ceding astronomers, and examined all the hypotheses they had devised, for the explanation of the phænomena of the heavens: the result of his labours was his great work, "De Revolutionibus Orbium Cœlestium," Libri VI. It was completed in 1530, but was not published till a very few days before his death in 1543. In the title-page is quoted the admonition of Plato, ἀγεωμέτρητος οὐδεὶς εἰσίτω. Copernicus was the author of a tract on Plane and Spherical Trigonometry, which contained a Table of Sines. It was first printed at Nuremberg, and afterwards at the end of the first book of his great work, De Revolu tionibus, &c. The publication of this work was superintended by George Joachim Rhæticus, the disciple and latterly the assistant of Copernicus, in his astronomical labours. He was born in 1514, and died in 1576 at Feldkirk in the Tyrol. With the view of making astronomical calculations more accurate, he commenced a table of sines, tangents, and secants for every ten seconds of the quadrant, to fifteen places of figures; which he did not live to complete. This work was completed and published by his disciple Valentine Otho in 1596. The table of sines for every ten seconds, and for every second in the first and last degrees of the quadrant, which he had completed, was published in 1613 by Pitiscus, who extended the value of some of the latter sines to twenty-two places of figures.

Nicholas Tartaglia was a celebrated mathematician, born at Brescia in 1479. He was the original discoverer of the solution of Cubic Equations, which he first effected in 1530, and which Cardan, who has generally had the merit of the discovery, surreptitiously obtained from him. The treatise of Tartaglia on the Theory and Practice of Gunnery, is the earliest which treated of the motion of projectiles. He also published an edition of Euclid's Elements, in Italian, with a Commentary. The last part of his great work, "Trattato de Numeri et Misure,” was published in 1558.

John Werner of Nuremberg, one of the most distinguished astronomers and geometers of his day, was born in 1468 and died in 1528: he was the first who attempted to restore the geometrical analysis of the ancient Greeks. In his "Opera Mathematica," which he published in 1522, will be found what he effected, both in the Conic Sections and in some solid problems. He also wrote a work on Triangles.

Zamberti made the first translation of Euclid's Elements, from the original Greek into Latin, which he published at Venice in 1505. The original Greek was first published at Basle in 1533, edited by Simon Grynæus. This edition of Euclid was the foundation of Commandine's translation in 1572, as it has been, in great measure, that of later editions.

During the sixteenth century Euclid was held in so high estimation among mathematicians, that no attempts appear to have been made to advance the science of Geometry, beyond the point at which he left it. Commentaries and translations seem to have been almost all they attempted.

Erasmus Reinhold wrote commentaries on Euclid's Elements and the great work of Copernicus. He was born at Salfeldt in Thuringia in 1511, and died in 1553, and was considered one of the most eminent mathematicians of his time. He wrote on Plane and Spherical Triangles, improved Muller's Tables of Sines and Tangents, and was the author of numerous other works on the sciences.

F. Maurolyco, of Messina, was born in 1494, and died in 1575; he made some improvements in Trigonometry, and edited the Spherics of Theodosius and Autolycus; he also published his "Emendatio et Restitutio Conicorum Apollonii Pergæi," in 1575.

Frederick Commandine was born at Urbino in 1509, and died in 1575: he is justly accounted one of the first geometers of his age. He composed several original works on the sciences; but is chiefly known for his translations of several of the Greek geometers, of whose works some had not previously been translated into Latin. He translated the geometrical writings of Archimedes, and wrote a commentary upon them; also four books of the Conics of Apollonius Pergæus, with the Lemmas of Pappus and the Commentaries of Eutocius, together with Serenus on Sections of the Cone and Cylinder. He was the first who translated the last six books of the Mathematical Collections of Pappus into Latin, which were published after his death, in 1588, by the munificence of the duke of Urbino. The edition most commonly met with is that of Manolessius, printed in 1660. translation of Pappus first directed the attention of mathematicians to the subjects of the lost works of the ancient geometers, and gave rise to various attempts for the restoration of several, which are described in the preface to the seventh book. Commandine's translation of Euclid with a commentary, was put forth in 1572. An Italian version was published under his direction in 1575. An English translation of the Latin version was published in 1715, by Dr John Kiel, at that time Savilian Professor of Astronomy at Oxford. Commandine also translated into Latin, an Arabic version of Euclid's tract on the Division of Surfaces.

This

To this period belongs John Dee, a man distinguished for his mathematical and astrological knowledge. He was educated at St John's College, Cambridge, where he chiefly devoted his attention to mathematical studies. He was made one of the fellows of Trinity College, at its foundation by Henry VIII., and afterwards, as we learn from Lilly's Memoirs, read lectures on Euclid at Rheims. He wrote a learned preface, of fifty folio pages, to Billingsley's Euclid: it bears the date of February 9, 1570, and was written at Mortlake. He translated into English the tract on Division of Surfaces, from the Latin version of Commandine.

To Henry Billingsley, a citizen of London, is due the merit of making the first English translation of Euclid's Elements of Geometry. It comprises the whole of the 13 books, with the 14th and 15th which were added by Hypsicles, and a 16th by Flussas. It was chiefly made from the Latin of Campanus, and contains a commentary, besides the preface, by John Dee, above-mentioned. It was first published in 1570 in a large folio volume; and a second edition was edited by Leeke and Serle in 1661.

Francis Vieta was born at Fontenoy in Lower Poitou in 1540, and was a man of original genius, as is manifest from his discoveries and improvements in different branches of the mathematical sciences. He introduced the use of letters into Algebra, and invented many theorems. He effected great improvements in Geometry, made considerable additions to the science of Trigonometry, and reduced it to a system. He wrote a treatise on Angular Sections, and restored the tract of Apollonius on Tangencies, which he published with the title of

Apollonius Gallus. His collected works were published at Leyden by Schooten.

Galileo Galilei, the cotemporary of Milton and friend of Kepler, was born at Pisa in Tuscany in 1564, and at a very early age, gave evidence of great genius for geometrical and philosophical pursuits. This became more evident while he was under the direction of Guido Ubaldi at the University of Pisa. From the time of Archimedes, a period of nearly 2000 years, little or nothing had been done in Mechanical Geometry, till Galileo first extended the bounds of that science by the application of Geometry to Motion. He first taught the true theory of uniformly accelerated and retarded motions, and of their composition, and proved that the spaces described by heavy bodies, falling freely from the beginning of their motion, are as the squares of the times. Contrary to the general belief, he maintained that all bodies, whether light or heavy, fall to the earth, through the same space in equal times; and attempted to verify the truth of his proposition by experiment. The two bodies, however, which he let fall from the top of the hanging tower of Pisa, did not reach the ground exactly at the same instant. The reason Galileo assigned was that the resistance of the air retarded the lighter body more than the heavier. He invented the cycloid, and the simple pendulum which he used in his astronomical experiments. The application of the pendulum to clocks was made by his son; and subsequently brought to perfection by Huygens. Galileo first proved that a body projected in any direction not perpendicular to the horizon describes a parabola; and it may be remarked that the Geometry of Galileo was wholly applied to explain and advance the science of Motion. The invention, in 1609, of the refracting telescope which still bears his name, disclosed new views of the solar system. By the aid of this he discovered that the moon is an opaque body with mountains and vallies, and that she receives her light by reflection from the sun. He also put forth the conjecture that the moon might be an inhabited world like the earth. He observed the different phases of the planet Venus, proving her motion round the sun. He discovered four of Jupiter's satellites, and caught an imperfect view of the ring of Saturn, which at the time of his observation appeared like two small stars, one on each side of the planet's disk. He was the first who discovered spots on the sun's disk, and from their varying position he inferred the motion of the sun on its axis. By his telescope he also discovered that the whiteness of the Milky Way is caused by innumerable stars apparently more close together than in the other parts of the heavens. His celebrated work, the "Dialogues on the Ptolemaic and Copernican Systems," was published at Florence, in 1632; and though dedicated to Ferdinand II., brought Galileo under the hatred of the Jesuits, and the power of the Inquisition. In June 1632, that court condemned him of heresy, for teaching that the sun is the centre of the solar system, and that the earth revolves on its own axis, and moves round the sun. He was obliged, on his knees, to abjure his belief of all he had advanced in his Dialogues, and to swear, that for the future, he would never assert or write any thing in favour of such heretical opinions. He was sentenced to imprisonment during the pleasure of the court; and for a certain time to recite daily the seven penitential psalms. Galileo is reported to have whispered in the ears of a friend, as he rose from his knees, "E pur se muove."

The influence of his powerful friends no doubt moderated the sentence of the Inquisition, whose proceedings in what they called heresy, were always of the most cruel, frequently of the most horrid description. Galileo was nevertheless kept strictly confined in the prison of the Inquisition for two years. Even when upwards of 70 years of age, new rigours were exercised against him, on account of some fresh suspicions of pope Urban VIII., which were inflamed by the philosopher's inveterate foes, the Jesuits. His health greatly suffered, and he was afterwards released from confinement; but became blind some years before his death. During this period he finished his dialogues on Motion. His death took place in 1642, at the age of 78. Most of the works of Galileo were collected and printed in 1656; they were translated into English by T. Salusbury, and published in 1661, in his Mathematical Collections. A more complete collection of his works was published at Milan, in 1811. At Florence, in 1674, was published a work of Galileo, under the title of " Quinto Libro de gli Elementi d'Euclidi, &c.," by Vincenzio Viviani, one of his distinguished pupils.

of

Christopher Clavius was born at Bamberg in Germany, in 1537, and died at Rome, in 1612: his writings on the mathematical sciences were collected and printed at Mayence, in five folio volume, in 1612: he superintended the reformation of the Calendar under the direction pope Gregory XIII. He was skilled in the ancient Geometry, and edited several of the Greek mathematical writings, and on some of them he wrote commentaries, among which may be mentioned the Elements of Euclid. He was the author of a work on Practical Geometry, and of a commentary on Sacro Bosco's treatise on the Sphere, which works were first published in 1570. He was a zealous cultivator of the sciences; but no discoveries or improvements are attributed to him.

Willebrod Snell was a man of original genius: he was born at Leyden, in 1591, and died in 1626. To him is attributed by Huygens the discovery of the law of the refraction of light, before it was made known by Descartes. He was a skilful geometer, and published in 1608, with the title of Apollonius Batavus, his attempted restoration of three tracts of Apollonius "De Sectione Determinata," "De Sectione Rationis," and "De Sectione Spatii." The tract "De Sectione Determinata" was translated into English by the Rev. J. Lawson, and published in 1772. This tract is imperfectly restored, as there is omitted the distinction of the situation of the points; and there is no complete exposition of the determinations. He also wrote a tract on the Circle; in which are given various approximations, both geometrical and arithmetical. Ludolph Van Ceulen is also noted for having calculated the ratio of the diameter to the circumference of the circle, to thirty-five places of decimals.

Sir Henry Savile, an accomplished scholar, founded two professorships at Oxford; one of Geometry, and the other of Astronomy, in 1619. As the first Savilian professor of Geometry, he delivered thirteen lectures on the first book of Euclid's Elements of Geometry, in 1620, which he published in Latin during the following year. In 1585 he appointed Warden of Merton College, and Provost of Eton in 1596; the former appointment he held for thirty-six years. In 1613 he published the works of Chrysostom in Greek, and was the author and editor of several other works.

was

Leonard Digges was a mathematician of some note, and the author of a treatise on Geometry, in three books, which he called Pantometria. It was published by his son in 1591, with a supplementary book on the five regular solids.

The ancient Geometry of the Greeks was considered perfect, as indeed within the bounds of its legitimate province it is; and no attempts were made to improve or extend the methods handed down from the ancients till the time of Kepler. He was born at Wiel, in 1571, and died in 1630: he introduced the new principle of infinity into Geometry. He conceived a circle to be made up of an infinite number of infinitely small triangles with their vertices in the centre, and their bases coinciding with the circumference of the circle. A cone, in the same manner, was supposed to consist of an infinite number of indefinitely small pyramids. This idea of Kepler lies at the foundation of the higher analysis. He published his views in a work entitled "Nova Stereometria," in 1615, which have been discussed under the names of "Infinitesimals;" "Fluxions and Fluents;" "Differential and Integral Calculus."

Pierre de Fermat was born in 1595, and died in 1665; though a person of extraordinary vanity, he was a mathematician of original genius. He attempted the restoration of the two books of Apollonius on Plane Loci: he has given the synthesis, but omitted the analysis of the propositions; he has also omitted the distinction of cases of each proposition, and has not ascertained the determinations. He wrote a treatise on Spherical Tangencies, in which he has demonstrated, in the case of spheres, properties analogous to those which Vieta had before demonstrated in his restoration of Apollonius on Tangencies. He was the author of a treatise on Geometric Loci, both plane and solid. Fermat had acquired some general notion of the Porisms of Euclid. He was the author of a method of maxima and minima, and of the quadrature of parabolas of all orders, besides several discoveries in the properties of numbers, one of which still bears the name of Fermat's theorem. His collected works were, after his death, published at Toulouse, in 1679.

Descartes was born in 1596, and died at Stockholm in 1650. He was the cotemporary of Galileo, Fermat, Roberval, and many other celebrated mathematicians. He has been cited as the inventor of the New Geometry, or, as it is called, Analytical Geometry; the foundation of which, however, had certainly been laid before his time. Algebra had been applied to Geometry by Vieta, and, to some extent, by other mathematicians. But though he did not originate, he certainly extended the limits and powers of Analytical Geometry, by the discovery of a new principle. The use of co-ordinates appears for the first time, though under a different name, in the second book of his Geometry, which was published in 1637. He first taught the method of expressing curves by equations. The simple conception of expressing curve lines by means of equations between the two variable co-ordinates of a point in the curve; and curve surfaces by means of equations which contain the three co-ordinates of any variable point in the surface, has led to a new science, and entitles the discoverer to be classed amongst men of profound genius. This discovery and its applications by means of the higher analysis, have given a power to Geometry before unknown. In the Epistles of Descartes, which were printed in 1683, he