 1817. Her influence in prison reform extended throughout Europe, and she visited the prisons of many countries in her efforts to improve the conditions of penal servitude. The friendship of Mrs. Fry with the De Morgans began in 1837. Her scheme for a female benefit society proved worthless from the actuarial standpoint, and would have been disastrous to all concerned if it had been carried out, and it was therefore fortunate that De Morgan was consulted in time. Mrs. De Morgan speaks of the consultation in these words: "My husband, who was very sensitive on such points, was charmed with Mrs. Fry's voice and manner as much as by the simple self-forgetfulness with which she entered into this business; her own very uncomfortable share of it not being felt as an element in the question, as long as she could be useful in promoting good or preventing mischief. I can see her now as she came into our room, took off her little round Quaker cap, and laying it down, went at once into the matter. 'I have followed thy advice, and I think nothing further can be done in this case; but all harm is prevented.' In the following year I had an opportunity of seeing the effect of her most musical tones. I visited her at Stratford, taking my little baby and nurse with me, to consult her on some articles on prison discipline, which I had written for a periodical. The baby—three months old—was restless, and the nurse could not quiet her, neither could I entirely, until Mrs. Fry began to read something connected with the subject of my visit, when the infant, fixing her large eyes on the reader, lay listening till she fell asleep." Memoirs, p. 91.

Mrs. Fry certainly believed that the writer was the old actuary of the Equitable, when she first consulted him upon the benevolent Assurance project; but we were introduced to her by our old and dear friend Lady Noel Byron, by whom she had been long known and venerated, and who referred her to Mr. De Morgan for advice. An unusual degree of confidence in, and appreciation of each other, arose on their first meeting between the two, who had so much that was externally different, and so much that was essentially alike, in their natures.—S. E. De M.

Anne Isabella Milbanke (1792-1860) married Lord Byron in 1815, when both took the additional name of Noel, her mother's name. They were separated in 1816.

An obscure writer not mentioned in the ordinary biographies.

Not mentioned in the ordinary biographies, and for obvious reasons.

"Before" and "after."

On Bishop Wilkins see note on page .

Provision for a journey.

See note on page .

Thomas Bradwardine (1290-1349), known as Doctor Profundus, proctor and professor of theology at Oxford, and afterwards Chancellor of St. Paul's and confessor to Edward III. The English ascribed their success at Crécy to his prayers.

He was consecrated Archbishop of Canterbury by the Pope at Avignon, July 13, 1349, and died of the plague at London in the same year.

"One paltry little year."

The title is carelessly copied, as is so frequently the case in catalogues, even of the Libri class. It should read: Arithmetica thome brauardini || Olivier Senant || Venum exponuntur ab Oliuiario senant in vico diui Jacobi sub signo beate Barbare sedente. The colophon reads: Explicit arithmetica speculatiua thōe brauardini bn reuisa et correcta a Petro sanchez Ciruelo aragonensi mathematicas legēte Parisius, īpressa per Thomā anguelart. There were Paris editions of 1495, 1496, 1498, s. a. (c. 1500), 1502, 1504, 1505, s. a. (c. 1510), 1512, 1530, a Valencia edition of 1503, two Wittenberg editions of 1534 and 1536, and doubtless several others. The work is not "very rare," although of course no works of that period are common. See the editor's Rara Arithmetica, page 61.

This is his Tractatus de proportionibus, Paris, 1495; Venice, 1505; Vienna, 1515, with other editions.

The colophon of the 1495 edition reads: Et sic explicit Geometria Thome brauardini cū tractatulo de quadratura circuli bene reuisa a Petro sanchez ciruelo: operaqz Guidonis mercatoris diligētissime impresse parisio in cāpo gaillardi. Anno dni. 1495. die. 20, maij.

This Petro Ciruelo was born in Arragon, and died in 1560 at Salamanca. He studied mathematics and philosophy at Paris, and took the doctor's degree there. He taught at the University of Alcalà and became canon of the Cathedral at Salamanca. Besides his editions of Bradwardine he wrote several works, among them the Liber arithmeticae practicae qui dicitur algorithmus (Paris, 1495) and the Cursus quatuor mathematicarum artium liberalium (Alcalà, 1516).

Star polygons, a subject of considerable study in the later Middle Ages. See note on page .

"A new theory that adds lustre to the fourteenth century."

There is nothing in the edition of 1495 that leads to this conclusion.

The full title is: Nouvelle théorie des parallèles, avec un appendice contenant la manière de perfectionner la théorie des parallèles de A. M. Legendre. The author had no standing as a scientist.

Adrien Marie Legendre (1752-1833) was one of the great mathematicians of the opening of the nineteenth century. His Eléments de géométrie (1794) had great influence on the geometry of the United States. His Essai sur la théorie des nombres (1798) is one of the classics upon the subject. The work to which Kircher refers is the Nouvelle théorie des parallèles (1803), in which the attempt is made to avoid using Euclid's postulate of parallels, the result being merely the substitution of another assumption that was even more unsatisfactory. The best presentations of the general theory are W. B. Frankland's Theories of Parallelism, Cambridge, 1910, and Engel and Stäckel's Die Theorie der Parallellinien von Euclid bis auf Gauss, Leipsic, 1895. Legendre published a second work on the theory the year of his death, Réflexions sur ... la théorie des parallèles (1833). His other works include the Nouvelles méthodes pour la détermination des orbites des comètes (1805), in which he uses the method of least squares; the Traité des fonctions elliptiques et des intégrales (1827-1832), and the Exercises de calcul intégral (1811, 1816, 1817).

Johann Joseph Ignatz von Hoffmann (1777-1866), professor of mathematics at Aschaffenburg, published his Theorie der Parallellinien in 1801. He supplemented this by his Kritik der Parallelen-Theorie in 1807, and his Das eilfte Axiom der Elemente des Euclidis neu bewiesen in 1859. He wrote other works on mathematics, but none of his contributions was of any importance.

Johann Karl Friedrich Hauff (1766-1846) was successively professor of mathematics at Marburg, director of the polytechnic school at Augsburg, professor at the Gymnasium at Cologne, and professor of mathematics and physics at Ghent. The work to which Kircher refers is his memoirs on the Euclidean Theorie der Parallelen in Hindenburg's Archiv, vol. III (1799), an article of no merit in the general theory.

Wenceslaus Johann Gustav Karsten (1732-1787) was professor of logic at Rostock (1758) and Butzow (1760), and later became professor of mathematics and physics at Halle. His work on parallels is the Versuch einer völlig berichtigten Theorie der Parallellinien (1779). He also wrote a work entitled Anfangsgründe der mathematischen Wissenschaften (1780), but neither of these works was more than mediocre.

Johann Christoph Schwab (not Schwal) was born in 1743 and died in 1821. He was professor at the Karlsschule at Stuttgart. De Morgan's wish was met, for the catalogues give "c. fig. 8," so that it evidently had eight illustrations instead of eight volumes. He wrote several other works on the principles of geometry, none of any importance.

Gaetano Rossi of Catanzaro. This was the libretto writer (1772-1855), and hence the imperfections of the work can better be condoned. De Morgan should have given a little more of the title: Solusione esatta e regolare ... del ... problema della quadratura del circolo. There was a second edition, London, 1805.

This identifies Rossi, for Joséphine Grassini (1773-1850) was a well-known contralto, prima donna at Napoleon's court opera.

William Spence (1783-1860) was an entomologist and economist of some standing, a fellow of the Royal Society, and one of the founders of the Entomological Society of London. The work here mentioned was a popular one, the first edition appearing in 1807, and four editions being justified in a single year. He also wrote Agriculture the Source of Britain's Wealth (1808) and Objections against the Corn Bill refuted (1815), besides a work in four volumes on entomology (1815-1826) in collaboration with William Kirby.

"That used to be so, but we have changed all that."

"Meet the coming disease."

George Douglas (or Douglass) was a Scotch writer. He got out an edition of the Elements of Euclid in 1776, with an appendix on trigonometry and a set of tables. His work on Mathematical Tables appeared in 1809, and his Art of Drawing in Perspective, from mathematical principles, in 1810.

See note , on page .

John Playfair (1748-1848) was professor of mathematics (1785) and natural philosophy (1805) at the University of Edinburgh. His Elements of Geometry went through many editions.

"Tell Apella" was an expression current in classical Rome to indicate incredulity and to show the contempt in which the Jew was held. Horace says: Credat Judæus Apella, "Let Apella the Jew believe it." Our "Tell it to the marines," is a similar phrase.

As De Morgan says two lines later, "No mistake is more common than the natural one of imagining that the"—University of Virginia is at Richmond. The fact is that it is not there, and that it did not exist in 1810. It was not chartered until 1819, and was not opened until 1825, and then at Charlottesville. The act establishing the Central College, from which the University of Virginia developed, was passed in 1816. The Jean Wood to whom De Morgan refers was one John Wood who was born about 1775 in Scotland and who emigrated to the United States in 1800. He published a History of the Administration of J. Adams (New York, 1802) that was suppressed by Aaron Burr. This act called forth two works, a Narrative of the Suppression, by Col. Burr, of the 'History of the Administration of John Adams' (1802), in which Wood was sustained; and the Antidote to John Wood's Poison (1802), in which he was attacked. The work referred to in the "printed circular" may have been the New theory of the diurnal rotation of the earth (Richmond, Va., 1809). Wood spent the last years of his life in Richmond, Va., making county maps. He died there in 1822. A careful search through works relating to the University of Virginia fails to show that Wood had any connection with it.

There seems to be nothing to add to Dobson's biography beyond what De Morgan has so deliciously set forth.

"Give to each man his due."

Hester Lynch Salusbury (1741-1821), the friend of Dr. Johnson, married Henry Thrale (1763), a brewer, who died in 1781. She then married Gabriel Piozzi (1784), an Italian musician. Her Anecdotes of the late Samuel Johnson (1786) and Letters to and from Samuel Johnson (1788) are well known. She also wrote numerous essays and poems.

Samuel Pike (c. 1717-1773) was an independent minister, with a chapel in London and a theological school in his house. He later became a disciple of Robert Sandeman and left the Independents for the Sandemanian church (1765). The Philosophia Sacra was first published at London in 1753. De Morgan here cites the second edition.

Pike had been dead over forty years when Kittle published this second edition. Kittle had already published a couple of works: King Solomon's portraiture of Old Age (Edinburgh, 1813), and Critical and Practical Lectures on the Apocalyptical Epistles to the Seven Churches of Asia Minor (London, 1814).

See note , on page .

William Stukely (1687-1765) was a fellow of the Royal Society and of the College of Physicians and Surgeons. He afterwards (1729) entered the Church. He was prominent as an antiquary, especially in the study of the Roman and Druidic remains of Great Britain. He was the author of numerous works, chiefly on paleography.

William Jones (1726-1800), who should not be confused with his namesake who is mentioned in note on page . He was a lifelong friend of Bishop Horne, and his vicarage at Nayland was a meeting place of an influential group of High Churchmen. Besides the Physiological Disquisitions (1781) he wrote The Catholic Doctrine of the Trinity (1756) and The Grand Analogy (1793).

Robert Spearman (1703-1761) was a pupil of John Hutchinson, and not only edited his works but wrote his life. He wrote a work against the Newtonian physics, entitled An Enquiry after Philosophy and Theology (Edinburgh, 1755), besides the Letters to a Friend concerning the Septuagint Translation (Edinburgh, 1759) to which De Morgan refers.

A writer of no importance, at least in the minds of British biographers.

Alexander Catcott (1725-1779), a theologian and geologist, wrote not only a work on the creation (1756) but a Treatise on the Deluge (1761, with a second edition in 1768). Sir Charles Lyell considered the latter work a valuable contribution to geology.

James Robertson (1714-1795), professor of Hebrew at the University of Edinburgh. Probably De Morgan refers to his Grammatica Linguae Hebrææ (Edinburgh, 1758; with a second edition in 1783). He also wrote Clavis Pentateuchi (1770).

Benjamin Holloway (c. 1691-1759), a geologist and theologian. He translated Woodward's Naturalis Historia Telluris, and was introduced by Woodward to Hutchinson. The work referred to by De Morgan appeared at Oxford in two volumes in 1754.

His work was The Christian plan exhibited in the interpretation of Elohim: with observations upon a few other matters relative to the same subject, Oxford, 1752, with a second edition in 1755.

Duncan Forbes (1685-1747) studied Oriental languages and Civil law at Leyden. He was Lord President of the Court of Sessions (1737). He wrote a number of theological works.

Should be 1756.

Edward Henry Bickersteth (1825-1906), bishop of Exeter (1885-1900); published The Rock of Ages; or scripture testimony to the one Eternal Godhead of the Father, and of the Son, and of the Holy Ghost at Hampstead in 1859. A second edition appeared at London in 1860.

Thomas Sadler (1822-1891) took his Ph.D. at Erlangen in 1844, and became a Unitarian minister at Hampstead, where Bickersteth's work was published. Besides writing the Gloria Patri (1859), he edited Crabb Robinson's Diaries.

This was his Virgil's Bucolics and the two first Satyrs of Juvenal, 1634.

Possibly in his Twelve Questions or Arguments drawn out of Scripture, wherein the commonly received Opinion touching the Deity of the Holy Spirit is clearly and fully refuted, 1647. This was his first heretical work, and it was followed by a number of others that were written during the intervals in which the Puritan parliament allowed him out of prison. It was burned by the hangman as blasphemous. Biddle finally died in prison, unrepentant to the last.

The first edition of the anonymous Ἁιρεσεων ἀναστασις (by Vicars?) appeared in 1805.

Possibly by Thomas Pearne (c. 1753-1827), a fellow of St. Peter's College, Cambridge, and a Unitarian minister.

Thomas Wentworth, Earl of Strafford, was borne in London in 1593, and was executed there in 1641. He was privy councilor to Charles I, and was Lord Deputy of Ireland. On account of his repressive measures to uphold the absolute power of the king he was impeached by the Long Parliament and was executed for treason. The essence of his defence is in the sentence quoted by De Morgan, to which Pym replied that taken as a whole, the acts tended to show an intention to change the government, and this was in itself treason.

The name assumed by a writer who professed to give a mathematical explanation of the Trinity, see farther on.—S. E. De M.

Sabellius (fl. 230 A.D.) was an early Christian of Libyan origin. He taught that Father, Son, and Holy Spirit were different names for the same person.

Sir Richard Phillips was born in London in 1767 (not 1768 as stated above), and died there in 1840. He was a bookseller and printer in Leicester, where he also edited a radical newspaper. He went to London to live in 1795 and started the Monthly Magazine there in 1796. Besides the works mentioned by De Morgan he wrote on law and economics.

It was really eighteen months.

While he was made sheriff in 1807 he was not knighted until the following year.

James Mitchell (c. 1786-1844) was a London actuary, or rather a Scotch actuary living a good part of his life in London. Besides the work mentioned he compiled a Dictionary of Chemistry, Mineralogy, and Geology (1823), and wrote On the Plurality of Worlds (1813) and The Elements of Astronomy (1820).

Richarda Smith, wife of Sir George Biddell Airy (see note , page ) the astronomer. In 1835 Sir Robert Peel offered a pension of £300 a year to Airy, who requested that it be settled on his wife.

Mary Fairfax (1780-1872) married as her second husband Dr. William Somerville. In 1826 she presented to the Royal Society a paper on The Magnetic Properties of the Violet Rays of the Solar Spectrum, which attracted much attention. It was for her Mechanism of the Heavens (1831), a popular translation of Laplace's Mécanique Céleste, that she was pensioned.

Dominique François Jean Arago (1786-1853) the celebrated French astronomer and physicist.

For there is a well-known series

1 + 1/2 + 1/3 + ... = π/6.

If, therefore, the given series equals 1, we have

2 = 1/6 π

or π = 12,

whence π = 2 √3.

But c = πd, and twice the diagonal of a cube on the diameter is 2d √3.

There was a second edition in 1821.

London, 1830.

He was a resident of Chatham, and seems to have published no other works.

Richard Whately (1787-1863) was, as a child, a calculating prodigy (see note , page ), but lost the power as is usually the case with well-balanced minds. He was a fellow of Oriel College, Oxford, and in 1825 became principal of St. Alban Hall. He was a friend of Newman, Keble, and others who were interested in the religious questions of the day. He became archbishop of Dublin in 1831. He was for a long time known to students through his Logic (1826) and Rhetoric (1828).

William King, D.C.L. (1663-1712), student at Christ Church, Oxford, and celebrated as a wit and scholar. His Dialogues of the Dead (1699) is a satirical attack on Bentley.

Thomas Ebrington (1760-1835) was a fellow of Trinity College, Dublin, and taught divinity, mathematics, and natural philosophy there. He became provost of the college in 1811, bishop of Limerick in 1820, and bishop of Leighlin and Ferns in 1822. His edition of Euclid was reprinted a dozen times. The Reply to John Search's Considerations on the Law of Libel appeared at Dublin in 1834.

Joseph Blanco White (1775-1841) was the son of an Irishman living in Spain. He was born at Seville and studied for orders there, being ordained priest in 1800. He lost his faith in the Roman Catholic Church, and gave up the ministry, escaping to England at the time of the French invasion. At London he edited Español, a patriotic journal extensively circulated in Spain, and for this service he was pensioned after the expulsion of the French. He then studied at Oriel College, Oxford, and became intimate with men like Whately, Newman, and Keble. In 1835 he became a Unitarian. Among his theological writings is his Evidences against Catholicism (1825). The "rejoinder" to which De Morgan refers consisted of two letters: The law of anti-religious Libel reconsidered (Dublin, 1834) and An Answer to some Friendly Remarks on "The Law of Anti-Religious Libel Reconsidered" (Dublin, 1834).

The work was translated from the French.

J. Hoëné Wronski (1778-1853) served, while yet a mere boy, as an artillery officer in Kosciusko's army (1791-1794). He was imprisoned after the battle of Maciejowice. He afterwards lived in Germany, and (after 1810) in Paris. For the bibliography of his works see S. Dickstein's article in the Bibliotheca Mathematica, vol. VI (2), page 48.

Perhaps referring to his Introduction à la philosophie des mathématiques (1811).

Read "equation of the."

Thomas Young (1773-1829), physician and physicist, sometimes called the founder of physiological optics. He seems to have initiated the theory of color blindness that was later developed by Helmholtz. The attack referred to was because of his connection with the Board of Longitude, he having been made (1818) superintendent of the Nautical Almanac and secretary of the Board. He opposed introducing into the Nautical Almanac anything not immediately useful to navigation, and this antagonized many scientists.

Isaac Milner (1750-1820) was professor of natural philosophy at Cambridge (1783) and later became, as De Morgan states, president of Queens' College (1788). In 1791 he became dean of Carlisle, and in 1798 Lucasian professor of mathematics. His chief interest was in chemistry and physics, but he contributed nothing of importance to these sciences or to mathematics.

Thomas Perronet Thompson (1783-1869), fellow of Queens' College, Cambridge, saw service in Spain and India, but after 1822 lived in England. He became major general in 1854, and general in 1868. Besides some works on economics and politics he wrote a Geometry without Axioms (1830) that De Morgan includes later on in his Budget. In it Thompson endeavored to prove the parallel postulate.

De Morgan's father-in-law. See note , page .

Johann Friedrich Herbart (1776-1841), successor of Kant as professor of philosophy at Königsberg (1809-1833), where he established a school of pedagogy. From 1833 until his death he was professor of philosophy at Göttingen. The title of the pamphlet is: De Attentionis mensura causisque primariis. Psychologiae principia statica et mechanica exemplo illustraturus.... Regiomonti,... 1822. The formulas in question are given on pages 15 and 17, and De Morgan has omitted the preliminary steps, which are, for the first one:

β (φ - z) δt = δz

unde βt= Const / (φ - z).

Pro t = 0 etiam z = 0; hinc βt = log φ/(φ - z).

z = φ (1 - ε-βt);

et δz/δt = βφε-βt

These are, however, quite elementary as compared with other portions of the theory.

See note , page .

William Law (1686-1761) was a clergyman, a fellow of Emanuel College, Cambridge, and in later life a convert to Behmen's philosophy. He was so free in his charities that the village in which he lived became so infested by beggars that he was urged by the citizens to leave. He wrote A serious call to a devout and holy life (1728).

He was a curate at Cheshunt, and wrote the Spiritual voice to the Christian Church and to the Jews (London, 1760), A second warning to the world by the Spirit of Prophecy (London, 1760), and Signs of the Times; or a Voice to Babylon (London, 1773).

His real name was Thomas Vaughan (1622-1666). He was a fellow of Jesus College, Oxford, taking orders, but was deprived of his living on account of drunkenness. He became a mystic philosopher and gave attention to alchemy. His works had a large circulation, particularly on the continent. He wrote Magia Adamica (London, 1650), Euphrates; or the Waters of the East (London, 1655), and The Chymist's key to shut, and to open; or the True Doctrine of Corruption and Generation (London, 1657).

Emanuel Swedenborg, or Svedberg (1688-1772) the mystic. It is not commonly known to mathematicians that he was one of their guild, but he wrote on both mathematics and chemistry. Among his works are the Regelkonst eller algebra (Upsala, 1718) and the Methodus nova inveniendi longitudines locorum, terra marique, ope lunae (Amsterdam, 1721, 1727, and 1766). After 1747 he devoted his attention to mystic philosophy.

Pierre Simon Laplace (1749-1827), whose Exposition du système du monde (1796) and Traité de mécanique celeste (1799) are well known.

See note , page .

John Dalton (1766-1844), who taught mathematics and physics at New College, Manchester (1793-1799) and was the first to state the law of the expansion of gases known by his name and that of Gay-Lussac. His New system of Chemical Philosophy (Vol. I, pt. i, 1808; pt. ii, 1810; vol. II, 1827) sets forth his atomic theory.

Howison was a poet and philosopher. He lived in Edinburgh and was a friend of Sir Walter Scott. This work appeared in 1822.

He was a shoemaker, born about 1765 at Haddiscoe, and his "astro-historical" lectures at Norwich attracted a good deal of attention at one time. He traced all geologic changes to differences in the inclination of the earth's axis to the plane of its orbit. Of the works mentioned by De Morgan the first appeared at Norwich in 1822-1823, and there was a second edition in 1824. The second appeared in 1824-1825. The fourth was Urania's Key to the Revelation; or the analyzation of the writings of the Jews..., and was first published at Norwich in 1823, there being a second edition at London in 1833. His books were evidently not a financial success, for Mackey died in an almshouse at Norwich.

Godfrey Higgins (1773-1833), the archeologist, was interested in the history of religious beliefs and in practical sociology. He wrote Horae Sabbaticae (1826), The Celtic Druids (1827 and 1829), and Anacalypsis, an attempt to draw aside the veil of the Saitic Isis; or an Inquiry into the Origin of Languages, Nations, and Religions (posthumously published, 1836), and other works. See also page , infra.

The work also appeared in French. Wirgman wrote, or at least began, two other works: Divarication of the New Testament into Doctrine and History; part I, The Four Gospels (London, 1830), and Mental Philosophy; part I, Grammar of the five senses; being the first step to infant education (London, 1838).

He was born at Shandrum, County Limerick, and supported himself by teaching writing and arithmetic. He died in an almshouse at Cork.

George Boole (1815-1864), professor of mathematics at Queens' College, Cork. His Laws of Thought (1854) was the first work on the algebra of logic.

Oratio Grassi (1582-1654), the Jesuit who became famous for his controversy with Galileo over the theory of comets. Galileo ridiculed him in Il Saggiatore, although according to the modern view Grassi was the more nearly right. It is said that the latter's resentment led to the persecution of Galileo.

De Morgan might have found much else for his satire in the letters of Walsh. He sought, in his Theory of Partial Functions, to substitute "partial equations" for the differential calculus. In his diary there is an entry: "Discovered the general solution of numerical equations of the fifth degree at 114 Evergreen Street, at the Cross of Evergreen, Cork, at nine o'clock in the forenoon of July 7th, 1844; exactly twenty-two years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science."

"It has been ordered, sir, it has been ordered."

Bartholomew Prescot was a Liverpool accountant. De Morgan gives this correct spelling on page . He died after 1849. His Inverted Scheme of Copernicus appeared in Liverpool in 1822.

Robert Taylor (1784-1844) had many more ups and downs than De Morgan mentions. He was a priest of the Church of England, but resigned his parish in 1818 after preaching against Christianity. He soon recanted and took another parish, but was dismissed by the Bishop almost immediately on the ground of heresy. As stated in the text, he was convicted of blasphemy in 1827 and was sentenced to a year's imprisonment, and again for two years on the same charge in 1831. He then married a woman who was rich in money and in years, and was thereupon sued for breach of promise by another woman. To escape paying the judgment that was rendered against him he fled to Tours where he took up surgery.

Herbert Marsh, Bishop of Peterborough. See note on page .

"Argument from the prison."

Richard Carlile (1790-1843), one of the leading radicals of his time. He published Hone's parodies (see note , page ) after they had been suppressed, and an edition of Thomas Paine (1818). He was repeatedly imprisoned, serving nine years in all. His continued conflict with the authorities proved a good advertisement for his bookshop.

Wilhelm Ludwig Christmann (1780-1835) was a protestant clergyman and teacher of mathematics. For a while he taught under Pestalozzi. Disappointed in his ambition to be professor of mathematics at Tubingen, he became a confirmed misanthrope and is said never to have left his house during the last ten years of his life. He wrote several works: Ein Wort über Pestalozzi und Pestalozzismus (1812); Ars cossae promota (1814); Philosophia cossica (1815); Aetas argentea cossae (1819); Ueber Tradition und Schrift, Logos und Kabbala (1829), besides the one mentioned above. The word coss in the above titles was a German name for algebra, from the Italian cosa (thing), the name for the unknown quantity. It appears in English in the early name for algebra, "the cossic art."

See note , page .

See note , page .

He seems to have written nothing else.

See note on page . The name is here spelled correctly.

Joseph Jacotot (1770-1840), the father of this Fortuné Jacotot, was an infant prodigy. At nineteen he was made professor of the humanities at Dijon. He served in the army, and then became professor of mathematics at Dijon. He continued in his chair until the restoration of the Bourbons, and then fled to Louvain. It was here that he developed the method with which his name is usually connected. He wrote a Mathématiques in 1827, which went through four editions. The Epitomé is by his son, Fortuné.

He wrote on educational topics and a Sacred History that went through several editions.

"All is in all."

"Know one thing and refer everything else to it," as it is often translated.

A writer of no reputation.

Sir John Lubbock (1803-1865), banker, scientist, publicist, astronomer, one of the versatile men of his time.

See note , page .

"Those about to die salute you."

Georges Louis Leclerc Buffon (1707-1788), the well-known biologist. He also experimented with burning mirrors, his results appearing in his Invention des miroirs ardens pour brûler à une grande distance (1747). The reference here may be to his Resolution des problèmes qui regardent le jeu du franc carreau (1733). The prominence of his Histoire naturelle (36 volumes, 1749-1788) has overshadowed the credit due to him for his translation of Newton's work on Fluxions.

See page . This article was a supplement to No. 14 in the Athenæum Budget.—A. De M.

There are many similar series and products. Among the more interesting are the following:

π
/
2 = 2·2·4·4·6·6·8...
/
1·3·3·5·5·7·7... ,

π-3
/
4 = 1
/
2·3·4 - 1
/
4·5·6 + 1
/
6·7·8 -...,

π
/
6 = 1
/
3 · 1 - 1
/
3·3 + 1
/
3·5 - 1
/
3·7 + 1
/
3·9 - ... ,

π
/
4 = 4 1
/
5 - 1
/
3·5 + 1
/
5·5 - 1
/
7·5 + ... - 1
/
239 - 1
/
3·239 + 1
/
5·239 - ... .

"To a privateer, a privateer and a half."

Joshua Milne (1776-1851) was actuary of the Sun Life Assurance Society. He wrote A Treatise on the Valuation of Annuities and Assurances on Lives and Survivorships; on the Construction of tables of mortality; and on the Probabilities and Expectations of Life, London, 1815. Upon the basis of the Carlisle bills of mortality of Dr. Heysham he reconstructed the mortality tables then in use and which were based upon the Northampton table of Dr. Price. His work revolutionized the actuarial science of the time. In later years he devoted his attention to natural history.

See note , page . He also wrote the Theory of Parallels. The proof of Euclid's axiom looked for in the properties of the equiangular spiral (London, 1840), which went through four editions, and the Theory of Parallels. The proof that the three angles of a triangle are equal to two right angles looked for in the inflation of the sphere (London, 1853), of which there were three editions.

For the latest summary, see W. B. Frankland, Theories of Parallelism, an historical critique, Cambridge, 1910.

Joseph Louis Lagrange (1736-1813), author of the Mécanique analytique (1788), Théorie des functions analytiques (1797), Traité de la résolution des équations numériques de tous degrés (1798), Leçons sur le calcul des fonctions (1806), and many memoirs. Although born in Turin and spending twenty of his best years in Germany, he is commonly looked upon as the great leader of French mathematicians. The last twenty-seven years of his life were spent in Paris, and his remarkable productivity continued to the time of his death. His genius in the theory of numbers was probably never excelled except by Fermat. He received very high honors at the hands of Napoleon and was on the first staff of the Ecole polytechnique (1797).

"I shall have to think it over again."

Henry Goulburn (1784-1856) held various government posts. He was under-secretary for war and the colonies (1813), commissioner to negotiate peace with America (1814), chief secretary to the Lord Lieutenant of Ireland (1821), and several times Chancellor of the Exchequer. On the occasion mentioned by De Morgan he was standing for parliament, and was successful.

On Drinkwater Bethune see note , page .

Charles Henry Cooper (1808-1866) was a biographer and antiquary. He was town clerk of Cambridge (1849-1866) and wrote the Annals of Cambridge (1842-1853). His Memorials of Cambridge (1874) appeared after his death. Thompson Cooper was his son, and the two collaborated in the Athenae Cantabrigiensis (1858).

William Yates Peel (1789-1858) was a brother of Sir Robert Peel, he whose name degenerated into the familiar title of the London "Bobby" or "Peeler." Yates Peel was a member of parliament almost continuously from 1817 to 1852. He represented Cambridge at Westminster from 1831 to 1835.

Henry John Temple, third Viscount of Palmerston (1784-1865), was member for Cambridge in 1811, 1818, 1820, 1826 (defeating Goulburn), and 1830. He failed of reelection in 1831 because of his advocacy of reform. This must have been the time when Goulburn defeated him. He was Foreign Secretary (1827) and Secretary of State for Foreign Affairs (1830-1841, and 1846-1851). It is said of him that he "created Belgium, saved Portugal and Spain from absolutism, rescued Turkey from Russia and the highway to India from France." He was Prime Minister almost continuously from 1855 to 1865, a period covering the Indian Mutiny and the American Civil War.

William Cavendish, seventh Duke of Devonshire (1808-1891). He was member for Cambridge from 1829 to 1831, but was defeated in 1831 because he had favored parliamentary reform. He became Earl of Burlington in 1834, and Duke of Devonshire in 1858. He was much interested in the promotion of railroads and in the iron and steel industries.

Richard Sheepshanks (1794-1855) was a brother of John Sheepshanks the benefactor of art. (See note , p. .) He was a fellow of Trinity College, Cambridge, a fellow of the Royal Society and secretary of the Astronomical Society. Babbage (See note , p. ) suspected him of advising against the government support of his calculating machine and attacked him severely in his Exposition of 1851, in the chapter on The Intrigues of Science. Babbage also showed that Sheepshanks got an astronomical instrument of French make through the custom house by having Troughton's (See note , page ) name engraved on it. Sheepshanks admitted this second charge, but wrote a Letter in Reply to the Calumnies of Mr. Babbage, which was published in 1854. He had a highly controversial nature.

See note , page . The work referred to is Passages from the Life of a Philosopher, London, 1864.

Drinkwater Bethune. See note , page .

Siméon-Denis Poisson (1781-1840) was professor of calculus and mechanics at the Ecole polytechnique. He was made a baron by Napoleon, and was raised to the peerage in 1837. His chief works are the Traité de mécanìque (1811) and the Traité mathématique de la chaleur (1835).

"As to M. Poisson, I really wish I had a thousandth part of his mathematical knowledge that I might prove my system to the incredulous."

This list includes most of the works of Antoine-Louis-Guénard Demonville. There was also the Nouveau système du monde ... et hypothèses conformes aux expériences sur les vents, sur la lumière et sur le fluide électro-magnétique, Paris, 1830.

Paris, 1835.

Paris, 1833.

The second part appeared in 1837. There were also editions in 1850 and 1852, and one edition appeared without date.

Paris, 1842.

Parsey also wrote The Art of Miniature Painting on Ivory (1831), Perspective Rectified (1836), and The Science of Vision (1840), the third being a revision of the second.

William Ritchie (1790-1837) was a physicist who had studied at Paris under Biot and Gay-Lussac. He contributed several papers on electricity, heat, and elasticity, and was looked upon as a good experimenter. Besides the geometry he wrote the Principles of the Differential and Integral Calculus (1836).

Alfred Day (1810-1849) was a man who was about fifty years ahead of his time in his attempt to get at the logical foundations of geometry. It is true that he laid himself open to criticism, but his work was by no means bad. He also wrote A Treatise on Harmony (1849, second edition 1885), The Rotation of the Pendulum (1851), and several works on Greek and Latin Grammar.

Walter Forman wrote a number of controversial tracts. His first seems to have been A plan for improving the Revenue without adding to the burdens of the people, a letter to Canning in 1813. He also wrote A New Theory of the Tides (1822). His Letter to Lord John Russell, on Lord Brougham's most extraordinary conduct; and another to Sir J. Herschel, on the application of Kepler's third law appeared in 1832.

Lord John Russell (1792-1878) first Earl Russell, was one of the strongest supporters of the reform measures of the early Victorian period. He became prime minister in 1847, and again in 1865.

Lauder seems never to have written anything else.

See note , page .

The names of Alphonso Cano de Molina, Yvon, and Robert Sara have no standing in the history of the subject beyond what would be inferred from De Morgan's remark.

Claude Mydorge (1585-1647), an intimate friend of Descartes, was a dilletante in mathematics who read much but accomplished little. His Récréations mathématiques is his chief work. Boncompagni published the "Problèmes de Mydorge" in his Bulletino.

Claude Hardy was born towards the end of the 16th century and died at Paris in 1678. In 1625 he edited the Data Euclidis, publishing the Greek text with a Latin translation. He was a friend of Mydorge and Descartes, but an opponent of Fermat.

That is, in the Bibliotheca Realis of Martin Lipen, or Lipenius (1630-1692), which appeared in six folio volumes, at Frankfort, 1675-1685.

See note , page .

Baldassare Boncompagni (1821-1894) was the greatest general collector of mathematical works that ever lived, possibly excepting Libri. His magnificent library was dispersed at his death. His Bulletino (1868-1887) is one of the greatest source books on the history of mathematics that we have. He also edited the works of Leonardo of Pisa.

He seems to have attracted no attention since De Morgan's search, for he is not mentioned in recent bibliographies.

Joseph-Louis Vincens de Mouléon de Causans was born about the beginning of the l8th century. He was a Knight of Malta, colonel in the infantry, prince of Conti, and governor of the principality of Orange. His works on geometry are the Prospectus apologétique pour la quadrature du cercle (1753), and La vraie géométrie transcendante (1754).

See note , page .

See note , page .

Lieut. William Samuel Stratford (1791-1853), was in active service during the Napoleonic wars but retired from the army in 1815. He was first secretary of the Astronomical Society (1820) and became superintendent of the Nautical Almanac in 1831. With Francis Baily he compiled a star catalogue, and wrote on Halley's (1835-1836) and Encke's (1838) comets.

See Sir J. Herschel's Astronomy, p. 369.—A. De M.

Captain Ross had just stuck a bit of brass there.—A. De M.

Sir James Clark Ross (1800-1862) was a rear admiral in the British navy and an arctic and antarctic explorer of prominence. De Morgan's reference is to Ross's discovery of the magnetic pole on June 1, 1831. In 1838 he was employed by the Admiralty on a magnetic survey of the United Kingdom. He was awarded the gold medal of the geographical societies of London and Paris in 1842.

John Partridge (1644-1715), the well-known astrologer and almanac maker. Although bound to a shoemaker in his early boyhood, he had acquired enough Latin at the age of eighteen to read the works of the astrologers. He then mastered Greek and Hebrew and studied medicine. In 1680 he began the publication of his almanac, the Merlinus Liberatus, a book that acquired literary celebrity largely through the witty comments upon it by such writers as Swift and Steele.

See note on page .

William Woodley also published several almanacs (1838, 1839, 1840) after his rejection by the Astronomical Society in 1834.

It appeared at London.

The first edition appeared in 1830, also at London.

See note , page .

Thomas Kerigan wrote The Young Navigator's Guide to the siderial and planetary parts of Nautical Astronomy (London, 1821, second edition 1828), a work on eclipses (London, 1844), and the work on tides (London, 1847) to which De Morgan refers.

Jean Sylvain Bailly, who was guillotined. See note , page .

See note , page .

Laurent seems to have had faint glimpses of the modern theory of matter. He is, however, unknown.

See note , page .

Francis Baily (1774-1844) was a London stockbroker. His interest in science in general and in astronomy in particular led to his membership in the Royal Society and to his presidency of the Astronomical Society. He wrote on interest and annuities (1808), but his chief works were on astronomy.

If the story is correctly told Baily must have enjoyed his statement that Gauss was "the oldest mathematician now living." As a matter of fact he was then only 58, three years the junior of Baily himself. Gauss was born in 1777 and died in 1855, and Baily was quite right in saying that he was "generally thought to be the greatest" mathematician then living.

Margaret Cooke, who married Flamsteed in 1692.

John Brinkley (1763-1835), senior wrangler, first Smith's prize-man (1788), Andrews professor of astronomy at Dublin, first Astronomer Royal for Ireland (1792), F.R.S. (1803), Copley medallist, president of the Royal Society and Bishop of Cloyne. His Elements of Astronomy appeared in 1808.

See note , page .

See note , page .

See note , page .

"It becomes the doctors of the Sorbonne to dispute, the Pope to decree, and the mathematician to go to Paradise on a perpendicular line."

See note , page .

See note , page .

Sylvain van de Weyer, who was born at Louvain in 1802. He was a jurist and statesman, holding the portfolio for foreign affairs (1831-1833), and being at one time ambassador to England.

Henry Crabb Robinson (1775-1867), correspondent of the Times at Altona and in the Peninsula, and later foreign editor. He was one of the founders of the Athenæum Club and of University College, London. He seems to have known pretty much every one of his day, and his posthumous Diary attracted attention when it appeared.

Was this Whewell, who was at Trinity from 1812 to 1816 and became a fellow in 1817?

Tom Cribb (1781-1848) the champion pugilist. He had worked as a coal porter and hence received his nickname, the Black Diamond.

John Finleyson, or Finlayson, was born in Scotland in 1770 and died in London in 1854. He published a number of pamphlets that made a pretense to being scientific. Among his striking phrases and sentences are the statements that the stars were made "to amuse us in observing them"; that the earth is "not shaped like a garden turnip as the Newtonians make it," and that the stars are "oval-shaped immense masses of frozen water." The first edition of the work here mentioned appeared at London in 1830.

Richard Brothers (1757-1824) was a native of Newfoundland. He went to London when he was about 30, and a little later set forth his claim to being a descendant of David, prince of the Hebrews, and ruler of the world. He was confined as a criminal lunatic in 1795 but was released in 1806.

Charles Grey (1764-1845), second Earl Grey, Viscount Howick, was then Prime Minister. The Reform Bill was introduced and defeated in 1831. The following year, with the Royal guarantees to allow him to create peers, he finally carried the bill in spite of "the number of the beast."

The letters of obscure men, the Epistolæ obscurorum virorum ad venerabilem virum Magistrum Ortuinum Gratium Dauentriensem, by Joannes Crotus, Ulrich von Hutten, and others appeared at Venice about 1516.

The lamentations of obscure men, the Lamentationes obscurorum virorum, non prohibete per sedem Apostolicam. Epistola D. Erasmi Roterodami: quid de obscuris sentiat, by G. Ortwinus, appeared at Cologne in 1518.

The criticism was timely when De Morgan wrote it. At present it would have but little force with respect to the better class of algebras.

Thomas Ignatius Maria Forster (1789-1860) was more of a man than one would infer from this satire upon his theory. He was a naturalist, astronomer, and physiologist. In 1812 he published his Researches about Atmospheric Phenomena, and seven years later (July 3, 1819) he discovered a comet. With Sir Richard Phillips he founded a Meteorological Society, but it was short lived. He declined a fellowship in the Royal Society because he disapproved of certain of its rules, so that he had a recognized standing in his day. The work mentioned by De Morgan is the second edition, the first having appeared at Frankfort on the Main in 1835 under the title, Recueil des ouvrages et des pensées d'un physicien et metaphysicien.

Zadkiel, whose real name was Richard James Morrison (1795-1874), was in his early years an officer in the navy. In 1831 he began the publication of the Herald of Astrology, which was continued as Zadkiel's Almanac. His name became familiar throughout Great Britain as a result.

See note , page .

Sumner (1780-1862) was an Eton boy. He went to King's College, Cambridge, and was elected fellow in 1801. He took many honors, and in 1807 became M.A. He was successively Canon of Durham (1820), Bishop of Chester (1828), and Archbishop of Canterbury (1848). Although he voted for the Catholic Relief Bill (1829) and the Reform Bill (1832), he opposed the removal of Jewish disabilities.

Charles Richard Sumner (1790-1874) was not only Bishop of Winchester (1827), but also Bishop of Llandaff and Dean of St. Paul's, London (1826). He lost the king's favor by voting for the Catholic Relief Bill.

John Bird Sumner, brother of Charles Richard.

Thomas Musgrave (1788-1860) became Fellow of Trinity in 1812, and senior proctor in 1831. He was also Dean of Bristol.

Charles Thomas Longley (1794-1868) was educated at Westminster School and at Christ Church, Oxford. He became M.A. in 1818 and D.D. in 1829. Besides the bishoprics mentioned he was Bishop of Ripon (1836-1856), and before that was headmaster of Harrow (1829-1836).

Thomson (1819-1890) was scholar and fellow of Queen's College, Oxford. He became chaplain to the Queen in 1859.

This is worthy of the statistical psychologists of the present day.

The famous Moon Hoax was written by Richard Adams Locke, who was born in New York in 1800 and died in Staten Island in 1871. He was at one time editor of the Sun, and the Hoax appeared in that journal in 1835. It was reprinted in London (1836) and Germany, and was accepted seriously by most readers. It was published in book form in New York in 1852 under the title The Moon Hoax. Locke also wrote another hoax, the Lost Manuscript of Mungo Park, but it attracted relatively little attention.

It is true that Jean-Nicolas Nicollet (1756-1843) was at that time in the United States, but there does not seem to be any very tangible evidence to connect him with the story. He was secretary and librarian of the Paris observatory (1817), member of the Bureau of Longitudes (1822), and teacher of mathematics in the Lycée Louis-le-Grand. Having lost his money through speculations he left France for the United States in 1831 and became connected with the government survey of the Mississippi Valley.

This was Alexis Bouvard (1767-1843), who made most of the computations for Laplace's Mécanique céleste (1793). He discovered eight new comets and calculated their orbits. In his tables of Uranus (1821) he attributed certain perturbations to the presence of an undiscovered planet, but unlike Leverrier and Adams he did not follow up this clue and thus discover Neptune.

Patrick Murphy (1782-1847) awoke to find himself famous because of his natural guess that there would be very cold weather on January 20, although that is generally the season of lowest temperature. It turned out that his forecasts were partly right on 168 days and very wrong on 197 days.

He seems to have written nothing else. If one wishes to enter into the subject of the mathematics of the Great Pyramid there is an extensive literature awaiting him. Richard William Howard Vyse (1784-1853) published in 1840 his Operations carried on at the Pyramids of Gizeh in 1837, and in this he made a beginning of a scientific metrical study of the subject. Charles Piazzi Smyth (1819-1900), astronomer Royal for Scotland (1845-1888) was much carried away with the number mysticism of the Great Pyramid, so much so that he published in 1864 a work entitled Our Inheritance in the Great Pyramid, in which his vagaries were set forth. Although he was then a Fellow of the Royal Society (1857), his work was so ill received that when he offered a paper on the subject it was rejected (1874) and he resigned in consequence of this action. The latest and perhaps the most scholarly of all investigators of the subject is William Matthew Flinders Petrie (born in 1853), Edwards professor of Egyptology at University College, London, whose Pyramids and Temples of Gizeh (1883) and subsequent works are justly esteemed as authorities.

As De Morgan subsequently found, this name reversed becomes Oliver B...e, for Oliver Byrne, one of the odd characters among the minor mathematical writers of the middle of the last century. One of his most curious works is The first six Books of the Elements of Euclid; in which coloured diagrams and symbols are used instead of letters (1847). There is some merit in speaking of the red triangle instead of the triangle ABC, but not enough to give the method any standing. His Dual Arithmetic (1863-1867) was also a curious work.

Brenan also wrote on English composition (1829), a work that went through fourteen editions by 1865; a work entitled The Foreigner's English Conjugator (1831), and a work on the national debt.

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Sir William Rowan Hamilton (1805-1865), the discoverer of quaternions (1852), was an infant prodigy, competing with Zerah Colburn as a child. He was a linguist of remarkable powers, being able, at thirteen years of age, to boast that he knew as many languages as he had lived years. When only sixteen he found an error in Laplace's Mécanique céleste. When only twenty-two he was appointed Andrews professor of astronomy, and he soon after became Astronomer Royal of Ireland. He was knighted in 1835. His earlier work was on optics, his Theory of Systems of Rays appearing in 1823. In 1827 he published a paper on the principle of Varying Action. He also wrote on dynamics.

"Let him not leave the kingdom,"—a legal phrase.

Probably De Morgan is referring to Johann Bernoulli III (1744-1807), who edited Lambert's Logische und philosophische Abhandlungen, Berlin, 1782. He was astronomer of the Academy of Sciences at Berlin.

Jacob Bernoulli (1654-1705) was one of the two brothers who founded the famous Bernoulli family of mathematicians, the other being Johann I. His Ars conjectandi (1713), published posthumously, was the first distinct treatise on probabilities.

Johann Heinrich Lambert (1728-1777) was one of the most learned men of his time. Although interested chiefly in mathematics, he wrote also on science, logic, and philosophy.

Joseph Diez Gergonne (1771-1859), a soldier under Napoleon, and founder of the Annales de mathématiques (1810).

Gottfried Ploucquet (1716-1790) was at first a clergyman, but afterwards became professor of logic at Tübingen.

"In the premises let the middle term be omitted; what remains indicates the conclusion."

Probably Sir William Edmond Logan (1789-1875), who became so interested in geology as to be placed at the head of the geological survey of Canada (1842). The University of Montreal conferred the title LL.D. upon him, and Napoleon III gave him the cross of the Legion of Honor.

"So strike that he may think himself to die."

"Witticism or piece of stupidity."

A very truculently unjust assertion: for Sir W. was as great a setter up of some as he was a puller down of others. His writings are a congeries of praises and blames, both cruel smart, as they say in the States. But the combined instigation of prose, rhyme, and retort would send Aristides himself to Tartarus, if it were not pretty certain that Minos would grant a stet processus under the circumstances. The first two verses are exaggerations standing on a basis of truth. The fourth verse is quite true: Sir W. H. was an Edinburgh Aristotle, with the difference of ancient and modern Athens well marked, especially the perfervidum ingenium Scotorum.—A. De M.

See note , p. . There was also a Theory of Parallels that differed from these, London, 1853, second edition 1856, third edition 1856.

The work was written by Robert Chambers (1802-1871), the Edinburgh publisher, a friend of Scott and of many of his contemporaries in the literary field. He published the Vestiges of the Natural History of Creation in 1844, not 1840.

Everett (1784-1872) was at that time a good Wesleyan, but was expelled from the ministry in 1849 for having written Wesleyan Takings and as under suspicion for having started the Fly Sheets in 1845. In 1857 he established the United Methodist Free Church.

Smith was a Primitive Methodist preacher. He also wrote an Earnest Address to the Methodists (1841) and The Wealth Question (1840?).

He wrote the Nouveau traité de Balistique, Paris, 1837.

Joseph Denison, known to fame only through De Morgan. See also page .

The radical (1784?-1858), advocate of the founding of London university (1826), of medical reform (1827-1834), and of the repeal of the duties on newspapers and corn, and an ardent champion of penny postage.

I. e., Roman Catholic Priest.

Murphy (1806-1843) showed extraordinary powers in mathematics even before the age of thirteen. He became a fellow of Caius College, Cambridge, in 1829, dean in 1831, and examiner in mathematics in London University in 1838.

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Sir John Bowring (1792-1872), the linguist, writer, and traveler, member of many learned societies and a writer of high reputation in his time. His works were not, however, of genuine merit.

Joseph Hume (1777-1855) served as a surgeon with the British army in India early in the nineteenth century. He returned to England in 1808 and entered parliament as a radical in 1812. He was much interested in all reform movements.

Sir Robert Harry Inglis (1786-1855), a strong Tory, known for his numerous addresses in the House of Commons rather than for any real ability.

Sir Robert Peel (1788-1850) began his parliamentary career in 1809 and was twice prime minister. He was prominent in most of the great reforms of his time.

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John Taylor (1781-1864) was a publisher, and published several pamphlets opposed to Peel's currency measures. De Morgan refers to his work on the Junius question. This was done early in his career, and resulted in A Discovery of the author of the Letters of Junius (1813), and The Identity of Junius with a distinguished living character established (1816), this being Sir Philip Francis.

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See page .

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Sir Nicholas Harris Nicolas (1799-1848) was a reformer in various lines,—the Record Commission, the Society of Antiquaries, and the British Museum,—and his work was not without good results.

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In the Companion to the Almanac for 1845 is a paper by Prof. De Morgan, "On the Ecclesiastical Calendar," the statements of which, so far as concerns the Gregorian Calendar, are taken direct from the work of Clavius, the principal agent in the arrangement of the reformed reckoning. This was followed, in the Companion to the Almanac for 1846, by a second paper, by the same author, headed "On the Earliest Printed Almanacs," much of which is written in direct supplement to the former article.—S. E. De Morgan.

It may be necessary to remind some English readers that in Latin and its derived European languages, what we call Easter is called the passover (pascha). The Quartadecimans had the name on their side: a possession which often is, in this world, nine points of the law.—A. De M.

Socrates Scholasticus was born at Constantinople c. 379, and died after 439. His Historia Ecclesiastica (in Greek) covers the period from Constantine the Great to about 439, and includes the Council of Nicæa. The work was printed in Paris 1544.

Theodoretus or Theodoritus was born at Antioch and died about 457. He was one of the greatest divines of the fifth century, a man of learning, piety, and judicial mind, and a champion of freedom of opinion in all religious matters.

He died in 417. He was a man of great energy and of high attainments.

He died in 461, having reigned as pope for twenty-one years. It was he who induced Attila to spare Rome in 452.

He succeeded Leo as pope in 461, and reigned for seven years.

Victorinus or Victorius Marianus seems to have been born at Limoges. He was a mathematician and astronomer, and the cycle mentioned by De Morgan is one of 532 years, a combination of the Metonic cycle of 19 years with the solar cycle of 28 years. His canon was published at Antwerp in 1633 or 1634, De doctrina temporum sive commentarius in Victorii Aquitani et aliorum canones paschales.

He went to Rome about 497, and died there in 540. He wrote his Liber de paschate in 525, and it was in this work that the Christian era was first used for calendar purposes.

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Johannes de Sacrobosco (Holy wood), or John of Holywood. The name was often written, without regard to its etymology, Sacrobusto. He was educated at Oxford and taught in Paris until his death (1256). He did much to make the Hindu-Arabic numerals known to European scholars.

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The Julian year is a year of the Julian Calendar, in which there is leap year every fourth year. Its average length is therefore 365 days and a quarter.—A. De M.

Ugo Buoncompagno (1502-1585) was elected pope in 1572.

He was a Calabrian, and as early as 1552 was professor of medicine at Perugia. In 1576 his manuscript on the reform of the calendar was presented to the Roman Curia by his brother, Antonius. The manuscript was not printed and it has not been preserved.

The title of this work, which is the authority on all points of the new Calendar, is Kalendarium Gregorianum Perpetuum. Cum Privilegio Summi Pontificis Et Aliorum Principum. Romæ, Ex Officina Dominici Basæ. MDLXXXII. Cum Licentia Superiorum (quarto, pp. 60).—A. De M.

Manuels-Roret. Théorie du Calendrier et collection de tous les Calendriers des Années passées et futures.... Par L. B. Francœur,... Paris, à la librairie encyclopédique de Roret, rue Hautefeuille, 10 bis. 1842. (12mo.) In this valuable manual, the 35 possible almanacs are given at length, with such preliminary tables as will enable any one to find, by mere inspection, which almanac he is to choose for any year, whether of old or new style. [1866. I may now refer to my own Book of Almanacs, for the same purpose].—A. De M.

Louis Benjamin Francœur (1773-1849), after holding positions in the Ecole polytechnique (1804) and the Lycée Charlemagne (1805), became professor of higher algebra in the University of Paris (1809). His Cours complet des mathématiques pures was well received, and he also wrote on mechanics, astronomy, and geodesy.

Albertus Pighius, or Albert Pigghe, was born at Kempen c. 1490 and died at Utrecht in 1542. He was a mathematician and a firm defender of the faith, asserting the supremacy of the Pope and attacking both Luther and Calvin. He spent some time in Rome. His greatest work was his Hierarchiæ ecclesiasticæ assertio (1538).

This was A. F. Vogel. The work was his translation from the German edition which appeared at Leipsic the same year, Entdeckung einer numerischen General-Auflösung aller höheren endlichen Gleichungen von jeder beliebigen algebraischen und transcendenten Form.

The latest edition of Burnside and Panton's Theory of Equations has this brief summary of the present status of the problem: "Demonstrations have been given by Abel and Wantzel (see Serret's Cours d'Algèbre Supérieure, Art. 516) of the impossibility of resolving algebraically equations unrestricted in form, of a degree higher than the fourth. A transcendental solution, however, of the quintic has been given by M. Hermite, in a form involving elliptic integrals."

There was a second edition of this work in 1846. The author's Astronomy Simplified was published in 1838, and the Thoughts on Physical Astronomy in 1840, with a second edition in 1842.

This was The Science of the Weather, by several authors... edited by B., Glasgow, 1867.

This was Y. Ramachandra, son of Sundara Lāla. He was a teacher of science in Delhi College, and the work to which De Morgan refers is A Treatise on problems of Maxima and Minima solved by Algebra, which appeared at Calcutta in 1850. De Morgan's edition was published at London nine years later.

Abraham de Moivre (1667-1754), French refugee in London, poor, studying under difficulties, was a man with tastes in some respects like those of De Morgan. For one thing, he was a lover of books, and he had a good deal of interest in the theory of probabilities to which De Morgan also gave much thought. His introduction of imaginary quantities into trigonometry was an event of importance in the history of mathematics, and the theorem that bears his name, (cos φ + i sin φ)n = cos nφ + i sin nφ, is one of the most important ones in all analysis.

John Dolland (1706-1761), the silk weaver who became the greatest maker of optical instruments in his time.

Thomas Simpson (1710-1761), also a weaver, taking his leisure from his loom at Spitalfields to teach mathematics. His New Treatise on Fluxions (1737) was written only two years after he began working in London, and six years later he was appointed professor of mathematics at Woolwich. He wrote many works on mathematics and Simpson's Formulas for computing trigonometric tables are still given in the text-books.

Nicholas Saunderson (1682-1739), the blind mathematician. He lost his eyesight through smallpox when only a year old. At the age of 25 he began lecturing at Cambridge on the principles of the Newtonian philosophy. His Algebra, in two large volumes, was long the standard treatise on the subject.

He was not in the class with the others mentioned.

Not known in the literature of mathematics.

Probably J. Butler Williams whose Practical Geodesy appeared in 1842, with a third edition in 1855.

Benjamin Gompertz (1779-1865) was debarred as a Jew from a university education. He studied mathematics privately and became president of the Mathematical Society. De Morgan knew him professionally through the fact that he was prominent in actuarial work.

Referring to the contributions of Archimedes (287-212 B.C.) to the mensuration of the sphere.

The famous Alexandrian astronomer (c. 87-c. 165 A.D.), author of the Almagest, a treatise founded on the works of Hipparchus.

Dr. Whewell, when I communicated this song to him, started the opinion, which I had before him, that this was a very good idea, of which too little was made.—A. De M.

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The common epithet of rank: nobilis Tycho, as he was a nobleman. The writer had been at history.—A. De M.

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He lost it in a duel, with Manderupius Pasbergius. A contemporary, T. B. Laurus, insinuates that they fought to settle which was the best mathematician! This seems odd, but it must be remembered they fought in the dark, "in tenebris densis"; and it is a nice problem to shave off a nose in the dark, without any other harm.—A. De M.

Was this T. B. Laurus Joannes Baptista Laurus or Giovanni Battista Lauro (1581-1621), the poet and writer?

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Referring to Kepler's celebrated law of planetary motion. He had previously wasted his time on analogies between the planetary orbits and the polyhedrons.—A. De M.

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"It does move though."

As great a lie as ever was told: but in 1800 a compliment to Newton without a fling at Descartes would have been held a lopsided structure.—A. De M.

Jean-le-Rond D'Alembert (1717-1783), the foundling who was left on the steps of Jean-le-Rond in Paris, and who became one of the greatest mathematical physicists and astronomers of his century.

Leonhard Euler (1707-1783), friend of the Bernoullis, the greatest of Swiss mathematicians, prominent in the theory of numbers, and known for discoveries in all lines of mathematics as then studied.

See notes , , page .

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The siderial day is about four minutes short of the solar; there are 366 sidereal days in the year.—A. De M.

The founding of the London Mathematical Society is discussed by Mrs. De Morgan in her Memoir (p. 281). The idea came from a conversation between her brilliant son, George Campbell De Morgan, and his friend Arthur Cowper Ranyard in 1864. The meeting of organization was held on Nov. 7, 1864, with Professor De Morgan in the chair, and the first regular meeting on January 16, 1865.

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John Russell Hind (b. 1823), the astronomer. Between 1847 and 1854 he discovered ten planetoids.

Sir Roderick Impey Murchison (1792-1871), the great geologist. He was knighted in 1846 and devoted the latter part of his life to the work of the Royal Geographical Society and to the geology of Scotland.

Friedrich Wilhelm Bessel (1784-1846), the astronomer and physicist. He was professor of astronomy at Königsberg.

This was the Reduction of the Observations of Planets made ... from 1750 to 1830: computed ... under the superintendence of George Biddell Airy (1848). See note , page .

The expense of this magnificent work was defrayed by Government grants, obtained, at the instance of the British Association, in 1833—A. De M.

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Franz Friedrich Ernst Brünnow (1821-1891) was at that time or shortly before director of the observatory at Dusseldorf. He then went to Berlin and thence (1854) to Ann Arbor, Michigan. He then went to Dublin and finally became Royal Astronomer of Ireland.

Johann Gottfried Galle (1812-1910), at that time connected with the Berlin observatory, and later professor of astronomy at Breslau.

George Bishop (1785-1861), in whose observatory in Regent's Park important observations were made by Dawes, Hind, and Marth.

James Challis (1803-1882), director of the Cambridge observatory, and successor of Airy as Plumian professor of astronomy.

On Leverrier and Arago see note , page , and note , page .

Robert Grant's (1814-1892) History of Physical Astronomy from the Earliest Ages to the Middle of the Nineteenth Century appeared in 1852. He was professor of astronomy and director of the observatory at Glasgow.

John Debenham was more interested in religion than in astronomy. He wrote The Strait Gate; or, the true scripture doctrine of salvation clearly explained, London, 1843, and Tractatus de magis et Bethlehemæ stella et Christi in deserto tentatione, privately printed at London in 1845.

More properly the Sydney Smirke reading room, since it was built from his designs.

The Antinomians were followers of Johannes Agricola (1494-1566). They believed that Christians as such were released from all obligations to the Old Testament. Some went so far as to assert that, since all Christians were sanctified, they could not lose this sanctity even though they disobeyed God. The sect was prominent in England in the seventeenth century, and was transferred to New England. Here it suffered a check in the condemnation of Mrs. Ann Hutchinson (1636) by the Newton Synod.

Aside from this work and his publications on Reeve and Muggleton he wrote nothing. With Joseph Frost he published A list of Books and general index to J. Reeve and L. Muggleton's works (1846), Divine Songs of the Muggletonians (1829), and the work mentioned on page . The works of J. Reeve and L. Muggleton (1832).

About 1650 he and his cousin John Reeve (1608-1658) began to have visions. As part of their creed they taught that astronomy was opposed by the Bible. They asserted that the sun moves about the earth, and Reeve figured out that heaven was exactly six miles away. Both Muggleton and Reeve were imprisoned for their unitarian views. Muggleton wrote a Transcendant Spirituall Treatise (1652). I have before me A true Interpretation of All the Chief Texts ... of the whole Book of the Revelation of St. John.... By Lodowick Muggleton, one of the two last Commissioned Witnesses & Prophets of the onely high, immortal, glorious God, Christ Jesus (1665), in which the interpretation of the "number of the beast" occupies four pages without arriving anywhere.

In 1652 he was, in a vision, named as the Lord's "last messenger," with Muggleton as his "mouth," and died six years later, probably of nervous tension resulting from his divine "illumination." He was the more spiritual of the two.

William Guthrie (1708-1770) was a historian and political writer. His History of England (1744-1751) was the first attempt to base history on parliamentary records. He also wrote a General History of Scotland in 10 volumes (1767). The work to which Frost refers is the Geographical, Historical, and Commercial Grammar (1770) which contained an astronomical part by J. Ferguson. By 1827 it had passed through 24 editions.

George Fox (1624-1691), founder of the Society of Friends; a mystic and a disciple of Boehme. He was eight times imprisoned for heresy.

If they were friends they were literary antagonists, for Muggleton wrote against Fox The Neck of the Quakers Broken (1663), and Fox replied in 1667. Muggleton also wrote A Looking Glass for George Fox.

John Conduitt (1688-1737), who married (1717) Newton's half niece, Mrs. Katherine Barton. See note , page .

Probably Peter Mark Roget's (1779-1869) Thesaurus of English Words (1852) is not much used at present, but it went through 28 editions in his lifetime. Few who use the valuable work are aware that Roget was a professor of physiology at the Royal Institution (London), that he achieved his title of F. R. S. because of his work in perfecting the slide rule, and that he followed Sir John Herschel as secretary of the Royal Society.

See note , page . This work went into a second edition in the year of its first publication.

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George Jacob Holyoake (1817-1906) entered into a controversial life at an early age. In 1841 he was imprisoned for six months for blasphemy. He founded and edited The Reasoner (Vols. 1-26, 1846-1861). In his later life he did much to promote cooperation among the working class.

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William Thomas Lowndes (1798-1843), whose Bibliographer's Manual of English Literature, 4 vols., London, 1834 (also 1857-1864, and 1869) is a classic in its line.

Jacques Charles Brunet (1780-1867), the author of the great French bibliography, the Manuel du Libraire (1810).