IV. Geometrical Theorems, and Formulce, particularly applicable to some 

 Geodetical Problems. By William Wallace, A.M. F.R.S. Edin., 

 F.R.A.S. Lond., Member of the Cambridge Philosophical Society, 

 and Professor of Mathematics in the University of Edinburgh. 



[Read Nov. 30, 1835.] 



Art. 1. The Geometrical Theorems and Trigonometrical Formulae 

 which are given in this paper are peculiarly applicable to the solution 

 of some Geodetical Problems, in particular to this which follows. 



" Three stations being given in position, or else the angles made 

 by the lines which join them ; also the angles which these lines sub- 

 tend at a fourth station in the plane of the others; to determine the 

 position of that fourth station." 



This problem is remarkable on account of its antiquity, and the 

 object to which it was applied. Hipparchus made use of it to deter- 

 mine the position of the INIoon's apogee and the radius of her epicycle, 

 and Ptolemy actually resolved it by a trigonometrical computation in 

 his Mathematical Syntaxis*. Vieta has given a geometrical construction 

 in his Apollonius Gallusf. He had in view the solution of Hipparchus' 

 problem ; but the fiction of epicycles being now rejected, Ptolemy's 

 application of the problem is merely an interesting fact in the history 

 of the ancient Astronomy. 



• Histoire de I'Astronoraie Ancienne, par Delambre. T. II. p. 150—164. 

 t Vietae Opera Mathematica, p. 344. Edit. 1646. 



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