By J M Blackledge; A K Evans; Martin J Turner; Institute of Mathematics and Its Applications
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Extra resources for Fractal geometry : mathematical methods, algorithms, applications
Since then D measures AB, BC, it will also measure the whole AC But it measures AB, Be also; therefore D measures AB, BC, A C ; therefore A C is commensurable with each of the magnitudes AB, BC [x. Def. I] N ext, let A C be commensurable with AB ; I say that AB, BC are also commensurable. For, since A C, AB are commensurable, some magnitude will measure them. Let it measure them, and let it be D. Since then D measures CA, AB, it will also measure the remainder Be. But it measures AB also; therefore D will measure AB, BC; therefore AB, BC are commensurable.
4J There is a difficulty in the text of the enunciatIOn of this propOSitIOn. Lirpwv KUTa. " Now straight lines can only be commensurable £n length in olle way, the degrees of commensurability being commensurability in length and commensurability in square only. But a straight line may be rational in two ways in relation to a give1l rational straight line, since it may be either commensurable ill length, or commensurable £n square only, with the latter. TWV with PYJTWV, translating" straight lines commensurable in length and rational in any of the aforesaid ways," and this agrees with the expression in the next proposition "a straight line once more rational in any of the aforesaid ways"; but the order of words in the Greek seems to be fatal to this way of translating the passage.
3 that it is proz1ed that there are an infinite number of straight lines (a) incommensurable in length only, or commensurable in square only, and (b) incommensurable in square, with any given straight line. But in truth the proposition could well be dispensed with; and the positive objections to its genuineness are considerable. In the first place, it depends on the following proposition, x. I I; for the last step concludes that, since and a, x are incommensurable in length, therefore a2, y2 are incommensurable.