III. If Two Sides of a Triangle be given, with an Angle oppofite to one of them, the Angle oppofite to the other of them is alfo Given.

Theorems Extraordinary.

I. If in a Circle Two Right Lines be infcribed, 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.

IHI. If in a Circle Three Right Lines fhall be infcribed, one of them cutting the other Two: Then the Rectangles of the Segments of each Line, fo cut, are directed proportional to the Rectangles of the refpective Segments of the Cutter.

IV. If a Plain Triangle be infcribed in a Circle, the Angles are one half of what their oppofite Sides do fubtend: And if it hath one Right Angle, then the longeft Side of that Triangle fhall be the Diameter of the Circle.

V. If in a Circle, any Plain Triangle be infcribed, 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; fo is the other adjacent Side, to the Diameter of the cir cumfcribing Circle.

The End of the First Part.

JANUA

THE

GATE

ΤΟ ΤΗΕ

Mathematical Sciences

OPENED.

PART II.

O F

TRIGONOMETRY.

WHEREΙΝ

The Doctrine of the Dimension of PLAIN and SPHERICAL TRIANGLES is Succinctly Handled, Geometrically Demonstrated,

And

Arithmetically,

Geometrically, Performed.
Inftrumentally,

By William Leybourn, Philomathemat.

LONDON:

Printed Anno Domini, MDCCİV.