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Euclid: Elementa
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Proposition 46. 
PROBL. 14. PROPOS. 46. 
第四十六題 
On a given straight line to describe a square. 
A DATA recta linea quadratum describere. 
一直線上。求立直角方形。 
Let AB be the given straight line; thus it is required to describe a square on the straight line AB. 
SIT data recta A B, super quam oporteat quadratum describere. 
. 
Let AC be drawn at right angles to the straight line AB from the point A on it [I. 11], and let AD be made equal to AB;  through the point D let DE be drawn parallel to AB, and through the point B let BE be drawn parallel to AD. [I. 31]  Therefore ADEB is a parallelogram;  therefore AB is equal to DE, and AD to BE. [I. 34]  But AB is equal to AD;  therefore the four straight lines BA, AD, DE, EB are equal to one another;  therefore the parallelogram ADEB is equilateral.  I say next that it is also right-angled.  For, since the straight line AD falls upon the parallels AB, DE, the angles BAD, ADE are equal to two right angles. [I. 29]  But the angle BAD is right; therefore the angle ADE is also right.  And in parallelogrammic areas the opposite sides and angles are equal to one another; [I. 34]  therefore each of the opposite angles ABE, BED is also right.  Therefore ADEB is right-angled.  And it was also proved equilateral. 
Ex A, & B, educantur A D, B C, perpendiculares ad A B, sintque ipsi A B, æquales,  & connectatur recta C D.  Dico A B C D, esse quadratum.  Cum enim anguli A, & B, sint recti, erunt A D, B C, parallelæ: Sunt autem & æquales, quod utraque æqualis sit ipsi A B. Igitur & A B, D C, parallelæ sunt & æquales: & ideo parallelogrammum est A B C D, in quo, cum A D, D C, C B, æquales sint ipsi A B,     omnes quatuor lineæ æquales existunt:  Sunt autem & omnes quatuor anguli recti, cum C, & D, æquales sint oppositis rectis A, & B. Quadratum igitur est A B C D, ex definitione;  s  s  s  s  s  s   
.  .  .  .  .  .  .  .  .  .  .  .  .  . 
Therefore it is a square; and it is described on the straight line AB.  Q. E. F. 
Ac proinde a data recta linea quadratum descripsimus.  Quod faciendum erat.

EX PROCLO
LINEARVM æqualium æqualia sunt quadrata: & quadratorum æqualium æquales sunt lineæ.
SINT primum rectæ A B, C D, æquales. Dico eorum quadrata A B E F, C D G H, æqualia quoque esse. Ductis enim diametris B F, D H, erunt duo latera B A, A F, trianguli B A F, duobus lateribus D C, C H, trianguli D C H, æqualia, utrumque utrique, cum ex definitione quadrati rectæ A F, C H, æquales sint rectis A B, C D. Sunt autem & anguli A, & C, æquales, nempe recti. Igitur triangula B A F, D C H, æqualia erunt. Quæ cum sint dimidia quadratorum, erunt & quadrata tota æqualia. Quod est propositum.
SINT deinde quadrata A B D E, B C F G, æqualis. Dico lineas quoque ipsorum A B, B C, æquales esse. Coniungantur enim quadrata ad angulum B, ut rectæ A B, B C, in directum constituantur. Et quoniam anguli A B G, A B D, sunt recti, erunt & rectæ G B, B D, in directum constitutæ. Ducantur diametri A D, C G, iunganturque rectæ A G, C D. Quoniam igitur quadrata A B D E, B C F G, æqualia sunt, erunt & triangula A B D, B C G, eorum dimidia, æqualia. Addito ergo communi triangulo B C D, fiet totum triangulum A C D, toti triangulo G D C, æquale. Quare triangula A C D, G D C, cum eandem habeant basin C D, ad easdemque sint partes, in eisdem sunt parallelis: ideoque parallelæ sunt A G, C D. Et quoniam, ut in scholio propositio 34. ostendimus, diameter in quadrato secat angulos quadrati bifariam, erunt anguli D A C, G C A, alterni semirecti, ideoque æquales. Quamobrem, & parallelæ sunt A D, C G. Igitur parallelogrammum est A D C G: ac propterea rectæ A D, C G, æquales. Quoniam ergo in triangulis A B D, B C G, latera A D, C G, æqualia sunt, & anguli quibus ea latera adiacent, inter se etiam æquales, cum sint semirecti, ut in scholio propositio 34. ostensum fuit: erunt reliqua latera æqualia, nempe A B, ipsi B C, &c. Quod est propositum.

SCHOLION
POSSENT hæc omnia multo breuius probari per superpositionem quadrati unius super aliud. Nam si lineæ sint æquales, si unæ alteri superponatur, congruent ipsæ inter se. Cum ergo & anguli sint æquales, nempe recti, conuenient quoque ipsi inter se, ideoque totum quadratum toti quadrato congruet. Quod si quadrata sint æqualia, congruent ipsa inter se, propter æqualitatem angulerum. Igitur & lineæ; alias unum quadratum alio maius esset. 
.  . 
 
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