326 POPULAR SCIENCE MONTHLY. 



observer; a sound mathematician; an expert and careful computer; 

 and he introduced marked improvements in the methods of calcula- 

 tion. He holds the very first rank in the Arabian school. In his 

 trigonometry he substituted sines for chords; reduced the calculation 

 of spherical right triangles to four cases; and possessed a general rule 

 for the solution of oblique spherical triangles in the cases (I.) given 

 a, b, c, required A; (II.) given a, h, C, required c; was acquainted 

 with the doctrine of tangents and cotangents, though he made no 

 useful application of it ; and seems to have known something of secants 

 and cosecants. To understand his exact merits as an observer, it would 

 be necessary to go into details that have no place here. 



Ibn Yunus, the scion of a noble family, was the astronomer-royal 

 of the Fatimite Caliphs of Cairo, where he constructed the Hakemite 

 tables, in 1008, from his own observations. Comparing his own ob- 

 servations with those determined by Hipparchus or Ptolemy, he ob- 

 tained accurate values of the changes that had supervened. They 

 were accurate for two reasons : In the first place, the modern observation 

 was very near to the truth; and in the second, the annual change was 

 better determined the greater the interval of elapsed years. Albategnius 

 and Ibn Yunus were 800 years after Ptolemy, while Ptolemy was but 

 263 years after Hipparchus, and Hipparchus but two centuries after 

 Timocharis. The divisors increased with the lapse of time.* 



"At Nishapur lived and died (early in the twelfth century) Omar 

 Khayyam busied in winning knowledge of every kind, and especially 

 in astronomy, wherein he attained to high preeminence. When Malik 

 Shah determined to reform the calendar, Omar was one of the eight 

 learned men required to do it; the result was the Jalili era, 'a com- 

 putation of time,' says Gibbon, 'which surpassess the Julian and ap- 

 proaches the accuracy of the Gregorian style.' He is also the author 

 of astronomical tables, and of a treatise on algebra" (Fitzgerald). 



It is interesting to note that the Bagdad astronomers observed an 

 eclipse of the sun by its reflection in water. The obliquity of the 

 ecliptic for the year 1000 Ibn Yunus found to be 23° 33' (the true 

 value is 23° 34' 16"). The latitude of Cairo he determined to be 



* It may be remarked, in passing, that the foregoing explains how it is 

 that Copernicus and Kepler had such accurate values of the periods of revolu- 

 tion of the diflferent planets. Hipparchus noted, for example, that Mars was in 

 conjunction with a certain star — Sirius for instance, on a certain day. Tycho, 

 1700 years later, observed that Mars was again in conjunction with Sirius on 

 a certain day, at a certain hour. In the seventeen centuries that had elapsed, 

 Mars had made about 860 revolutions. The interval of time between the two 

 epochs, divided by the number of revolutions, gave the time of revolution with 

 great exactness. On the other hand, the distance of Mars from the sun was only 

 roughly known, even to Kepler. Of the dimensions of the planets nothing was 

 known until their apparent angular diameters had been measured with the 

 telescope. Anaxagoras held that the sun was about the same size as the 

 Peloponesus. 



