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The Theory of Continuants in the Historical Order of its Development up to 1870

Published online by Cambridge University Press:  15 September 2014

Thomas Muir
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Extract

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’ Arithmetica 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 undertaken by Euler in his Specimen Algorithmi Singularis of the year 1764, in which denoting by

the convergents to

he established a long series of identities, such as

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 25 , Issue 1 , 1906 , pp. 129 - 159
Copyright
Copyright © Royal Society of Edinburgh 1906

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References

note * page 129 For the early history see Favaro's Notizie storiche sulle frazioni continue dal secolo decimoterzo al decimosettimo published in vol. vii. of Boncompagni's Bollettino: and as regards Bombelli see a paper by G. Wertheim in the Abhandl. zur Gesch. d. Math., viii. pp. 147–160.

note * page 130 The state of the theory in 1833 can best be gathered from Stern's monograph published in vol. x. of Crelle's Journal.Google Scholar

note * page 134 This is the author's date at the end of the paper (p. 381). The first two parts of the volume, however, are dated 1855, and the remaining two 1856.Google Scholar

note * page 137 An interesting extension of this is given by Brioschi, in the Nouvelles Annales de Math., xiv. (Jan. 1854) p. 20.Google Scholar

note * page 139 v. The theory of orthogonants … in Proc. Roy. Soc. Edinburgh, xxiv. p. 261.

note * page 142 It is in this mode of writing Aκι, viz., with the negative sign, that Jacobi's peculiarity consists. Not content with removing from Rn the row and column in which α κι occurs and prefixing to the minor thus obtained the sign factor (−l)ι+κ, he takes the further step of moving the row with the index κ over κ − ι +1 rows, thus arriving at

Of course there is at the second step the option of moving the column with the index ι over κ-ι + 1 columns, and this Ramus does.

note * page 154 To obtain the cofactor of the product of a number of a set of elements in a determinant Worpitzky puts a 1 in the determinant in place of each element occurring in the said product, O's in all the other places of the rows to which these elements belong, and O's for all the other elements of the set.

note * page 157 In giving to Ns+1, s Ns+2,s, Ns + 3,s the values 1, 1, 0 which are necessitated by assuming the generality of the recursion-formula

Worpitzky forgets to note that in these cases the proposition Nk,n = Nn,k;, used by him in the demonstration, does not hold.