Geometrical constructions equivalent to non-linear algebraic transformations of the plane in Newton's early papers

Summary The theory of constructive formation of plane algebraic curves in Newton's writings is discussed in § 1: the apparatus by which Newton forms the curves, Newton's theorems on forming unicursal curves, his theory of conics, and his theory of (m, n) correspondence. Special Cremona plane and space transformations obtained by Newton's organic method are dealt with in § 2. The article ends with § 3, which shows two different directions in the theory of the constructive formation of plane algebraic curves in the XVIII-XIXth centuries. A synopsis is appended.Abbreviations MPN The Mathematical Papers of Isaac Newton, edited by D. T. Whiteside, Vols. 1–3, Cambridge, 1967–1969 - Hudson H. Hudson, Cremona Transformations in Plane and Space, Cambridge, 1927 - PT (abridged) Philosophical Transactions of the Royal Society 1665–1800 (abridged), London, 1809 - Andreev ₁ K. A. Andreev, On geometrical correspondences ... (in Russian), Moscow, 1879 - Andreev ₂ K. A. Andreev, On the Geometrical Formation of Plane Curves (in Russian), Kharkov, 1875     

The problem of the invariance of dimension in the growth of modern topology,part II

Summary This work examines the historical origins of topological dimension theory with special reference to the problem of the invariance of dimension. Part I, comprising chapters 1–4, concerns problems and ideas about dimension from ancient times to about 1900. Chapter 1 deals with ancient Greek ideas about dimension and the origins of theories of hyperspaces and higher-dimensional geometries relating to the subsequent development of dimension theory. Chapter 2 treatsCantor's surprising discovery that continua of different dimension numbers can be put into one-one correspondence and his discussion withDedekind concerning the discovery. The problem of the invariance of dimension originates with this discovery. Chapter 3 deals with the early efforts of 1878–1879 to prove the invariance of dimension. Chapter 4 sketches the rise of point set topology with reference to the problem of proving dimensional invariance and the development of dimension theory. Part II, comprising chapters 5–8, concerns the development of dimension theory during the early part of the twentieth century. Chapter 5 deals with new approaches to the concept of dimension and the problem of dimensional invariance. Chapter 6 analyses the origins ofBrouwer's interest in topology and his breakthrough to the first general proof of the invariance of dimension. Chapter 7 treatsLebesgue's ideas about dimension and the invariance problem and the dispute that arose betweenBrouwer andLebesgue which led toBrouwer's further work on topology and dimension. Chapter 8 offers glimpses of the development of dimension theory afterBrouwer, especially the development of the dimension theory ofUrysohn andMenger during the twenties. Chapter 8 ends with some concluding remarks about the entire history covered. Dedicated to Hans Freudenthal     

P.S. Laplace's work on probability

Taken together with my previous articles [77], [80] devoted to the history of finite random sums and to Laplace's theory of errors, this paper sheds sufficient light on the whole work of Laplace in probability. Laplace's theory of probability is subdivided into theory of probability proper, limit theorems and mathematical statistics (not yet distinguished as a separate entity). I maintain that in its very design Laplace's theory of probability is a discipline pertaining to natural science rather than to mathematics. I maintain also the idea that the so-called Laplacian determinism was no hindrance to applications of his theory of probability to natural science and that one of his utterances in this connection could have well been made by Maxwell's contemporaries.Two possible reasons why the theory of probability stagnated after Laplace's work are singled out: the absence of new fields of application and, also, the insufficient level of mathematical abstraction used by Laplace. For all his achievements, I reach the general conclusion that he did not originate the theory of probability as it is now known. Dedicated to the memory of my Father, Boris A. Sheynin (1898–1975), the first generation of the Russian revolution Cette inégalité [Lunaire] quoique indiquée par les observations, était négligée par le plus grand nombre des astronomes, parce qu'elle ne paraissait pas résulter de la théorie de la pesanteur universelle. Mais, ayant soumis son existence au Calcul des Probabilités, elle me parut indiqués avec une probabilité si forte, que je crus devoir en rechercher la cause.(P. S. Laplace (Théor. anal. prob., p. 361))     

J. Zaragosa's centrum minimum,an early version of barycentric geometry

Summary Using the properties of the Centre of Gravity to obtain geometrical results goes back to Archimedes, but the idea of associating weights to points in calculating ratios was introduced by Giovanni Ceva in De lineis rectis se invicem secantibus: statica constructio (Milan, 1678). Four years prior to the publication of Ceva's work, however, another publication, entitled Geometria Magna in Minimis (Toledo, 1674), 2 appeared stating a method similar to Ceva's, but using isomorphic procedures of a geometric nature. The author was a Spanish Jesuit by the name of Joseph Zaragoza.Endeavouring to demonstrate an Apollonius' geometrical locus, Zaragoza conceived his idea of centrum minimum — a point strictly defined in traditional geometrical terms — the properties of which are characteristic of the Centre of Gravity. From this new concept, Zaragoza developed a theory that can be considered an early draft of the barycentric theory that F. Mobius was to establish 150 years later in Der barycentrische Calcul (Leipzig, 1827).Now then, whereas Ceva's work was rediscovered and due credit was given him, to this day Zaragoza's work has remained virtually unnoticed.     

The affinity of mitochondria for Ca++

Riassunto E' stata studiata l'affinità dei mitocondri di fegato di ratto e di cuore di coniglio per il Ca⁺⁺, nel sistema dipendente da energia. Si sono usati tamponi EGTA-Ca⁺⁺ per mantenere la concentrazione del Ca⁺⁺ costante durante l'esperimento. Si è dimostrato che laK _m apparente per il Ca⁺⁺ non è lontana da 10^–6 M.

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We are indebted to Dr.B. Chance for many stimulating discussions and for facilitating the progress of the work with the generous hospitality in his Institute.     

Die anorganischen Bestandteile des Honigtaues vonMegoura viciae Buckt

Summary In the honeydew ofMegoura viciae Buckt., sucking on the sieve tube sap ofVicia faba, the following cations are present in measurable quantities: potassium (13.0–14.1 mg/ml), sodium (0.04–0.051 mg/ml), magnesium (1.8–2.3 mg/ml) and calcium (0.07–0.09 mg/ml). There are only traces of copper, iron, manganese, zinc, cobalt and molybdenum. Amongst the anions, phosphate was found at 1.9–2.5 and chloride at 0.02–0.05 mg/ml, whereas nitrate and sulphate could not be detected.     

Le “Commentaire” d'Hipparque II. Position de 78 étoiles

Summary The following paper is the second part of a study devoted to the only preserved work of Hipparchus, his Commentary to Aratus. The first part, published in 1984, was based on the hypothesis that Hipparchus wrote his treatise with the help of a mobile sphere and led us to the determination of some features of that instrument. The coordinates of 78 stars, which are given below, have been obtained on the basis of the same hypothesis and using results already published. The stars selected are those for which the text connects numerical data to at least two events out of rising, setting or culmination. An analysis of the positions shows that the stars have been located on the sphere in an equatorial frame slightly different from the rotation axis frame. Statistical data processing shows that the stars were plotted on the sphere as circles rather than as points: the celestial coordinates of the stars are then those of the center of the disc, whose radius was determined. The position of a few stars could be evaluated thanks to the determination of an approximate value of the cubit used by Hipparchus. Lastly, the time of the observations was confirmed to be –140±25 years.The study confirms the basic hypothesis and throws light on the part played by the mobile sphere when the heavens are being described. It gives a better idea of the difficulties of constructing the instrument and of the ways of using it; a more detailed model of Hipparchus' mobile sphere will allow a more thorough investigation of all stars mentioned in the Commentary.

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Présenté par J. North     

Spontaneous occurrence of aneuploidy in somatic cells ofRana tigrina (Ranidae: Anura)

Zusammenfassung Es wurde Aneuploidie bis zu drei Chromosomen in den männlichen Körperzellen vonRana tigrina (Ranidae: Anura: Amphibia) beobachtet. Die Variationen betreffen:2n–3, 2n–2, 2n–1, 2n, 2n+1, 2n+2 und2n+3.

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Acknowledgments. My sincere thanks are due to Dr.S. P. Sharma, Zoology Department, for facilities; to Mr.R. K. Pillai and MissMadhu for technical assistance with preparations and Drs.K. K. Rishi andL. K. Vats for helpful suggestions.     

Isaac Newton and the problem of the earth's shape

In this article I discuss the theory of the earth's shape presented by Isaac Newton in Book III of his Principia. I show that the theory struck even the most reputable continental mathematicians of the day as incomprehensible. I examine the many obstacles to understanding the theory which the reader faced — the gaps, the underived equations, the unproven assertions, the dependence upon corollaries to practically incomprehensible theorems in Book I of the Principia and the ambiguities of these corollaries, the conjectures without explanations of their bases, the inconsistencies, and so forth. I explain why these apparent drawbacks are, historically considered, strengths of Newton's theory of the earth's shape, not weaknesses.     

Gamma aminobutyric acid induced changes in the spontaneous firing rates of insect neurons

Zusammenfassung -Aminobuttersäure (GABA) vermindert die Impulsfrequenz der spontan-aktiven Nervenzellen beim Insekt. Wirksame Konzentrationen betragen 0,1–25 · 10^–4 M. Ein unerwartetes Ergebnis war die Beobachtung, dass nach Applikation von GABA eine Erhöhung der Impulsfrequenz vor der Verminderung stattfindet.

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This investigation was supported in part by the NSF College Teacher Research Participation Program, Grant No. GY-682, and by NSF Undergraduate Research Participation Program, Grant No. GY-151. — We thank Mr.A. McAfee, our undergraduate participant, for his assistance in this investigation.     

Anastácio da Cunha and the concept of convergent series

Summary Anastácio da Cunha's definition of convergent series (Principios Mathematicos, Lisboa 1790, p. 106) was analysed by the Portuguese mathematician J. Vicente Gonçalves (Análise do Livro VIII dos Principios Mathematicos de José Anastácio da Cunha, Congresso do Mundo Português Vol. XII, Tomo I, Lisboa 1940, 123–140) and more recently by the historian A. P. Youschkevitch (J. A. da Cunha et les fondements de l'analyse infinitésimale Revue d'Histoire des Sciences Tomme XXVI, N. 1, 1973, 9–22). This latter author contests Gonçalves' claim to the effect that Cunha anticipated in his definition the well known criteria of convergence commonly attributed to A. Cauchy. However, Youschkevitch's otherwise deeper analysis is based primarily on a French translation of Principios published in Bordeaux in 1811. In this paper, that translation is shown to be misleading at crucial places. Cunha's definition is further analysed, and an interpretation in terms of the potential infinity is proposed which results in a redress of Cunha's originality in this matter.     

Das Parallelenproblem im Corpus Aristotelicum

Summary In the Corpus Aristotelicum are numerous items suggesting that the assertion of the fifth postulate in Euclid's Elements had been preceded by attempts to demonstrate this postulate itself, or some equivalent fundamental proposition, within the rigorous frame of Absolute Geometry in Bolyai's sense. Thus geometers contemporary with Aristotle tried to solve the problem which became known commonly in later centuries as the Problem of Parallels.Probably these geometers first attempted a direct solution. Only one text at our disposal supports this hypothesis: (1) Anal. Prior. 65 a 4–7. My analysis below in Chapter I shows that a mathematical meaning can be read from this somewhat obscure text only if it is interpreted as an allusion by Aristotle to those geometers who believe they are demonstrating, obviously in an absolute way, the proposition Elem. I 29, equivalent to the fifth postulate, but do not realize that in the process they are using lemmas which result themselves from the proposition to be demonstrated. Such a lemma would assert the uniqueness of the parallels, existence of which was shown in an absolute way in Elem. I 27. My conjecture and reconstruction afford a natural explanation for an inconsequence singular for Book I of the Elements, namely, the presence of the proposition Elem. I 31 in the purely Euclidean part of the book, in spite of the fact that the assertion merely repeats the absolute proposition Elem. I 27 without explicitly containing any Euclidean element.It is probable that the failure of these direct attempts led to an indirect approach to the problem through reductio ad absurdum of some hypothesis contrary to what was to become Postulate V or to some equivalent proposition. Numerous texts survive from which it is clear that geometers contemporary with Aristotle followed fairly far the consequences of an hypothesis contrary to the fifth postulate, obtaining important results which are partly identical with some theorems of Saccheri. Some of these texts attest first of all that what Saccheri called the Hypothesis of the Obtuse Angle had been stated in an independent and explicit way and that the fundamental result, identical with Prop. 14 of Saccheri's Euclides ab omni naevo vindicatus (1733), had been obtained, namely, that within Bolyai's Absolute Geometry this hypothesis leads to the remarkable formal contradiction that parallels intersect. This conclusion followed from two different formulations of the Obtuse Angle Hypothesis: (2) Anal. Prior. 66a 11–14, if the exterior angle (formed by a secant which intersects two parallel straight lines) is smaller than the interior angle (opposite and situated on the same side of the secant), and (3) 66a 14–15, if the sum of the angles in a triangle is greater than 2R. Finally, an item in (4) Ethica ad Eudemum 1222b 35–36 shows us that by investigating the Obtuse Angle Hypothesis, the Greek geometers also discovered the quadrilateral in which the sum of the angles is equal to 8R; this quadrilateral, which does not appear even in Saccheri's book, is the maximal quadrilateral of the Riemann geometry, a quadrilateral degenerated into a straight line closed upon itself (Chapter IV 20).Nowhere in the Corpus does the Hypothesis of the Acute Angle appear in an independent formulation. Nevertheless in (5) Anal. Poster. 90a 33–34 this Hypothesis is mentioned along with the other two: namely, Aristotle states that the essence of the triangle consists in the sum of its angles' being equal to, greater than or less than 2R (Chapter V 27). The formulation of the fifth postulate in the Elements allows greater probability to the conjecture of independent existence of the Acute Angle Hypothesis as well. Indeed, in its original formulation the fifth postulate is redundant, since it unnecessarily specifies in which of the half-planes (bounded by the secant) the intersection of the two straight lines occurs; this specification is itself a theorem. The Acute Angle Hypothesis must have been formulated not only symmetrically to (3) Anal. Prior. 66 a 14–15, that is, the sum of the angles of the triangle is less than 2R, as results from (5) Anal. Poster. 90 a 33–34, but also symmetrically to (2) Anal. Prior. 66 a 11–14. In the latter case the following final conclusion should have been reached in order to reduce to absurdity the Acute Angle Hypothesis: Two straight lines cut by a secant are incident if the sum of the interior angles (on the same side of the secant) is smaller than 2R, and the incidence occurs on that side of the secant where the sum of the angles is less than 2R. In the frame of the Acute Angle Hypothesis, this end conclusion is relevant only if this final specification (concerning the half-plane where the incidence occurs) is explicitly emphasised. According to my conjecture, it was precisely the practical impossibility of reaching this conclusion as a theorem of Absolute Geometry that later determined Euclid to transpose this decisive end conclusion from the Acute Angle Hypothesis, without changing its wording, and to include it among the postulates (Chapter II 13).A queer passage of Proklos (In primum Euclidis Elementorum, ed. Friedlein p. 368, 26–369, 1) in which the Acute Angle Hypothesis is presented in the form of a Zenonian paradox reinforces the conjecture that this hypothesis was studied independently by the ancient geometers (Chapter VI 33). Thus failure to solve the Problem of Parallels preceded not only the later Non-Euclidean geometry but also Euclidean geometry itself.The general undifferentiated Contra-Euclidean Hypothesis appears in the following form in all the other texts examined: The sum of the angles in the triangle is not equal to 2R. This hypothesis is nowhere qualified by Aristotle as being absurd or impossible: On the contrary, he takes it always as being just as much justified a priori as is the Euclidean theorem Elem. I 32 which contradicts it. For instance in (6) Anal. Poster. 93 a 33–35 Aristotle puts the problematical alternative: Which of the two propositions is right (or, which of the two constitutes the Logos, the raison d'être of the triangle), the one that states that the sum of the angles in the triangle is equal to 2R, or on the contrary, the one that states that the sum of the angles in the triangle is not equal to 2R (Chapter V 28)?In a number of texts the theorem Elem. I 32 itself and the general Contra-Euclidean Hypothesis are treated as being a sort of principle, and stress is laid on the idea that the logical consequences of each of these items invariantly preserve its specific (Euclidean or non-Euclidean) geometrical content [(7) 1187 a 35–38 (Chapter IV 18); (8) 1222 b 23–26 (Chapter IV 19); (9) 1187 b 1–2 (Chapter IV 18); (10) 1222 b 41–42 Chapter IV 21); (11) 1187 b 2–4 (Chapter IV 18)]; (12) Physica 200 a 29–30: If the sum of the angles in the triangle is not equal to 2R, then the principles of geometry cannot remain the same (Chapter V 25); (13) Metaph. 1052 a 6–7: It is impossible that the sum of the angles in the triangle be sometimes equal to 2R and sometimes not equal to 2R (Chapter V 24). Finally, the most important item of this sort is to be found in (14) De Caelo 281 b 5–7: If we accept as a starting hypothesis that it is impossible for the sum of the angles in the triangle to be equal to 2R, then the diagonal of the square is commensurable with its side (Chapter III).Another group of texts reveal Aristotle's attitude as regard these Contra-Euclidean theorems: (15) 1222 b 38–39 (Chapter IV 20); (16) 200 a 16–19 (Chapter VI 30); (17) 402 b 18–21 (Chapter VI 31); (18) 171 a 12–16 (Chapter VI 32); (19) 77 b 22–26 (Chapter V 26); (20) 101 a 15–17 (Chapter VI 31); (21) 76 b 39–77 a 3 (Chapter VI 31). These passages reveal Aristotle's conviction that these paradoxical Contra-Euclidean propositions (which cannot be annihilated by reductio ad absurdum) are nevertheless inacceptable as bad, probably because their graphical construction requires curved lines for representing the concept of straight lines.Finally, another group of texts show that Aristotle sensed in a way the necessity of adding to the foundations of Geometry a new postulate, from which the proposition Elem. I 32 should follow rigorously.

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Aram Frenkian zum Gedächtnis

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Vorgelegt von J. E. Hofmann     

Der qualitative und quantitative Nachweis der «fixen Luft» (1756), ein Wendepunkt in der Geschichte der Chemie und Biologie

Summary Reference is made to a treatise published in 1756 byJoseph Black (1728–1799), which was the first work to contain conclusive evidence of a gas bound to solid bodies; and in this connection the historical significance of the earliest studies on carbon dioxide is emphasised. Attention is drawn in particular to a subject about which little has hitherto been known,i.e., the use whichBlack and his contemporaries (notablyDavid Macbride) made of this discovery by applying it to animal and human physiology.

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Eine ausführlichere Würdigung dieses geschichtlichen Sachverhalts erscheint in Gesnerus (Schweiz)1956, Heft 3/4.     

Steroid stereochemistry

Zusammenfassung Ein Überblick über die angewandten Methoden zur Bestimmung der absoluten Konfiguration in Steroidseitenketten wurde gegeben. Die allgemeine Anwendung derR- undS-Regel wurde empfohlen.

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In part, a summary of recommendations presented by the author to a National Academy of Sciences and National Research Councilad hoc Committee on Steroid Nomenclature (R. C. Elderfield, Chairman); October 13–15, 1961, Columbus (Ohio). The meeting of thead hoc committee was made possible by a grant from the U.S. Air Force Office of Scientific Research.

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Part XV of a series entitledSteroid and Related Natural Products. Refer toG. R. Pettit andP. Hofer, Exper.19, 67 (1963), for the preceding contribution.     

Leonardo da Vinci's solution to the problem of the pinhole camera

The longstanding challenge of the pinhole camera for medieval theorists was explaining why luminous bodies cast onto a screen different images at different distances from the screen.I argue that this problem was first solved not by Francesco Maurolico, as David Lindberg concludes in his influential series of articles on the camera, but by Leonardo da Vinci. In studies in the Codex Atlanticus dating c. 1508–14, Leonardo explains the changes in screen patterns with distance by applying a key perspective principle to two kinds of projection pyramids that figure into pinhole camera imaging.In contrast, Maurolico's later conclusions about the pinhole camera are only partly correct. Maurolico gives a mistaken account of why pinhole images change with distance. He also introduces the erroneous notion that similar superimposed parts of the camera image actually fuse as the screen withdraws.     

Zur Wirkung von Colchiceinamiden an Zellenin vitro undin vivo

Summary Colchiceine-N-mono-methylamide is 3–4 times as active as colchicine as shown by its effect on the mouse-ascites-tumor (Ehrlich) and on fibroblastsin vitro.     

Kepler,Tycho, and the ‘Optical Part of Astronomy’: the genesis of Kepler's theory of pinhole images

In the last half of the 16^th century, the method of casting a solar image through an aperture onto a screen for the purposes of observing the sun and its eclipses came into increasing use among professional astronomers. In particular, Tycho Brahe adapted most of his instruments to solar observations, both of positions and of apparent diameters, by fitting the upper pinnule of his diopters with an aperture and allowing the lower pinnule with an engraved centering circle to serve as a screen. In conjunction with these innovations a method of calculating apparent solar diameters on the basis of the measured size of the image was developed, but the method was almost entirely empirically based and developed without the assistance of an adequate theory of the formation of images behind small apertures. Thus resulted the unsuccessful extension of the method by Tycho to the quantitative observation of apparent lunar diameters during solar eclipses. Kepler's attention to the eclipse of July 1600, prompted by Tycho's anomalous results, gave him occasion to consider the relevant theory of measurement. The result was a fully articulated account of pinhole images. Dedicated to the memory of Ronald Cameron Riddell (29.1.1938–11.1.1981)     

Sur l'action de l'acide 2,4-dichlorophénoxyacétique sur la croissance des racines deZea mays et dePisum cultivéesin vitro

Summary (1) A response ofZea mays andPisum roots cultured aseptically to 2,4-dichlorophenoxyacetic acid has been observed.Depending on the concentrationZea mays shows an increase of the growth rate (optimal concentration 10^–11 mol) which turns over to an inhibition (above 10^–7 mol). The curve is similar to that obtained by 3-indoleacetic acid, which proves the phytohormonal character of 2,4-D.(2) ThePisum root is more sensitive than theZea mays root. A concentration of 10^–7 molinhibitsPisum in a high degree, whileZea mays is no more inhibited, thus demonstrating the selective herbicidal action of 2,4-D against isolated roots of Dicotyledons culturedin vitro.     

Dissociation-constants of metal-ion-complexes with alkaline phosphatase from pig kidney

Summary Using metal-ion buffers it was possible to remove Zn²⁺, Mg²⁺ and Mn²⁺ ions of pig kidney alkaline phosphatase reversibly. The dissociation constants obtained are K_EMg:4·10^–7 M, K_EMn:4·10^–8 M and K_EZn:8·10^–13 M (22°C, pH:9.6, :0.07).Acknowledgement: The authors thank Dr.H. U. Wolf for helpful suggestions and valuable discussion and MissH. Köth for technical assistance.     

Neue Gesichtspunkte zum 5. Buch Euklids

Summary The author's purpose is to read the main work of Euclid with modern eyes and to find out what knowledge a mathematician of today, familiar with the works of V. D. Waerden and Bourbaki, can gain by studying Euclid's theory of magnitudes, and what new insight into Greek mathematics occupation with this subject can provide.The task is to analyse and to axiomatize by modern means (i) in a narrower sense Book V. of the Elements, i.e. the theory of proportion of Eudoxus, (ii) in a wider sense the whole sphere of magnitudes which Euclid applies in his Elements. This procedure furnishes a clear picture of the inherent structure of his work, thereby making visible specific characteristics of Greek mathematics.After a clarification of the preconditions and a short survey of the historical development of the theory of proportions (Part I of this work), an exact analysis of the definitions and propositions of Book V. of the Elements is carried out in Part II. This is done word by word. The author applies his own system of axioms, set up in close accordance with Euclid, which permits one to deduce all definitions and propositions of Euclid's theory of magnitudes (especially those of Books V. and VI.).In this way gaps and tacit assumptions in the work become clearly visible; above all, the logical structure of the system of magnitudes given by Euclid becomes evident: not ratio — like something sui generis — is the governing concept of Book V., but magnitudes and their relation of having a ratio form the base of the theory of proportions. These magnitudes represent a well defined structure, a so-called Eudoxic Semigroup with the numbers as operators; it can easily be imbedded in a general theory of magnitudes equally applicable to geometry and physics.The transition to ratios — a step not executed by Euclid — is examined in Part III; it turns out to be particularly unwieldy. An elegant way opens up by interpreting proportion as a mapping of totally ordered semigroups. When closely examined, this mapping proves to be an isomorphism, thus suggesting the application of the modern theory of homomorphism. This theory permits a treatment of the theory of proportions as developed by Eudoxus and Euclid which is hardly surpassable in brevity and elegance in spite of its close affinity to Euclid. The generalization to a classically founded theory of magnitudes is now self-evident.

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Vorgelegt von J. E. Hofmann