The following is as given in Sir Thomas L. Heath's translation, which can be found in the book The Thirteen Books of The Elements, Vol. 1.

Book I | Book II | Book IX |
---|---|---|

Definitions, Postulates, and Common Notions | Definitions | Proposition 20 |

Proposition 1, Proposition 3, | Proposition 14 | Proposition 36 |

Proposition 5, Proposition 6, | ||

Proposition 29, Proposition 47 |

*To construct a square equal to a given rectilineal figure.*

Let `A` be the given rectilineal figure;

thus it is required to construct a square equal to the rectilineal figure `A`.

For let there be constructed the rectangular parallelogram `BD` equal to the rectilineal figure `A`.
[I. 45]

Then, if `BE` is equal to `ED`, that which was enjoined will have been done; for a square `BD` has been constructed equal to the rectilineal figure `A`.

But, if not, one of the straight lines `BE`, `ED` is greater.

Let `BE` be greater, and let it be produced to `F`;

let `EF` be made equal to `ED`, and let `BF` be bisected at `G`.

With centre `G` and distance one of the straight lines `GB`, `GF` let the semicircle `BHF` be described; let `DE` be produced to `H`, and let `GH` be joined.

Then, since the straight line `BF` has been cut into equal segments at `G`, and into unequal segments at `E`,

the rectangle contained by `BE`, `EF` together with the square on `EG` is equal to the square on `GF`.

But `GF` is equal to `GH`;

therefore the rectangle `BE`, `EF` together with the square on `GE` is equal to the square on `GH`.

But the squares on `HE`, `EG` are equal to the square on `GH`;
[I. 47]

therefore the rectangle `BE`, `EF` together with the square on `GE` is equal to the squares on `HE`, `EG`.

Let the square on `GE`be subtracted from each;

therefore the rectangle contained by `BE`, `EF` which remains is equal to the square on `EH`.

But the rectangle `BE`, `EF` is `BD`, for `EF` is equal to `ED`;

therefore the parallelogram `BD` is equal to the square on `HE`.

And `BD` is equal to the rectilineal figure `A`.

Therefore the rectilineal figure `A` is also equal to the square which can be described on `EH`.

Therefore a square, namely that which can be described on `EH`, has been constructed equal to the given rectilineal figure `A`.

Q.E.F.