Showing papers in "Archive for History of Exact Sciences in 1974"

New light on Frobenius' creation of the theory of group characters

Abstract: In an earlier paper [1971] I sketched the background to Frobenius' creation of the theory of group characters and sought to determine the role played therein by Richard Dedekind. An important source f insight was provided by Dedekind's letters to Frobenius, portio s of which were publi hed in Dedekind's Werke [1931 : 414-442]. Unfortunately, my efforts to locate Frobenius' part of th correspondence were unsuccessful, and it seemed t be ir etrievably lost. Recently, h wever, it was expectedly discover d by Kim rling [1972 : § 8] and is now located in the Clifford Memorial Library, University of Evansville, Indiana.1 The purpose of this paper is to co plete the historical anal sis of the origins of the t ory of group characters begun in my paper [1971]. There the empha is was necessarily upon Dedekind and the u ber-theoretic tradition which motivated his work. The Evansville ma uscripts now mak it possible o portray Fr benius1 role in much greater detail. The correspondence between Dedekind and Frobenius runs to more than 300 pages and falls in the periods I882-83, 1895-98, I90I. Frobenius' share amounts to 1 78 pages, many of them written in April of 1 8 6 when he was creating the theory of group ch racters as we now k ow iti.e. as opposed to the rlier, li i ed form of the theory developed by Dedekind for Abelian groups. These pages are virtually reports on " ork in progress" to Dedekind and provide invaluabl insight into the manner in which Frob nius was led to his gene alized characters and their funda e tal properties (Sections 3 and 4). It turns out that

Hipparchus on the distances of the sun and moon

Abstract: "Hipparchus' investigation of this topic is based principally on the sun. For since it is a consequence of certain other properties of the sun and moon (which we shall discuss below) that when the distance of one of the luminaries is given that of the other is also determined, he tries to demonstrate the distance of the moon by guessing the distance of the sun. He first assumes that the sun has only the least perceptible parallax, in order to find its distance, and afterwards uses the solar eclipse which he adduces. At one time he assumes that the sun has no perceptible parallax, at another that it has a significant [ixavov] parallax : as a result the ratios of the moon's distance [to the earth's radius] came out different for him for each of the hypotheses he set out. For in the case of the sun it is quite uncertain not merely how great a parallax it has, but whether it has any at all".

The differential: Nineteenth and twentieth century developments

Abstract: In this essay I shall be concerned principally with the way in which mathematicians have dealt with the concept of the differential and the meaning of differentiability of functions of more than one real variable or of a vector variable. For the nineteenth century the principal concern is with realvalued functions of several real variables. In the twentieth century attention shifts to the differential in functional analysis, and specifically to the Frechet differential for functions from one normed vector space to another. The classical function of several real variables is subsumed under functional analysis. The exposition is arranged in three parts: In Part I, I recount briefly how the concept of the differential for ordinary analysis was given essentially its modern formulation by Stolz in 1893 and how Frechet, in 1911 and 1912, pointed the way to the concept of the differential appropriate to functional analysis. In Part II, I review the treatments of differentials in the nineteenth century. In Part III, I survey the conceptual developments of the differential in the twentieth century.

Two letters from Einstein concerning his distant parallelism field theory

Abstract: There have been many attempts to produce a satisfactory macroscopic unified field theory for gravitation and electromagnetism. Yet none have aroused the general interest, expectations and even excitement, all generated to a large extent by the popular press, as the celebrated theory of Einstein that was published in its first definitive form in 1929 [1]. Previously Einstein had quite successfully interpreted gravitation in his theory of general relativity which employed a Riemannian metric having curvature, but no torsion. The non-RiEMANNiAN metric has, in general, both curvature and torsion. It was, in particular, that nonRiemannian metric having no curvature, but still having torsion, that fascinated Einstein for a long time and provided the basis for a unified field theory for gravitation and electricity. Due to the absence of curvature in that metric, directions at two separate points, any distance apart, could be compared without having to consider the parallel displacement of a vector along any particular curve joining those points, as in Levi-Civita parallelism in Riemannian geometry. Thus one referred to this metric as having distant parallelism ("Fern-Parallelismus").

Clausius' erste Anwendung der Wahrscheinlichkeitsrechnung im Rahmen der atmosphärischen Lichtstreuung

Abstract: Mainly depending on Gibbs' judgement, most historians of science concerned with the history of statistical mechanics attributed to Maxwell the merit of having introduced probabilistic methods into physics. Partly on the basis of unpublished material it is shown that Clausius in his early work on meteorological optics not only used probabilistic ideas but also applied the mathematical methods of the theory of probability in order to eliminate an hypothesis. Clausius' understanding of the role of hypotheses in physics apparently comes from Lambert's Photometria. Lambert seems to be responsible also for the concept of probability applied by Clausius in his early work. Around 1849 Clausius justified mean values as a description of the behaviour of a great many particles by the “laws of probability”. His first application of the calculus of probabilities shows parallels to Young's solution of the matching problem in 1819.

The mensuration of quadrilaterals and the generation of Pythagorean triads: A mathematical, heuristical and historical study with special reference to Brahmagupta's rules

Abstract: The ancient rule for the area of a quadrilateral,\(Q = \tfrac{1}{4}(a + c)(b + d)\), is examined to show its inaccuracy and its arbitrariness, and in order to see how those using it might have become aware of its shortcomings. Consideration ofBrahmagupta's vastly more sophisticated rule,\(Q = \sqrt {(s - a)(s - b)(s - c)(s - d)}\), and of the Hindu schedule of recognized quadrilateral types, leads to the question of the proper assessment to be made of this strand of Indian mathematics. The work on quadrilaterals was intimately connected with the generation of “Pythagorean” triangles out ofbīja (=“seed”) numbers by the same method as had probably been used by the mathematicians of Old Babylonia. Ways in which this procedure could have arisen by induction from particular numerical calculation are shown and the rest of the study consists primarily of an investigation as to whether the same kind of approach might not also have been used to obtain the remarkable rules on quadrilaterals given byBrahmagupta. It is found that there is no compelling need to adopt the common assumption thatBrahmagupta must have had access to Alexandrian mathematics. But even if he did chance to learn of some helpful items from the works ofHero orPtolemy, it still seems necessary for a proper appreciation of his understanding of the mensuration of quadrilaterals to suppose that he would have worked such items into the contemporary Indian mathematical context in some such way as is indicated here. In particular, it is argued thatBrahmagupta, far from indulging in reckless generalization, could well have proceeded with great caution, from one amenable type of quadrilateral to another, verifying his inductions by comparing them with results given by alternative calculation methods.

Early antecedents of error theory

Abstract: Galileo is the originator not only of the new mechanics which underlay the scientific revolution, but also of so much else in science that it was long the fashion that on discovering Galilean roots for various theories one would look no further on the assumption that there was a complete break between medieval science and what came afterward. Since Duhem, more recent scholarship has tended to correct this one-sided view and has sought to explore the medieval background to Galileo's work and a new perspective has thus emerged. Because of the characteristically modern nature of Error Theory it seemed hardly to be expected that significant medieval antecedents exist. Error Theory cannot arise without both the practical techniques of experimentation and the theoretical developments of the mathematics of probability. Though quite in keeping with the revolutionary nature of his work, it was therefore no mean accomplishment which is ascribed to him in saying that " Galileo had formulated the main propositions of the probabilistic error theory ".1 Galileo's discussion on error analysis occurs in connection with his presentation of the widely differing observational data reported on a nova.2 Now astronomy is the one science which already in antiquity had a long history of careful observation and collecting and comparing data. It is not unreasonable, therefore, to suppose that consideration should have been given long before Galileo to questions of error and the reliability of data. We shall see that this was indeed the case and in fact it seems likely that Galileo's 'error theory' consists of propositions which were more or less commonly assumed, although his is apparently the first known systematic presentation.

The Recto of the Rhind mathematical papyrus how did the ancient Egyptian scribe prepare it

Abstract: The Recto of the RMP of the British Museum, now nearly 4,000 years old, occupies about one-third of the whole of the 1 8-foot roll, and is the most extensive of all the arithmetical tables to be found among the ancient Egyptian papyri. It is inscribed in hieratic characters, the cursive form of hieroglyphics, and normally reads from right to left. The table gives the values of 2 divided by the fifty odd numbers from 3 to 101, all expressed as the sums of unit fractions; for example, 2 -r7 is ' ?. Unit fractions have unity for numerators, which with the solitary exception of the fraction §, were the only fractions the Egyptians used, or ever could use, because of their notation. Thus the number 7 in hieratic was ^, and the same number with a stroke above it, >?, represented the fraction *. The Recto table was of great importance to the scribes, because of its frequent use in ordinary multiplication and division, comparable perhaps in modern days, with a full set of multiplication tables. But one asks, why always 2 divided by odd numbers? The reason is that in Egyptian multiplication and division, only the twice times table was used, by constant doubling, and by occasionally finding | of a fraction from their rule, (RMP 61 B), or their tables. It was the very useful circumstance, of which the scribes were aware, that the powers of 2, namely, 1, 2, 4, 8, 16, 32, . . . , will produce, by properly choosing and adding them, every possible integer, entirely uniquely. Thus to multiply 147 by 43, the scribe wrote 147, doubled it (147X2), doubled the answer (147x4), doubled again (147 X 8), and so on, so that the addition of the products for 147 X (1, 2, 8, 32), gives the product 147x43Now if the multiplicand contained unit fractions, then their constant doubling presented difficulties which the Recto table helped to solve. There was no need to include the doubling of even unit fractions in the Recto table, because, for example, ? x 2 is obviously ', but a second doubling would present ' x 2, or 2 divided by 9, which the table provides as (' ?), but not as (I I), since two equal unit fractions were never written together, except perhaps as part of a calculation. Division was much the same, because if the scribe wished to divide 147 by 43, he did it by finding out what 43 must be multiplied by, to obtain 147, which is multiplication again but often more difficult. Addition of the integral parts of the various multiple products was easy enough, but the addition of the various unit fractions was a different proposition.

Pappus in the thirteenth century in the Latin west

Abstract: It has traditionally been the "communis opinio doctorum" that Pappus of Alexandria's Collectio Mathematica (late 3rd century A.D.) was not known during the Latin Middle Ages and that the first translation of Pappus's magnum opus into Latin was that undertaken by Federicus Commandinus, published at Pesaro in 15881 (reprinted at Venice the same and the next year).2 Indeed, this is the conclusion one reaches after examining the standard sources on the history of medieval mathematics. Thus, to mention one instance only, Moritz Cantor in his mammoth Vorlesungen uber Geschichte der Mathematik says :

Das Problem der Mischkristallisation von Vitriolen vor der Entdeckung des Isomorphismus

Abstract: One of the principal difficulties which R. J. Hauy's theory of the definite and exclusive relation between the chemical composition and the shape of crystals had to encounter should have been the fact that mixtures of different salts were discovered, combining in various combinations and still showing the same crystal form. In connection with their research on vitriols this phenomenon has been particularly studied by N. Leblanc (1786/87/88, 1802) and F. S. Beudant (1817, 1818). In addition to their concepts of the Surcomposition (Leblanc) and the Melange chimique (Beudant)—including a criticism by W. H. Wollaston which forced Beudant to a greater precision of his ideas—both scientists came close to the concept of a solid solution. On the other hand, those researches helped to sustain Hauy's theory since the seemingly puzzling phenomena could easily be explained if one accepted the additional hypothesis that even traces of certain substances have the power to force their particular crystal form on the whole chemical substance. Iron sulfate seemed to be a striking example for this hypothesis as less than 2% of this salt in a mixture of other sulfates is capable of giving its form to the whole mixture.