1819.] Royal Academy of Sciences. 377 



imitators. Alphonso caused new tables to be calculated, which 

 Were employed for 200 years. Regiomontanus declared several 

 times that they were erroneous ; but he had neither time nor the 

 means to compose better. He engaged much in trigonometry, 

 but did not proceed so far as Aboul-Wesa, or Ebn-Jounis ; he 

 did not perceive the usefulness of tangents, the formulae for 

 which he found in Albategnius. Our trigonometrical system 

 was completed by Vieta, who published a table in which the 

 tangents and secants were for the first time united with the sines, 

 and who afterwards published the four analytical formulae which 

 are sufficient for the solution of every case of oblique-angled 

 spherical triangles. To him we owe some curious and very 

 useful formulae for tangents, secants, and even for sines ; he 

 created that theory which he called angular sections ; lastly, 

 from his theorems, although frequently very obscure, the expres- 

 sion of the first and second differences of sines may be drawn, 

 and more expeditious and surer methods for the construction of 

 trigonometrical tables than those he has himself mentioned. 



We have only been able to indicate, in a very brief manner, 

 the subjects that are treated in this astronomy of the middle ages, 

 which also includes an analysis of all the works that are any 

 wavs remarkable, which appeared in this interval of more than 

 600 years, ending with the appearance of Copernicus. This 

 restorer of astronomy, with Tycho, Kepler, and Galileo, will 

 furnish abundant matter for the first volume of Modern Astro- 

 nomy, which is in the press. 



We mentioned in the History of 1817, the obligations we were 

 under to M. Sedillot for what concerns the Arabs and the 

 Tartars. 



Notice on the Operations which are undertaken to determine the 

 Figure of the Earth ; by M. Biot. — This hasty sketch* of every 

 thing that has been done for this 150 years, in regard to deter- 

 mining the figure of the earth, was listened to with great 

 interest in the public meeting of the four Academies. The 

 description of the Shetland islands, so new to us, where the 

 author charged with these delicate and difficult operations 

 found the most cordial hospitality and every attention that could 

 ensure the success of his enterprise, was particularly applauded. 

 The whole of this recital will be found in our Memoirs. We 

 confine ourselves at present to that which constitutes the 

 object of the voyage, and announce with pleasure that M. Biot 

 has brought home with him 38 series of observations on the 

 pendulum, of five or six hours each; 1,400 observations of 

 latitude in 55 series, taken as many to the north as to the south 

 of the zenith, and about 1200 heights of the sun, to determine 

 the going of the clock. Such part of the calculations as time 

 hstt permitted to be executed proves that the results will be 

 conformable to those already deduced from the theory of the 



