. 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. le at the centre, and Book III,BAC an angle at the circumference, which have the same cir- <^v^cumference BC for their base ; the angleBEC is double of the angle BAC. First, let E the centre of the circle bewithin the angle BAC, and join AE, andproduce it to F: because EA is equal toEB, the angle EAB i
. 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. le at the centre, and Book III,BAC an angle at the circumference, which have the same cir- <^v^cumference BC for their base ; the angleBEC is double of the angle BAC. First, let E the centre of the circle bewithin the angle BAC, and join AE, andproduce it to F: because EA is equal toEB, the angle EAB is equal ^ to theangle EBA; therefore the angles EAB,EBA are double of the angle EAB; butthe angle BEF is equal to the anglesEAB, EBA; therefore also the angle BEFis double of the angle EAB: for the samereason, the angle FEC is double of the angle EAC: therefore the whole angle BEC is double of thewhole angle BAC. Again, Let E the centre of the A circle be without the angle BDC, andjoin DE, and produce it to G. Itmay be demonstrated, as in the firstcase, that the angle GEC is doubleof the angle GDC, and that GEB apart of the first is double of GDB apart of the other; therefore the re-maining angle BEC is double of theremaining angle BDCr Therefore,the angle at the centre, 8cc, Q. E. PROP. XXL THEOR. THE angles in the same segment of a circle are Sec nequal to one another. Let ABCD be a circle, and BAD,BED angles in the same segmentBAED: the angles BAD, BED areequal to one another. Take F the centre of the circleABCD: and, first, let the segmentBAED be greater than a semicircle,and join BF, FD : and because theangle BED is at the centre, and theangle BAD at the circumference,and that they have the same part of
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
