. 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. D C the centre is in CA. There- fore, if a straight line, Sec. Q. E. D, PROP. XX. THEOR. See N. THE angle at the centre of a circle is double ofthe angle at the circumference, upon the same base,that is, upon the same part of the circumference. OF Let ABC be a circle, and BEC an angle at th
. 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. D C the centre is in CA. There- fore, if a straight line, Sec. Q. E. D, PROP. XX. THEOR. See N. THE angle at the centre of a circle is double ofthe angle at the circumference, upon the same base,that is, upon the same part of the circumference. OF Let ABC be a circle, and BEC an angle 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 Therefo
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
