Elements of geometry / from the Latin translation of Commandine. To which is added, a treatise of the nature of arithmetic of logarithms, likewise another of the elements of plain and spherical trigonometry; with a preface ... By Doctor John Keil ... The whole revised ... [with a] preface, by Samuel Cunn.

  • Euclid
Date:
1741
Catalogue details

Licence: Public Domain Mark

Credit: Elements of geometry / from the Latin translation of Commandine. To which is added, a treatise of the nature of arithmetic of logarithms, likewise another of the elements of plain and spherical trigonometry; with a preface ... By Doctor John Keil ... The whole revised ... [with a] preface, by Samuel Cunn. Source: Wellcome Collection.

    361/465 (page 310)
    3io PROPOSITION XXXV. I - • • - • cHhe Sines of the Sides B C, DC, are proportioned to the Sines of the oppofite Angles D and B. BEcaufe by the 29th of this, S, B C : R :: S, C A: S, of the Angle B. And by the fame, inverting R : S, D C':: S, Angle D : S, of C A • whence, by Equality of perturbate Ratio, S, B C : S, D C ;; S, D : S, B- PROPOSITION XXXVI. In any fpherical Triangle ABC, the Kell angle CF x AE, or FM x AE, contained under the Sines of the Legs, BC, BA, is to the Square of the Ra¬ dius, aslLor I A—~LAtheDijferenceof theverf- ed Sines of the Bafe C A, andthe Difference of the hegs AMj^GNj/fo verfed S inc of the Angle B, LET a great Circle PN be deferibed from the Pole Band let BP, BN, be Quadrantsj and then PN is the Meafure of the Angle B; alfo deferibe from the fame Pole B a lefTer Circle CFM thro’ C • the Planes of fhefe Circles fhall be perpendicular to the Plain B ON (by the id of this). And P G, C H, being perpendicular in the fame Plain, fall on the common Sections ON, FM- fuppofe in G, H. Again draw H I, perpendicular to AO ; and then the Plain draw thro’ CH, HI, fhall be perpendicular to the Plain A O B. Whence A I, which is perpendicu¬ lar to B I, will be perpendicular to the Right Line CI (by Def 4, El. 11.) • and lo AI is the verfed Sine of the Arc AC, and AL the verfed Sine of the Arc A M = B M — B A = B C — BA. The Ifofceles T ri- angles CFM, PON, are equiangular, fince MF? NO, as alfo CF, PO, (by 16. El. 11.) are parallel. Wherefore, if Perpendiculars CH, P G, be drawn to the Sides FM, ON, the Triangles will be divided fimilarly, and we fhall have F M : O N :: MH:GN, Alfo, becaufe the Triangles AOE, PIH, DLM, are equiangular, we fhall have AE : AO :: IL : MH. But