# Who first gave a definition of congruent triangles?

Who was the first to define congruent triangles? I couldn't find the definition in Euclid's Elements.

• It’s in Common Notion 4 in Book I of Euclid. – Michael E2 Dec 25 '19 at 6:38
• @MichaelE2 "Things which coincide with one another equal one another" is not exactly a definition of congruent triangles. The closest he has to a definition is written into Proposition 4 of the same book:"If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides." – Conifold Dec 25 '19 at 6:47
• I would have added some more but I’m on a phone. it messed up my message. The phrase is epharmozonta ep’ allela that corresponds to congruent – Michael E2 Dec 25 '19 at 6:51

It depends on how the question is interpreted. Common notion 4, "things which coincide with one another equal one another", is not exactly a definition of congruent triangles or even of congruence generally. Likely due to Platonist strictures, Euclid deliberately avoids using congruence in his demonstrations. The idea of moving and superimposing figures Plato judged as "corrupting the good of geometry", which is why we do not find "workman tools" like straightedge and compass, or mechanically generated curves, in the Elements either, see When were the concepts of pure and applied Mathematics introduced?

Many Euclidean demonstrations display proliferation of auxiliary triangles where the use of congruence would have simplified the clutter. But at the very beginning Euclid has no choice, as he needs tests of equality for his auxiliary triangles. The first such test is Proposition I.4:

"If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides."

All sides and angles are equal to each other is a way to define "congruence of triangles" without congruence. Euclid could have just made I.4 another postulate, but as he wanted to give a demonstration he had to resort to superimposing. Even so, the use of imposition is buried inside the demonstration and unacknowledged. Joyce comments on this in a note to I.26 of his translation of the Elements:

"Euclid’s congruence theorems are I.4 (side-angle-side), I.8 (side-side-side), and this one, I.26 (side and two angles). Calling them congruence theorems is anachronistic, since Euclid did not explicitly use the concept of congruence. We would say that two triangles ABC and DEF are congruent if the angles A, B, and C equal the angles D, E, and F respectively, and the sides AB, BC, and AC equal the sides DE, EF, and DF respectively, and the triangle ABC equals the triangle DEF (by which is meant that they have the same area).

The word "congruent" for superimposing (superpono) appears in 16-17th century Latin translations of Euclid, whose authors had no preconceptions about motion. "Congruent" replaces "coincide" in Common notion 4. Clavius in Euclidis Elementorum (1589) writes, for example:

"Hinc enim fit, ut aequalitas angulorum ejusdem generis requirat eandem inclinationem linearum, ita ut lineae unius conveniant omnino lineis alterius, si unus alteri superponatur. Ea enim aequalia sunt, quae sibi mutuo congruunt."

"For want of this, equality of the angles of the same kind requires the same inclination of the lines, and lines to lines coincide when placed on top of each other. For those things are equal, that are congruent with each other." [translation mine]

According to Miller's Earliest Known Uses the word is first used systematically in something like the modern sense only by Leibniz in 1679:

"As a more technical term for a relation between figures, congruent seems to have originated with Gottfried Wilhelm Leibniz (1646-1716), writing in Latin and French. His manuscript "Characteristica Geometrica" of August 10, 1679, is in his Gesammelte Werke, dritte Folge: mathematische Schriften, Band 5. On p. 150 he says that if a figure can be applied exactly to another without distortion, they are said to be congruent...

His Figure 39 shows two radii of a circle, with the center labelled both A and C. Later (p. 154) he points out that "congruent" is the same as "similar and equal." He used "congruent" in the modern (Hilbert) sense, applied to line segments and various other things as well as triangles.

Shortly afterwards, on September 8, 1679, he included a similar definition in a letter to Hugens (sic) van Zulichem. In his ges. Werke etc. as above, volume 2, p. 22, he illustrates congruence with a pair of triangles, and says that they "peuvent occuper exactement la meme place, et qu'on peut appliquer ou mettre l'un sur l'autre sans rien changer dans ces deux figures que la place." [Ken Pledger and Julio González Cabillón]".