By Hubert Stanley, Wall
The speculation of endured fractions has been outlined through a small handful of books. this is often one in all them. the focal point of Wall's booklet is at the learn of persisted fractions within the idea of analytic features, instead of on arithmetical facets. There are prolonged discussions of orthogonal polynomials, energy sequence, limitless matrices and quadratic kinds in infinitely many variables, convinced integrals, the instant challenge and the summation of divergent sequence. ``In penning this publication, i've got attempted to remember the scholar of really modest mathematical training, presupposing just a first path in functionality concept. hence, i've got integrated things like an evidence of Schwarz's inequality, theorems on uniformly bounded households of analytic capabilities, houses of Stieltjes integrals, and an advent to the matrix calculus. i've got presupposed an information of the hassle-free houses of linear fractional alterations within the complicated airplane. ``It has now not been my purpose to put in writing an entire treatise as regards to endured fractions, masking the entire literature, yet quite to offer a unified conception correlating convinced elements and purposes of the topic inside of a bigger analytic constitution ... '' --from the Preface
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Tannery3 pointed out that the former title corresponds well enough to the long extract' which Proclus gives in his first prologue, and also to the fragments contained in the Anonymi variae collectiones published by H ultsch at the end of his edition of Herons; but it does not suit most of the other passages borrowed by Proclus. 1 the lllathematics; and Pappus probably refers to one particular portion of the work, say the first Book. If the sixth Book treated of conics, as we may conclude from Eutocius, there must have been more Books to follow, because Proclus has preserved us details about higher curves, which must have come later.
P. 250, 20. ti Proc]us, p. 2Il, '9-23. 10 ibid. p. 283, 7-10. 11 ibid. pp. 304, "-305, 3. 13 ibid. p. 426, 6-912 ibid. pp. 428, 7-429,9, 3 CH. IV] PROCLUS AND HIS SOURCES 37 so far as the story of the sacrifice is concerned, the same thing is told of Thales in connexion with his discovery that the angle in a semicircle is a right angle l , and Plutarch is not certain whether the ox was sacrificed on the discovery of I. 47 or of the problem about application of areas". Plutarch's doubt suggests that he knew of no evidence for the story beyond the vague allusion in the distich of Apollodorus "Logisticus" (the "calculator") cited by Diogenes Laertius also s ; and Proclus may have had in mind this couplet with the passages of Plutarch.
CH. D. Attempts have however been made in two recent tracts to overthrow almost the whole of these arguments 1. (1) It is asserted that the olive-press of Mecltanics III. 20 is not the same as that referred to by Pliny. C. , while Heron quotes Posidonius and is therefore later, the intervals are al1 too short to make it probable that Heron would be mentioned in Geminus' historical work; and I think that the name of Heron may well have been inserted after that of Ctesibius by Proclus himself. (3) The view that Vitruvius did not use Heron's ,work is attacked, and the contrary sought to be proved, on the basis apparently of three passages.
Analytic Theory of Continued Fractions, by Hubert Stanley, Wall