. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. egment b 11. ; and they are upon equal straight lines BC, EF; but simi- segments of circles upon equal straight lines are equal <= to * one another: therefore the segment BAC is equal to the segment £. 94 EDF : but the whole circle ABC is equal to the whole DEF; ^ M 90 THE ELEMENTS Booklll
. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. egment b 11. ; and they are upon equal straight lines BC, EF; but simi- segments of circles upon equal straight lines are equal <= to * one another: therefore the segment BAC is equal to the segment £. 94 EDF : but the whole circle ABC is equal to the whole DEF; ^ M 90 THE ELEMENTS Booklll. therefore-the remaining segment BKC is equal to the remaining^^yi^ segment ELF, and the circumference BKC to the circumferenceELF. Wherefore, in equal circles, &;c. Q. E. D. PROP. XXVn. THEOR. IN equal circles, the angles which stand upon equalcircumferences are equal to one another, whether theybe at the centres or circumferences. a 20. 3. Let the angles BGC, EHF at the centres, and BAG, EDF atthe circumferences of the equal circles ABC, DEF stand uponthe equal circumferences BC, EF: the angle BGC is equal tothe angle EHF, and the angle BAG to the angle EDF. If the angle BGC be equal to the angle EHF, it is manifest *that the angle BAG is also equal to EDF: but, if not, one of. them is the greater: let BGC be the greater, and at the pointb G, in the straight line BG, make^ the angle BGK equal to thec26. 3. angle EHF; but equal angles stand upon equal circumferences Swhen they are at the centre; therefore the circumference BK isequal to the circumference EF: but EF is equal to BC; there-fore also BK is equal to BC, the less to the greater, which isimpossible: therefore the angle BGC is not unequal to the an-gle EHF; that is, it is equal to il: and the angle at A is half ofthe angle BGC, and the angle at D half of the angle EHF : there-fore the angle at A is equal to the angle at D. Wherefore, inequal circles, S^c Q- E. D.
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
