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A previous writer, Rafaello Bombelli, had used them in his treatise on Algebra (about 1579), and it is quite possible that Cataldi may have got his ideas from him. They next appear to have been used by Daniel Schwenter (1585-1636) in a Geometrica Practica published in 1618. The theory, however, starts with the publication in 16J5 by Lord Brouncker of the continued fraction I 23252 i 2 2 2 . This he is supposed to have deduced, no one knows how, from Wallis' formula for ?? Huygens (Descriptio automati planetarii, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his Planetarium. Nicol Saunderson (1682-1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler's Algebra (1795). Stern wrote at length on the subject in Crelle's Journal (x., 1833; xi., 1834; xviii., 1838).
A new waterfront area at Cardiff Bay contains the Senedd building, home to the Welsh Assembly and the Wales Millennium Centre arts complex.
Current developments include the continuation of the redevelopment of the Cardiff Bay and city centre areas with projects such as the Cardiff International Sports Village, a BBC drama village, Sporting venues in the city include the Millennium Stadium (the national stadium for the Wales national rugby union team), SWALEC Stadium (the home of Glamorgan County Cricket Club), Cardiff City Stadium (the home of Cardiff City football team), Cardiff International Sports Stadium (the home of Cardiff Amateur Athletic Club) and Cardiff Arms Park (the home of Cardiff Blues and Cardiff RFC rugby union teams).
Caer is Welsh for fort and -dyf is in effect a form of Taf (Taff), the river which flows by Cardiff Castle, with the ⟨t⟩ showing consonant mutation to ⟨d⟩ and the vowel showing affection as a result of a (lost) genitive case ending. The antiquarian William Camden (1551–1623) suggested that the name Cardiff may derive from "Caer-Didi" ("the Fort of Didius"), a name supposedly given in honour of Aulus Didius Gallus, governor of a nearby province at the time when the Roman fort was established.
Although some sources repeat this theory, it has been rejected on linguistic grounds by modern scholars such as Professor Gwynedd Pierce.
The Cardiff Urban Area covers a slightly larger area outside the county boundary, and includes the towns of Dinas Powys and Penarth.
A small town until the early 19th century, its prominence as a major port for the transport of coal following the arrival of industry in the region contributed to its rise as a major city.Cardiff was made a city in 1905, and proclaimed the capital of Wales in 1955.Since the 1980s, Cardiff has seen significant development.We have seen that the simple infinite continued fraction converges. The tests for convergency are as follows: Let the continued fraction of the first class be reduced to the form dl d2 d3 d4 then it is convergent if at least one of the series. diverges, and oscillates if both these series converge. In fact, a continued fraction ai a2 an can be constructed having for the numerators of its successive convergents any assigned quantities pi, P2, P3, , p ,,, and for their denominators any assigned quantities ql, q2, q 2, The partial fraction b n /a n corresponding to the n th convergent can be found from the relations pn = anpn -I bnpn -2 1 qn = anq,i l bngn-2; and the first two partial quotients are given by b l =pi, a1 = ql, 1)102=1,2, a1a2 b2= q2. n l which we can transform into u1 u2 utu3 u2114 un -2u,, -u1 u2-u2 u3-u3 u4- ... There is, however, a different way in which a series may be represented by a continued fraction. It is practically identical with that of finding the greatest common measure of two polynomials. We have F(n i,x) -F(n,x) = (y n)(y n I) F (n 2,x), whence we obtain F(i,x) _ i / y (y I) x /(y I)(y 2) which may also be written y 7 I-1-7 2 - By .The infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents. For the convergence of the continued fraction of the second class there is no complete criterion. Perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in Lagrange's investigation of the vibrations of a stretched string (see Lord Rayleigh, Theory of Sound, vol. If we form then the continued fraction inwhich pi, p2, p3 9 ..., pn are u1, u1 u 2, ul u2 u3, , /41 u2 un, and ql, q2, q3, , qn are all unity, we find the series u 1 u2 . un equivalent to the continued fraction un u l ul un - 1 ? We may require to represent the infinite convergent power series ao alx a2x 2 ... As an instance leading to results of some importance consider the series x x2 F(n,x) =I (y n)I! By putting =x 2 /4 for x in F(o,x) and F(i,x), and putting at the same time y =1/2, we obtain x 2 x 2 x2 x 2 x2 tan x x x tanh x x x x I - 3 - 5-7-...The Cardiff metropolitan area makes up over a third of the total population of Wales, with a mid-2011 population estimate of about 1,100,000 people.