are less than the Angles DAB, DBA; Wherefore the Rays of Light DA, DB, approach nearer the Perpendiculars E A, FB, than do the first Rays CA, C B. Suppose at an infinite Distance be removed the Point, or Eye D; Then will the Angle D AB, differ infinitely little from a right one: Confequently the Rays DA, DB, will differ infinitely little from E A, FB, the two Parallel Rays. Q, E.D. THEOREM II. A Point in the Surface of the Sphere is there projected into a Point, where a perpendicular Ray, paffing through the Point to be projected, meets with the Plane of the Projection. Draw the Perpendicular Ray RD, B Then 'tis evident by Infpection, the Point in the Sphere C, will be feen, or projected into the Point D, where the Ray of Light RD, falls perpendicular on AB, the common Section of the Plane. 2. E. D. VOL. H 7 C 2 THE THEOREM III. A Right Line, perpendicular to the Plane of the Projection, is there projected into a Point, where it cuts the faid Plane. precisely over the Right Line CD, will perceive no more thereof, than what covers the Point D in the common Section AB; Into which Point it is therefore projected. 2.E.D. THEOREM IV. A Right Line Parallel, or Oblique, to the Plane of the Projection, will be projected into a Right Line. Demonftration. Let the common Section of the Plane be AB, and the given Right Line be CD; By Theo. 2. the Points C, D, will be projected into the Points E, F, in the Section of the Plane A B. And 'tis plain, by Theo. 3. that all the intermediate Points 0, 0, 0, 0, 0, &c. will be projected into the Points q, q, q, q, q, &c. The Projection of any right Line is then greatest, when 'tis parallel to the Plane of the Projection. Demonftration. Let the Section of the Plane be EH, then (by Theor. 4.) the will be projected into the and the Oblique Line CD Now because of the Right A B, EH, the Angles E AB, ABH, A B D EF GH are right ones alfo ; fo E A, HB, will touch the Circle in A, and B ; Wherefore all other Lines CF, DG, drawn perpendicular to the Line E H, or AB; is greater than the Line FG. 2, E. D. Corollary. Hence a right Line parallel to the Plane of the Pro jection, is projected into a right Line and a Line oblique to the faid Plane, lefs than it felf. equal to it self; into one that is THE THEOREM VI. A Semicircle ftanding at right Angles with the Plane of the Projection, is projected into that right Line (viz. its Diameter) in which it cuts the faid Plane. Demonftration. Let the Semicircle be Ao B, from which let fall the Perpendiculars oq, oq, oq oq, &c. to the Section of the Plane AB; Then because the given Semicircle Ao D is at right Angles to the Plane, by Hypothesis; fo therefore are the Perpendiculars oq, oq, oq, &c. Hence (by Theor. 2.) all the Points 0, 0, 0, 0, 0, &c. will be projected into the Points q, q, q, q, q, &c. which are in the common Section A B. Confequently the whole Curve A o B, will be projected into the right Line AB, in which it cuts the Plane of the Projection. QE.D. Corollary, Hence alfo it follows, that a Circle ftanding at right Angles with the Plane of the Projection, is alfo projected into a Right Line equal to its Diameter, fince the Extremities, or Diameter, of a Semicircle and the Whole Circle are all one, THEOREM VII. Any Arch of a Circle at Right Angles with the Plane of the Projection, is projected into the Right Sine of that Arch; and the Complemental Arch is projected into the Verfed Sine of the faid Arch. A Circle parallel to the Plane of the Projection, is projected into a Circle equal to it felf in the faid Plane. Demonftration. This Propofition is felf evident to any one who understands what has been hitherto faid; and will rather be obscured than made clearer by any Geometrical Diagram or Demonstration. THEOREM IX. A Circle oblique to the Plane of the Projection, is projected into an Ellipfis on the faid Plane. Demon |