# "Species" and "terms" meaning polynomials and monomials

I found in some old Latin texts and their translations that polynomials were once called "species" (if I understand correctly that they meant the same thing, but it looks like it). And their constituent parts are called "terms" to this day.

My question is: Why they called them "species" and "terms"? What was the reason behind choosing these particular names for those notions?

• The title misled me into thinking that this question was about Flajolet's work in combinatorics. Would "Species" and "terms" meaning polynomials and monomials express your intent suitably? Oct 4, 2018 at 10:14
• Yes, it can work this way too. Thank you for clarifying this up. I corrected the title.
– SasQ
Oct 4, 2018 at 10:20
• Newton uses "species" to refer to classes of polynomials (see e.g. hsm.stackexchange.com/questions/5224/…), which makes the term natural. Perhaps this is the meaning in your source too. Oct 4, 2018 at 11:28
• Hmm... So the original meaning was to make a classification system, similar to those in botany or zoology? Seems legit.
– SasQ
Oct 4, 2018 at 11:59
• It might be useful to add an example (or multiple) of the usage of these terms, for concreteness.
– Danu
Oct 8, 2018 at 16:26

The source of species (in mathematics) seems to be François Viète.

See Logistice speciosa (algebra) in contrast to Logistice numerosa (arithmetic), into his In artem analyticem isagoge (1591), page 19 :

Logistice numerosa est quae per numeros, Speciosa quae per species seu formas exhibitur [Numerical logistic is (a logistic) that employs numbers, symbolic logistic one that employs symbols or signs for things].

See Introduction (page 13) for comments :

Samuel Jeake in 1696, has it that this "name ... with the Latins serveth for the Figure, Form or shape of any thing" and that, accordingly, "Species are Quantities or Magnitudes, denoted by Letters, signifying Numbers, Lines, Lineats, Figures Geometrical, &c." Alexandre Saverein's Dictionnaire Universel de Mathematique et de Physique (Paris, 1753), vol. I, p. 17, says that the expression "algebre specieuse" derives from that fact that quantities are represented by letters which designate "leur forme et leur espece," adding "d'ou vient le mot spécieuse."

Smith thinks Diophantus "the most likely source for Vieta's use of the word 'species' " and that it is, in effect, his substitute for Diophantus' εἶδος : form, image, shape.

• “species seu formas” seems especially relevant to the title question, as (homogeneous) polynomials were certainly long called forms. Is species older, and might form actually have arisen as a synonym or preferred translation of it? Oct 4, 2018 at 19:46
• @Mauro: Interesting trail to investigate. I definitely have to take a closer look at Viète's works then. But I'm pretty sure I've seen the word "species" used in older works too (unles they were later translations and that word has been used there post factum, not by the original authors). However, here's an example excerpt from Euler's "Introductio in Analysin Infinitorum" I.I.2:
– SasQ
Oct 4, 2018 at 23:25
• "Quemadmodum scilicet ex ideis individuorum formantur ideae specierum et generum, ita quantitas variabilis est genus, sub quo omnes quantitates determinatae continentur." There are two different English translation of this passage I have. One by J.D.Blanton (emphasis mine): "Just as from the ideas of individuals the ideas of species and genus are formed, so a variable quantity is a genus in which are contained all determined quantities."
– SasQ
Oct 4, 2018 at 23:32
• Another one by Ian Bruce: "Evidently to the extent that ideas of the appearances and kinds of variables are formed from notions of their indivisibility, thus a variable quantity is a kind, within which all the determined magnitudes may be contained." (But this translation doesn't seem to match the original very well, even with my limited knowledge of Latin I can tell that it's totally off :q )
– SasQ
Oct 4, 2018 at 23:36
• Long story short, @Viktor Blasjo's remark about Newton's attempt at classification of polynomials may be on point here as well, since Euler seems to make a taxonomy-like system starting from "individuals", then "species" and "generi" (Lat. "specierum et generum" was explicitly stated in the original text), and it is in the context of categorizing functions into polynomials, rational functions, irrational functions and transcendental functions.
– SasQ
Oct 4, 2018 at 23:40