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ABCD apotome applied base binomial straight line Book breadth called circle commensurable in length commensurable in square common cone construction contained corresponding cylinder definition diameter divided double drawn equal Euclid figure follows fourth given greater half height Hence incommensurable inscribed irrational straight line joined Lemma less magnitudes measure medial area medial straight line meet parallel parallelepipedal parallelogram pentagon perpendicular plane polygon prism produces proof proportional PROPOSITION proved pyramid rational straight line reference remainder respectively right angles roots segment side similar Similarly solid sphere square number squares on AC straight lines commensurable Suppose Take third triangle twice the rectangle whence whole
Page 12 - 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 12 - 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 335 - 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 295 - 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 26 - 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 287 - 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 252 - 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 24 - 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