. 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. .and the circle is less than the square described about it; there- ^-^-v-i-^fore the square EFGH is greater than half of the circle. Divide a circumferences EF, FG, GH, HE each into two equal partsin the points K, L, M, N, and join EK, KF, FL, LG, GM, WH,HN, NE: therefore each of the triangles EK
. 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. .and the circle is less than the square described about it; there- ^-^-v-i-^fore the square EFGH is greater than half of the circle. Divide a circumferences EF, FG, GH, HE each into two equal partsin the points K, L, M, N, and join EK, KF, FL, LG, GM, WH,HN, NE: therefore each of the triangles EKF, FLG, GMH,HNE is greater than half of the segment of the circle it standsin ; because, if straight lines touching the circle be drawn throughthe points K, L, M, N, and parallelograms upon the straightlines EF, FG, GH, HE be completed ; each of the triangles EKF,FLG, GMH, HNE shall be the half a of the parallelogram inwhich it is: but every segment is less than the parallelogram inwhich it is: wherefore each of the triangles EKF, FLG, GMH,HNE is greater than half the segment of the circle which con-tains it: and if these circumferences before named be dividedeach into two equal parts, and their extremities be joined bystraight lines, by continuing to do this, there will at length re-. main segments of the circle which, together, shall be less thanthe excess of the circle EFGH above the space S: because, bythe preceding lemma, if from the greater of two unequal raag^nitudes there be taken more than its half, and from the rem in-der more than its half, and so on, there shall at length remair. amagnitude less than the least of the proposed magnitudes. Letthen the segments EK, KF, FL, LG, GM, MH, HN, NF bethose that remain and are together less tl)an the excess of thecircle EFGH above S: therefore the rest of the circle, viz. thepolygon EKFLGMHN, is greater than the space S. Describelikewise in the circle AliCD the polygon AXBOCPDR similarto the polygon EKFLGMHN: as, therefore, th
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
