Tycho Brahe: a picture of scientific life and work in the sixteenth century/Chapter 12

by John Louis Emil Dreyer

 

CHAPTER XII.

Conclusion.

TYCHO BRAHE'S SCIENTIFIC ACHIEVEMENTS.

Among the most important instruments in use at Alexandria were the so-called spheres or armillæ. These are said to have been used in China at an early date,^[1] but the invention has doubtless been made independently by the Greek astronomers. They were probably known at the time of Timocharis and Aristillus (about 300 B.C.), and it is certain that Hipparchus employed them. In the complicated form used by him and his successors (called by Tycho "armillæ zodiacales") the instrument consisted of six circles, of which the largest represented the meridian, and was carefully placed in position on a solid stand. On the inner rim it was furnished with two pivots, representing the north and south poles, on which turned a slightly smaller circle, the solstitial colure, to which was fixed immovably and at right angles another of the same size, representing the ecliptic. The colure was furnished with pivots representing the poles of the ecliptic, and on these turned two circles, one larger than the colure, another smaller than it, while the latter enclosed a sixth circle, which could slide inside it, and was furnished with two sights diametrically opposite to each other. With this instrument the difference of longitude and latitude could be measured, the circles being divided to one-sixth degree, while half that quantity could be estimated.

The Arabs constructed similar instruments, and already Mashallah, who lived about the year 775 (before the time of Al Mamun), wrote about astrolabes and armillæ, and these were used by Ibn Yunis, Abul Wefa, and others.^[2]

Gemma's Astronomical Ring.

Alhazen also made use of armillæ for his investigations on refraction, and it has even been assumed that he is the inventor of the far simpler equatorial armillæ, which are generally ascribed to Tycho Brahe, who also considers himself as their inventor.^[3] But in any case, it is certain that equatorial armillæ were not known in Europe; that Walther, the principal observer before Tycho, only knew zodiacal armillæ; and that the principle of equatorial ones was first described in 1534 by Gemma Frisius, who, however, only designed an instrument of very small dimensions, intended to be held in the hand.^[4] Tycho remarks that the instruments of the ancients were of solid metal, and as they had to be very large to allow spaces of 10′ to be marked on them, they must have been very cumbersome;^[5] and it is deserving of particular notice that he has an open eye to the importance of perfect symmetry in the instrument. He points out that the poles of the ecliptic at different times occupy different positions with regard to the meridian, and that the instrument therefore must be subject to severe strains, which would seriously affect the accuracy of the observations, even if the circles are of moderate dimensions and not too heavy. For the same reasons he rejected the clumsy "Torquetum" of Regiomontanus, which had never been much used.^[6]

Although Tycho possessed a zodiacal instrument which had the advantage of consisting only of four circles, he chiefly made use of equatorial armillæ, which instruments represent a great step forward, on account of their comparative simplicity and perfect symmetry. He constructed three instruments of this kind, which are all figured in his Mechanica. The first one, which was mounted in the small northern observatory of Uraniborg, consisted of three circles of steel, of which the meridian and the equator were firmly joined together, and both the equator and the movable declination circle were furnished with sights (made of brass), which could be moved along the circles, and to which the observer applied his eye, while a small cylinder in the centre of, and perpendicular to, the polar axis served as objective sight. The second instrument was placed in the small southern observatory, and only differed from the first one by the equator being movable and attached to a revolving (but not graduated) declination circle, while a smaller and graduated declination circle carried sights. The undivided circle might very well have been left out and the graduated one fixed to the equator. The outer circle (meridian) was nearly five feet in diameter.^[7]

The third and most important instrument of this kind was mounted in the largest crypt of Stjerneborg and was far more extensively used by Tycho, who considered it one of his most accurate instruments. It consisted merely of a declination circle 91/2 feet in diameter, and a semicircle, which represented the part of the equator below the horizon, and rested on eight stone piers. The former has two pointers turning round a small cylinder in the centre of the polar axis, and perpendicular to the plane of the circle, and each furnished with an eye-piece sight, while a third sight slides along the equator. The polar axis (of iron, but hollow) could be adjusted in inclination and azimuth by screws, which acted on a square plate in a hole in which the lower pointed end of the axis fitted. By reversing the circle double observations of declination might be taken, using first one sight and then the other, and Tycho remarks that this instrument had the further advantage over the two others that stars near the equator were as easily observed as those more distant from it, as the equatorial arc was at some distance behind the declination circle.

Circles and semicircles had naturally been in use from a very early date. We need only refer to the astrolabium of Ptolemy, which consisted of a graduated circle inside which another circle could slide, carrying two small cylinders diametrically opposite to each other, while the instrument was kept vertical by a plumb-line. This astrolabium was imitated by many successive astronomers; among others, by Abul Wefa, who has described a meridian circle for observing the sun,^[8]

Armillæ a Equatoriæ Maximæ.

while smaller circles became extensively used, particularly by navigators. Nonius suggested attaching the pointer or alidade to some point on the circumference instead of to the centre, as the divisions on this plan might be made twice as large as usual.^[9] I only mention this proposal because Tycho (who does not allude to Nonius) constructed a large semicircle revolving round a vertical axis, with a long ruler turning on a pivot at one end of the horizontal diameter.

In addition to complete circles, quadrants were also used long before Tycho's time, though not extensively. Ptolemy describes a meridian quadrant attached to a cube of stone or wood, with a small cylinder in the centre of the arc, of which the shadow indicated the altitude of the sun on the graduation.^[10] Among the numerous instruments which Nasir al-din Tusi erected in the splendid observatory at Meragha, in the north-west of Persia (about A.D. 1260), was a Ptolemean mural quadrant, made of hard wood, and with a radius of about twelve feet. The limb was of copper, on which three arcs were drawn; the middle one was divided into degrees, while of the others, one showed every fifth degree, the other the minutes (all of them?). Nasir al-din mentions the instrument of Ptolemy, which evidently had served him as a model.^[11]

Tycho Brahe is therefore not (as supposed by some writers) the inventor of the mural quadrant, an instrument which up to the end of the eighteenth century was the most important one in astronomical observatories. Of course he cannot have known anything of the Arabian quadrants, but the description of Ptolemy must have been familiar to him. The advantage of meridian observations for many purposes was also well known before his time, particularly for finding the declination of the sun, which gave its place in the zodiac by the tables. Hagecius had even observed the altitude and time of transit of the new star over the meridian,^[12] but nobody had erected an instrument permanently in the meridian. The great superiority as to stability which a mural quadrant possessed over the armillæ did not escape Tycho; and as he was the first thoroughly to perceive the influence of refraction in altering the apparent positions of stars, the wish naturally arose to observe the stars at their greatest altitude on the meridian, where that influence was smallest.^[13]

From the meridian quadrant to quadrants which could be placed in any azimuth the transition was simple enough, and we find accordingly among the instruments at Meragah an "instrument des quarts de circles mobiles." This consisted of an azimuth circle on which were two quadrants turning on a common vertical axis, by which two observers could find the altitudes and difference of azimuth of two objects.^[14] In Europe an azimuth instrument seems to have been first used by Landgrave Wilhelm IV., who observed altitudes and azimuths of the new star of 1572, apparently by setting the instrument to a certain whole or half degree of azimuth, and measuring the altitude when the star reached that azimuth.^[15] Quadrants capable of being turned round a vertical axis had been known long before,^[16] but as it was so much easier to graduate a straight line than an arc, the triquetrum continued to be the favourite instrument for measuring altitudes down to the end of the sixteenth century. Tycho did not think much of this instrument, which he calls "instrumentum parallacticum sive regularum," and he did not make much use of the two he had constructed, and one of which was of large dimensions, and furnished with an azimuth circle 16 feet in diameter.^[17] He preferred the "quadrans azimuthalis," and constructed four instruments of this kind, which were extensively used, though chiefly for merely observing altitudes, while the azimuths were rarely taken, especially during his later years. The largest quadrant (quadrans magnus chalibeus) was enclosed in a square (also of steel), of which the side was equal to the radius of the quadrant. Two of the sides were graduated, and the alidade pointed to these graduations as well as to those on the arc, so that the instrument was a combination of a quadrant and the "quadratum geometricum" of Purbach (which the Arabians had also known), which increased the solidity of the instrument.

An important use to which the quadrants were put at Uraniborg was the determination of time. At Alexandria the beginning, middle, or end of the hour was generally the only indication of time which accompanied the observations of planets, which was perhaps sufficient, owing to the limited accuracy of the observations. The time was found by water- or sand-clocks, which were verified by observing the culmination of some of the forty-four stars which Hipparchus had selected so well that the time could be determined with an error not much exceeding a minute.^[18] An important step forward as regards the accurate determination of time was made by the Arabs in the ninth century. Ibn Yunis mentions a solar eclipse observed at Bagdad on the 30th November 829 by Ahmed Ibn Abdallah, called Habash, who at the beginning of the eclipse found the altitude of the sun to be 7°, while at the end the altitude was 24°. This seems to have been the earliest though crude attempt to use observations of altitude to indicate time, but the advantage of the method was evident, and at the lunar eclipse on the 12th August 854 the altitude of Aldebaran was measured equal to 45° 30′. Ibn Yunis adds that he from this made out the hour-angle to be 44° by means of a planisphere. Ibn Yunis communicates a number of other instances from the tenth century,^[19] but the instruments used were very small, and only divided into degrees; and though Al Battani gave formulæ for the computation of the hour-angle, the Arabians generally contented themselves with the approximate graphical determination by the so-called astrolabe or planisphere.

In Europe the use of observations of altitude for determining time was introduced in 1457 by Purbach, who, at the beginning and end of the lunar eclipse on the 3rd September, measured the altitude of "penultima ex Plejadibus."^[20] Bernhard Walther was the first to introduce in observatories the use of clocks driven by weights. Thus we find among his observations one of the rising of Mercury. At the time of rising he attached the weight to a clock of which the hour-wheel had fifty-six teeth, and as one hour and thirty-five teeth passed before the sun rose, he concluded that the interval had been one hour thirty-seven minutes. Walther adds that this clock was a very good one, and indicated correctly the interval between two successive noons; but all the same he must have seen how unreliable it was, for though he used the clock during the lunar eclipse in 1487, he at the same time measured some altitudes.^[21]

In Tycho Brahe's observatory the clocks never played an important part. Though he possessed three or four clocks, he does not anywhere describe them in detail, while he in several places remarks that he did not depend on them, as their rate varied considerably even during short intervals, which he attributed to atmospherical changes (although he kept them in heated rooms in winter), as well as to imperfections in the wheels. At the side of the mural quadrant he had placed two clocks, indicating both minutes and seconds, in order that one might control the other, and in the southern observatory was a large clock (horologium majus) with all the wheels of brass. Whether Bürgi, during Tycho's residence at Prague, supplied him with a pendulum clock, as stated by a later writer,^[22] must remain very doubtful, but that Tycho did not possess such a clock at Uraniborg seems certain, as he would not have neglected to describe so important an addition to his stock of instruments. As he found the clocks so uncertain, Tycho also tried time-keepers similar to the clepsydræ of the ancients, which measured time by a quantity of mercury flowing out through a small hole in the bottom of a vessel, in which the mercury was kept at a constant height, in order that the outflow might not vary with the varying weight of the mercury. By ascertaining the quantity of mercury which flowed out in twenty-four hours, it was easy to make out the interval which passed between the culmination of the sun and a star by starting the time-keeper when the former passed the meridian, and letting it run until the latter passed, and then weighing the amount which had flowed out. Instead of mercury, Tycho also tried lead monoxide powder, and adds to his account of these experiments some remarks about Mercury and Saturn (lead), and their astrological relations, which naturally suggested themselves to his mind.^[23] But he does not seem to have used these clepsydræ except by way of experiment, and his methods of observing made him in most cases independent both of them and of the clocks. In addition to the altitudes (about which he justly remarks that they must not be taken too near the meridian, where they vary very slowly, nor near the horizon, where they are much affected by refraction), he observed hour-angles of the sun or standard stars with the armillæ to control the indications of his clocks, and his observations of the moon, comets, eclipses, &c., where accurate time determinations are indispensable, were thereby doubly valuable. Occasionally azimuths were also observed for the same purpose, the zero of the azimuth circle having been found by observing the east and west elongation of the Pole Star.^[24]

For observations of altitude Tycho also used a sextant of 51/2 feet radius, turning on a vertical axis, with one end-radius kept horizontal by means of a plumb-line attached to the centre of the radius. We have already mentioned that Tycho, driven by necessity, had observed altitudes of the new star with a sextant, and as the planets never could attain an altitude above 60°, he found a sextant a convenient instrument for many purposes, and specially mentions that it was easily taken asunder and transported wherever required. Though Tycho believed himself to be the inventor of this instrument, he had been anticipated by the Arabs, as Al Chogandi in 992 at Bagdad erected a sextant (which is even said to have been of sixty feet radius) for measuring the inclination of the ecliptic.^[25]

The sextant was with Tycho Brache a favourite instrument, which he had already constructed for Paul Hainzel in 1569 for measuring the angular distances of stars. At Hveen he constructed three large sextants for this purpose. One of these, which was placed in the great northern observatory, and was made entirely of brass, was on the same plan as the Augsburg instrument, the arc being attached to the end of one arm, the two arms being placed at the proper angle by a screw, the eye of the observer being at the hinge on which the arms turned.^[26] The second was placed in one of the crypts of Stjerneborg, and was a solid sextant of wood, covered with painted canvas, and a brass arc 51/2 feet radius, braced with stays and supported on a globe sheeted with copper, which enabled the two observers to place it in the plane through the two stars to be measured, while they steadied it in the position required by two long rods with pointed ends which rested on the floor. One of the observers sighted one star through a fixed sight at one end of the arc (C), while the other observer pointed to the other star through a sight at the end of a movable radius. Both observers employed

Sextans Trigonicus.

the same object-sight, a small cylinder (A) at the centre of the arc and perpendicular to the plane of the sextant. As the observers when measuring very small distances would get their heads too close together, there was for that purpose a second cylinder on one of the end-radii (F), and a removable sight on the arc (G), placed so that the line through them was parallel to a line from the centre to the middle of the arc. One observer then sighted along these, and the other along the movable arm as usual.^[27]

For measuring small distances (less than 30°) Tycho also constructed an "arcus bipartitus," consisting of a rod 51/2 feet long, with a cross-bar at one end, having at each extremity a small cylinder, and two arcs of 30° at the other end, of which the cylinders occupied the centres. With this instrument, which was placed in the great northern observatory, the distances of the principal stars of Cassiopea inter se were measured in order to fix the position of the new star by the measures taken in 1572–73.^[28]

The size of these various instruments, as well as their solid construction, would not have been sufficient to ensure the accuracy in the observations which Tycho actually attained, and which so much exceeded that reached by previous observers, if he had not added special contrivances for that purpose. Before Tycho's time there was only one way of making small fractions of a degree distinguishable—by making the instrument as large as possible. In addition to Al Chogandi's 60-foot sextant, a quadrant of 21 feet radius is said to have been constructed by Al Sagani (about the year 1000), and the value which the Arabs were obliged to attach to large instruments was expressed in the remark of Ibn Carfa, that if he were able to build a circle which was supported on one side by the Pyramids and on the other by the Mocattam mountain, he would do it.^[29]

The first to suggest a method of subdividing an arc of moderate dimensions was Pedro Nunez, whose work, De Crepusculis, was published in 1542. He proposed inside the graduated arc of a quadrant to draw 44 concentric arcs, and divide them respectively into 89, 88, 87 . . . 46 equal parts, so that the alidade in any position would (more or less accurately) touch a division mark on one of the 45 circles. The indication of this mark was multiplied by 90°/n where n is the number of divisions on the arc on which the mark touched by the alidade is. But however ingenious this proposal was, it was anything but a practical one, as it is not easy to divide an arc into 87 or 71 equal parts, and the observer would generally be in doubt which division was nearest the alidade.^[30] Tycho Brahe tried this plan on three of the instruments first constructed at Hveen (the two smallest quadrants and a sextant), but abandoned it again as far inferior to the one he subsequently adopted.^[31] By a strange misunderstanding, the name of Nonius is even at the present day often applied to the beautiful and practical invention of Vernier (1631), with which it has nothing whatever in common. A step towards the idea of Vernier was made by Christopher Clavius and the Vice-Chancellor Curtius, and the latter communicated this plan to Tycho in 1590, but it was not much more practical than that of Nunez, and was probably never carried out in practice.^[32]

We have seen that Tycho Brahe in his youth followed the example of the Arabians by constructing a large quadrant at Augsburg, with a radius of 19 feet. But already before that time he had in 1564 obtained a cross-staff divided by transversals.^[33] He says himself that Bartholomæus Scultetus had got the idea of this method of subdivision from his teacher Homilius, and now taught it to him; but in a letter to the Landgrave (of 1587) Tycho states that he was seventeen years of age when he at Leipzig learned the use of transversals for subdividing a straight line from Homilius.^[34] The latter died, however, in July 1562, when Tycho was only 151/2 years old, and had only been a few months at Leipzig, and it is therefore more probable that Scultetus really was the means of imparting the idea to Tycho. At all events, Tycho did not attempt to claim the invention for himself, though it was afterwards often attributed to him. But whether Homilius really was the first inventor is more than doubtful, and Scultetus himself has even stated that the method was already known to Purbach and Regiomontanus.^[35] We can, however, scarcely believe this to have been the case, as it would be difficult to explain why the method had never come to light, even though Walther notoriously guarded the belongings of Regiomontanus with a curious fear of their being known; and in the Scripta of Regiomontanus there is no trace of his having used so excellent a method. Curiously enough, there are two other names mentioned in connection with this invention. In his book Alæ seu scalæ mathematicæ^[36] Digges states that transversals were first applied to the divisions on the cross-staff by the English instrument-maker Richard Chanzler, and Reymers Bär mentions that the method was described in Puehler's Geometry, published in 1561.^[37]

Under any circumstances, it was Tycho Brache who introduced the use of transversals on the graduated arcs of astronomical instruments. He did not use transversal lines, such as afterwards became universally used, but rows of dots, which were fully as convenient, and he showed that the error

Transversal Divisions.

introduced by employing these rectilinear transversals for the division of arcs would not exceed 3″, which would be imperceptible.^[38] When Wittich had brought the news of this contrivance to Cassel, Bürgi modified it a little by using lines instead of the rows of dots, and adding a scale on the alidade, the section of which with a transversal line showed the number of minutes to be added to the indication of the preceding division line, while on Tycho's instruments each of the dots corresponded to 1′.

But it was not sufficient to find means to read off the measured angle accurately; it was also of great importance to point the instrument to a star with greater precision than hitherto, and here Tycho had nobody to show the way. Up to his time an alidade had been furnished with two pinnules, one at each end, consisting of a brass plate with a small hole in the middle, and if this hole was made too small, a faint star could not be seen, while a larger hole made the observation too uncertain. To meet this difficulty Tycho introduced a special pinnule at the eye-end of the alidade, consisting of a square plate with a narrow slit close to the side next the alidade, while there were three other slits between the three other sides and small movable pieces of metal parallel to them. By moving these pieces the slits could be made wide or narrow according as faint or bright stars were observed. At the object-end was a small square plate exactly of the same size as the plate at the eye-end. When the alidade was pointed to a star, and the latter through the four slits was seen to touch the three sides of the object-pinnule and shine through a slit along the side next the alidade, the observer knew that the alidade accurately and without any parallax represented the straight line between his eye and the star. For observations of the sun there was in the centre of the objective pinnule a round hole through which the light fell on a small circle on the eye-pinnule, and the sunlight was generally conducted "through a canal" to keep off extraneous light. In many cases, Tycho (as we have already seen) modified the arrangement by substituting a small cylinder (perpendicular on the alidade) for the objective pinnule. On the armillæ this cylinder was placed in the centre of the axis, while the eye-pinnules could slide along the graduated circles.

Like the transversal divisions, the improved sights were introduced at Cassel by Paul Wittich, and the value of these improvements was found to be so great, that while the observers could formerly scarcely observe within 2′, the attainable accuracy was now 1/2′ or 1/4′. It appears that Wittich had not described the pinnules accurately, as he had only given them two slits instead of four, which Rothmann (or probably Bürgi) soon found preferable.^[39]

Before Tycho settled at Hveen he had never regularly observed the sun,^[40] but (as we have already mentioned) from his birthday, the 14th December 1576, he took regular observations of the meridian altitude of the sun, and later, when his stock of instruments increased, several quadrants were simultaneously employed for this purpose. Above all, he employed from March 1582 the great mural quadrant for observing the sun on the meridian, while the declination was also very frequently measured with the armillæ. These observations were made with the object of improving the theory of the sun's apparent motion. The equinoxes of the years 1584–88 were carefully determined, but owing to the difficulty of fixing the moment of solstice on account of the very slow change of declination at maximum and minimum, he did not make use of the solstices to find the position of the apogee and the amount of the excentricity of the orbit, but determined the time when the sun was 45° from the equinoxes, in the centre of the signs of Taurus and Leo. Copernicus had followed the same plan, but had made use of the signs of Scorpio and Aquarius, while Tycho objected to these that the sun was too low in the sky, and the influence of refraction and parallax too great. He found the longitude of the apogee = 95° 30′, with an annual motion of 45″ (should be 61″, Copernicus had only found 24″), and the excentricity of the solar orbit = 0.03584, the greatest equation of centre being 2° 31/4′. In the determination of the apogee he was more successful than Copernicus; but while the latter made the equation of the centre too small, Tycho made it too great. The length of the tropical year he found by combining some observations by Walther (reduced anew after determining the latitude of Nürnberg) with his own to be equal to 365^d 5^h 48^m 45^s, only about a second two small. With his new numerical data he computed tables for the apparent motion of the sun, which he remarks are worthy of considerable confidence, as they depend on observations made with three or four large instruments made of metal, and capable of determining the position of the sun within 10″, or at most 20″; and by comparing the tables with observations by Regiomontanus, Walther, the Landgrave, and Hainzel, he shows that they represent the observed places within a small fraction of a minute, while the Alphonsine tables and those of Copernicus are often 15′ or 20′ in error.^[41]

The solar observations at Uraniborg led to a result which Tycho does not seem to have anticipated. The colatitude, as found by the greatest and smallest altitude of the sun at the solstices, differed from that deduced from observations of the Pole Star by a considerable quantity, which sometimes amounted to 4′. Having ascertained that the discrepancy did not arise from instrumental errors, he was led to attribute it to the effect of refraction. As soon as the great armillæ at Stjerneborg were finished, he instituted systematic observations to prove this, and to determine the amount of refractions at various altitudes. Having first found, by following the sun throughout the day with the armillæ, that the declination apparently varied, as stated by Alhazen in his book on optics, he repeatedly in the years 1585 to 1589 devoted a whole day, generally in June, when the declination of the sun changed very slowly, to investigations on refraction. With an altazimuth quadrant he measured at frequent intervals the altitude and azimuth, and from the latitude of the observatory, the azimuth and the decimation, he computed the altitude, which, deducted from the observed altitude, gave the amount of refraction. Another method was by observing simultaneously with the quadrant and with the armillæ. In the triangle between the pole and the true and apparent places of the sun, two sides (real and apparent declination) and one angle (180° minus the parallactic angle) were known, from which the third side could be computed, which was the effect of refraction in altitude. This is a most inconvenient and troublesome method, and must have given the computers plenty to do, if the observations were really extensively used in this way for the construction of his refraction tables. For these investigations he assumed the real declination (i.e., corrected for refraction) as equal to the declination as observed on the meridian, as he thought the refraction at the meridian altitude at summer solstice (571/2°) insensible. He assumed, in fact, that refraction disappeared already at 45°, where it in reality amounts to 58″. Unluckily Tycho spoiled the refraction table which he constructed from his solar observations by assuming, with all previous astronomers, from Ptolemy down to Copernicus, that the horizontal parallax of the sun was equal to 3′. It is remarkably strange that Tycho should not have endeavoured to deduce this important constant from new observations which ought to have shown him that it was for his instruments insensible. This was the only astronomical quantity which he borrowed from his predecessors, and it was a wrong one.^[42] The refractions, as given by him, must therefore be diminished by 3′ × cosine of altitude, and it is interesting to see that he was well aware of the fact that the refractions as found by the stars were different from those which he had mixed up with the imaginary solar parallax, as he gives a separate table of stellar refraction, in which the quantities are smaller than those in the solar refraction table by 4′ 30″; so that according to him refraction becomes insensible in the case of stars at 20° altitude (where it is in reality 2′ 37″). Possibly the refraction of stars was not as carefully looked into as that of the sun, though the observations "pro refractionibus fixarum indagandis" are numerous, particularly in the year 1589, and are similar to those of the sun.^[43]

Imperfect though Tycho's researches on refraction were, they represent a great step forward, as he was the first to determine from observations the actual amount of refraction, and to correct his results for it. This was among the earlier achievements at Uraniborg, and showed the great superiority of the new instruments over the earlier ones. Though not unknown to the ancients, and theoretically examined to some extent by Alhazen and Vitello (whom Tycho quotes, though he doubts whether they really carried out the experiments mentioned by them, as their armillæ could not have been large or accurate), the only astronomer who had practically noticed the effect of refraction was Walther. He found, by observing the sun when setting, by means of his zodiacal armillæ, that the sun seemed to be outside the ecliptic, and explained this as being caused by refraction; but he thought this could only be appreciable very near the horizon, and did not attempt to investigate its laws, for which his instruments would hardly have been accurate enough.

As to the cause of refraction, Tycho did not think that the difference of density of the ether and the atmosphere was of much importance, as he points out that in that case refraction should not disappear except at the zenith, while he imagined it to become insensible half-way towards the zenith. He therefore ascribed refraction chiefly to atmospheric vapours, though he believed that the atmosphere gradually decreased in density and was essentially different from the ether, and he naturally rejected as absurd the Aristotelean idea of a sphere of fire encircling the earth. He had in his correspondence with Rothmann several times discussed questions connected with refraction, not only because the observer at Cassel only made the quantity of refraction about half as great as Tycho did, but also because Rothmann thought that there was no difference between the celestial ether and the air except density.^[44]

Tycho recognised as an effect of neglected refraction various discrepancies between the elements of the solar orbit determined by Copernicus and his own results. We have already mentioned^[45] that he sent one of his pupils to Frauenburg, and found that the latitude had been assumed 23/4′ too small, which, together with the neglect of refraction, accounted for the errors in Copernicus' determination of the obliquity and the other elements of the solar orbit, as the longitude concluded from the erroneous declinations would be as much as 13′ in error at 45° from the equinox.

Among Tycho's "puerile and juvenile" observations there are very few indeed of the moon; only now and then the approach of the moon to some bright star is mentioned, and the distance measured with the "radius" or sextant. At Hveen he gradually came to devote more attention to the moon, and from 1582 his lunar observations are very regular, and become year by year more numerous. They include distances from fixed stars, altitudes, declinations, and differences of right ascension from fixed stars, and as often as practicable the moon was observed in the nonagesimal or that point of her daily course in which the effect of parallax took place only in latitude. Eclipses were carefully attended to whenever they occurred;^[46] but, unlike the ancient astronomers, Tycho did not confine himself to observing the moon at the syzygies and quadratures, but followed her throughout her monthly course year after year, determining her position both on and off the meridian, and not forgetting to observe her at apogee, or, as he called it, "in maxima remotione utriusque epicycli." He thus succeeded in detecting the third inequality in the motion in longitude, the variation, which reaches its maximum of 39′.5 (Tycho found 40′.5) in the octants, when the difference of longitude of sun and moon is 45°, 135°, &c. But apart from this, he could not be satisfied with the way in which Ptolemy had represented the motion in longitude (by a deferent and one epicycle, the centre of the former moving in a circle round the earth in a retrograde direction), because it represented the apparent diameters of the moon very badly. In fact, the moon ought, according to the theory of Ptolemy, to appear nearly twice as great at perigee as at apogee. This had not escaped Copernicus, who avoided it by making the deferent concentric with the earth, and adding a second epicycle with a motion twice as rapid as the first one.^[47] Tycho chose another way of representing the motion in longitude. The deferent (radius = 1) according to him had its centre in a circle with radius 0.02174, in the circumference of which the earth was placed, so that the centre of the deferent was in the earth in the syzygies, and farthest from it at the quadratures. There were two epicycles with radii 0.058 and 0.029, the period in the former being the anomalistic month, and the moon moving in the latter twice as rapidly and in the opposite direction, in such a manner that at apogee the moon was 0.029 outside the deferent, at perigee 0.058 + 0.029 = 0.087 inside it. The effect of the two epicycles gave the maximum of the first inequality 4° 59′ 30″, while the circle through the earth gave an equation of 1° 14′ 45″ (evection), not differing much from Ptolemy's values, though somewhat more accurate.^[48]

So far Tycho had not made much advance, but the discovery of the third and fourth inequalities was a very great step in advance. He probably thought that there were epicycles enough in his theory, and therefore he did not attempt to account for the variation by adding another. He merely let the centre of the first epicycle oscillate (librate) backwards and forwards on the deferent to the extent of 40′.5 on each side of its mean position, the latter moving along the deferent with the moon's mean motion in anomaly, and the centre of the epicycle being in its mean position at the syzygies and quadratures, and farthest from it at the octants, the period of a complete libration being half a synodical revelation.^[49] At the same time Tycho's observations showed the existence of another inequality, the fourth one in longitude, of which the solar year was the period, so that the observed place was behind the computed one, while the sun moved from perigee to apogee, and before it during the other six months. We have already mentioned that the solar and lunar eclipses continued to be carefully observed by Tycho, and at the latest during his stay at Wittenberg, he had clearly grasped the peculiarity in the lunar motion just described, since Herwart von Hohenburg wrote to Kepler (in July 1600) that he had probably heard from Brahe himself how the latter in the paper he had printed at Wittenberg^[50] had introduced a "circellum annuæ variationis, cujus initium statuitur sole versante in principio Cancri, ita ut in priori semicirculo hujus circelli verus locus Lunæ promoveatur in consequentia, et in posteriori retrotrahatur in præcedentia." Kepler also bears witness to the introduction of this circellus during Tycho's stay at Wittenberg.^[51] But the representation of the lunar motion had become so complicated that Tycho shrank from introducing more circles (for which reason he had adopted a mere libratory motion to account for the variation), and the idea of a really unequal motion was too much opposed to the time-honoured conception of uniform circular motion. He (or rather Longomontanus) therefore ultimately allowed for the annual equation by using a separate equation of time for the moon, differing by 8m. 13s. multiplied by sine of the solar anomaly from the ordinary one, even though this left 5′ or 6′ of the irregularity unaccounted for.^[52] The correct amount of the equation (11′, and not 4′.5) was found by Horrox, but he applied it in the same manner as Tycho had done.

It is very interesting to see that Kepler had independently discovered the annual equation about the same time as Tycho did. In the calendar for 1598, which he had to prepare as provincial mathematician for Styria, Kepler had in detail described the solar eclipse of the 7th March (25th February) 1598, making use of Magini's tables.^[53] But the phenomenon turned out very different from what he expected, as the eclipse not only was very far from being total (or nearly so), but occurred an hour and a half later than expected. As the only reservation taken by Kepler had been that the eclipse might possibly occur half an hour earlier, he had to say something about the cause of this error in the calendar for 1599. In this he therefore stated that the solar eclipse, as well as the lunar eclipse in February and the Paschal full moon, had been more than an hour late; but the lunar eclipse in August had been too early, and it appeared to him that one would have to assume "that a natural month or period of the moon with regard to the sun in winter, ceteris paribus, is a little longer and slower than in summer, and the fault is the moon's and not the sun's, as nothing can be reformed as to the latter without great confusion; but whether the inequality is to be applied to the moon itself or to the length of the day, and what cause it may have in nature and the Copernican philosophy, cannot be explained in a few words."^[54] A letter to Mästlin of December 1598 shows that Kepler had not thought further about the matter, and

merely threw out this solution because he thought it easier to defend than one founded on corrections to the solar theory, and he adds that his calendar was not written for learned men, and would never be seen outside Styria.^[55] It happened, however, that the calendar was read by Herwart von Hohenburg, who in January 1599 requested Kepler to give him further information about the solar eclipse. Being thus obliged to consider the matter more fully, Kepler did so in his reply, in which his wonderful genius displays itself by the way in which he suggests that the moon might be retarded in its motion by a force emanating from the sun, which would be greatest in winter, when the moon and earth are nearer to the sun than they are in the summer.^[56] At the same time he suggests that the phenomenon might also be caused by an irregularity in the rotatory velocity of the earth, and in after years he accepted this idea, and did not consider the phenomenon as caused by an equation in the lunar motion.^[57]

Tycho Brahe's discoveries as regards the lunar motion in latitude were as important as those he made of inequalities in longitude. The inclination of the lunar orbit to the ecliptic had by Hipparchus been found to be 5°, which value had been retained even by Copernicus. Several of the Arabian astronomers had, however, noticed that this was not correct. Thus Abul Hassan Ali ben Amadjour early in the tenth century stated that he had often measured the greatest latitude of the moon, and found results greater than that of Hipparchus, but varying considerably and irregularly. Ibn Yunis, who quotes this, adds that he had himself found 5° 3′ or 5° 8′. Other Arabians are, however, said to have found from 4° 45′ to 4° 58′, which does not speak well for the accuracy of their observations.^[58] Tycho first began to suspect that the value of Hipparchus was wrong when examining an observation of the comet of 1577. On November 13 he had measured the distance of the comet from the moon, and found 18° 30′, while the observed distances of the comet from stars by computation gave its distance from the moon equal to 18° 9′, allowing for the lunar parallax. At first he attributed the difference to refraction, but in 1587, when the moon attained its greatest latitude about Cancer, so that neither errors in the parallax nor refraction could influence the result much, he found the lunar inclination to be 5° 15′, and thought it might have increased since the days of Ptolemy, just as the obliquity of the ecliptic had diminished.^[59] The examination of all his observations showed him, however, later, that the inclination varied between 4° 58′ 30″ and 5° 17′ 30″, while the retrograde motion of the nodes was found not to be uniform, so that the true places of the nodes were sometimes as much as 1° 46′ before or behind the mean ones. This inequality of the nodes had not been detected by the ancients, because it disappears in the syzygies and quadratures, where they alone observed the moon. Tycho explained this and the change of inclination by assuming that the true pole of the lunar orbit described a circle round the mean pole with a radius of 9′ 30″, so that the inclination reaches its minimum at syzygy and its maximum at quadrature.^[60] He applied corrections separately to the latitude for equation of node and for change of inclination, a form which was retained even by Newton and Euler, until Tobias Mayer showed that the two equations can be combined into one, varying with the double distance of the moon from the sun, less the argument of latitude of the moon.^[61]

It would lead us too far if we were in this place to enter into a description of Tycho's lunar tables, or of his precepts for finding the longitude from his theory.^[62] We shall only mention that he was the first to tabulate the reduction, or the difference between the moon's motion along its orbit, and the same referred to the ecliptic. The table of parallax makes this quantity vary between 66′ 6″ and 56′ 21″, the apparent diameter varying from 32′ to 36′ at full moon, while he believed to have found from his observations of eclipses that the diameter appears less at new moon (25′ 36″ to 28′ 48″), owing to the limb being "extenuated" by the solar rays. He therefore denied the possibility of a total solar eclipse, to some extent also misled by the accounts of the luminosity seen round the sun at the eclipses of 1560, 1567, and 1598.^[63]

The planets had been, favourite objects with Tycho from his youth. His very first attempts at observing had been sufficient to show him how imperfectly the existing theories of the planetary motions agreed with the actually observed positions of the planets, and throughout his life he never neglected to take regular observations of the five planets.^[64] His early observations of planets were of course similar to those made by his predecessors. The ancients had generally fixed the position of a planet by mere alignment, or, if the distance from a star was small, by expressing it in lunar diameters, while conjunctions of planets inter se or near approaches to fixed stars were greatly valued as tests of theory. As long as Tycho only possessed few and small instruments, he naturally often had recourse to these old methods, but he commenced also very early to adopt the method, first used by Walther, of measuring the distance of a planet from two well-known fixed stars.^[65] At Hveen he never quite gave up this method, but he chiefly depended on meridian altitudes and observations with the armillæ, and even the difficult planet Mercury was carefully watched for and observed on every opportunity.^[66]

Though Tycho did not live long enough to try his hand seriously at the theory of the planetary motions, we have seen that he was at Benatky occupied with the theory of Mars, and succeeded in representing the longitudes well (Kepler says within 2′), while the latitudes gave more trouble.^[67] But already at Uraniborg he had not contented himself with a mere accumulation of material, but had drawn some conclusions from the comparison of his results with the tabular places of the planets. We have seen that Tycho, like Ptolemy and Copernicus, assumed the solar orbit to be simply an excentric circle with uniform motion. But already in 1591, he might have perceived from the motion of Mars that this could not be sufficient, as he wrote to the Landgrave that "it is evident that there is another inequality, arising from the solar excentricity, which insinuates itself into the apparent motion of the planets, and is more perceptible in the case of Mars, because his orbit is much smaller than those of Jupiter and Saturn."^[68] He concluded (strangely enough) that his own planetary system alone could account for this, and he can therefore not have had a clear idea of the cause of the phenomenon. Again, in his letter to Kepler of April 1, 1598, he mentioned that the annual orbit of Mars (according to Copernicus) or the epicycle of Ptolemy was not always of the same size with regard to the excentric, but varied to the extent of 1° 40′.^[69] This eventually led Kepler to the discovery of the elliptic orbits, but it showed him already in Tycho's lifetime that the solar excentricity was only half as great as hitherto supposed, and that the remainder of the equation of centre would have to be accounted for by a uniform motion round a punctum æquans (that is, as long as only circular orbits were admissible).^[70] There was another important matter in which Kepler's suggestion was acted upon. Soon after his arrival at Benatky, he found that Tycho, like his predecessors, referred all the planetary motions to the mean place of the sun, while he had himself in his Mysterium Cosmographicum referred them to the actual place of the sun. He gave the impulse to this being done in the lunar theory by Longomontanus, and he mentions in the Appendix to the Progymnasmata that the necessity of this step had also become evident in the case of Mars.

With regard to Tycho's observations and researches on comets, we need only refer to Chapter VII., where they have been examined in sufficient detail. It is not among the least of Tycho's scientific merits that he finally proved comets to be celestial bodies.

That a new catalogue of accurate positions of fixed stars was urgently needed had early been felt by Tycho Brahe. The Ptolemean catalogue of stars was fourteen hundred years old, and was probably little more than a reproduction of the still older catalogue of Hipparchus. None of the Arabian astronomers had observed fixed stars, but had contented themselves with adding the precession to the longitudes of Ptolemy; and the only independent catalogue, that of Ulugh Beg, was not yet known in Europe. The co-ordinates of stars given in Ptolemy's catalogue were known at Tycho's time through the two Latin editions of the Almegist of 1515 and 1528 and the Greek edition of 1558; but to the original errors of observation had been added a goodly number of errors of copying, so that the discrepancies of the various editions inter se were numerous and large. The observations of the new star and of the successive comets made Tycho feel the necessity of getting accurate places for his stars of comparison, and when his observatory was complete, he took up the work of forming a new star catalogue with great energy.

By Hipparchus the longitudes of stars had been deduced from the longitude of the sun by using the moon as intermediate link, which method is described by Ptolemy, who gives a full account of the manipulation of the zodiacal armillæ. Unfortunately Ptolemy does not say a word about the manner in which the standard stars (Regulus and Spica) were connected with the other stars, nor does he give any details about the actual observations on which the adopted places of the stars were founded. It is, therefore, not known whether every single star was connected with a standard star, or whether he perhaps also made use of conjunctions of stars with the moon (which had been of great value for the deduction of stellar positions for earlier epochs for the determination of the constant of precession),^[71] and nothing but a slight sketch of the method was handed down to posterity. The Arabs, as already remarked, did not observe fixed stars, and here, as in several other branches of practical astronomy, Walther was the first to recommence work. At the Nürnberg observatory he introduced a very important improvement on the method of Hipparchus by substituting Venus for the moon, as the small diameter, slow motion, and very small parallax made the planet far more suitable for the purpose than the moon. Among the observations published in the Scripta of Regiomontanus, the first observation of this kind is from the 6th March 1489, and there are several from the following years; and as the book was published in 1544, Tycho Brahe has known Walther's plan, while the further development of it is due to himself.^[72] The method recommended itself to Tycho because it did not involve the accurate knowledge of time by clocks or clepsydræ, while he made this objection to the method followed by the Landgrave of observing the altitude of stars, together with their transits over the meridian or a certain azimuth. The meridian method had been used by Tycho to determine the places of twelve stars observed with the comet of 1577, and for this purpose he made use of α Aquilæ as fundamental star, determining its right ascension by observing the meridian transits of it and the moon when not too far apart. He knew, therefore, by experience how undesirable it was to trust to the clocks.^[73]

In the spring of 1582 Venus was most favourably situated, and from the 26th February it was for about six weeks clearly visible in full daylight even before it passed the meridian, so that it could be observed at a sufficient height above the horizon to make errors in the adopted refractions harmless.^[74] With the sextans trigonicus two observers measured the distance between Venus and the sun, the shadow of the little cylinder at the centre of the arc falling on the movable pinnule; at the same time the altitudes of the two celestial bodies were measured, and occasionally their azimuths, while their declinations were observed by the armillæ, and their meridian altitudes as often as opportunity offered. After sunset the same sextant was employed to measure the distance of Venus from certain conspicuous stars near the zodiac (Aldebaran, Pollux, and some others in the same constellations), while, as before, altitudes and declinations were also observed. In deducing the positions of the stars observed, the motion of Venus and the sun in the interval between the day and night observations was taken into account. By simple trigonometrical operations the difference of right ascension between the sun and a zodiacal star was computed, and as the right ascension of the sun was known from the solar tables, the absolute right ascension of the star was thus found from the observations, while the declination was directly measured. All the stars thus determined were connected by distance measures with the star α Arietis, which he preferred to γ Arietis, which by Copernicus had been adopted as principal standard star, as being nearest to the vernal equinox, but which Tycho found too faint to be conveniently observed by moonlight. Each observation thus gave a value for the right ascension of α Arietis. During the following six years Tycho repeated these observations as often as an opportunity offered, and, in order to eliminate the effect of parallax and refraction, he combined the results in groups of two, so that one was founded on an observation of Venus while east of the sun, the other on an observation of Venus west of the sun; while the observations were selected so that Venus and the sun as far as possible had the same altitude, declination, and distance from the earth in the two cases. From the observations of 1582 Tycho selects three single determinations, and from the years 1582–88 twelve results, each being the mean of two results found in the manner just described. The fifteen values of the right ascension of α Arietis agree wonderfully well inter se, the probable error of the mean being only ±6″, but the twenty-four single results in the twelve groups show rather considerable discordances, the greatest and smallest differing by 16′ 30″. But anyhow the final mean adopted by Tycho is an exceedingly good one, agreeing well with the best modern determinations. He adopts for the end of the year 1585 26 0′ 30″, the modern value for the same date being 26° 0′ 45″.^[75]

From the absolute right ascension of α Arietis thus determined, and the directly observed declination, Tycho determined the co-ordinates of other stars by measuring the distance from α Arietis and the declination, after which the spherical triangle between the pole and the two stars (in which the three sides were known) gave the angle at the pole or the difference of right ascension. Proceeding thus from one star to another round the heavens, Tycho determined the right ascensions first for four, then for six, and finally for eight principal standard stars; and as the sums of the differences of right ascension in the three cases only differ a few seconds from 360°, he imagined that he had proved his results to be extremely accurate. It is needless to say that the accuracy cannot be so great as Tycho fondly hoped, as the errors of observation would be increased by neglect of refraction and by his ignorance of the existence of aberration and nutation. But it must be conceded that Tycho's results were an immense improvement on the positions of fixed stars as previously known, as the comparison with the best modern star-places for the nine stars reduced to the end of 1585 gives the probable error of Tycho's standard right ascensions equal to 24″.1, and that of his standard declinations (after correcting them for refraction) = ± 25″.9.^[76]

It is interesting to see that observations of absolute right ascension were made at Cassel about the same time, and by the same method, except that Jupiter was at first used instead of Venus. As Jupiter could not be observed with the sun above the horizon, this involved trusting to the rate of the clocks for many hours, which perhaps was more feasible at Cassel, where Bürgi introduced the use of the pendulum for controlling the clocks. In 1587 Venus was, however, made use of, the altitude and azimuth of the sun, Venus, and Aldebaran being observed in succession. The results thus found for the right ascension of the latter star agreed well inter se, fixing it at 63° 10′ for the beginning of 1586, or more than 6′ greater than that found by Tycho. This systematic error, with which all the right ascensions determined by means of Aldebaran became affected, and which also, with nearly the same amount, entered into the longitudes, was discussed in several letters between the Landgrave, Rothmann, and Tycho. The Landgrave thought 5′ or 6′ a very trifling quantity, not worth mentioning, as nobody hitherto had been able to determine longitudes with that accuracy.^[77] Tycho at first suggested that the discrepancy might be caused by an error in the solar declination, caused by a faulty suspension of the plumbline which marked the zero point on the quadrant at Cassel, and to which Rothmann had referred in a former letter.^[78] Afterwards he concluded that the error was caused by all the observations being made in the evening, when refraction would tend to make the longitude of Venus appear greater.^[79] It seems, however, that the real cause was the unlucky solar parallax of 3′ which Rothmann (like Tycho) had borrowed from the ancients, and which would act particularly injuriously on his results, as his observations were all made in winter, and at low altitudes of both the sun and Venus, and not combined, like those at Hveen, to eliminate errors as much as possible.^[80]

On the basis of the nine standard stars and twelve additional stars near the zodiac, Tycho Brahe built up his star catalogue. Of a star to be determined, the declination was measured directly by the armillæ or a meridian quadrant, and the distance from a known star was measured with a sextant. This furnished, as before, a spherical triangle, with the three sides known, from which the angle at the pole or the difference of right ascension could be computed. Generally the star was connected with two known stars, one preceding and one following it, which gave two results for the right ascension as a control. Tycho communicates twelve examples of this double determination, the results always agreeing within a minute.^[81] For stars in higher declinations the additional precaution was taken of connecting them with three stars, as in the case of the constellation of Cassiopea, in which Tycho was specially interested on account of the new star, and which he observed in 1578 and 1583. The other constellations were all observed in the years 1586 to 1591. It is needless to say that the twenty-one standard stars were not sufficient, but that it became necessary to build further on the stars determined by them. Magnitudes were frequently noted, and in the final star catalogue they were entered, occasionally with two dots added (:) or one (.), to show that the star was slightly brighter or fainter than indicated by the figure. But these estimates of magnitude were probably not made with particular care, so that it would be risky to draw conclusions from a comparison of them with the more systematically made observations of relative brightness of Ptolemy, Al Sûfi, and astronomers of the nineteenth century.^[82]

In reducing his observations, Tycho adopted 51″ as the value of the constant of precession, which he deduced from a comparison of his own places for Regulus and Spica with those found by Hipparchus, Al Battani, and Copernicus.^[83] Although the places of Spica recorded by Timocharis and Ptolemy gave respectively 491/4″ and 531/4″, he had sense enough to attribute this to the crudeness of earlier observations, and pointed out that these often erred very greatly as to the relative positions of stars which were supposed to have been well observed, so that there was no need of assuming any irregularity in the precession of the equinoxes in order to reconcile discrepancies in the absolute longitudes. The origin of this old idea, that the equinoxes did not recede with uniform velocity on the ecliptic, but were also subject to an oscillating motion, is shrouded in mystery. The name of Tabit ben Korra (who lived in the second half of the ninth century) is usually associated with this trepidatio, but the idea seems to be very old, and is first mentioned by Theon, the commentator of Ptolemy, according to whom "some ancient astrologers" had found that the stars had an oscillating motion 8° backwards and forwards in 672 years; and according to Al Batraki (Alpetragius in the twelfth century), the erroneous value of 36″ which Ptolemy had found for the constant of precession, gave rise to the whole mischief, as his successors could not believe that he had found an erroneous value. Al Battani was the only Arabian astronomer of note who was not an implicit believer in trepidation, but from the time of Al Zerkali of Cordova (about 1060) the theory of this wholly imaginary phenomenon was developed minutely. In the Alphonsine tables the period of the inequality of precession was assumed to be 7000 years, though King Alphonso personally seems to have believed precession to be uniform. From these tables and the Arabian authors the theory was spread to Europe, and was further investigated by Purbach and Regiomontanus, who assumed with Tabit that the apparent equinox moved in a small circle with a radius of 4° 18′ round the mean equinox, whereby the annual precession was sometimes accelerated and sometimes retarded. In the sixteenth century trepidation was made the subject of two treatises by Johannes Werner of Nürnberg, and in the third book of his great work Copernicus has also examined it in detail, and showed how annual precession had always varied from the time of Timocharis (300 B.C.) till his own time. It was a natural consequence of the belief in the motion of the equinox on a small circle that the obliquity of the ecliptic should also vary irregularly; and though it had been steadily diminishing since the days of Eratosthenes, even Copernicus considered such irregularities proved by the observations of the ancients and the Arabians. The first to see that the obliquity of the ecliptic had always diminished at a regular rate since the commencement of history seems to have been Fracastoro (1538), after whom the same was asserted by Egnazio Danti in 1578.^[84]

The authority of Tycho Brahe was so great, that the mere fact of his having ignored the phenomenon of trepidation was sufficient to lay this spectre, which had haunted the precincts of Urania for a thousand years, and possibly much longer. Though he had expressed himself somewhat guardedly (promising to discuss the matter further in the great work which he did not live to write), he had done enough by making his contemporaries aware of the vast difference between the accuracy of ancient observations and that of his own, and trepidation was never again heard of.^[85]

It would not convey a correct idea of the accuracy which Tycho attained in his observations if we were to compare the positions of stars given in his catalogue with those resulting from modern observations. It would certainly be possible to reconstruct his catalogue from his original observations, but as this considerable labour would not benefit modern astronomy, for which a recurrence to Tycho Brahe's observations would hardly ever be of value except in very special cases, it is not likely to be undertaken. We are, however, able to form a conception of the accuracy of his results in other ways. First, the star of 1572 was, as we have seen, connected by distance measures with nine stars in Cassiopea. Computing the positions of these from modern data, Argelander found the probable error of one distance of the new star (with the sextant of 1572) to be ±18″.2, while the distances between the stars of Cassiopea measured with the arcus bipartitus gave ±41″.0.^[86] The first result seems rather too small, but as we do not possess the original individual observations of Nova, we have no way of knowing how many such are embodied in the mean results. From the distance measures of the comet of 1577 Woldstedt found the probable error of one observed distance = ± 4′.2,^[87] but as he mixed the sextant measures with those obtained with the cross-staff, which Tycho always mentions as an untrustworthy instrument, this large probable error is not surprising. The most valuable investigation which we possess concerning Tycho's instruments is the discussion of the observations of the comet of 1585 by C. A. F. Peters.^[88] When this comet appeared, Tycho's collection of instruments was complete, and we may assume that the observations are typical. Tycho states that his indications of time have been corrected by the observed hour-angles of stars, and by recomputing these the mean correction of + 22^s.5 was found, with a probable error of ± 37^s. This only shows, as Tycho merely gave the time in whole minutes, that the great armillæ of Stjerneborg were well adjusted. But a very much better proof of this is furnished by the observations. By the armillæ the comet was compared in right ascension with certain standard stars, while its declination was observed with the same instrument. From the total of these observations Peters found that the polar axis of the armillæ was inclined to the horizon by an angle which exceeded the latitude by only 65″ ± 33″, and formed with the meridian an angle of only 36″ ± 13″. The probable error of one observation of declination was ± 49″, that of one right ascension = 81″, and consequently that of one observed hour-angle = ± 57″. The error of collimation (or parallax, as Tycho called it) was − 30″.1, by which amount the observed declinations were too large. The comet was also observed with the sextans trigonicus, and the probable error of one observed distance was found equal to ± 45″, the collimation error being − 114″.6.^[89] These results are sufficient to show that Tycho's instruments were really made with the great care which he declares he had always bestowed on them,^[90] and in connection with the above results as to Tycho's standard stars, they exhibit the vast stride forward which observing astronomy made at Uraniborg, and which but for the invention of the telescope could hardly have been much exceeded by his successors.^[91]

It will not be out of place to say a few words here about a time-honoured absurdity which has attributed great carelessness to Tycho Brahe in the adjustment of his instruments in azimuth. In 1671, Picard, when determining the latitude of Uraniborg, measured the azimuths of the principal church spires in Seeland and Scania visible from the site of Uraniborg. At Copenhagen he found among Tycho's manuscripts similar observations which showed considerable differences from his own.^[92] Picard did not lay any stress on this discrepancy when mentioning it in the account of his journey, probably because he saw from the MS. that Tycho had merely measured these approximate azimuths for the sole purpose of constructing a map of the island. By others the matter was, however, misunderstood; and by some the discrepancy was even supposed to prove a shifting of the meridian line between the times of Tycho and Picard; while others have pointed to Tycho as a blunderer in comparison with the builder of the Great Pyramid, who was able to orient the sides of that remarkable structure with considerable accuracy.^[93] It was, however, shown by a Danish writer, Augustin, that Tycho and Picard had in two cases pointed to different spires. At Elsinore Tycho had pointed to St. Mary's Church, while Picard had pointed to the taller spire of the church of St. Olaus, built in 1614; and the cathedral of Lund has two towers, of which Tycho had taken the southern one, while Picard pointed midway between the two. This accounted for the most serious differences, and the remaining measures would agree well by assuming an error of 14′, by which amount Tycho's meridian line should have deviated from the true south point towards the east. Augustin even imagined that he had found in the printed observations the proof that Tycho detected this error on the 2nd November 1586.^[94] It is, however, evident from the words used by Tycho that he must on this occasion have referred to a recent readjustment (in novo meridiano) of the instruments at Stjerneborg only, and not to some meridian line adopted since 1579, at which time (at the latest) the azimuths of the church spires were measured.^[95] The observations of the comet of 1585, as we have just seen, prove conclusively that in that year the great armillæ were in excellent adjustment, so that Tycho cannot have made use of any badly placed meridian mark. I have also computed a number of observed altitudes and azimuths of stars from 1582, and from these it is evident that the zero line of the azimuth circle was within 1′ of the meridian.^[96] As Tycho never once alludes to the use of meridian marks or terrestrial azimuth marks (which he could not possibly have seen from the subterranean observatory, where stars near the horizon could only be observed with portable instruments in the open air), while he frequently states that he verified his instruments by observations, it is impossible that he can, even before 1586, have made a mistake of 14′ in azimuth in the adjustment of his numerous instruments.

The astronomical work in Tycho Brahe's observatory must have involved a considerable amount of computing, even though the great globe, no doubt, was very often used for the solution of spherical triangles. Trigonometry had made considerable advances in the fifteenth and sixteenth centuries, and Tycho could build on the labours of Purbach, Regiomontanus, Copernicus, and others, both as regards the solution of triangles and tables of sines and tangents. But

logarithms had not yet been invented, and great inconvenience was therefore felt whenever it became necessary to multiply or divide trigonometrical quantities. To obviate this difficulty a method was invented, the so-called Prostaphæresis,^[97] by which addition and subtraction were substituted for multiplication and division, and in the history of this invention, which was made independently by several mathematicians, the name of Tycho is also mentioned. The Arabs had had an idea of this method; at least, Ibn Tunis makes use of the formula^[98]

cos A cos B = 1/2 [cos (A − B) + cos (A + B)]

but, like many other discoveries of the Arabs, this formula had to be deduced anew in Europe. It was found by Viète, as well as the corresponding formula:

sin A sin B = 1/2 [cos (A − B) − cos (A + B)]

but as Viète's Canon Mathematicus, which was published in 1579, seems only to have been printed in a few copies at his own expense, it is very possible that Tycho Brahe never saw it, or at least that he had not seen it in 1580, when, according to Longomontanus, he and Wittich invented Prostaphæresis.^[99] This was among the inventions which Wittich a few years later brought to Cassel, where Bürgi soon developed the method further. It appears that Wittich merely had shown him the above formula for sin A sin B; but Bürgi applied the principle to the formulæ of spherical trigonometry, and ultimately was led to discover logarithms years before Napier did; but, as is always the case with that remarkable man, without securing the priority by a timely publication.^[100] At Uraniborg the method did not make any progress after the departure of Wittich, and it is therefore more likely that it was he, and not Tycho, who was the inventor, as he is known to us (through the repeated testimony of Tycho) as an able mathematician. In 1591 a short treatise on plane and spherical trigonometry was drawn up at Uraniborg, but it does not indicate that Tycho had developed trigonometry in any way, as the rules are similar to those given in other treatises of that day, and are frequently expressed in even clumsier language than usual at that time.^[101] The demand for the facilities offered by the Prostaphæresis was, however, so great, that Reymers Bär, Clavius, Joestelius, Magini, and others, with more or less success, continued to work in this direction, until the method was driven from the field by the discovery of Napier.

We have followed Tycho Brahe through his chequered career, and we have reviewed his scientific labours. No doubt his contemporaries were not uninfluenced in their estimation of him by his princely residence, with its tasteful decoration and wonderful observatories, and also by its singular situation on the little island, which contributed to exhibit the noble astronomer in a romantic light. But while these circumstances threw a halo over Tycho even before his works had become known beyond a limited circle, posterity has hardly been influenced by considerations like these when affirming the judgment of his time. He not only conceived the necessity of supplying materials for the discovery of the true motions of the heavenly bodies, and by his improvement of instruments and accumulation of observations made it possible for Kepler to reach this goal, but in almost all the branches of practical and spherical astronomy he opened new paths, and made the first serious advance since the days of the Alexandrian school. Hereby he showed his superiority to the Landgrave; for though the latter had perceived the necessity of systematic observations at least as early as Tycho did, he confined his attention almost entirely to the fixed stars, and had to borrow the improvements in instruments from Tycho, and let them be worked out by the great mechanical talents of his assistant, Bürgi, before his observations could rival those of Tycho in accuracy. It was, therefore, not at Cassel, but at Uraniborg that the reform of practical astronomy was carried out, and posterity has not thought it an exaggeration when one of the greatest astronomers of the nineteenth century spoke of Tycho Brahe as a king among astronomers.^[102]

1. About the armillæ (equatorial) of the Chinese, see Observations mathematiques, astronomiques, &c. tirées des anciens livres Chinois, red. par Souciet, ii. p. 5, iii. p. 105, and my paper on the instruments at Peking in Copernicus, vol. i. (1881), p. 134 et seq.
2. Sédillot, Prolégomènes des tables astron. d'Oloug-Beg, Paris, 1847, p. xvi. Mémoire sur les instruments astron. des Arabes, Paris, 1841, p. 198. Abul Wefa used only five circles, the smaller latitude circle being crossed diametrically by a pointer or by a tube carrying the sights.
3. Sédillot, Prolégomènes, p. cxxxiv.; Mémoire sur les instr., p. 198; but the "armille équatoriale" mentioned in the latter place is evidently nothing but Ptolemy's instrument for observing the solstices, i.e., a graduated circle in the plane of the meridian.
4. Tractatus de annulo astronomico. The author possesses a scarce little book in which the various uses of simple circles, quadrants, and systems of circles (including Gemma's rings) are described. Annuli astronomici, instrumenti cum certissimi tum commodissimi usus, ex variis authoribus. Lutetiæ, 1557, small 8vo., 159 ff.
5. Progymn., i. p. 140; Epist., p. 9.
6. Mechanica, fol. C. 4; Progymn., i. p. 141.
7. Also figured in Progymn., i. p. 251.
8. Sédillot, Mémoire, p. 196; Wolf's Geschichte der Astronomie, p. 132.
9. Delambre, Astr. du moyen age, p. 399.
10. As remarked by Delambre (Astr. ancienne, ii. p. 75), it appears doubtful whether Ptolemy ever actually used an instrument of this kind, as he only quotes one observation made with it, the difference between the sun's altitude at the two solstices, for which he gives exactly the same value as had been found by Eratosthenes; and as his latitude was 15′ wrong, his quadrant (if he used it) must have been very small.
11. Monatliche Correspondenz, xxiii. (1811), p. 346. A perfectly similar description from an Arabian MS. by Muvayad al-Oredhi of Damascus is given by Sédillot, Memoire, p. 194.
12. Progym., p. 521. Tycho did not approve of this method, as it involved the use of clocks.
13. For a description of the Tychonic quadrant, see above, p. 101.
14. Mon. Corresp., xxiii. p. 355. The instrument is doubtless the same as described by Sédillot, Mémoire, p. 200. An azimuth circle of copper, 10 cubits in diameter, was in the year 513 after Hedschra erected at Cairo for observations of the sun. Caussin, Le livre de la grande Table Hakémite (Notices des manuscrits, tom. vii.), Paris, an. xii. p. 21.
15. Progymn., p. 491. At Kremsmünster observatory there is a small azimuth circle with a vertical semicircle of ivory, dating from 1570. Wolf's Geschichte der Astronomie, p. 112.
16. The "instrument aux deux piliers" at Méragah was a modification of Ptolemy's quadrant (Sédillot, l. c., fig. 113), but it could also be arranged so as to be movable in azimuth (see also Monatl. Corresp., xxiii. p. 359).
17. Progymn., pp. 142 and 636; Eplstolæ, p. 75.
18. Schjellerup, Sur le chronomètre céleste d'Hipparque (Copernicus, vol. i., 1881, p. 25).
19. Caussin, l. c., p. 100 et seq. About some errors of copying, by which some of the observations were affected, see Knobel's paper on Ulugh Beigh's Catalogue, in the Monthly Notices of the Roy. Astr. Soc., xxxix. p. 339.
20. Scripta cl. mathematici J. Regiomontani, Norimb., 1544, fol. 36.
21. Ibid., fol. 50 et seq. The mere statement, what degree of the zodiac was on the meridian (medium cœli) when an observation was made, was, however, still very often the only indication of time given, even by Walther. See, for instance, Tycho's first observation at Hveen, above, p. 86 footnote.
22. Joachim Becher, De nova temporis demetiendi ratione theoria (1680), quoted by R. Wolf, Geschichte d. Astr., p. 370.
23. "Est in Mercurio, quicquid quærunt Sapientes . . . Sicque Saturnus et Mercurius coniunctis operibus hanc inquisitionem expedirent: cum & secundum Astrologos, illorum coniunctio aut benevola invicem radiatio . . . aut etiam intuitus beneuolentior, præ ceteris aliis significationibus ad ingenii et solertiæ contemplationisque profunditatem, laborisque invictam constantiam, plurimum conducere credantur." Progym., p. 151.
24. Epist., p. 73. Rothmann was, therefore, not the inventor of this method of finding the meridian, as supposed by Wolf (Geschichte, pp. 374 and 598). Tycho had already used the Pole Star for azimuth in 1578, as appears from his MS. journals and Observ. comet., p. 16.
25. L. A. Sédillot, Mémoire, p. 204; Matériaux pour servir à l'hist. des sciences chez les Grecs et les Orientaux, i. p. 358. Prolégomènes (1847), p. xlii. Sarafedaula, who founded the Bagdad observatory, was not a Chaliph, as supposed by Bailly and Wolf, but Emir-ul-umara.
26. Sextans chalybeus, used already in 1577; Mechanica, fol. E.; De mundi aeth. rec. phen., p. 460. The sextant at Cassel (constructed from Wittich's description) also required one observer only, who placed his eye at the centre of the arc.
27. Sextans trigonicus. Mech., fol. D. 5; Progymn, p. 248.
28. Arcus bipartitus. Mech., fol. D. 4.
29. Sédillot, Prolégomènes, pp. lvii. and cxxix. The 180-foot quadrant of Ulugh Bey was doubtless a kind of sundial, such as are also found in India. Ibn Yunis quotes an observation of the autumnal equinox of 851 at Nisapur (Khorassan) with a great armilla which showed single minutes (Caussin, p. 148).
30. The limited accuracy attainable is shown in tabular form by Delambre, Moyen Age, p. 404.
31. Mechanica, fol. A. 2; Epist., p. 62. The quadrans mediocris was, in addition to the arcs of Nonius, divided by transversals, and on the sextant Tycho removed the Nonian division altogether.
32. Mechanica, fol. G. 5; compare Chr. Clavii Opera (1612), t. iii. p. 10.
33. Mechanica, fol. G., 2nd page; Progymn., p. 671.
34. Epist., p. 62.
35. "Von allerlei Solarien, das ist, himmlischen Circeln und Uhren . . . durch Bart. Scultetum, Görlitz, 1572," quoted by R. Wolf, Astr. Mittheilungen, xxxiii. p. 90.
36. London, 1573, fol. I. 3, where there is a drawing of a rectilinear scale with transversals.
37. Kästner, Gesch. der Math., iii. p. 479; Delambre, Astr. mod., i. p. 299.
38. De mundi aeth. rec. phen., p. 461; Mechanica, fol. I. 2. According to R. Wolf (l. c.), Rothmann has in an unpublished MS. made the same investigation.
39. Epist., pp. 21, 28–29; T. Brahei et doct. vir. Epist., p. 100.
40. Only in 1574 he had at Heridsvad observed the meridian altitude of the sun on seven days in March and on two days in May.
41. Progym., pp. 1–78.
42. Among Tycho's original observations there is at the beginning of the year 1581 a table of solar parallax for every degree (beginning with 2′ 58″ at 1°) up to 60° altitude. Compare Progymn., p. 80.
43. Table of solar refraction, Progymn., p. 79. For comparison with modern refractions (after deducting Tycho's parallax) see Delambre, Astr. mod., i. p. 156. The table of stellar refraction in Progymn., p. 280. On p. 124 Tycho gives a table of lunar refraction, not differing much from the solar one. In Barretti Hist. Cœl., p. 221, there is a table of refraction in A. R. and Decl. for the star Spica Virginis.
44. Progymn., p. 91; Epist., pp. 83, 91, 106. Compare above, p. 206.
45. See above, p. 123.
46. The materials at Tycho's disposal included observations of twenty-one lunar and nine solar eclipses.
47. For further details of Ptolemy's lunar theory, see, in particular, P. Kempf, Untersuchung über die Ptolemäische Theorie der Mondbewegung, Inaugural Dissertation, Berlin, 1878. Godfray's Lunar Theory (chap, viii.) gives short sketches of Ptolemy's and Copernicus' theories.
48. We have mentioned (p. 272) that Tycho had got part of the appendix on the lunar theory printed at Hamburg, but did not make use of the sheets thus printed, giving as reason that the printer had done his work badly. Tengnagel had given a copy to Magini, who in 1600 pointed out some discrepancies, the two first inequalities being stated to amount at most to 7° 41′ 15″, while the dimensions of the circles, so far as Magini could make out, gave 11′ or 14′ less. Tycho replied that the whole had been recast, partly at Wittenberg, partly in Bohemia, and that new tables had been calculated (Carteggio, pp. 232 and 238). In his Astronomia Danica, 2nd edit., Amstel., 1640, p. 242, Longomontanus talks of the lunar hypothesis described above as one "quam anno Salvatoris nostri 1600 apud Nobilissimum et omnium præstantissimum astronomum Dn. Tychonem Brahe invenimus."
49. I shall not here enter into a discussion of the well-nigh thrashed out question whether Abul Wefa's mohadzat is the lunar variation or not, but only point out the utter absurdity of the suggestion of L. A. Sédillot Matériaux, i. p. 216) that Tycho might possibly have seen a translation of the Almegist from the Arabian, in which some abstract from Abul Wefa's book might have been given. If so, why has nobody else known this book until the present century? Tycho's discovery was not, as Sédillot believed, found among his papers and published by Kepler in 1610; it is distinctly announced in his Mechanica (fol. G. 2 verso), published in 1598, as a new inequality: "Nam & aliam quandam habet ea inæqualitatis insinuationem secundum Longitudinem, quam ab iis animadversum est." Kepler in many places mentions Tycho as the discoverer of the variation, and the insinuation that Tycho himself did not claim the discovery, but merely called his lunar theory "hypothesis redintegrata," is groundless, as Tycho used the same expression of his planetary system, which he most assuredly did claim as his own (e.g., in a letter to Mästlin, Kepleri Opera, i. p. 45).
50. "In dem deliquio Lunæ, so sie zu Wittenberg drucken lassen" (Keperi Opera, iii. p. 28). We have seen (p. 272) that Tycho gave up the idea of printing the lunar theory at Wittenberg.
51. Kepleri Opera, viii. p. 627.
52. Ibid. In a letter to Archduke Ferdinand, written early in July 1600, Kepler gives an account of Tycho's researches on the moon, and alludes to the annual equation in the following words (ii. p. 9): "Solent ceteri astronomi non experientia sed ratione moniti optima tempus aequare propter duas causas, primo propter inaequales partium signiferi ascensiones rectas, deinde propter motus Solis diurnos inaequales. Hanc posteriorem Tycho negligit, causam afferens experientiam, qua deprehendatur in collatione eclipsium aequalitatis rationem iniri non posse, nisi aut haec negligatur aequatio, aut annuus circellus tot epicyclis Lunæ insuper adjiciatur." In 1603 Kepler had also to explain to Fabricius that experience had shown Tycho the necessity of omitting part of the equation of time in the lunar motion (ii. p. 96).
53. Opera, i. p. 396.
54. Ibid., i. p. 409.
55. Ibid., pp. 409–411.
56. Ibid., p. 412 et seq. Compare another letter from Kepler to Herwart of April 1599, published in Ungedruckte wissenschaftliche Correspondenz zwischen J. Kepler und H. von Hohenburg, 1599. Edirt von C. Anschütz. Prag (Altenburg, S. A.), 1886.
57. This idea is particularly developed in Epitome Astr. Copern., Liber IV. (Opera, vi. pp. 359 et seq.). See also an interesting paper by Anschütz in Zeitschrift für Mathematik und Physik, Jahrgang xxxi. and xxxii., 1886–87. In this the author maintains that Tycho Brahe cannot be considered as the discoverer of the annual equation, because he did not distinctly announce it as a separate inequality like the variation, but allowed for the effect of it by leaving out part of the equation of time. I confess myself unable to follow this reasoning. Tycho clearly perceived the effect of the annual equation, and only adopted the peculiar dodge about the equation of time for fear of making his theory too complicated. We might as well deny that Columbus discovered America because he lived and died in the belief that he had merely come to the eastern extremity of Asia.
58. L. A. Sédillot, Matériaux pour scrvir, &c., t. i. p. 283 et seq. The sons of Musa ben Schaker (about 850) seem to have been the first to find a value differing from that of the ancients. Some Chinese observers found 5° 2′. Copernicus, an Internat. Journal of Astronomy, vol. ii. (1882), p. 128.
59. De mundi aeth. rec. phen., p. 40.
60. Copernicus had employed a similar construction to explain the trepidatio or (imaginary) oscillation of the equinoxes.
61. In Godfray's Lunar Theory, chap, viii., Tycho's hypothesis is described as if he supposed the lunar pole to move in the small circle with double the synodical velocity of the node. Though this, of course, is the correct representation of the perturbations in latitude, it is not Tycho's idea, as he took no notice what- ever of the position of the node with regard to the sun, but let the pole move with double the synodical velocity of the moon. In the well-known term 9′ sin (☾ − 2☉ + ☊), if we, instead of the quantity within the bracket, write 2(☾ − ☉) − (☾ − ☊), we get Tycho's period, as the inclination will vary by − 9′ cos 2 (☾ − ☉). But if we put (☾ − ☊) − 2 (☉ − ☊), the inclination will vary by + 9′ cos 2 (☉ − ☊), and the period is 173 days. That Kepler had remarked the importance of the position of the sun with regard to the node may be seen from Tab. Rudolph., pp. 89–90; Opera, vi. pp. 588 and 648. Of modern authors, Montucla seems to be the only one who has remarked that Tycho paid no attention to the node (Histoire des Math., i. p. 666).
62. For an account of these, see Delambre, Hist. de l'Astr. mod., i. p. 164.
63. Progym., p. 134; Kepler, Ad Vitell. Pared., chap. viii. (Opera, ii. p. 309); Riccioli, Almag. novum, ii. p. 372. See also Tycho's letter to Mästlin in 1598 (Opera, i. p. 46). About Tycho's observations of the solar and lunar diameters, see above, chapter viii. p. 191.
64. In a letter to Rothmann (Epist., p. 114) Tycho expressed his regret that so little attention was paid to the planets at Cassel, since the positions of fixed stars were principally of interest by enabling an observer to follow the course of the planets.
65. At first the youthful observer generally only measured the distance from one star; but from December 1564 two stars are often, and from 1569 always employed.
66. The earliest observation of Mercury seems to be of April 17, 1574, at Heridsvad.
67. Above, p. 303.
68. Epist. Astr., p. 206. Magini had also noticed this apparent inequality in Mars; see above, chap. ix. p. 213. Tycho also alludes to it in Mechanica, fol. Gr. 3: "Tum quoque circuitum illum annuum, quem Copernicus per motum Terræ in orbe magno, veteres secundum Epicycles excusarunt, variationi cuidam obnoxium esse perspeximus."
69. Opera, i. p. 44, iii. p. 267 (De Stella Martis, xxii.).
70. Progymn., p. 821. In the Tubulæ Rudolph., p. 57, Kepler says of Tycho: "De solis quidem Eccentrico simplici, cum videret, illum non tolerari ab observationibus planetarum caeterorum, desciscere ultimis temporibus cepit, eumque parem caeteris planetis concessit; quacunque ea res explicanda esset Hypothesi."
71. The statement by Copernicus (De Revolut., lib. ii.), that Menelaus used lunar conjunctions to determine a number of star-places, arises perhaps from a mixing up of two circumstances, viz., the observations by Menelaus of two conjunctions in A.D. 98 (recorded by Ptolemy, vii. cap. 3), and the tradition mentioned by several authors, according to which Menelaus in the first year of Trajan had compiled a star catalogue which Ptolemy had adopted, after adding 25′ to the longitudes (Schjellerup's Al Sûfi, p. 42; Albohazzin, quoted by Riccius, Delambre, Moyen Age, p. 380).
72. Tycho does not allude to Walther, but mentions that Cardan had in 1537 determined the place of α Libræ by means of Venus (though apparently without reference to the sun), which he found absurd. Copernicus and Werner had determined the place of a few fixed stars (particularly of Spica) by measuring the declination, borrowing the latitude from the catalogue of Ptolemy, and from these calculating the longitude and right ascension. Progymn., i. p. 146.
73. De mundi æth. rec. phen., p. 32.
74. Therefore Tycho gladly turned from the morose Saturn and the deceitful Mercury (i.e., from the use of timekeepers regulated by lead or mercury) to the charming Venus (Progymn., p. 153).
75. For details see Note E.
76. See Note E. Adopting the star-places given in Woldstedt's paper on the comet of 1577, the probable errors in longitude and latitude of the stars on p. 32 of De mundi æth. rec. phen. turn out to be ± 1′.i8. and ± 1′.25.
77. Epist. Astron., p. 78.
78. Ibid., p. 45; compare p. 33. Rothmann suggested that perhaps the appreciable size of Venus might have something to do with it (p. 88).
79. Progymn., p. 274.
80. R. Wolf, Astron. Mitth., xlv. (1878), p. 131. The Hessian star catalogue was to contain 1032 stars, but was never finished. In its incomplete state it is published in Barretti Historia Cœlestis, under the year 1593, which the editor has erroneously assumed to be the epoch (instead of 1586), probably because the longitudes are about 6′ too great (as 7 × 50″ = 5′ 50″). Compare Flamsteed's Hist. Cœl. Brit., vol. iii., Proleg., p. 90, and p. 21 et seq., where Tycho's and the Landgrave's star-places are given side by side.
81. Progymn., p. 224 et seq.
82. Tycho's star catalogue was reprinted by Kepler in the Tabulæ Rudolphinæ (1627), and by Baily in Memoirs R. Astron. Soc., vol. xiii.
83. Progymn., pp. 253–255.
84. Primo volume dell' Uso e Fabrica dell' Astrolabio e del Planisferio. Firenze, 1578.
85. About the successive development of the ideas on trepidation, see Delambre, Moyen Age, passim, particularly pp. 53, 73, 186, 250, 264; Kästner, Gesch. d. Math., ii. p. 60; Mittheilungen des Coppernicus Vereins zu Thorn, ii. (1880), p. 3 et seq.
86. Astron. Nachrichten, lxii. p. 273 (1864).
87. F. Woldstedt, De gradu praecisionis positionum cometæ 1577. Helsingfors, 1844.
88. Astron. Nachr., xxix. p. 209 et seq. (1849).
89. By computing the orbit from the sextant observations alone, Peters found the probable error of one distance = 110″.5, which result, however, is less certain than the one given above.
90. "Plura enim hic quam ipsa magnitudo necessaria sunt. Nam et materiæ soliditas, aëris mutationi nihil cedens, & preparationis concinnitas, diuisionum subtilitas, pinnacidiorum atque perpendiculi iusta applicatio, firma fulcra, debita dispositio, conueniens & obsecundans tractatio, accurata collimatio & numeratio: & pleraque eiusmodi, adesse oportet. Quorum tamen vix omnia instrumento ligneo, quantæcunque magnitudinis, competere, aut sane non diu in eo sarta tecta perdurare possunt. Longe igitur præferendum censeo e solida metallica materia confectum instrumentum." . . . Proyymn., p. 635.
91. By using verniers, improved pinnules, &c.,Hevelius (without using telescopes) reduced the probable error of a distance measure to 18″, to the amazement both of contemporaries and of posterity (Lindelöf, Ueber die Genauigkeit der von Hevelius gemess. Sternabstände, St. Petersburg Bulletin, 1853).
92. They occur in a rough volume of observations, 1578–81, and are copied into the volume for 1563–81, so often quoted above in Chapter ii. They are entered at the end of the year 1578, but it is not stated when they were made. There are also azimuths measured from a hill and from the church at Hveen, probably with a cross-staff, and they are headed, "Observations geographiæ in insula Huena factæ."
93. In his éloge of Chazelles, Fontenelle had already in 1710 remarked the absurdity of attributing such an error to Tycho, and Montucla had expressed himself to the same effect. Hist. des Math., i. p. 669.
94. Skrifter som udi det Kong. Videnskabernes Selskab ere fremlagte, xii., 1779, p. 191 et seq.; résume in the Connaissance des Temps pour l'an 1820, p. (385). Compare Corresp. astron. du Baron de Zach, vol. i. p. 402.
95. Tycho's words are (Hist. Cœl., p. 170): "In novo meridiano nionstrabant armillæ 15 M. ante verum meridianum. Quare omnia tempora hactenus observata uno minuto tardiora sunt debito, non tamen ubique unius minuti est differentia, quia non semper eodem modo se habuit; ubique dimidii." The instrument here referred to is the great equatorial at Stjerneborg; the hour circle had probably been found to be set 15′ wrong. On p. 210 (same date) Tycho adds to some observations with the quadrans volubilis (also at Stjerneborg) the remark: "Azimutha sunt ex nova restitutione meridiani ante biduum facta." In the Connaissance des Temps for 1816, p. (230), Delambre quotes the note to the observation with the mural quadrant of 3rd December 1582 (Hist. Cœl., p. 4), and assumes from this that Tycho in 1582 had found an error in his azimuths. The note in question has, however, nothing to do with this matter, as it only explains that the recently mounted quadrant had not yet been properly fixed to the wall.
96. See Note F. at the end of this volume.
97. Astronomers need hardly be reminded that this word (formed from πρόσθεσις, addition, and ἄφαίρεσις, subtraction) had originally signified the equation of the centre, in which sense it was still used by Tycho.
98. Delambre, Astr. du Moyen Age, pp. 112 and 164.
99. Si autem de hujus compendii inventore quis quærat, nec Arabes aut Joannem Regiomontanum fuisse, scripta eorum analemmatica declarent; neminem certe habeo Tychone nostro & Vitichio Vratislaviensi antiquiorem: quorum scilicet mutua opera primum anno 1582 [should be 1580] in Huæna, sphærica quædam triangula tali pragmatiæ pro studiosis Vranicis sunt subjecta."—Longomontani Astr. Danica, p. 8.
100. R. Wolf, Astr. Mittheilungen, No. 32; Gesch. d. Astronomie, p. 348 et seq.
101. In the University Library at Prague, published in facsimile by Studnicka at Prague in 1886: "[Tychonis Brahe] Triangulorum planorum et sphaericorum Praxis arithmetica." The original is written on twenty leaves, inserted at the end of a copy of Rhetici Canon doctrinæ triangulorum. Tycho has written his name under the title of the MS., but the handwriting of the remainder does not seem to be his.
102. Bessel, Populäre Vorlesungen, p. 422.