I would also like to know who first used the term in English. What I can do
here is give references to a reproduction of Wallis' Latin text, and one or
two other things.
Wallis' Arithmetica infinitorum first appeared in 1656 (or 1665; but my
photocopy has 1665 on its title page) and then, reset, in his Opera
mathematicorum pars altera, 1695. (There is a facsimile reproduction of
this 3-volume Opera, with an introduction by CJ Scriba, published, I think,
by Ohlms, but it is so expensive that our library refused to buy it.) There
is a cleaned-up reproduction of a bit of the page where he talks of a
"fractio, quae denominatorem habeat continue fractum" and writes it out as
a descending sequence of fractions, in
Pi: A Source Book, ed L Berggren, J & P Borwein, Springer 1997, pp.78-80
on p,80 where (I think) it has been added to the passage extracted from
A Source Book in Mathematics, 1200-1800, ed DJ Struik, Harvard University
Press, 1969, pp.244-251.
This reproduction is taken from the Opera, and its pages are chopped up a
bit: p.70 comes from Wallis, 462-3, page break after the Table, after
'Totus'; p.79 from pp.467-8, page break after 'Item'; and then on p.80, the
beginning of the section where Wallis describes Brouncker's continued
fraction expansion for Pi, from Wallis p.469. (The extracted passage from
Struik's commentary says nothing of Brouncker's cf; does the book itself?)
Pi: A Source Book also contains a dirty reproduction from Wallis' book on
p.76, where we see a step in his derivation of 'Wallis' product' for 4/pi.
Aside: Wallis also introduced the symbol for infinity in this book and a
pamphlet published (perhaps/perhaps not) the previous year, and there is
another reproduction of a page from the 1st ed containing this symbol (but
not the first such page), together with his comments on doing arithmetic
with it, in my
The Binomial Coefficient Function, American Mathematical Monthly 103 1(996)
1-17.
There is an earlier use of almost the same layout of cfs in Cataldi,
Trattato del modo brevissimo ... (1613), reproduced with translation in my
article on continued fractions in
Companion Encyclopedia of the History and Philosophy of the Mathematical
Sciences, 2 vols, ed I Grattan-Guinness, Routledge, 1994,
on pp.734-5, and this article also contains on p.736 an reproduction of the
calculation of convergents from
D Schwenter, Deliciae physico-mathematicae... 1636, p.113.
There are continued fractions all over Euler's works, and it would be
interesting to see a reproduction of some manuscript by him containing one
such - or were all of these passages dictated to secretaries when he was
blind?
But, as I said, there is no information here on the first English use of
the term! I know of no translation, even a modern one, of the Wallis
passage about Brouncker's continued fraction, apart from 3 lines in
C Brezinski, History of Continued Fractions and Pade Approximants,
Springer,1991, pp.79-80 (not in the index!).
It is possible that the information on English usage may be in this book,
but I'm not offering to go through to look for it! In any case, the book
has a very strong French bias.
David Fowler
> The web page (at http://members.aol.com/jeff570/mathword.html) quotes Morris
> Kline in Mathematical Thought From Ancient to Modern Times, page 255, as
> attributing the coinage of the term to Wallis in Opera Mathematica vol. I in
> 1695. However Drazin has found the phrase "continue fracta" in Arithmetica
> Infinitorum Prop. CXCI. So perhaps Wallis coined the term in 1655.
> However, Drazin is interested in knowing who coined the term in English or
> who translated it into English. I can forward any posts to him, but his
> address is:
>
> Philip G Drazin
> P.Drazin@bristol.ac.uk
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From: David Fowler <david.fowler@warwick.ac.uk>
Subject: Re: [HM] S.W. Steen
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My little addition to this topic:
I went to a few - not many! - lectures by Steen when I was an
undergraduate. He would come in with an ancient, yellowing, and battered
wodge of notes and copy out what were to me incomprehensible extracts from
them. I soon gave up his course.
As I understand it, his book was a worked-up version of those notes. A
malicious friend relates how he found himself sitting next to Steen at a
feast in Christs College, their old Cambridge college. He asked how the
book was doing; Steen replied, rather sadly: "Not very well. It's only sold
six copies so far."
I can't remember the actual number; it may have been even less, and I will
check when I see him next. But it might indicate that it will now be
difficult to locate a copy.
David Fowler