876 Proceedings of Philosophical Societies. [Nov. 



it easier and far more varied solutions ; and they invented a 

 number of dials, both fixed and portable. Aboul-llassan 

 deduced from the conic sections very curious rules for tracing 

 the arcs of the signs independently of the hour lines. Neverthe- 

 less his method had neither the simplicity nor the generalisation 

 which it was possible to give to his needlessly complicated 

 problems. He did not see that in every dial that can be 

 described on a plane, the parameter of the section is always the 

 same, being equal to twice the co-tangent of the declination of 

 the sun, and that this depended, neither on the section of the 

 cone, nor on the height of the pole upon the plane. No one as 

 yet had perceived this theorem ; no one had dreamed of dispos- 

 ing the general equation of the conic sections in a manner pecu- 

 liarly adapted to dialling. This equation was at length published, 

 depending only on the height of the pole, the declination of the 

 sun, and lastly on the height of the gnomon, which it is conve- 

 nient to assume as unity. There results from this formula an 

 extremely simple mechanical method of describing the arcs of 

 the signs, the trigonometrical calculation of which is always 

 very long, and also requires the horary angle of the plane. 

 Aboul-Hassan first made the curious remark that every plane 

 may be considered as the horizon of a place, whose geographical 

 longitude and latitude might be easily determined : he first gave 

 a projection of the pole on the plane, and consequently a point 

 common to all the equinoctial hour lines : he also first spoke of 

 substituting these equal hours in place of the ancient unequal 

 hours, of which the Arabs then made use in imitation of the 

 Chaldeans and Greeks. 



Dialling, on passing into Europe, underwent other changes ; 

 instead of the straight style, an axis was substituted, the entire 

 shadow of which covered successively all the hour lines; instead 

 of the arcs of the signs, mechanical constructions were invented, 

 which, being reduced to formulae, are found to be identically the 

 same as the modern methods. But the Europeans demonstrated 

 nothing, neither did the Arabians : their works are frequently 

 unintelligible ; and in order to demonstrate their obscure prac- 

 tices, their historian has found it necessary to include in about 

 60 trigonometrical formulae every thing that concerns this part 

 of astronomical knowledge; he finds among them the demonstra-* 

 tions which Clavius was not able to discover, and also draws from 

 them new methods of tracing, either mechanically, or by calcula 

 tion, the arcs of the signs, the meridian line of mean time, and 

 all the hour lines of dials without any centre, whether Babylo- 

 nian, Italian, or French. 



lhe Persians and tartars showed the same respect to the 

 Greek theory ; they adopted the Arabian trigonometry ; they 

 improved our knowledge of the motion of the sun; and v.e owe 

 to the latter a new catalogue of the stars; namely, that of Ulugh- 

 beigh. The first European astronomers were equally servile 



