. 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. 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 : but AG is equal to
. 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. 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 : but AG is equal to GD; therefore GD is greater than GH, the less than thegreater, which is impossible. Therefore the straight line whichjoins the points F, G cannot fall otherwise than upon the pointA, that is, it must pass through it. Therefore, if two circles, E. D. PROP. XIL THEOR. IF two circles touch each other externally, thestraight line which joins their centres shall passthrough the point of contact. Let the two circles ABC, ADE touch each other externallyin the point A ; and let F be the centre of the ciix;le ABC, andG the centre of ADE: the straight line which joins the pointsF, G shall pass through the point of contact A. Far, if not, let it pass otherwise, if possible, as FCDG, and. 78 THE ELEMENTS Book III. join FA, AG: and because F is the centre of the circle ABC,*—Y-. AF is equal to FC : also, ^ E because G is the centre of the circle ADE, AG is e- qual to GD: therefore FA, AG are equal to FC, DG; wherefore the whole FG is greater than FA,a AG: but it is also less*; which is impossible : therefore the straight line which joins the points F, G shall not pass otherwise than through the point of contact A, that is, it must pass through it. Therefore, if two circles, Sec. Q. E. D.
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