5, which are the subject Matter of the foregoing Lions ; yet if he hath no other Expedient than zading of their Definitions, and viewing proSchemes, he must have an exceeding happy ever to acquire a perfect Notion of them by this only. therefore it behoves every one, who would any tolerable Proficiency in this part of Know to quality himself, as directed in the Preface to pok, at least fo far as he is capable and hath tunity. are less than the Angles DAB, DB A; Wherefore the Rays of Light D A, D B, approach nearer the Perpendiculars E A, FB, than do the first Rays C A, C B. Suppose at an infinite Distance be removed the Point, or Eye Di Then will the Angle D AB, differ infinitely little from a right one: Confequently the Rays D A, DB, will differ infinitely little from E A, FB, the two Parallel Rays. 2.E.D. THEOREM II. CH A P. 11. ems serving to the Orthographical Protion of the Sphere, called the Ana nma. A Point in the Surface of the Sphere is there projected into a Point, where a perpendicular Ray, passing through the point to be projected, meets with the Plane of the Projection, Demonstration, Suppose the common Sec tion of the Plane of the Projection be AB, and a great Circle of a Sphere be ACB, in which let the given Point be C. Draw the Perpendicular Ray RD, Then 'tis evident by Inspection, the Point in the Sphere C, will be seen, or projected into the Point D, where the Ray of Light RD, falls perpendicular on A B, the common Section of the Plane 2. E. D. DI B THEOREM I. HE Rays of Light by which the Eye, placed at an infinite Distance, beholds an Object, differ infinitely little from Parallel Rays. Demonstration. F Pse any Object AB, n by an Eye first in C, afterwards more remote D; Prop. 1 Lib. Euclid. the ngle ACB, er than the Angle ADB; ently, the Angles AB, GBA, A are THEOREM III. A Right Line, perpendicular to the Plane of the Projection, is there projected into a Point, where it cuts the said Plane. Demonstration. R C be CD, ?Tis evident, an Eye D D placed in R 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. Q.E.D. THEOREM IV. A Right Line Parallel, or Oblique, to the Plane of the Projection, will be projected into a Right Line. Demonstration. 208OOD and the given Right Line be CD; B E999999F D B E999999F that all the intermediate Points 0, 0, 0, 0, 0, &c. will be projected into the Points 9,9, 9, 9, 9, &c. THEOREM V. Demonstration. Plane be EH, A. B will be projected into the Line EH, D} will be projected into the Line FG. E F G H and the Parallel Lines A B, EH, the Angles E AB, AB H, are right ones also ; so 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; will fall within the Perpendiculars A E, and B H, and consequently the Line EH, is greater than the Line FG. 2. E. D. Corollary. Hence a right Line parallel to the Plane of the Projection, is projected into a right Line equal to it self ; and a Line oblique to the said Plane, into one that is less than it self. THE 1 UB 2, Plane AB ; THEOREM VI. A Semicircle standing at right Angles with the Plane of the Projection, is projected into that right Line ( viz. its Diameter) in which it cuts the said Plane, Demonstration. Let the Semicircle be A0B, from which let fall the Perpendiculars 09, 09, 09, oq, &c. to the Section of the 999 7777 Then because the given Semicircle A D is at right Angles to the Plane, by Hypothesis ; so 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 9,979, 9, 9, &c. which are in the common Section AB. Consequently the whole Curve A o B, will be projected into the right Line AB, in which it cuts the Plane of the Projection. 2.E.D. Corollary, Hence also it follows, that a Circle standing at right Angles with the Plane of the Projection, is also projected into a Right Line equal to its Diameter, since 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 Versed Sine of the said Arch. bo, co, which is the Right Sine of the Arch Ao, A Circle parallel to the Plane of the Projection, is projected into a Circle equal to it self in the said Plane. D. at Demonstration. This Proposition is self evident to any one who understands what has been hitherto said ; 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 Ellipsis on the said Plane. t Demon |