**Congruent **The Latin word *congruere* meant "coming together" or "working together". I learned from Glen Woodburn recently that, "Actually, *gruere* comes from the latin word *grui* which means to be in harmony with. So congruent translates to mean together in harmony with." Whether applied to a geometric shape, or a military unit, it meant that all the parts fit together. According to a message from Nathan Sidoli, in Euclid's __Elements__ the "word that Heath translates as "coincides" is *efarmo^zein* - to fit exactly" . Nathan refers to Common Notion 4 in Book one, which Heath translates as "Things which coincide with one another, are equal to one another."

During the 16^{th} century translations of Euclid into Latin began to use the Latin term for Common Notion 4. In a note to the Math Hisotry list J. Cabilon wrote that "Christoph Clavius (1537?-1612) wrote: '...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**.' (vol. I, p. 363)"[emphasis added]

At Jeff Miller's web site there are several notes on the development of the term congruence. In particular he says that, "In English, writers commonly refer to geometric figures as equal as recently as the nineteenth century. In 1828, Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre) has:

Two triangles are equal, when an angle and the two sides which contain it, in the one, are respectively equal to an angle and the two sides which contain it, in the other."

The modern symbol for congruence common to most US high school texts, which combines the tilde ~ above an equal sign, =, @ was first used by many writers for similarity as well. It is sometimes used with the wave inverted also. Leibniz used a tilde with a single underline as a unique symbol for congruence, but so many symbols were in use that it did not catch on. According to Cajori, the use of the modern symbol for congruence became the accepted practice around the beginning of the 20th century. He suggests the first use was by G. A. Hill and George B Halstead. The symbol is still not universally accepted and was not used in England at the time of his writing because of confusion with the tilde symbols use for difference.

I recently (Feb, 2004) posted a request for information to the Historia Matematica discussion group and received the following update on the use of the **~** symbol in England, and the related question of a symbol for congruence. According to a post from Herbert Prinz,

"In modern English texts on navigation, nautical astronomy or its history, the tilde is frequently used to express the function | a - b |, where |x| stands for absolute value. E. g. Cotter, The Complete Nautical Astronomer, 1969. I am not sure when this practice started. In older texts on the same subject, say, Moore, The Practical Navigator, 1800, one does not find the tilde used in this way. For one, because instructions were given mostly verbally without the use of any symbols at all. And second, the distinction from '-' was unnecessary, as it was always understood, if not explicitly stated, that one must subtract the smaller from the larger value."

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Tony Mann pointed out that in England the @ symbol was, "commonly used for 'is isomorphic to', and is used colloquially for 'is essentially equivalent to'." John Harper of Victoria University in New Zealand added that, "Geometric congruence was indicated by 3 parallel equal lines: ≡, an equals sign with a straight underline. It still is, according to Borowski & Borwein "Collins Dictionary of Mathematics" (HarperCollins, Great Britain 1989) who give ~ on top of = only for approximate equality." Cajori credits the creation of the ≡ symbol for geometric congruence to Reimann and was used by Bolyai.

Gauss used the term congruent in modular arithmetic to refer to numbers which had the same remainder upon divison, for example 12 ≡ 7 mod 5 since each has a remainder of two when divided by five.

**Conjugate** is the union of the common Latin prefix *com* (together) and the root *juge *(yoke) and means to bind together in a pair. Mathematically it is often used for things that are opposites in some way, as in the complex conjugates. The same word in grammar refers to words of a common origin and related meaning, and in biology to an act of sexual union, for which the more common term is conjugal relations.

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