. 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. en pratio to D. Because EB has a given ratio to C, and theexcess of C above a given magnitude has a giv-en ratio to D ; therefore d the excess of EBabove a given magnitude has a given ratio toD : let this given magnitude be EF : thereforeFB the excess of EB above EF has a given ra- ^ C Dtio to D : and AF
. 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. en pratio to D. Because EB has a given ratio to C, and theexcess of C above a given magnitude has a giv-en ratio to D ; therefore d the excess of EBabove a given magnitude has a given ratio toD : let this given magnitude be EF : thereforeFB the excess of EB above EF has a given ra- ^ C Dtio to D : and AF is given, because AE, F^ DATA. 387 are given : therefore FB the excess of AB above a given mag-nitude AF has a given ratio to D, PROP. XXVIII. 25. IF two lines given in position cut one another,the point or points in which they cut one another aiegiven. Let two lines AB, CD given in position cut one another inthe point E ; the point E is gi- C ven. Because the lines AB, CDare given in position, they havealways the same situations, andtherefore the point, or points,in which they cut one anotherhave always the same situation :and because the lines AB, CD canbe found a, the point, or points,in which they cut one another,are likewise found; and thereforeare given in position a. See a 4. def. PROP. XXIX. 26. IF the extremities of a straight line be given in po-sition ; the straight line is given in position and mag-nitude. Because the extremities of the straight lin& are given, theycan be found a ; let these be the points A, B, between which a 4. straight line AB can be drawnb ; b 1 Post- this has an invariable position, be- A ? ? -B ulate cause between two given points there can be drawn but one straight line: and when the straightline AB is drawn, its magnitude is at the same time exhibited^or given: therefore the straight line AB is given in positiopand magnitude. J8b EUCLIDS gr. PROP. XXX, IF one of the extremities of a straight Hne given inposition and magnitude b
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