adopted in England, but Briggs underlined the decimal figures, and would have printed a number such as 25-379 in the form 25379. Subsequent writers added another line, and would have written it as 25379; nor was it till the beginning of the eighteenth century that the current notation was generally employed, and even yet the notation varies slightly in different countries. A knowledge of the decimal notation became general among practical men with the introduction of the French decimal standards.

CHAPTER XII.

THE MATHEMATICS OF THE RENAISSANCE.1

CIRC. 1450-1637.

THE last chapter is a digression from the chronological arrangement to which, as far as possible, I have throughout adhered, but I trust by references in this chapter to keep the order of events and discoveries clear. I return now to the general history of mathematics in western Europe. Mathematicians had barely assimilated the knowledge obtained from the Arabs, including their translations of Greek writers, when the refugees who escaped from Constantinople after the fall of the eastern empire brought the original works and the traditions of Greek science into Italy. Thus by the middle of the fifteenth century the chief results of Greek and Arabian mathematics were accessible to European students.

The invention of printing about that time rendered the dissemination of discoveries comparatively easy. It is almost a truism to remark that until printing was introduced a writer appealed to a very limited class of readers, but we are perhaps apt to forget that when a medieval writer "published" a work the results were known to only a few of his contemporaries.

1 Where no other references are given, see parts xii, xiii, xiv, and the early chapters of part xv of Cantor's Vorlesungen; on the Italian mathematicians of this period see also G. Libri, Histoire des sciences mathématiques en Italie, 4 vols., Paris, 1838-1841.

This had not been the case in classical times, for then and until the fourth century of our era Alexandria was the recognized centre for the reception and dissemination of new works and discoveries. In medieval Europe, on the other hand, there was no common centre through which men of science could communicate with one another, and to this cause the slow and fitful development of medieval mathematics may be partly ascribed.

The introduction of printing marks the beginning of the modern world in science as in politics; for it was contemporaneous with the assimilation by the indigenous European school (which was born from scholasticism, and whose history was traced in chapter VIII) of the results of the Indian and Arabian schools (whose history and influence were traced in chapters IX and x), and of the Greek schools (whose history was traced in chapters II to v).

The last two centuries of this period of our history, which may be described as the renaissance, were distinguished by great mental activity in all branches of learning. The creation of a fresh group of universities (including those in Scotland), of a somewhat less complex type than the medieval universities above described, testify to the general desire for knowledge. The discovery of America in 1492 and the discussions that preceded the Reformation flooded Europe with new ideas which, by the invention of printing, were widely disseminated; but the advance in mathematics was at least as well marked as that in literature and that in politics.

During the first part of this time the attention of mathematicians was to a large extent concentrated on syncopated algebra and trigonometry; the treatment of these subjects is discussed in the first section of this chapter, but the relative importance of the mathematicians of this period is not very easy to determine. The middle years of the renaissance were distinguished by the development of symbolic algebra: this is treated in the second section of this chapter. The close of the sixteenth century saw the creation of the science of dynamics:

this forms the subject of the first section of chapter XIII. About the same time and in the early years of the seventeenth century considerable attention was paid to pure geometry: this forms the subject of the second section of chapter XIII.

The development of syncopated algebra and trigonometry.

Regiomontanus.1 Amongst the many distinguished writers of this time Johann Regiomontanus was the earliest and one of the most able. He was born at Königsberg on June 6, 1436, and died at Rome on July 6, 1476. His real name was Johannes Müller, but, following the custom of that time, he issued his publications under a Latin pseudonym which in his case was taken from his birthplace. To his friends, his neighbours, and his tradespeople he may have been Johannes Müller, but the literary and scientific world knew him as Regiomontanus, just as they knew Zepernik as Copernicus, and Schwarzerd as Melanchthon. It seems as pedantic as it is confusing to refer to an author by his actual name when he is universally recognized under another: I shall therefore in all cases as far as possible use that title only, whether latinized or not, by which a writer is generally known.

Regiomontanus studied mathematics at the university of Vienna, then one of the chief centres of mathematical studies in Europe, under Purbach who was professor there. His first work, done in conjunction with Purbach, consisted of an analysis of the Almagest. In this the trigonometrical functions sine and cosine were used and a table of natural sines was introduced. Purbach died before the book was finished: it was finally published at Venice, but not till 1496. As soon as this was completed Regiomontanus wrote a work on astrology,

His life was written by P. Gassendi, The Hague, second edition, 1655. His letters, which afford much valuable information on the mathematics of his time, were collected and edited by C. G. von Murr, Nuremberg, 1786. An account of his works will be found in Regiomontanus, ein geistiger Vorläufer des Copernicus, by A. Ziegler, Dresden, 1874; see also Cantor, chap. lv.

which contains some astronomical tables and a table of natural tangents this was published in 1490.

Leaving Vienna in 1462, Regiomontanus travelled for some time in Italy and Germany; and at last in 1471 settled for a few years at Nuremberg, where he established an observatory, opened a printing-press, and probably lectured. Three tracts on astronomy by him were written here. A mechanical eagle, which flapped its wings and saluted the Emperor Maximilian I. on his entry into the city, bears witness to his mechanical ingenuity, and was reckoned among the marvels of the age. Thence Regiomontanus moved to Rome on an invitation from Sixtus IV. who wished him to reform the calendar. He was assassinated,

shortly after his arrival, at the age of 40.

Regiomontanus was among the first to take advantage of the recovery of the original texts of the Greek mathematical works in order to make himself acquainted with the methods of reasoning and results there used; the earliest notice in modern Europe of the algebra of Diophantus is a remark of his that he had seen a copy of it at the Vatican. He was also well read in the works of the Arab mathematicians.

The fruit of his study was shewn in his De Triangulis written in 1464. This is the earliest modern systematic exposition of trigonometry, plane and spherical, though the only trigonometrical functions introduced are those of the sine and cosine. It is divided into five books. The first four are given up to plane trigonometry, and in particular to determining triangles from three given conditions. The fifth book is devoted to spherical trigonometry. The work was printed at Nuremberg in 1533, nearly a century after the death of Regiomontanus.

As an example of the mathematics of this time I quote one of his propositions at length. It is required to determine a triangle when the difference of two sides, the perpendicular on the base, and the difference between the segments into which the base is thus divided are given [book 11, prop. 23]. The following is the solution given by Regiomontanus.