. 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. rior angle of the triangleABC, is equal ^ to the two anglesABC, ACB; therefore the angleBAC is equal to the angle FAC,and each of them is therefore arights angle : wherefore the angleBAC in a semicircle is a rightangle. And because the two anglesABC, BAC of the triangle ABC are together less<i than two
. 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. rior angle of the triangleABC, is equal ^ to the two anglesABC, ACB; therefore the angleBAC is equal to the angle FAC,and each of them is therefore arights angle : wherefore the angleBAC in a semicircle is a rightangle. And because the two anglesABC, BAC of the triangle ABC are together less<i than two d angles, and that Bx\C is a right angle, ABC must be lessthan a right angle; and therefore the angle in a segment ABCgreater than a semicircle, is less than a right angle. And because ABCD is a quadrilateral figure in a circle, anytwo of its opposite angles are equal* to two right angles ; there- e 22. the angles ABC, ADC are equal to two right angles; andABC is less than a right angle; wherefore the other ADC isgreater than a right angle. Besides, it is manifest, that the circumference of the greatersegment ABC falls without the right angle CAB, but thecircumference of the less segment ADC falls within the rightangle CAF. And this is all that is meant, when in the. b c 1. 94 THE ELEMENTS Book III. Greek text, and the translations from it, the angle of the greater??•V-—^ segment is said to be greater, and the angle of the less segment is said to be less, than a right angle. Cor. From this it is manifest, that if one angle of a triangle be equal to the other two, it is a right angle, because the angle adjacent to it is equal to the; same two; and when the adjacent angles are equal, they are right angles. PROP. XXXII. THEOR* IF a straight line touches a circle, and from thepoint of contact a straight line be drawn cutting thecircle, the angles made by this line with the line touch-ing the circle, shall be equal to the angles which arein the a
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
