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Additional resources for An Introduction to Lorentz Surfaces (De Gruyter Expositions in Mathematics 22)
GENERALIZATION, SPECIALIZATION, ANALOGY we remove arbitrary angles a, 13 a restriction, namely the restriction that < a < 90. We often generalize in passing from just one object to a whole class containing that object. 3. Specialization is passing from the consideration of a given set of For example, objects to that of a smaller set, contained in the given one. we when we pass from the consideration of polygons to polygons, and we specialize still further when we pass from specialize regular polygons with n sides to the regular, that that of regular equilateral, triangle.
Number by 1 1 ; in fact, of any number, af These laws are the same for various classes of numbers; we may consider here rational numbers, or real numbers, or complex numbers. In general, systems of objects subject to the same fundamental laws (or axioms) may be con- sidered as analogous to each other, clear meaning. and this kind of analogy has a completely The addition of real numbers is analogous to the multiplication of in still another sense. , (2) = log p. ) By virtue of each positive number corresponds a perfectly determined real number, and to each real number a perfectly determined positive this relation, to GENERALIZATION, SPECIALIZATION, ANALOGY number.
Predicting from E" means deriving under the assumption that E is true, A fact "agrees with "verifying" means deriving without this assumption. j" if it can be (easily) derived from the assumption that E is true. In the following we take for granted the elements of the calculus (which, from the formal side, were completely known to Euler at the time of his discovery) including the rigorous concept of limits (about which Euler never attained full clarity) We shall use only limiting processes which can be justified (most of them quite easily) but we shall not enter into detailed .