By Thomas L. Heath, 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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**Additional info for The Thirteen Books of the Elements, Vol. 3: Books 10-13 **

**Example text**

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.

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

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