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.

    118/465 (page 87)
    > ' ■ * \ *'■*» > * Book III. Euclid's Elements. a Semicircle, becaufe the Centre thereof is without the fame. But if the Angle A B D be equal to the Angle BAD: then if AD be made equal to BD, or DC, the three Right Lines AD, DB, DC, are equal between themfelves, and D will be the Centre of the Circle to be defcribed • and the Segment ABC is a Semi¬ circle. But if the Angle A B D is lefs than the Angle B A D, let the Angle BAE be made, at the given Point A with the Right Line B A, within the Segment ABC, equal to the Angle A B D. Then the Point E, in the Right Line D B, will be the Centre, and A B C a Segment greater than a Semi¬ circle. Therefore a Circle is defcribed, whereof a Segment is given j 'which nwas to he done. PROPOSITION XXVL Theorem. i J# eftal Circles, equal Angles ft and upon equal Circumferences^whether they leat their Centrej, or at their Circumferences. T ET A B C, D E F, be equal Circles, and let B G C, EHF, be equal Angles at their Centres, and, BAC, EDh, equal Angles at their Circumferen¬ ces. I fay, the Circumference BKC is equal to the Circumference ELF. For, let B C, E F, be joined. Becaufe ABC, D E F, are equal Circles, the Lines drawn from their Centres i will be equal. Therefore the two Sides BG, GC, are equal to the two Sides EH, HF; and the Angle G is equal to the Angle at H. Wherefore the Bale I BC is* equal to the Bafe EF. Again, becaufe the *4-1. i Angle at A is equal to that at D, the Segment BAC will be f fimilar to the Segment EDF; and they are iDef-IJ- i upon equal Right Lines B C, E F. But thofe fimilar Segments of Circles, that are upon equal Right Lines, ] are + equal to each other. Therefore theSegment i&oftbh. I BAC will be f equal to the Segment EDF. But f Drf u. j the whole Circle A B C is equal to the whole Circle ; E)EF. Therefore the remainingCircumference BKC G 3 ihall