Mathematical recreations and essays . as a foundation for his reasoning. Attempted Demonstrations of the Postulate. I proceed nowto describe a few of these attempts to prove the postulate or theproposition. Ptolemy s Proof of the Postulate, One of the earliest ofthese efforts to prove the postulate was due to Ptolemy, theastronomer, in the second century after Christ. It is as follows*.Let the straight line EFGH meet the two straight lines AB andCD so as to make the sum of the angles BFG and FGD equal totwo right angles. It is required to prove that AB and CD areparallel. If possible let them
Mathematical recreations and essays . as a foundation for his reasoning. Attempted Demonstrations of the Postulate. I proceed nowto describe a few of these attempts to prove the postulate or theproposition. Ptolemy s Proof of the Postulate, One of the earliest ofthese efforts to prove the postulate was due to Ptolemy, theastronomer, in the second century after Christ. It is as follows*.Let the straight line EFGH meet the two straight lines AB andCD so as to make the sum of the angles BFG and FGD equal totwo right angles. It is required to prove that AB and CD areparallel. If possible let them not be parallel, then they willmeet when produced say at M (or j^). But the angle AFGis the supplement of BFG and is therefore equal to the angle FGG is equal to BFG, Hence the sumof the angles AFG and FGG is equal to two right angles, andtherefore the lines BA and DC, if produced, will meet at M(or N). But two straight lines cannot enclose a space, thereforeAB and CD cannot meet when produced, that is they Conversely, if AB and CD be parallel, then AF and CG arenot less parallel than FB and GD; and therefore whatever bethe sum of the angles AFG and FGG, such also must be the * Proclus, ed. G. Friedlein, Leipzig, 1873, pp. 362—368. 314j the parallel postulate [CH. XV sum of the angles FOD and BFG, But the sum of the fourangles is equal to four right angles, and therefore the sum ofthe angles BFQ and FGD must be equal to two right angles. This proof is not valid. Apart from all considerations aboutthe nature of space, no reason is given why the sums of theangles on either side of the secant should be assumed to beequal. The whole question turns on whether the straight lineswould not meet, even though the sum of the angles on one sideis a little more than two right angles, and on the other a littleless. It is conceivable that parallels might open out as they areprolonged, and thus that a straight line inclined at a smallangle to one of them should ne
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