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

By Thomas L. Heath, Euclid

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

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A B 0-- D--- F--- E-- F or let A be divided into as many equal parts as there are units in D, and let C be equal to one of them; and let F be made up of as many magnitudes equal to C as IO there are units in E. Since then there are in A as many magnitudes equal to C as there are units in D, whatever part the unit is of D, the same part is C of A also; therefore, as C is to A, so is the unit to D. [VII. Def. 20] I5 But the unit measures the number D ; therefore C also measures A. And since, as C is to A, so is the unit to D, therefore, inversely, as A is to C, so is the number D to the unit.

Let A be the assigned straight line; thus it is required to find two straight lines incommensurable, the one in length only, and the other in square also, with A. Let two numbers B, C be set out which have not to one [x. BOOK X 10 another the ratio which a square number has to a square number, that is, which are not similar plane A---numbers; D----and let it be contrived that, E----as B is to C, so is the square on A to B--the square on D c----for we have learnt how to do this[x. J therefore the square on A is commensurable with the square on D.

5J But it has not; therefore the magnitudes A, B are incommensurable. Therefore etc. PROPOSITION 9. in length. But the squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number,. and squares which have not to one another the ratio which a square number has to a square mtmber will not have thezr sides commensurable in length either. X·9] PROPOSITIONS 7-9 29 F or let A, B be commensurable in length; I say that the square on A B A has to the square on B the _0 __ ratio which a square number o has to a square number.

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


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