Download The Thirteen Books of the Elements, Vol. 3: Books 10-13 by Thomas L. Heath, Euclid PDF

By Thomas L. Heath, Euclid

This is the definitive version of 1 of the very maximum classics of all time—the complete Euclid, no longer an abridgement. using the textual content tested by means of Heiberg, Sir Thomas Heath encompasses virtually 2500 years of mathematical and old learn upon Euclid.
This unabridged republication of the unique enlarged version includes the complete English textual content of all thirteen books of the Elements, plus a severe gear which analyzes each one definition, postulate, and proposition in nice element. It covers textual and linguistic concerns; mathematical anayses of Euclid's rules; classical, medieval, Renaissance, smooth commentators; refutations, helps, extrapolations, reinterpretations, and old notes, all given with large quotes.
"The textbook that shall fairly change Euclid has now not but been written and possibly by no means will be," Encyclopaedia Britannica.

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Extra info for The Thirteen Books of the Elements, Vol. 3: Books 10-13

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P2 and A2,jk. p2 (or ,\Jk. p2) to a rational straight line CT. J k . r!. CT r!. 2 J k . (or AJk . are in the ratio of the CT CT 2 areas Jk. p~ and A Jk. p2 (or AJk. p2) themselves and are therefore comThe breadths mensurable. Now [x. 22 Therefore J J k . CT is rational but incommensurable with CT. ,\2 Jk . ~ (or AJk. ~) is so also; PROPOSITIONS 23, 24 55 whence the area A2Jk. p2 (or AJk. p2) is contained by two rational straight lines commensurable in square only, so that U 1p (or JA.. k1p) is a medial straight line.

Nes commensurable £11. length £s med£al. -------,0 I say that A C is medial. For on AB let the square AD be described; therefore AD is medial. AI----IB And, since AB is commensurable in length with BC, '-----JD while AB is equal to BD, therefore DB is also commensurable in length with BC; [VI. I, X. II] so that DAis also commensurable with A C. But DAis medial; [x. ] therefore A C is also medial. Q. E. D. There is the same difficulty in the text of this enunciation as in that of x. 19. e. they can be either commensurable in length or commensurable in square only with a givm medial straight line.

But, by (2), x is rational: which is impossible. Therefore pz is not rational. PROPOSITION 27. To find medz"al straight 1£1zes comme1zsurable z"n square only whzCh contaz"n a rati01zaI1'eetangle. Let two rational straight lines A, B commensurable In square only be set out; let C be taken a mean proportional between o A, B, [VI. 13J c and let it be contrived that, as A is to B, so is C to D. [VI. 12J A B Then, since A, B are rational and commensurable in square only, the rectangle A, B, that is, the square on C [VI.

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