## The Elements, Volume 2 |

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Page 315

For let the

any other number EF -£— ? — + + — ? the same number of times ; D I say that,

alternately also, the IE K ± f

For let the

**unit**A measure any number BC, and let another number D measureany other number EF -£— ? — + + — ? the same number of times ; D I say that,

alternately also, the IE K ± f

**unit**A measures the number D the same number of ...Page 386

For let C, the side of A, be taken, and let C by multiplying itself make D. It is then

manifest that C by multiplying a D has made A. B Now, since C by multiplying

itself has c- d — made D, therefore C measures D according to the

For let C, the side of A, be taken, and let C by multiplying itself make D. It is then

manifest that C by multiplying a D has made A. B Now, since C by multiplying

itself has c- d — made D, therefore C measures D according to the

**units**in itself.Page 392

If as many numbers as we please beginning from an

proportion, and the number after the

square. And, if the number after the

there be as ...

If as many numbers as we please beginning from an

**unit**be in continuedproportion, and the number after the

**unit**be square, all the rest will also besquare. And, if the number after the

**unit**be cube, all the rest will also be cube. Letthere be as ...

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### Common terms and phrases

ABCD angle ABC angle BAC antecedent Archimedes Aristotle base bisected centre circle ABC circumference construction continued proportion corresponding sides cube number definition diameter drawn enunciation equal angles equiangular equilateral equimultiples Euclid Euclid's proof Eutocius ex aequali four magnitudes fourth proportional geometrical progression given circle given straight line greater ratio greatest common measure Heiberg hypothesis Iamblichus inscribed joined less mean proportional number measures the number multiple multitude Nicomachus odd number parallel parallelogram pentagon perpendicular polygon Porism prime number Proclus Prop proper fraction proposition Proposition 14 proved rect rectangle rectangle contained rectilineal figure reductio ad absurdum remaining angle right angles segment semicircle similar and similarly similar plane numbers Simson solid numbers square number subtracted taken Theon of Smyrna theorem touches the circle triangle ABC unit