Page:Philosophical Transactions - Volume 004.djvu/44

 or diminish the unknown Root of Equation, as to render it a whole number (or lesse differing therefrom, than any Error assign'd,) and by Albert Gerards Method of Aliquot parts to find the same, and thereby the Root sought, although it be a Mixt Number, Fraction, or Surd.

Probably this may sympathise with what is promised by the Learned Huddenis in Annexis Geometriæ Cartesianæ. where he saith, he intended not then to publish certain Rules, he had ready; whereof one was, To find out all the irrational Roots both of Literal and Numeral Equations. This must be understood when such Roots are possible; for 'tis certain, there are infinite Equations, whose Roots are noways explicable, either in whole or mixt numbers, Fractions or Surds, and can be no otherwise explain'd, but by a quàm proxime.

9. The Author of this Narrative considering, that the Conick Sections may be projected from lesser Circles placed on the Sphere, and thence easily (otherwise than hitherto hath been handled) described by Points, and that by their Intersections same Spherick Problem is determined, accordingly he found, that this following Problem according to the various Scituation of the Eye, and of the Projecting Plain, would take in all Cases.

The Distances of an unknown Star are given from two Stars of known Declination and Right Ascension; the Declination and Right Ascension of the unknown Star is required.

And saith, he hath observed, that, admitting the Mechanisme of dividing the Periphery of a Circle into any number of equal parts, or (which is equivalent) the Use of a Line of Chords, that this Problem, wherever the Eye be plac'd, may be resolved by Plain Geometry, and yet the Ey shall be so plac'd, as to determin it by the Intersections of the Conick Sections; consequently those Points of Intersection (the Species and Position of the figures being given) may be found without describing any more Points than those sought; and the Lengths of Ordinates falling from thence on the Axes of either figure calculated by mixt Trigonometry, and hence likewise the Roots of all Cubick and Bi-quadratick Equations found by Trigonometry.

For giving from the Mesolabe mention'd the Scheme that finds these Roots, it will then be required to fit those Sections into Cones, which have their Vertex either in the Center; or an assigned point in the Surface of the Sphere, to which they Rh