observer; a sound mathematician; an expert and careful computer; and he introduced marked improvements in the methods of calculation. 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, b, 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 observations with those determined by Hipparchus or Ptolemy, he obtained 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.[1]
"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 computation of time,' says Gibbon, 'which surpassess the Julian and approaches 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 revolution of the different 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.
