. 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. a 9. 3. PROP. X. THEOR. ONE circumference of a circle cannot cut anotherin more than two points. If it be possible, let the circumfe-rence FAB cut the circumferenceDEF in more than two points, B, G, F; take the centre K of thecircle ABC, and join KB, KG, KF:and because within the circle DEFthere is
. 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. a 9. 3. PROP. X. THEOR. ONE circumference of a circle cannot cut anotherin more than two points. If it be possible, let the circumfe-rence FAB cut the circumferenceDEF in more than two points, B, G, F; take the centre K of thecircle ABC, and join KB, KG, KF:and because within the circle DEFthere is taken the point K, fromwhich to the circumference DEFfall more than two equal straightlines KB, KG, KF, the point K is». OF EUCLID. 77 the centre of the circle DEF: but K is also the centre of the Book ABC; therefore the same point is the centre of two cir ^ .r-mjcles that cut one another, which is impossible. Therefore one b5. of a circle cannot cut another in more than twopoints. Q. E. D. PROP. XL THEOR. IF two circles touch each other internally, thestraight line which joins their centres being producedshall pass through the point of contact. Let the two circles ABC, ADE touch each other internallyin the point A, and let F be the centre of the circle ABC, andG the centre of the circle ADE; the ^ straight line which joins the centresF, G, being produced, passes throughthe point A. For, if not, let it fall otherwise, if pos^sible, as FGDH, and join AF, AG: and because AG, GF are greater^ than |y ^ 7?\^ / i ^, that is, than FH, for FA is equal toFH, both being from the same centre;take away the common part FG; there-fore the remainder AG is greater thanthe remainder GH
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
