Download Johannes de Tinemue's Redaction of Euclid's Elements, the by H. L. L. Busard PDF

By H. L. L. Busard

Euklids Hauptwerk, die Elemente, gilt als dasjenige wissenschaftliche Werk, das am häufigsten bearbeitet und benutzt wurde; es conflict ueber 2000 Jahre lang nicht nur das mathematische Lehrbuch schlechthin, sondern es beeinfluáte auch die Entwicklung anderer wissenschaftlicher Disziplinen. Das Werk wurde im 12. Jahrhundert aus dem Arabischen ins Lateinische uebersetzt, u.a. von Adelhard von bathtub. Diese Übersetzung wurde der Ausgangspunkt fuer zahlreiche weitere Bearbeitungen, wie die Redaktion, die um 1200 wahrscheinlich von Johannes de Tinemue angefertigt wurde. Campanus, der in den Jahren 1255/59 die fuer Jahrhunderte maágebende Euklid-Ausgabe besorgte, hat diese Redaktion sehr wahrscheinlich auch gekannt. "It is understand that that the well known Euclid editor Busard has back discovered a masterly edition." Mathematical reports "àBusard's variation is critical for our knowing of excessive medieval arithmetic. " Centaurus.

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Extra info for Johannes de Tinemue's Redaction of Euclid's Elements, the So-Called Adelard III Version

Example text

This u0 evidently has the desired property. Suppose now that C is a nonempty closed bounded convex subset of X, e > 0, and x* is in X*. By the remark above there is a point x0 in C so that x*(xo) >~x*(x) - ellx - x0l[ for all x in C. Consider two convex sets KI and K2 in X • IR: K1 := {(x, t): x ~ C; x*(x) >1 t}, K2 := {(x, t): x E X; t ~ x*(xo) + ~ l l x - x011}. The set K2 has a nonempty interior which is disjoint from K1, so the separation theorem gives a nonzero point (u*, c~) in X* 9 • and a fl so that u* (x) + a t ~> fl for (x, t) in Kl and u*(x) + at <.

To see this, let f be a Lipschitz function into a separable conjugate space Z* and let {Zn}ne~=lbe dense in Z. t. At a point to where all of these functions are differentiable, f ( t o ) ( z ) is differentiable for every z in Z (observe that h - l ( f ( t o -+- h) f (to))(z) - k -1 ( f (to + k) - f (to))(z) --+ 0 as h, k --+ 0 because the difference quotient is uniformly bounded since f is Lipschitz and tends to zero on the dense set {Zn }n~=l). , but the limit is only in the weak* sense. This is all that can be said using just the separability of Z.

Given the subspace E, take positive vectors x l . . , xn in X whose span contains E and normalized so that u := max/xi has norm one. Then F C Xu and Xu is isometric to a C ( K ) space for which the clopen subsets of K form a base for the topology, which means that the span of indicator functions of clopen sets is dense in C ( K ) . Thus fixing a basis yl . . . yk for E and 3 > 0, we get a subspace F of Xu spanned by disjoint vectors and a vector u in F with ]lull = 1 so that for each 1 ~< i ~< k, there is a vector Xi SO that ]Xi -- Zi] ~ (~U (and hence Ilxi - zi [I ~< ~).

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