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time, when more accurate observations had been 

 made, the necessity of introducing epicycle on 

 epicycle, in order to bring the theory into accord- 

 ance with facts, made it very complicated. But De 

 Morgan has acutely observed that in so far as the 

 ancient astronomers supposed that it was necessary 

 to resolve every celestial motion into a series of 

 uniform circular motions they erred greatly ; but 

 that if the hypothesis be regarded as a convenient 

 way of expressing known facts, it is not only 

 legitimate but convenient. It was as good a theory 

 as, with their instruments and knowledge, it was 

 possible to frame, and, in fact, corresponds to the 

 expression of a given function as a sum of sines or 

 cosines, a method which is of frequent use in modern 

 analysis." (" A Short Account of the History of 

 Mathematics," p. 91.) No farther advance in the 

 theory of astronomy was made until the time of 

 Copernicus, whose book appeared in the year of his 

 death, 1543. It is very noteworthy that Ptolemy, 

 in the second book of his astronomy/ pointed out 

 that the explanations would be much simplified if 

 the earth were supposed to rotate on its own axis 

 once a day ; but he said that this was inconsistent 

 with known facts, whereby he meant that it could 

 not be brought into consistent relation with the 



* This work is generally known as the " Almagest," which is said 

 to be derived from an Arabic corruption of the Greek 



