MATHEMATICS. 325 



this treatise, Apollonius wrote others on several very general and 

 difficult problems of geometrical analysis, which he pursued into 

 all their detail of cases. Their titles and subjects are given us by 

 Pappus. Many of them have since exercised the ingenuity of the 

 most skilful of modern mathematicians. For instance, the problem of 

 4 Tactions,' of which the most difficult case is to draw a circle touch- 

 ing three given circles, has been solved by Vieta and Newton. ' The 

 Section of Ratio ' and ' The Section of Space ' have been restored by 

 Halley. This problem is to draw a line through a given point, 

 cutting segment from two given straight lines : in the first place so 

 that they may have a given ratio ; in the next place so that they may 

 contain a given rectangle. In * The Determinate Section ' it was 

 required to find a point in a straight line, such that the rectangles of 

 its distances from given points should have a given ratio : this w T as 

 resolved by Dr. Simson. The problem of ' Inclinations ' proposed 

 to draw through a given point a straight line, so that a given portion 

 of it should be intercepted between two given straight lines. Some 

 of these problems had been solved by Euclid, and Pappus blames 

 Apollonius for the harsh manner in which he speaks of the solution 

 of his predecessor, which did not pretend to be complete. 



Like the other mathematicians of his time he also applied to 

 astronomy, as we learn from his having, like Eratosthenes, a sobriquet 

 derived from a Greek letter. He was called Epsilon (e) from his 

 perpetual attention to the moon, which resembled the form in which 

 that letter was written. After his time, the principal progress of 

 Greek mathematicians was made in astronomy. Either that the 

 powers of the Greek geometry had reached their limit, or that 

 inventive genius became more scarce, succeeding generations con- 

 tented themselves almost entirely with commenting upon what had 

 been done by the giants in geometry who were the first race. Thus 

 Hypatia the daughter of Theon, Pappus, Serenus, and Eutocius wrote 

 commentaries upon Apollonius ; Eutocius also upon Archimedes ; 

 Theon upon Euclid : and it is from such of these as are still extant 

 that much of the preceding information is derived. They Had means 

 of knowledge which have since been lost; and we might have been 

 able to give a much more complete and accurate account of the 

 extensive series of inventions which the old mathematics exhibited, 

 if time had spared the histories of this science by Theophrastus and 

 Eudemus, from which later writers seem to have drawn the light 

 whose scattered rays reflected from them we have been attempting to 

 collect. 



translation from the Arabic in 1661. The comparison of the conjectural with the 

 ancient Apollonius is to the credit of both. Viviani's propositions are more varied 

 and extensive, but perhaps those of the ancient geometer are more recondite and 

 difficult 



