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Diophantus’ notation for higher powers of the unknown includes a superscript, usually written as a lowercase or uppercase upsilon - e.g., $\Delta^{\upsilon}$ as the square. It is not clear to me what the role of this superscript is. Neither $\Delta$ nor $K$ (cube) is used in the Greek number system so there is no risk of ambiguity (though curiously $\upsilon$ is part of the number system as $400$). The superscript appears to be superfluous.

Q: What is the role of the superscript here?

Diophantus includes a special symbol - $\mathring{M}$ - to identify the constant term in an expression. On the one hand, we can see why this is necessary formally in order to avoid ambiguity that may arise as a result of his practice of writing coefficients on the right and the absence of a symbol for addition. On the other hand, again from a formal point of view, it suggests that he is treating the constant term as a separate class of object, distinct from number, square, cube, etc., and with a separate designation.

Q: Did Diophantus’ inventory of types treat constants as a separate type?

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    $\begingroup$ Are you referring to particular passages in a particular edition? If so, could you please give details? $\endgroup$ Jan 21, 2020 at 22:43
  • $\begingroup$ @kimchilover I have not had the opportunity to directly view the ancient sources so I am using the symbolism and syntax that is standard in modern accounts - in particular Leo Corry’s A Brief History of Numbers, but also used in online sources such as wikipedia and others. I am aware that there are historiographic, methodological, and indeed typographic issues in relying on Islamicate sources. If you are aware of issues with the symbolism or syntax used in modern accounts, then perhaps you could post this as an answer.(...) $\endgroup$
    – nwr
    Jan 21, 2020 at 23:23
  • $\begingroup$ (...) I have seen some typographical differences in modern sources. E.g., Springer’s edition on the recently discovered Books IV-VII uses $Q^{\upsilon}$ for the square but maintains $K^{\upsilon}$ for the cube. $\endgroup$
    – nwr
    Jan 21, 2020 at 23:23

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The letter $\upsilon$ or $\Upsilon$ is the second letter of δύναμις, κύβος and δυναμοδύναμις (square, cube and bisquare) of the words the notation abbreviates. Being the same letter, it plays an additional role of identifying the power symbols as being of a kind, marking Diophantus's "numerical species" (squares, cubes, etc., multiplied by coefficients). Notably, his power symbols have the same exponents, whereas ours have the same base. In other words, he thinks of them not as we do, a power marker modifying the "unknown", but in reverse, the implicit "unknown" modifying a generic power, given by the base of the power symbols. As Hettle puts it in The Symbolic and Mathematical Influence of Diophantus’s Arithmetica: "Diophantus defined powers of unknowns not as a power of the unknown, but as an unknown with the property of being that power of a positive rational number; accordingly he refers to each power as a “species” of number".

In some sense then the letter $\upsilon$ is the mark of genericity, or of the unknown. However, the unknown by itself is denoted ς, a contraction of the first two letters of ἀριθμος, "number", according to Heath, which Diophantus uses to name it in the text. In contrast to the earlier tradition going back to Pythagoreans, he for the first time admits (positive) rationals as "numbers", and they are the only possible values of the unknown and legitimate solutions to equations he considers. Although it would seem natural to us to assimilate ς as marking "linear species", the difference in notation suggests that he does not think that way. Perhaps, this is a trace of Pythagoreans not viewing $1$ as a "number".

Numerical species are also sharply separated by Diophantus, notationally and terminologically, from free terms, to us, "zero power species", marked by $\mathring{M}$ for monad. They are Pythagorean and Euclid's "numbers" (positive integers greater than $1$, i.e. multitudes of monads) to which square, cube, etc., can be applied numerically. Apparently, he was concerned with possible confusion of his species with perfect squares, cubes, etc., even aside from the fact that zero concept, let alone zero power, was not available at the time.

Here is a more detailed commentary from Acerbi's Unaccountable Numbers:

"At the beginning of his treatise, Diophantus explains the notation that he will use throughout; he is the first Greek mathematician who consistently adopts a set of signs in order to make his text more concise and, in a sense, conducive to the kind of “algebraic” manipulations forming the technical core of his method for solving numerical problems. In particular, he establishes a terminology to denote what in algebraic language are the powers of the “unknown” $x^2, x^3, …$; in Diophantus’ theoretical framework, these are abstract numerical εἴδη “species,” namely generic square, cube, … numbers. The species introduced are assigned a denomination and a conventional sign; the sign is made of the first letter of each component of the denomination, possibly supplemented with the second letter (this always happens to be upsilon): to the generic square number (δύναμις) corresponds the sign $Δ^\Upsilon$, to the κύβος the sign $K^\Upsilon$, to the fourth power (δυναμοδύναμις) the sign $Δ^\UpsilonΔ$, etc... At the end of the list of species, Diophantus also assigns a denomination and a conventional sign to the most generic abstract number, namely one that neither is a particular number nor can be said to have the features characterizing one of the aforementioned species; I shall call it, with a slight abuse of language, the “1-species”; it corresponds to the “unknown” of present-day algebra.

[...] Capital $Δ$, $K$, and $\Upsilon$ are currently printed, but of course no indication to that effect is contained in the text. It is quite obvious that our notation owes very much, both in conception and in the form of the signs, to Diophantus’: note his use of the term δύναμις “power” and the idea of putting a part of the conventional sign “at the exponent.” One crucial difference is that we conceive of the species as powers of the “unknown,” whereas Diophantus draws a sharp distinction between these notions, as we shall see presently. This difference is made particularly conspicuous by the fact that Diophantus’ conventional signs all have the same exponent (the insignificant letter upsilon) and a variable “base” indicating the species (letters $Δ$ and $K$, possibly doubled), whereas modern algebraic signs all have the same base (the most significant “unknown” $x$) and a variable exponent indicating the power to which the base is to be raised.

[...] These species must not be confused with particular numbers that happen to be square, cube, fourth powers. On a terminological level, Diophantus settles the problem of separating particular square numbers from the species “square” by means of the opposition τετράγωνος/δύναμις; a lexical ambiguity (admittedly quite harmless) remains in the case of the κύβος, which may designate both a particular cube number (such as $8$) and the species “cube.”

Diophantus highlights this difference when he alludes to the Euclidean definition of number (Elem. 7.def.2) and when he defines a square number: in both cases he adds a τινος, either to πλήθους or to ἀριθμοῦ. This means that the object so qualified is particular, yet generic... The following considerations may help further clarify the point. Diophantine numerical species were invoked by the fourth-century mathematical polymath and commentator Theon of Alexandria (In Alm. 452.21–453.16 Rome) to explain the structure of orders within the sexagesimal system used by the astronomers. The sexagesimal orders are in fact numerical εἴδη; they correspond to the orders of magnitude in the decimal system... these numerical species do not coincide with particular numbers: indeed, $300$ is not a square, but $3$ items of the square εἶδος “hundreds”; conversely, the species “hundreds” is not a number (it does not even coincide with number $100$)."

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  • $\begingroup$ Very convincing. This plays well with Corry referring to the notation for the various numerical species as "abbreviated designators", I hadn't quite made that connection. Regarding monades, when you say they are sharply separate from numerical species, are you saying that they are not a numerical species, or simply that they are a distinct numerical species. $\endgroup$
    – nwr
    Jan 22, 2020 at 2:09
  • $\begingroup$ @Nick The difference in notation suggests that he does not assimilate linear and constant terms to the rest of the species, for constants this is also stressed terminologically. Our assimilation via the first and zero power is rather formalistic, and zero was not yet even a fully formed concept, although Ptolemy did use $o$ earlier for gaps in sexagesimals. Nonetheless, general rules concerning like terms, formulated for species, are applied to linear and constant terms as well. $\endgroup$
    – Conifold
    Jan 22, 2020 at 7:30
  • $\begingroup$ One more question, if I may. Corry writes that these expressions should be read as a "list" of species. On the other hand, modern writers, include Corry, translate these expressions into modern symbolism and in doing so insert a "+" sign in between the terms. Am I right in assuming that this would not make sense to the Greeks since it entails combining species of different type - e.g., adding a planar object to a solid object would not make sense? $\endgroup$
    – nwr
    Jan 22, 2020 at 17:49
  • $\begingroup$ @Nick Diophantus did not much work with geometric magnitudes, most problems are formulated as about "numbers" (positive rationals), so multiplying and adding them was unproblematic. Even when the problem comes from geometry he does not hesitate to write things like "To find a right-angled triangle such that the hypotenuse minus each of the sides gives a cube". $\endgroup$
    – Conifold
    Jan 22, 2020 at 18:50
  • $\begingroup$ Thanks. I probably should not have used terms like planar and solid. Corry describes them a figurate - as in figurate numbers - but obviously distinct from numbers. I was reading a bit about Al-Khayyam's insistence on "dimensional homogeneity" (via restrictions on the coefficients) in his expressions for cubics, thus allowing him to combine terms without concern for their species. Since I did not recall Diophantus having any issues with combining terms I was left wondering if I had missed something. Al-Khyaam did work with geometric magnitudes. $\endgroup$
    – nwr
    Jan 22, 2020 at 19:00

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