III. If Two Sides of a Triangle be given, with an Angle oppofite to one of them, the Angle opposite to the other of them is alfo Given. Theorems Extraordinary. I. If in a Circle Two Right Lines be inscribed, cutting each other, the Rectangles of the Segments of each Line, are equal: And the Angle at the Point of Interfection, is measured by the half Sum of its intercepted Arches. II. If to a Circle Two Right Lines be adfcribed from a Point without, the Rectangles of each Line from the Point affigned, to the Convex and Concave are equal: And the Angle at the affigned Point is measured by the half Difference of its intercepted Arches. III. If in a Circle Three Right Lines shall be inscribed, one of them cutting the other Two: Then the Rectangles of the Segments of each Line, so cut, are directed proportional to the Rectangles of the respective Segments of the Cutter. IV. If a Plain Triangle be inscribed in a Circle, the Angles are one half of what their opposite Sides do fubtend: And if it hath one Right Angle, then the longest Side of that Triangle shall be the Diameter of the Circle. V. If in a Circle, any Plain Triangle be inscribed, and a Perpendicular be let fall upon one of the Sides, from the oppofite Angular Point. Then, as the Perpendicular, to one of the adjacent Sides; so is the other adjacent Side, to the Diameter of the cir cumscribing Circle. ! | The End of the First Part. JANUA THE GATE TO THE Mathematical Sciences OPENED. PART II. OF TRIGONOMETRY. WHEREIN The Doctrine of the Dimension of PLAIN and SPHERICAL TRIANGLES is Succinctly Handled, Geometrically Demonstrated, And (Arithmetically, { Geometrically, Performed. By William Leybourn, Philomathemat. LONDON: Printed Anno Domini, MDCCİV. |