Euclid, Elements I 4©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 4: If two triangles have two sides respectively equal to two sides, and the angle contained by the straight lines equal to the angle, then they will have the base equal to the base and the triangle will be equal to the triangle, and the remaining angles will be respectively equal to the remaining angles which the equal sides subtend.

(general diagram)

Diagrams for book 1 proposition 4, following the display, construction, and demonstration(diagram 1) Let there be two triangles, ABG, DEZ, having the two sides, AB, AG, respectively equal to the two sides, DE, DZ, with AB to DE and AG to DZ, and an angle, that by BAG, equal to an angle, that by EDZ. (diagram 2) I say that a base, BG, will also be equal to EZ, and triangle ABG to triangle DEZ, and the remaining angles will respectively be equal to the remaining angles which the equal sides subtend, the angle by ABG to that by DEZ and the angle by AGB to that by DZE.


(diagram 3) For upon triangle ABG being fitted onto triangle DEZ and point A being positioned at point D and straight-line AB at DE, point B will also fit at E due to the fact that AB is equal to DE. In fact, with AB fitting at DE, straight-line AG will also fit at DZ since the angle by BAG is equal to that by EDZ. Thus, point G will also fit at Z since again AG is equal to DZ. But truly B also fit onto E. Thus, a base, BG will fit onto EZ.For if with B fitting onto E and G onto Z base BG does not fit onto EZ, two straight-lines will enclose a region, which is impossible. Therefore, base BG will fit onto EZ and will be equal to it. (diagram 4) Thus, a whole, triangle ABG, will fit onto a whole, triangle DEZ and will be equal to it, (diagram 5) and the remaining angles will fit onto the remaining angles and will be equal to them, the angle by ABG to that by DEZ and the angle by ABG to that by DZE. Therefore, if two triangles have the two sides respectively equal to the two sides, and the angle contained by the straight lines equal to the angle, then they will have the base equal to the base and the triangle will be equal to the triangle, and the remaining angles will be respectively equal to the remaining angles which the equal sides subtend, just what it was required to show.

Superposition, the fitting of one figure on another occurs rarely, at Ι 8, III 24 (cf. also Phainomena 2 for a different use). One needs an assumption of rigidity in superposition (equals can be superimposed), but not Common Notion 7, that things that fit onto one another are equal. Beyond that, it is unclear how Euclid can move objects around, as he has only established that straight-lines may be copied.

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