. 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. of the circle: thatis, the angle FBD is equal to the angle which is in the segmentDAB, and the angle DBE to the angle in the segment BCD. * a 11. 1. From the point B draw ^ BA at right angles to EF, and takeany point C in the circumference BD, and join AD, DC, CB;and because the straight line EF touches t
. 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. of the circle: thatis, the angle FBD is equal to the angle which is in the segmentDAB, and the angle DBE to the angle in the segment BCD. * a 11. 1. From the point B draw ^ BA at right angles to EF, and takeany point C in the circumference BD, and join AD, DC, CB;and because the straight line EF touches the circle ABCD inthe point B, and BA is drawn at ^ right angles to the touching linefrom the point of contact B, the b 19. 3. centre of the circle is ^ in BA ;therefore the angle ADB in a c 31. 3. semicircle is a rights angle, andconsequently the other two angles A 32. h BAD, ABD are equal d to a rightangle: but ABF is likewise aright angle ; therefore the angleABF is equal to the angles BAD,ABD : take from these equals thecommon angle ABD ; therefore the remaining angle DBF is equalto the angle BAD, which is in the alternate segment of the circle ;and because ABCD is a quadrilateral figure in a circle, the oppo- e 22 S. site angles BAD, BCD are equal« to two right angles ; therefore. OF EUCLID. 93 the angles DBF, DBE, being likewise equal* to two right angles, Book equal to the angles BAD, BCD ; and DBF has been proved *—v—equal to BAD : therefore the remaining angle DBE is equal to^^^. angle BCD in the alternate segment of the circle. Where-fore, if a straight line, Stc. Q. E. D. PROP. XXXIII. PROB. UPON a given straight line to describe a segment ^^^ ^of a circle, containing an angle equal to a given rec-tilineal angle. Let AB be the given straight line, and the angle at C thegiven rectilineal angle; it is required to describe upon the givenstraight line AB a segment of a circle, containing an angle equalto the angle C. First, Let the angle at C be aright a
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
