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area a medial base binomial straight line bisected circle ABCD circle EFGH commensurable in length commensurable in square cone cube cut in extreme cylinder decagon Deff diameter dodecahedron equilateral Eucl Euclid extreme and mean greater segment Hence icosahedron inscribed irrational straight line Lemma let the square magnitudes mean ratio measure medial area medial straight line medial whole parallelepipedal solids parallelogram pentagon perpendicular plane of reference polygon prism Proposition proved ratio triplicate rational and incommensurable rational area rational straight line rectangle AC rectangle contained right angles second apotome side similar Similarly sixth binomial solid angle sphere square number square on AB squares on AC straight lines commensurable surable theorem triangle twice the rectangle vertex whence
Page 14 - Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. the
Page 14 - than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the magnitude C. For C if multiplied
Page 347 - Solid parallelepipeds contained by parallelograms equiangular to one another, each to each, that is, of which the solid angles are equal, each to each, have to one another the ratio compounded of the ratios of their sides. The
Page 305 - a plane is at right angles to a plane, when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane.
Page 28 - squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.
Page 297 - Also, from a point above a plane there can be but one perpendicular to that plane; for, if there could be two, they would be parallel to one another [xi. 6], which is absurd.
Page 262 - 6. The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes.
Page 26 - PROPOSITION 6. If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable. For let the two magnitudes A, B have to one another the ratio which the number D has to the number E