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 activity as a civil servant to submit many works just about historic arithmetic, either well known and educational. First released in 1926 because the moment version of a 1908 unique, this publication comprises the 3rd and ultimate quantity of his three-volume English translation of the 13 books of Euclid's components, overlaying Books Ten to 13. This exact textual content may be of price to somebody with an curiosity in Greek geometry and the historical past of arithmetic.

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Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in· square also.

The steps are exactly the same as shown under (I) and (2) of the last note, with v instead of ", except only in the lines "x ,... 2X" and "2X '"' x" which are unaltered, while, in the references, x. 13, 16 take the place of x. 12, IS respectively. [LEMMA. Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in· square also.

II] But the square on CD is commensurable with the square on BF, for the straight lines are rational in square; and the rectangle DC, CB is commensurable with the rectangle FE, EG, for they are equal to the square on A ; therefore the square on CD is also incommensurable with the rectangle DC, CB. [x. r3] But, as the square on CD is to the. rectangle DC, CB, so is DC to CB; [Lemma] therefore DC is incommensurable in length with CB. [x. II] Therefore CD is rational and incommensurable in length with CB.