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As a non-native speaker of English, I have been perplexed by the phrase "in the sequel" as used in textbooks, lecture notes, and even research articles. None of my linguistically inclined native speaker friends have seen it outside of mathematical literature, and the relevant Oxford English Dictionary definition of "sequel" suggests mathematical usage is non-standard:

The ensuing narrative, discourse, etc.; the following or remaining part of a narrative, etc.; that which follows as a continuation; esp. a literary work that, although complete in itself, forms a continuation of a preceding one.

What I've inferred from context is that it means something along the lines of "for the rest of this book/paper/text", especially as it's usually used to introduce notation and/or convention.

Some examples:

We hope that the relation between linear transformations and matrices is by now sufficiently clear that the reader will not object if in the sequel, when we wish to give examples of linear transformations with various properties, we content ourselves with writing down a matrix.

--Paul Halmos, Finite-Dimensional Vector Spaces, p. 86

[Here, and in the sequel, Card(S) denotes the number of elements in the finite set S.]

--J.P. Serre, Local Fields, p. 64

In the sequel we shall denote by ∅ the empty set and by {pt} a set with one element.

-- Pierre Schapira, Algebra and Topology course notes, p. 8

In the sequel, we will denote by L(C ) the configuration space of any convergent CFG C.

--E. Goles et. al., Sandpile Models and Lattices: a comprehensive survey, Theoretical Computer Science, 2004, Vol 322, Issue 2, p. 398

So my actual question is two-fold.

  1. What does the phrase actually mean?

  2. When is its use warranted over the use of phrases such as "for the rest of the book/paper/text"?

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closed as off topic by Robin Chapman, Harry Gindi, Mariano Suárez-Alvarez, Akhil Mathew, Loop Space Jul 29 '10 at 8:00

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

It means exactly what the OED says it means. – Robin Chapman Jul 28 '10 at 23:32
.. This was also on mathunderflow, where Pete Clark gave a satisfactory answer to it. – Harry Gindi Jul 28 '10 at 23:38
I wonder why was this copied from Math.SE? – Mariano Suárez-Alvarez Jul 28 '10 at 23:43
Indeed, it is because of this brilliant answer… that I gained the ability to comment on posts at Math.SE. I agree that it is a little curious that this question appeared here less than a day after it got asked and answered on the other site. – Pete L. Clark Jul 28 '10 at 23:59
I don't understand why people are voting to close this question. – Kevin H. Lin Jul 29 '10 at 1:11

It means "from now on." (Such as, "in the sequel, $K$ will denote a perfect field...") This does, in fact, tend to appear only in older books, or in books in translation. I also was confused when I first saw the term.

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From Online etymology dictionary

sequel: early 15c., "train of followers," from O.Fr. sequelle, from L.L. sequela "that which follows, result, consequence," from sequi "to follow," from PIE base *sekw- (cf. Skt. sacate "accompanies, follows," Avestan hacaiti, Gk. hepesthai "to follow," Lith. seku "to follow," L. secundus "second, the following," O.Ir. sechim "I follow"). Meaning "consequence" is attested from late 15c. Meaning "story that follows and continues another" first recorded 1510s.

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So maybe you can also use it as "The sequel of this theorem is that this and that happen." or in "As a sequel of the intermediate value theorem you can ascertain the existence of solutions of many equations." – Mlazhinka Shung Gronzalez LeWy Jul 28 '10 at 23:42

I have always understood "in the sequel" to mean "in what follows". I have never checked but I assumed this was from the Latin as in "sequence" and "non sequitur"

In the usage you refer to it means what follows in the same article. The sequel to a film is the next film.

I see Franklin has checked the etymology.

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Related also is the Latin abbreviation et seq., which means "and what follows." Use it like this, "For a summary of this fact, see Lemma 2.11 et seq. in Fakename's article." – Willie Wong Jul 29 '10 at 0:24
Right. This is an abbreviation for "et sequitur"; similarly we have et al. for "et aliter" (and others) and etc. for "et cetera" (and things). – Bruce Westbury Jul 29 '10 at 1:22
Actually, it is 'et sequens'. The verb is 'sequor'. 'Sequitur' is the third person singular form from which 'sequitur' the noun is derived. 'Sequens' is the present active participle. This is not just a quibble on etymology: 'sequitur' the noun has a specific meaning as logically following, whereas 'sequens' the participle just means "following" in general. So as 'et seq.' is usually used to refer to the "narrative which follows", a reader thinking of "et sequitur" runs the risk of finding a non sequitur. =) Okay, this is getting too much to be LatinOverflow... – Willie Wong Jul 29 '10 at 1:48
(Okay, I can't resist: I was taught that 'et al.' expands to 'et alii/aliae/alia' for people, and 'et alibi' for places. I don't recall the word 'aliter', so I looked it up, and it seems to be the adverb meaning 'otherwise'.) – Willie Wong Jul 29 '10 at 1:57
Willie, Thanks. – Bruce Westbury Jul 29 '10 at 7:08

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