340 PHILOSOPHICAL TRANSACTIONS. [aNNO l66g. 



means of the subsequent dividend and divisor, it may certainly be known 

 whether the figure so assumed be too great or too little. 



After one root is obtained, the methods of Hudden and others will depress 

 the equations, so as to obtain more, and consequently all of them. 



The author of this narrative considers that the conic sections may be pro- 

 jected from lesser circles of the sphere, and thence easily described by points ; 

 and that by their intersections some spheric problem is determined. Accord- 

 ingly he found that this following problem, according to the various situation of 

 the eye, and of the projecting plain, would take in all cases. 



The distances of an unknown star being given from two stars of known de- 

 clination and right ascension; the declination and right ascension of the un- 

 known star is required. 



And he observes that, admitting the mechanism of dividing the periphery of 

 a circle into any number of equal parts, or, which is equivalent, the use of a line 

 of cords, that this problem, wherever the eye be placed, may be resolved by 

 plain geometry, and yet the eye be so situated as to determine it by the inter- 

 sections of the conic sections; consequently these points of intersection, the 

 species and position of the figures being given, may be found without describ- 

 ing any more points than those sought; and the lengths of ordinates falling 

 from thence on the axes of either figure be calculated by mixed trigonometry; 

 and hence likewise the roots of all cubic and biquadratic equations be found by 

 trigonometry. For having from the Mesolabe of Sluse the scheme that finds 

 these roots, it will then be required to fit those sections into cones, which have 

 their vertex either in the centre, or an assigned point in the surface of the 

 sphere, to which they relate as projected, and proceed to the resolution of 

 the problem proposed : and how to fit in those sections, see the seven books of 

 Apollonius or of Mydorgius, the third volume of Descartes's Letters, or Leo- 

 taudi Geometrica practica, or Andersonii Exercitat. Geometricae. 



As to the problem itself, it is determined on the sphere by the intersections 

 of the two lesser circles of distance, whose poles are the known stars. And this 

 problem has divers geometric ways of solution. As, 



1 . By plain geometry ; supposing a plain to touch the sphere at the north- 

 pole, if the eye be at the south-pole, projecting those circles on the said plain, 

 they are still circles, by reason of the sub-contrary sections of the visual cones 

 whose centres fall in the sides of the right-lined angle, made by the projected 

 meridians that pass through the known stars ; and thus the problem is easily 

 solved. 



2. By conic geometry; In one case it may be done, by placing the eye at the 

 centre of the sphere, and projecting as before; viz. when the longer axes of the 



