Euclid, Elements, volume 3, page 182

Home Collections/Texts Perseus Catalog Research Grants Open Source About Help

Hide browse bar Your current position in the text is marked in blue. Click anywhere in the line to jump to another position:

This text is part of:

Greek and Roman Materials

Search the Perseus Catalog for:

View text chunked by:

book : type : number
volume : page

Table of Contents:

view as XML

and, as BC is to CD, so is the square on FG to the square on GH. [X. 6, Por.]

Since then, as E is to BC, so is the square on A to the square on FG, therefore the square on A is commensurable with the square on FG. [X. 6]

But the square on A is rational; therefore the square on FG is also rational; therefore FG is rational.

And, since E has not to BC the ratio which a square number has to a square number, therefore neither has the square on A to the square on FG the ratio which a square number has to a square number; therefore A is incommensurable in length with FG. [X. 9]

Again, since, as BC is to CD, so is the square on FG to the square on GH, therefore the square on FG is commensurable with the square on GH. [X. 6]

But the square on FG is rational; therefore the square on GH is also rational; therefore GH is rational.

And, since BC has not to CD the ratio which a square number has to a square number, therefore neither has the square on FG to the square on GH the ratio which a square number has to a square number; therefore FG is incommensurable in length with GH. [X. 9]

And both are rational; therefore FG, GH are rational straight lines commensurable in square only; therefore FH is an apotome. [X. 73]

I say next that it is also a third apotome.

For since, as E is to BC, so is the square on A to the square on FG, and, as BC is to CD, so is the square on FG to the square on HG, therefore, ex aequali, as E is to CD, so is the square on A to the square on HG. [V. 22]

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

The National Science Foundation provided support for entering this text.

Purchase a copy of this text (not necessarily the same edition) from Amazon.com

view as XML

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 United States License.

An XML version of this text is available for download, with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted changes, storing new additions in a versioning system.