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1903-4.] Dr Muir on the Theory of Continuants. 
The Theory of Continuants in the Historical Order of its 
Development up to 1870. By Thomas Muir, LL.D. 
(MS. received October 5, 1903. Read November 2, 1903.) 
The more or less disguised use of continued fractions has been 
traced back to the publication of Bombelli’s Algebra in 1572, 
eighty-four years, that is to say, before the publication of Wallis’ 
Arithmetiea Infinitorum , in which Brouncker’s discovery was 
announced and the fractions explicitly expressed.* The study of 
the numerators and denominators of the convergents viewed as 
functions of the partial denominators was first seriously under- 
taken by Euler in his Specimen Algor ithmi Singularis of the year 
1764, in which denoting by 
l («. h ) («, t>, c) 
( (4) ’ (4, e) ’ 
the convergents to 
1 
a + T + J_ 
C + , 
he established a long series of identities, such as 
(a, b, c, d, . . . ) = a(b, c, d, . . . ) + (c, d, . . .) 
(a,) b, c, . . . 1) = (l, . . . , c, b } a ) , 
(a, b)(b, c) - (b)(a, b,c) = 1, 
(a, b, c,)(d , e, f) - ( a , b, c, d } e, f) = - (a, b)(e, f) , 
The study was pursued by Hindenburg and his followers during 
the last twenty years of the eighteenth century, but not with any 
great profit ; and, although in the first half of the nineteenth 
century considerable attention was given to the theory of con- 
tinued fractions as a whole, little advance was made in elucidating 
* For the early history see Favaro’s Notizie storiche sulle frazioni continue 
dal secolo decimoterzo at decimosettimo published in vol. vii. of Boncom- 
pagni’s Bollettino : and as regards Bombelli see a paper by G. Wertheim in 
the Abhandl zur Gesch. d. Math., viii. pp. 147-160. 
PROC. ROY. SOC. EDIN. — VOL. XXV. 
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