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I recently came across the term "triangle given in species" in Hatton's Projective Geometry. Searching in archive.org turned up other examples (such as this) of 19th century texts, and it is also used in Euclid. But I don't see it defined anywhere. By context, I think it refers to similar triangles, but it would be nice to see a definition somewhere. Also, does it have a meaning wider than "similar" in geometry?

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Such terms as “given in species” are defined in Euclid’s Data (Greek, English):

III. Rectilineal figures are said to be given in species, which have each of their angles given, and the ratios of their sides given. (English version, R. Simpson, 1810, p. 367)

[Species is the translation of eidos, shape or form; see LSJ, εἶδος, def. A.2.b.]

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    $\begingroup$ One speculates that, since everybody had studied Euclid in those days, it was not necessary to define the term -- everybody knew it. At some time it must have dropped out of usage though. $\endgroup$ – brainjam Aug 30 '20 at 0:10
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    $\begingroup$ Three observations. Firstly, the phrase plays a different role from "similar": "similar" means having the same angles and aspect ratios as some other shape; "given in species" means having prescribed angles and aspect ratios, no other shape required. Secondly, Google Scholar and Google Books reveal it's extremely rare for any work to use both phrases, "given in species" and "similar triangle". Thirdly, Thomas Heath, in his translation of Elements, claimed Proclus was confused about the meaning of "given in species": could this have contributed to the phrase dropping out of usage? $\endgroup$ – Daniel Hatton Aug 30 '20 at 11:23
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    $\begingroup$ @DanielHatton Thanks for the observations. I think I would say "no other shape given," but that's probably just quibbling. Another way to say it is that "similar" is a relation between two triangles and "given in species" is a property or assumption about a single triangle. Google Ngrams, for what its data is worth, show that "given in species" had a precipitous decline in usage around 1840. I don't know why. I suspect education reform. $\endgroup$ – Michael E2 Aug 30 '20 at 15:21
  • $\begingroup$ I wonder if this is the same root as the word "specific" in the sense of "specific heat", "specific gravity", "specific strength", etc. to mean a ratio of two quantities. $\endgroup$ – Michael Seifert Aug 30 '20 at 15:25
  • $\begingroup$ @MichaelSeifert Certainly "specific" and "species" have the same root in Latin, but I guess the question is whether scientists adopted the scientific terms based on the mathematical meaning of "given in species." $\endgroup$ – Michael E2 Aug 30 '20 at 15:41
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This terminology does appear in English 100+ years ago, and is arguably archaic, but just means "types" or "kinds".

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