By Thomas L. Heath
After learning either classics and arithmetic on the collage of Cambridge, Sir Thomas Little Heath (1861-1940) used his time clear of his task as a civil servant to put up many works just about old arithmetic, either renowned and educational. First released in 1926 because the moment version of a 1908 unique, this booklet includes the 3rd and ultimate quantity of his three-volume English translation of the 13 books of Euclid's parts, masking Books Ten to 13. This unique textual content could be of worth to a person with an curiosity in Greek geometry and the background of arithmetic.
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Extra resources for Thirteen Books of Euclid's Elements. Books X-XIII
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.