and AC to DE) then fhall the other Sides BC, CE, of thofe Triangles be in the fame freight Line. = = For the Ang. Aa ACDa = D; and AB': AC: DC: DE. therefore the Ang, B DCE.. therefore the Ang. B+ Ad ACE. But the Ang. B+A+ ACB 2 right Ang.Whence the Ang. ACE + ACB two right Ang. Therefore BCE is a right Line. Q. E. D. e d PROP. XXXIII. In equal Circles DBCA, HFGP, the Angles BDC, FHG, have the fame Proportion as the Circumferences of the Circles BC, FG, on which they ftand, whether the Angles be at the Centres (as BDC, FHG) or at the Circumferences A, E; and fo likewife are the Sectors BDC, FHG, as being at the Centres. f 28.3. 8 27.3. Draw the right Lines BC, FG, and apply CI CB; and GL FG = LP; and join DI, HL, HP. g The Arch BCCI, alfo the Arches FG, GL, LP are equal; therefore the Ang. BDC CDI, and the Ang. FHG = GHL = LHP. Whence the Arch BI is the fame Multiple of the Arch BC, as the Ang. BDI is of the Ang. BDC. In like manner the Arch FP is the fame Multiple of the Arch FG as the Ang. FHP is of the Ang. FHG. But if the Arch BI be be, a 27. 3. or than FP; in like manner a fhall the Ang. BDI be,=, or than FHP. therefore the Arch BC: FG :: Ang. BDC: FHG:: 6 def. 5. BDC FHG 2 2 ::AE QE. D. e = b CNI; and there- с 15.5. d 20.3. e 27. 3. f 24. 3. 4. 1. 8 h 2 ax. Again, the Ang. BMC fore the Segment BCM Triang. BDC CDI; BDCMh CDIN. In like manner the Sectors FHG, GHL, LHP, are equal; therefore fince when the Arch BI is,, or than FGP, the Sector BDI is likewife,, or than : 6 def. 5. FGP. therefore fhall the Sector BDC: FHG :: i the Arch BC: FG. Q. E. D. gle. CORO L. Hence, (1.) As Sector to Sector, fo is Angle to An (2.) The Angle BDC at the Centre is to four right Angles, as the Arch BC on which it ftands, is to the whole Circumference. For as the Ang. BDC to one right Angle, fo is the Arch BC to a Quadrant; therefore BDC is to four right Angles as the Arch BC to four Quadrants, that is, to the whole Circumference. Alfo the Ang. A two right Ang. :: Arch BC: Periph. Hence, (3.) The Arches, IL, BC, of unequal Circles which fubtend equal Angles, whether they be at the Centre, as IAL, BAC, or at the Periphery, are fimilar. For IL: Periph.:: Ang. IAL (BAC): four right Ang. Alfo the Arch. BC: Periph. Ang. BAC : four right Ang. Therefore IL: Periph.:: BC i BC: Periph. And fo the Arches IL, and BC are fimilar. Whence, B (4.) Two Semidiameters AB, AC, cut off fimilar Arches IL, BC, from concentrick Peripheries. End of the Sixth Book: E U V. The Inclination of a Right Line AB to a Plane CD, is the acute Angle ABE contain'd under that Line, and the Line BE, drawn in the Plane from that end B of the inclining Line, which is in the Plane to the Point E, where a Right Line AE falls from the other End A of the inclining Line perpendicular to the faid Plane. VI. The Inclination of a Plane AB to a Plane D CD is the acute Angle FHG contain'd under the Right Lines FH, GH, drawn in both the Planes AB, CD to the fame Point H of the common Section BE, ma king the Right Angles FHB, FHE. VII. Planes are faid to be inclined fimilarly, when the faid Angles of Inclination are equal. VIII. Parallel Planes are fuch, which being produced, never meet. IX. Similar folid Figures are fuch that are con tained under equal Numbers of fimilar Planes. |