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

By H. L. L. Busard

Euklids Hauptwerk, die Elemente, gilt als dasjenige wissenschaftliche Werk, das am häufigsten bearbeitet und benutzt wurde; es struggle 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 tub. 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 remember the fact that that the well known Euclid editor Busard has back learned a masterly edition." Mathematical stories "àBusard's version is critical for our realizing of excessive medieval arithmetic. " Centaurus.

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

Example text

This can be proved like in [KS,II], but we also can get this immediately by using a general t h e o r e m stated in [F1]. 3 If ~r : X --* B is semiuniversal in t E B for holomorphic deformations, then it there also is semiuniversal for real analytic deformations. , x l . th. , s ~ . th. ,~lu = X o & and d& is uniquely determined. By restricting a = &luna we have Zlcrnc = X o a . 0 and ~ to a the m a p do is uniquely determined. f. [Abi] or see below). We need some preparations: We put Y ' = Y \ { P 1 , .

Desired properties. So o u r d e f o r m a t i o n t~ : Z ---+ G h a s all W e n o w c a n p r o v e t h e following: 4 . 1). , s k ) -+ B' ) depend real analytically on B ' . , s ' ) depends real analytically on the fiber parameter ~ C B I . Theorem Proof. L e t t E B t b e a r b i t r a r y . 2 t h e r e e x i s t s a h o l o m o r p h i c s e m i u n i v e r s a l d e f o r m a t i o n 7r : X -+ B C (P~ w i t h X0 = X~ a n d n = dirn~ HI(Xo, ®~). 3 we k n o w t h a t lr : X -+ B is also s e m i u n i v e r s a l for r e a l a n a l y t i c deformations.

H o l o m o r p h i c S y m p l e c t i c S t r u c t u r e s . [9,12] We consider now K~ihler manifolds with holomorphic symplectic structure or hyperKiihler manifolds. While Atiyah's article in this volume treats hyper-K~ihler manifolds more from the real or quaternionic viewpoint, we shall emphasize the complex analytic viewpoint. 2, the reader is referred to Atiyah's article and references therein. A hyper-Kiihler manifold is a Kiihler manifold with a nongenerate holomorphic, parallel 2-form.

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