# MathAnnalen, Volumes 61-70

Alfred Clebsch, Carl Neumann, Felix Klein, Adolph Mayer, David Hilbert, Otto Blumenthal, Albert Einstein, Constantin Carathéodory, Erich Hecke, Bartel Leendert Waerden, Heinrich Behnke
J. Springer, 1911 - Electronic journals

### Popular passages

Page 185 - Whatever equivalent form is discoverable in arithmetical algebra considered as the science of suggestion, when the symbols are general in their form, though specific in their value, will continue to be an equivalent form when the symbols are general in their nature as well as in their form (Peacock 1833, pp.
Page 206 - But it is said that the symbol = here designates symbolical equivalence. The truth of this assertion depends on the definition of this phrase, and without doubt many arbitrary definitions might be given, in accordance with which the Binomial Theorem might be considered to hold for divergent series.
Page 185 - ... from arithmetical to symbolical algebra, we now make the following hypotheses : (1.) The symbols are unlimited, both in value and in representation. (2.) The operations upon them, whatever they may be, are possible in all cases. (3.) The laws of combination of the symbols are of such a kind as to coincide universally with those in arithmetical algebra when the symbols are arithmetical quantities, and when the operations to which they are subject are called by the same names as in arithmetical...
Page 582 - BOLZA. Die Lagrangesche Multiplikatorenregel in der Variationsrechnung für den Fall von gemischten Bedingungen, und die zugehörigen Grenzgleichungen bei variabeln Endpunkten.
Page 591 - Meyer (Königsberg), lieber eine Anwendung der Invariantentheorie auf die Entwicklung von Integralen, insbesondere rationaler, elliptischer und hyperelliptischer, in Reihen.
Page 187 - ... for such reasons, to notice it somewhat in detail. If the operations of algebra be considered as general, and the symbols which are subject to them as unlimited in value, it will be impossible to avoid the formation of divergent as well as of convergent series: and if such series be considered as the results of operations which are definable, apart from the series themselves, then it will not be very important to enter into such an examination of the relation of the arithmetical values of the...
Page 187 - It would necessarily lead to a great and embarrassing multiplication of cases ; it would deprive almost all algebraical operations of much of their certainty and simplicity...
Page 582 - ... Verschiebt man dann diese Punkte, indem man die Segmente (anb), (bpc), (cqd), (dma) ungeändert läßt, bis sie auf einem Kreise liegen, so hat man nach § 50, 1. die Fläche des Vierecks ab cd vergrößert; die übrigen Teile der Fläche F und der Umfang von s sind dadurch nicht geändert. Strenge Beweise des Satzes, daß der Kreis unter allen Figuren gleichen Umfanges den größten Inhalt hat, geben F. Edler, Göttinger Nachrichten 1882, Caratheodory und Study, Mathematische Annalen, Bd. 66....
Page 189 - ... within any conceivable distance of D. By the second part of the theorem (197), it is proved that arc BO : arc BF : : angle BCO : angle BCF. Now, although the arc BD is itself incommensurable with BF, yet it is the limit of the arcs BO, and the angle BCD is the limit of the angles BCO. Therefore, since whatever is true up to the limit is true at the limit, arc BD : arc BF : : angle BCD : angle BCF.
Page 188 - The motto which I should adopt against a course which seems to me calculated to stop the progress of discovery would be contained in a word and a symbol — remember...