Prelude to dimension theory: The geometrical investigations of Bernard Bolzano

This paper treats Bernard Bolzano's (1781–1848) investigations into a fundamental problem of geometry: the problem of adequately defining the concepts of line (or curve), surface, solid, and continuum. Bolzano's interest in this problem spanned most of his creative lifetime. In this paper a full discussion is given of the philosophical and mathematical motivation of Bolzano's problem as well as his two solutions to the problem. Bolzano's work on this part of geometry is relevant to the history of modern mathematics, because it forms a prelude to the more recent development of topological dimension theory.     

Newton and the classical theory of probability

Summary Probabilistic ideas and methods from Newton's writings are discussed in § 1: Newton's ideas pertaining to the definition of probability, his probabilistic method in chronology, his probabilistic ideas and method in the theory of errors and his probabilistic reasonings on the system of the world. Newton's predecessors and his influence upon subsequent scholars are dealt with in §2: beginning with his predecessors the discussion continues with his contemporaries Arbuthnot and De Moiver, then Bentley. The section ends with Laplace, whose determinism is seen as a development of the Newtonian determinism.An addendum is devoted to Lambert's reasoning on randomness and to the influence of Darwin on statistics. A synopsis is attached at the end of the article.Abbreviations PT abridged Philosophical Transactions of the Royal Society 1665–1800 abridged. London, 1809 - Todhunter I. Todhunter, History of the mathematical theory of probability, Cambridge, 1865 To the memory of my mother, Sophia Sheynin (1900–1970)     

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.

---

---

Aram Frenkian zum Gedächtnis

---

Vorgelegt von J. E. Hofmann     

Interactions between mechanics and differential geometry in the 19th century

Conclusion 79. This study of the interaction between mechanics and differential geometry does not pretend to be exhaustive. In particular, there is probably more to be said about the mathematical side of the history from Darboux to Ricci and Levi Civita and beyond. Statistical mechanics may also be of interest and there is definitely more to be said about Hertz (I plan to continue in this direction) and about Poincaré's geometric and topological reasonings for example about the three body problem [Poincaré 1890] (cf. also [Poincaré 1993], [Andersson 1994] and [Barrow-Green 1994]). Moreover, it would be interesting to find out how the 19^th century ideas discussed here influenced the developments in the 20^th century. Einstein himself is a hotly debated case.Yet, despite these shortcommings, I hope that this paper has shown that the interactions between mechanics and differential geometry is not a 20^th century invention. Klein's view (see my Introduction) that Riemannian geometry grew out of mechanics, more specifically the principle of least action, cannot be maintained. On the other hand, when Riemannian geometry became known around 1870 it was immediately used in mechanics by Lipschitz. He began a continued tradition in this field, which had several elements in common with the new view of mechanics conceived by the physicists and explicitly carried out by Hertz.Before 1870 we found only scattered interactions between differential geometry and mechanics and only direct ones for systems of two or three degrees of freedom. For more degrees of freedom the geometrical ideas were in some interesting cases taken over by analogy, but these analogies did not lead to formal introduction of geometries of more than three dimensions.     

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     

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.     

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.

---

---

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

“... in Christophorum Clavium de Contactu Linearum Apologia” Considerazioni attorno alla polemica fra Peletier e Clavio circa l'angolo di contatto (1579–1589)

Summary In this work 1 focus my attention upon the question of the angle of tangency in the XVI^th Century, especially in the polemic between J. Peletier and Chr. Clavius (1579–1589). The interest in the question favored deliberation about the theory of proportions, the principle of Eudoxus-Archimedes and the set of angles of tangency (this is a non-Archimedian set); there were problems about logical proofs and geometrical proofs.

---

---

Memoria presentata da H. Freudenthal     

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.

---

---

Vorgelegt von J. E. Hofmann     

Einstein,Nordström and the early demise of scalar,Lorentz-covariant theories of gravitation

Conclusion The advent of the general theory of relativity was so entirely the work of just one person — Albert Einstein — that we cannot but wonder how long it would have taken without him for the connection between gravitation and spacetime curvature to be discovered. What would have happened if there were no Einstein? Few doubt that a theory much like special relativity would have emerged one way or another from the researchers of Lorentz, Poincaré and others. But where would the problem of relativizing gravitation have led? The saga told here shows how even the most conservative approach to relativizing gravitation theory still did lead out of Minkowski spacetime to connect gravitation to a curved spacetime. Unfortunately we still cannot know if this conclusion would have been drawn rapidly without Einstein's contribution. For what led Nordström to the gravitational field dependence of lengths and times was a very Einsteinian insistence on just the right version of the equality of inertial and gravitational mass. Unceasingly in Nordström's ear was the persistent and uncompromising voice of Einstein himself demanding that Nordström see the most distant consequences of his own theory.     

On the interaction of hypoxanthine with adenine

Résumé Une conversion non-enzymatique d'hypoxanthine en adénine fut proposée sur la base d'observations spectrophotométriques obtenues de solutions contenant adénine et hypoxanthine dans 1–5M de phosphate d'ammonium. Les résultats fournis par les expériences sur la radioactivité ne confirme pas la conclusion émise parBowne ² que hypoxanthine et adénine agissent l'une sur l'autre en solution concentrée de phosphate d'ammonium pour former de l'adénine.

---

---

This work was supported by United States Public Health Service Grants No. AM-08897, No. AM-10126, and No. 2A-5124; a grant from The Arthritis Foundation (Northern California Chapter); and from the Committee on Research, University of California School of Medicine (San Francisco).     

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)     

The role of glycine in the biosynthesis of steroids

Zusammenfassung Während der Biosynthese von Cholesterol mit homogenisierter Rattenleber wird [2-¹⁴C] des Glycins viel besser eingebaut als [1-¹⁴C].Saccharomyces cerevisiae produziert radioaktives Squelen (ausser Ergosterol mit Radioaktivität des Ringsystems) mit [2-¹⁴C] Glycin und mit [3-¹⁴C] Serin, aber nicht mit [1-¹⁴C] Glycin.

---

---

Studies on Biosynthesis. Part VI. For Part V, seeA. K. Bose, K. S. Khanchandani andB. L. Hungund, Experientia,27, 1403 (1971). b) Presented at the 164th National Meeting of the American Chemical Society, New York, August, 1972.

---

The support of this research by Stevens Institute of Technology and Sandoz Foundation is gratefully acknowledged. We wish to thank Drs.P. T. Funke, M. S. Manhas, P. K. Bhattacharyya, M. Anchel andH. Levey for valuable discussions and help with some of the experiments.     

Francis Hauksbee's theory of electricity

Conclusion Historians of science have usually assumed that the science of electricity developed in the period prior to Franklin, or at least prior to Nollet, in what amounted to a theoretical vacuum. It has been my aim in this paper to demonstrate the falsity of that assumption. I have shown, I hope, that Hauksbee's important researches were guided throughout by strong theoretical considerations, and I have indicated that Dufay's even more important studies were guided by exactly the same considerations. Nor was their theory in any sense a stagnant one. As it was developed by Hauksbee, it could give a fairly adequate explanation of almost all the known electrical phenomena; it even enabled him to predict the outcome of experiments such as the one involving the rubbing of a globe while it was positioned near a second, exhausted, globe. With the discovery of so many new phenomena in the 1730's, the theory turned out to be no longer adequate, but it is not at all surprising that it was a few years before the full extent of its inadequacies was appreciated, nor is it surprising that a strong continuity is evident between it and the theory which eventually replaced it. In the meantime, the theory continued to serve a useful function by suggesting new lines of research to its adherents. The theory functioned, then, in the same way as any other scientific theory, and it deserves a more serious treatment than it has usually received. This paper, I hope, can serve as a beginning.     

Bolzano,Cauchy and the “new analysis” of the early nineteenth century

Summary In this paper I discuss the development of mathematical analysis during the second and third decades of the nineteenth century; and in particular I assert that the well-known correspondence of new ideas to be found in the writings of Bolzano and Cauchy is not a coincidence, but that Cauchy had read one particular paper of Bolzano and drew on its results without acknowledgement. The reasons for this conjecture involve not only the texts in question but also the state of development of mathematical analysis itself, Cauchy both as personality and as mathematician, and the rivalries which were prevalent in Paris at that time.     

A history of the axiomatic formulation of probability from Borel to Kolmogorov: Part I

This paper, the first of two, traces the origins of the modern axiomatic formulation of Probability Theory, which was first given in definitive form by Kolmogorov in 1933. Even before that time, however, a sequence of developments, initiated by a landmark paper of E. Borel, were giving rise to problems, theorems, and reformulations that increasingly related probability to measure theory and, in particular, clarified the key role of countable additivity in Probability Theory.This paper describes the developments from Borel's work through F. Hausdorff's. The major accomplishments of the period were Borel's Zero-One Law (also known as the Borel-Cantelli Lemmas), his Strong Law of Large Numbers, and his Continued Fraction Theorem. What is new is a detailed analysis of Borel's original proofs, from which we try to account for the roots (psychological as well as mathematical) of the many flaws and inadequacies in Borel's reasoning. We also document the increasing realization of the link between the theories of measure and of probability in the period from G. Faber to F. Hausdorff. We indicate the misleading emphasis given to independence as a basic concept by Borel and his equally unfortunate association of a Heine-Borel lemma with countable additivity. Also original is the (possible) genesis we propose for each of the two examples chosen by Borel to exhibit his new theory; in each case we cite a now neglected precursor of Borel, one of them surely known to Borel, the other, probably so. The brief sketch of instances of the Cantelli lemma before Cantelli's publication is also original.We describe the interesting polemic between F. Bernstein and Borel concerning the Continued Fraction Theorem, which serves as a rare instance of a contemporary criticism of Borel's reasoning. We also discuss Hausdorff's proof of Borel's Strong Law (which seems to be the first valid proof of the theorem along the lines sketched by Borel).In retrospect, one may ask why problems of geometric (or continuous) probability did not give rise to the (Kolmogorov) view of probability as a form of measure, rather than the study of repeated independent trials, which was Borel's approach. This paper shows that questions of geometric probability were always the essential guide to the early development of the theory, despite the contrary viewpoint exhibited by Borel's preferred interpretation of his own results.     

Quantum decoherence and the approach to equilibrium (II)

In a previous paper [Hemmo, M & Shenker, O (2003). Quantum decoherence and the approach to equilibrium I. Philosophy of Science, 70, 330–358] we discussed a recent proposal by Albert [(2000). Time and chance. Cambridge, MA: Harvard University Press. Chapter 7] to recover thermodynamics on a purely dynamical basis, using the quantum theory of the collapse of the quantum state of [Ghirardi, G, Rimini, A and Weber, T., (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review, D 34, 470–479]. We proposed an alternative way to explain thermodynamics within no collapse interpretations of quantum mechanics. In this paper some difficulties faced by both approaches are discussed and solved: the spin echo experiments, and the problem of extremely light gases. In these contexts, we point out several ways in which the above quantum mechanical approaches as well as some other classical approaches to the foundations of statistical mechanics may be distinguished experimentally.     

Becquerel and the choice of uranium compounds

Conclusion The common assumption that Becquerel had no special reason to study uranium compounds in his search for substances emitting penetrating radiation cannot explain (a) Becquerel's own accounts, which refer to his choice as due to the peculiar harmonic series of bands; (b) Becquerel's systematic test of all uranium compounds (and metallic uranium), in contrast to his neglect of other substances; and (c) Becquerel's belief in invisible phosphorescence as an explanation of the radiation emitted by uranium compounds, even after his discovery that non-luminescent and metallic uranium also emit penetrating radiation.By comparing Becquerel's older studies of uranium to his radioactivity research, this paper has presented a reconstruction that can explain all of these points above. According to the historical evidence presented here, it is likely that Becquerel concentrated his attention on uranium and its compounds because the mechanical theory of luminescence opened up the possibility that, precisely in the case of uranium and its compounds, a violation of Stokes's law could occur, and penetrating short-wavelength radiation could be emitted through a special type of phosphorescence.     

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.

---

---

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.

---

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