metic of Grammateus, a teacher at the University of Vienna. His pupil, Christoff Rudolff, the writer of the first text-book on algebra in the German language (printed in 1525), employs these symbols also. So did Stifel, who brought out a second edition of Rudolff's Coss in 1553. Thus, by slow degrees, their adoption became universal. There is another short-hand symbol of which we owe the origin to the Germans. In a manuscript published sometime in the fifteenth century, a dot placed before a number is made to signify the extraction of a root of that number. This dot is the embryo of our present symbol for the square root. Christoff Rudolff, in his algebra, remarks that "the radix quadrata is, for brevity, designated in his algorithm with the character √, as √4." Here the dot has grown into a symbol much like our own. symbol was used by Michael Stifel. Our sign of equality is due to Robert Recorde (1510–1558), the author of The Whetstone of Witte (1557), which is the first English treatise on algebra. He selected this symbol because no two things could be more equal than two parallel lines. The sign for division was first used by Johann Heinrich Rahn, a Swiss, in 1659, and was introduced in England by John Pell in 1668. This same Michael Stifel (1486 ?-1567), the greatest German algebraist of the sixteenth century, was born in Esslingen, and died in Jena. He was educated in the monastery of his native place, and afterwards became Protestant minister. The study of the significance of mystic numbers in Revelation and in Daniel drew him to mathematics. He studied German and Italian works, and published in 1544, in Latin, a book entitled Arithmetica integra. Melanchthon wrote a preface to it. Its three parts treat respectively of rational numbers, irrational numbers, and algebra. Stifel gives a table containing the numerical values of the binomial coefficients for powers below the 18th. He observes an advantage in letting a geometric progres sion correspond to an arithmetical progression, and arrives at the designation of integral powers by numbers. Here are the germs of the theory of exponents. In 1545 Stifel published an arithmetic in German. His edition of Rudolff's Coss contains rules for solving cubic equations, derived from the the writings of Cardan. We remarked above that Vieta discarded negative roots of equations. Indeed, we find few algebraists before and during the Renaissance who understood the significance even of negative quantities. Fibonacci seldom uses them. Pacioli states the rule that "minus times minus gives plus," but applies it really only to the development of the product of (ab) (cd); purely negative quantities do not appear in his work. The great German "Cossist" (algebraist), Michael Stifel, speaks as early as 1544 of numbers which are "absurd ” or "fictitious below zero," and which arise when "real numbers above zero" are subtracted from zero. Cardan, at last, speaks of a "pure minus"; "but these ideas," says Hankel, "remained sparsely, and until the beginning of the seventeenth century, mathematicians dealt exclusively with absolute positive quantities." The first algebraist who occasionally places a purely negative quantity by itself on one side of an equation, is Harriot in England. As regards the recognition of negative roots, Cardan and Bombelli were far in advance of all writers of the Renaissance, including Vieta. Yet even they mentioned. these so-called false or fictitious roots only in passing, and without grasping their real significance and importance. On this subject Cardan and Bombelli had advanced to about the same point as had the Hindoo Bhaskara, who saw negative roots, but did not approve of them. The generalisation of the conception of quantity so as to include the negative, was an exceedingly slow and difficult process in the development of algebra. We shall now consider the history of geometry during the Renaissance. Unlike algebra, it made hardly any progress. The greatest gain was a more intimate knowledge of Greek geometry. No essential progress was made before the time of Descartes. Regiomontanus, Xylander of Augsburg, Tartaglia, Commandinus of Urbino in Italy, Maurolycus, and others, made translations of geometrical works from the Greek. John Werner of Nürnberg published in 1522 the first work on conics which appeared in Christian Europe. Unlike the geometers of old, he studied the sections in relation with the cone, and derived their properties directly from it. This mode of studying the conics was followed by Maurolycus of Messina (1494-1575). The latter is, doubtless, the greatest geometer of the sixteenth century. From the notes of Pappus, he attempted to restore the missing fifth book of Apollonius on maxima and minima. His chief work is his masterly and original treatment of the conic sections, wherein he discusses tangents and asymptotes more fully than Apollonius had done, and applies them to various physical and astronomical problems. The foremost geometrician of Portugal was Nonius; of France, before Vieta, was Peter Ramus, who perished in the massacre of St. Bartholomew. Vieta possessed great familiarity with ancient geometry. The new form which he gave to algebra, by representing general quantities by letters, enabled him to point out more easily how the construction of the roots of cubics depended upon the celebrated ancient problems of the duplication of the cube and the trisection of an angle. He reached the interesting conclusion that the former problem includes the solutions of all cubics in which the radical in Tartaglia's formula is real, but that the latter problem includes only those leading to the irreducible case. The problem of the quadrature of the circle was revived in П this age, and was zealously studied even by men of eminence and mathematical ability. The army of circle-squarers became most formidable during the seventeenth century. Among the first to revive this problem was the German Cardinal Nicolaus Cusanus (died 1464), who had the reputation of being a great logician. His fallacies were exposed to full view by Regiomontanus. As in this case, so in others, every quadrator of note raised up an opposing mathematician: Orontius was met by Buteo and Nonius; Joseph Scaliger by Vieta, Adrianus Romanus, and Clavius; A. Quercu by Peter Metius. Two mathematicians of Netherlands, Adrianus Romanus and Ludolph van Ceulen, occupied themselves with approximating to the ratio between the circumference and the diameter. The former carried the value to 15, the latter to 35, places. The value of is therefore often named "Ludolph's number." His performance was considered so extraordinary, that the numbers were cut on his tomb-stone in St. Peter's church-yard, at Leyden. Romanus was the one who propounded for solution that equation of the forty-fifth degree solved by Vieta. On receiving Vieta's solution, he at once departed for Paris, to make his acquaintance with so great a master. Vieta proposed to him the Apollonian problem, to draw a circle touching three given circles. "Adrianus Romanus solved the problem by the intersection of two hyperbolas; but this solution did not possess the rigour of the ancient geometry. Vieta caused him to see this, and then, in his turn, presented a solution which had all the rigour desirable." 25 Romanus did much toward simplifying spherical trigonometry by reducing, by means of certain projections, the 28 cases in triangles then considered to only six. Mention must here be made of the improvements of the Julian calendar. The yearly determination of the movable feasts had for a long time been connected with an untold amount of confusion. The rapid progress of astronomy led to the consideration of this subject, and many new calendars were proposed. Pope Gregory XIII. convoked a large number of mathematicians, astronomers, and prelates, who decided upon the adoption of the calendar proposed by the Jesuit Lilius Clavius. To rectify the errors of the Julian calendar it was agreed to write in the new calendar the 15th of October immediately after the 4th of October of the year 1582. The Gregorian calendar met with a great deal of opposition both among scientists and among Protestants. Clavius, who ranked high as a geometer, met the objections of the former most ably and effectively; the prejudices of the latter passed away with time. He The passion for the study of mystical properties of numbers descended from the ancients to the moderns. Much was written on numerical mysticism even by such eminent men as Pacioli and Stifel. The Numerorum Mysteria of Peter Bungus covered 700 quarto pages. He worked with great industry and satisfaction on 666, which is the number of the beast in Revelation (xiii. 18), the symbol of Antichrist. reduced the name of the 'impious' Martin Luther to a form which may express this formidable number. Placing a = 1, b = 2, etc., k = 10, 20, etc., he finds, after misspelling the name, that M(30) A (1) R (80) T (100) I (9) N(40) L(20) V (200) T (100) E(5) R (80) A (1) constitutes the number required. These attacks on the great reformer were not unprovoked, for his friend, Michael Stifel, the most acute and original of the early mathematicians of Germany, exercised an equal ingenuity in showing that the above number referred to Pope Leo X., -a demonstration which gave Stifel unspeakable comfort.2 Astrology also was still a favourite study. It is well known that Cardan, Maurolycus, Regiomontanus, and many other eminent scientists who lived at a period even later than |