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rapid progress in algebra and allied branches of mathematics. Of the three great inventions which have so enormously diminished the labour of numerical calculations, one, the so-called Arabic notation (chapter III., § 64), was already familiar, the other two (decimal fractions and logarithms) were suggested in the 16th century and were in working order early in the 17th century.

97. The first important set of observations taken after the death of Regiomontanus and Walther (chapter III., § 68) were due to the energy of the Landgrave William IV. of Hesse (1532-1592). He was remarkable as a boy for his love of study, and is reported to have had his interest in astronomy created or stimulated when he was little more than 20 by a copy of Apian’s beautiful Astronomicum Caesareum, the cardboard models in which he caused to be imitated and developed in metal-work. He went on with the subject seriously, and in 1561 had an observatory built at Cassel, which was remarkable as being the first which had a revolving roof, a device now almost universal. In this he made extensive observations (chiefly of fixed stars) during the next six years. The death of his father then compelled him to devote most of his energy to the duties of government, and his astronomical ardour abated. A few years later, however (1575), as the result of a short visit from the talented and enthusiastic young Danish astronomer Tycho Brahe (§ 99), he renewed his astronomical work, and secured shortly afterwards the services of two extremely able assistants, Christian Rothmann (in 1577) and Joost Bürgi (in 1579). Rothmann, of whose life extremely little is known, appears to have been a mathematician and theoretical astronomer of considerable ability, and was the author of several improvements in methods of dealing with various astronomical problems. He was at first a Coppernican, but shewed his independence by calling attention to the needless complication introduced by Coppernicus in resolving the motion of the earth into three motions when two sufficed (chapter IV., § 79). His faith in the system was, however, subsequently shaken by the errors which observation revealed in the Prussian Tables. Bürgi (1552-1632) was originally engaged by the Landgrave as a clockmaker, but his remarkable mechanical talents were soon turned to astronomical account, and it then appeared that he also possessed unusual ability as a mathematician.59

98. The chief work of the Cassel Observatory was the formation of a star catalogue. The positions of stars were compared with that of the sun, Venus or Jupiter being used as connecting links, and their positions relatively to the equator and the first point of Aries (♈) deduced; allowance was regularly made for the errors due to the refraction of light by the atmosphere, as well as for the parallax of the sun, but the most notable new departure was the use of a clock to record the time of observations and to measure the motion of the celestial sphere. The construction of clocks of sufficient accuracy for the purpose was rendered possible by the mechanical genius of Bürgi, and in particular by his discovery that a clock could be regulated by a pendulum, a discovery which he appears to have taken no steps to publish, and which had in consequence to be made again independently before it received general recognition.60 By 1586 121 stars had been carefully observed, but a more extensive catalogue which was to have contained more than a thousand stars was never finished, owing to the unexpected disappearance of Rothmann in 159061 and the death of the Landgrave two years later.

99. The work of the Cassel Observatory was, however, overshadowed by that carried out nearly at the same time by Tycho (Tyge) Brahe. He was born in 1546 at Knudstrup in the Danish province of Scania (now the southern extremity of Sweden), being the eldest child of a nobleman who was afterwards governor of Helsingborg Castle. He was adopted as an infant by an uncle, and brought up at his country estate. When only 13 he went to the University of Copenhagen, where he began to study rhetoric and philosophy, with a view to a political career. He was, however, very much interested by a small eclipse of the sun which he saw in 1560, and this stimulus, added to some taste for the astrological art of casting horoscopes, led him to devote the greater part of the remaining two years spent at Copenhagen to mathematics and astronomy. In 1562 he went on to the University of Leipzig, accompanied, according to the custom of the time, by a tutor, who appears to have made persevering but unsuccessful attempts to induce his pupil to devote himself to law. Tycho, however, was now as always a difficult person to divert from his purpose, and went on steadily with his astronomy. In 1563 he made his first recorded observation, of a close approach of Jupiter and Saturn, the time of which he noticed to be predicted a whole month wrong by the Alfonsine Tables (chapter III., § 66), while the Prussian Tables (§ 94) were several days in error. While at Leipzig he bought also a few rough instruments, and anticipated one of the great improvements afterwards carried out systematically, by trying to estimate and to allow for the errors of his instruments.

In 1565 Tycho returned to Copenhagen, probably on account of the war with Sweden which had just broken out, and stayed about a year, during the course of which he lost his uncle. He then set out again (1566) on his travels, and visited Wittenberg, Rostock, Basle, Ingolstadt, Augsburg, and other centres of learning, thus making acquaintance with several of the most notable astronomers of Germany. At Augsburg he met the brothers Hainzel, rich citizens with a taste for science, for one of whom he designed and had constructed an enormous quadrant (quarter-circle) with a radius of about 19 feet, the rim of which was graduated to single minutes; and he began also here the construction of his great celestial globe, five feet in diameter, on which he marked one by one the positions of the stars as he afterwards observed them.

In 1570 Tycho returned to his father at Helsingborg, and soon after the death of the latter (1571) went for a long visit to Steen Bille, an uncle with scientific tastes. During this visit he seems to have devoted most of his time to chemistry (or perhaps rather to alchemy), and his astronomical studies fell into abeyance for a time.

100. His interest in astronomy was fortunately revived by the sudden appearance, in November 1572, of a brilliant new star in the constellation Cassiopeia. Of this Tycho took a number of extremely careful observations; he noted the gradual changes in its brilliancy from its first appearance, when it rivalled Venus at her brightest, down to its final disappearance 16 months later. He repeatedly measured its angular distance from the chief stars in Cassiopeia, and applied a variety of methods to ascertain whether it had any perceptible parallax (chapter II., §§ 43, 49). No parallax could be definitely detected, and he deduced accordingly that the star must certainly be farther off than the moon; as moreover it had no share in the planetary motions, he inferred that it must belong to the region of the fixed stars. To us of to-day this result may appear fairly commonplace, but most astronomers of the time held so firmly to Aristotle’s doctrine that the heavens generally, and the region of the fixed stars in particular, were incorruptible and unchangeable, that new stars were, like comets, almost universally ascribed to the higher regions of our own atmosphere. Tycho wrote an account of the new star, which he was ultimately induced by his friends to publish (1573), together with some portions of a calendar for that year which he had prepared. His reluctance to publish appears to have been due in great part to a belief that it was unworthy of the dignity of a Danish nobleman to write books! The book in question (De Nova ... Stella) compares very favourably with the numerous other writings which the star called forth, though it shews that Tycho held the common beliefs that comets were in our atmosphere, and that the planets were carried round by solid crystalline spheres, two delusions which his subsequent work did much to destroy. He also dealt at some length with the astrological importance of the star, and the great events which it foreshadowed, utterances on which Kepler subsequently made the very sensible criticism that “if that star did nothing else, at least it announced and produced a great astronomer.”

In 1574 Tycho was requested to give some astronomical lectures at the University of Copenhagen, the first of which, dealing largely with astrology, was printed in 1610, after his death. When these were finished, he set off again on his travels (1575). After a short visit to Cassel (§ 97), during which he laid the foundation of a lifelong friendship with the Landgrave, he went on to Frankfort to buy books, thence to Basle (where he had serious thoughts of settling) and on to Venice, then back to Augsburg and to Regensburg, where he obtained a copy of the Commentariolus of Coppernicus (chapter IV., § 73), and finally came home by way of Saalfeld and Wittenberg.

101. The next year (1576) was the beginning of a new epoch in Tycho’s career. The King of Denmark, Frederick II., who was a zealous patron of science and literature, determined to provide Tycho with endowments sufficient to enable him to carry out his astronomical work in the most effective way. He accordingly gave him for occupation the little island of Hveen in the Sound (now belonging to Sweden), promised money for building a house and observatory, and supplemented the income derived from the rents of the island by an annual payment of about £100. Tycho paid his first visit to the island in May, soon set to work building, and had already begun to make regular observations in his new house before the end of the year.

Fig. 51.—Uraniborg. From a collection of letters published by Tycho.


The buildings were as remarkable for their magnificence as for their scientific utility. Tycho never forgot that he was a Danish nobleman as well as an astronomer, and built in a manner suitable to his rank.62 His chief building (fig. 51), called Uraniborg (the Castle of the Heavens), was in the middle of a large square enclosure, laid out as a garden, the corners of which pointed North, East, South, and West, and contained several observatories, a library and laboratory, in addition to living rooms. Subsequently, when the number of pupils and assistants who came to him had increased, he erected (1584) a second building, Stjerneborg (Star Castle), which was remarkable for having underground observatories. The convenience of being able to carry out all necessary work on his own premises induced him moreover to establish workshops, where nearly all his instruments were made, and afterwards also a printing press and paper mill. Both at Uraniborg and Stjerneborg not only the rooms, but even the instruments which were gradually constructed, were elaborately painted or otherwise ornamented.

102. The expenses of the establishment must have been enormous, particularly as Tycho lived in magnificent style and probably paid little attention to economy. His income was derived from various sources, and fluctuated from time to time, as the King did not merely make him a fixed annual payment, but added also temporary grants of lands

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