We consider the relationship of the lattice-theoretic properties and the jump-theoretic properties satisfied by a recursively enumerable Turing degree. The existence is shown of a high2 r.e. degree which does not bound what we call the base of any Slaman triple.

We investigate the algebraic structure of the upper semi-lattice formed by the recursively enumerable Turing degrees. The following strong generalization of Lachlan's Nonsplitting Theorem is proved: Given n ≥ 1, there exists an r.e. degree d such that the interval $\lbrack\mathbf{d, 0'}\rbrack \subset\mathbf{R}$ admits an embedding of the n-atom Boolean algebra B n preserving (least and) greatest element, but also such that there is no (n + 1)-tuple of pairwise incomparable r.e. degrees above d which pairwise join to 0' (and (...) hence, the interval $\lbrack\mathbf{d, 0'}\rbrack \subset\mathbf{R}$ does not admit a greatest-element-preserving embedding of any lattice L which has n + 1 co-atoms, including B n + 1 ). This theorem is the dual of a theorem of Ambos-Spies and Soare, and yields an alternative proof of their result that the theory of R has infinitely many one-types. (shrink)

Up to now only three lunar tables by Leonhard Euler (1707–1783), published in 1745, 1746, and 1772, were known. For a long time, however, it was assumed that the first two of these tables were identical. The author compared these tables with each other and proved the contrary. This fact also transpires from an examination of their history, which was reconstructed using Euler’s correspondence. In addition, evidence has been found in Euler’s voluminous scientific correspondence and in contemporary publications of the (...) eighteenth century that, between 1742 and 1750, Euler published additional lunar tables anonymously. The author proved Euler’s authorship of at least five lunar tables containing some 100 pages, which were neither recorded in the list of his works nor published in the Leonhardi Euleri Opera Omnia. (shrink)