The Keplerian and mean motions. A geometrical study

Summary The point of reference to which the mean motion of the planets against Keplerian motion can optimally be applied and the behavior of its astronomical functions are analysed mathematically. To the results of our problem, which was solved by the first and second laws of Kepler, virtually all models of planetary motions seem to be related.     

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 Kepler Problem from Newton to Johann Bernoulli

Summary Newton solved what was called afterwards for a short time the directKepler problem (le problème direct): given a curve (e.g. an ellipse) and the center of attraction (e.g. the focus), what is the law of this attraction ifKepler's second law holds?The problème inverse (today: the problème direct) was attacked system-atically only later, first byJacob Hermann, then solved completely byJohann Bernoulli in 1710 and followingBernoulli byPierre Varignon. How didBernoulli solve the problem? What method did he use for this purpose and which of his accomplishments do we still follow today?In the second part various questions connected to the first part are dealt with from the point of view, Conflict and Cooperation, suggested byJ. van Maanen to the participants of the Groningen conference.     

Newton's early computational method for dynamics

On December 13, 1679Newton sent a letter toHooke on orbital motion for central forces, which contains a drawing showing an orbit for a constant value of the force. This letter is of great importance, because it reveals the state ofNewton's development of dynamics at that time. Since the first publication of this letter in 1929,Newton's method of constructing this orbit has remained a puzzle particularly because he apparently made a considerable error in the angle between successive apogees of this orbit. In fact, it is shown here thatNewton's implicitcomputation of this orbit is quite good, and that the error in the angle is due mainly toan error of drawing in joining two segments of the oribit, whichNewton related by areflection symmetry. In addition, in the letterNewton describes quite correctly the geometrical nature of orbits under the action of central forces (accelerations) which increase with decreasing distance from the center. An iterative computational method to evaluate orbits for central forces is described, which is based onNewton's mathematical development of the concept of curvature started in 1664. This method accounts very well for the orbit obtained byNewton for a constant central force, and it gives convergent results even for forces which diverge at the center, which are discussed correctly inNewton's letterwithout usingKepler's law of areas.Newton found the relation of this law to general central forces only after his correspondence withHooke. The curvature method leads to an equation of motion whichNewton could have solvedanalytically to find that motion on a conic section with a radial force directed towards a focus implies an inverse square force, and that motion on a logarithmic spiral implies an inverse cube force.     

Criticism of trepidation models and advocacy of uniform precession in medieval Latin astronomy

A characteristic hallmark of medieval astronomy is the replacement of Ptolemy’s linear precession with so-called models of trepidation, which were deemed necessary to account for divergences between parameters and data transmitted by Ptolemy and those found by later astronomers. Trepidation is commonly thought to have dominated European astronomy from the twelfth century to the Copernican Revolution, meeting its demise only in the last quarter of the sixteenth century thanks to the observational work of Tycho Brahe. The present article seeks to challenge this picture by surveying the extent to which Latin astronomers of the late Middle Ages expressed criticisms of trepidation models or rejected their validity in favour of linear precession. It argues that a readiness to abandon trepidation was more widespread prior to Brahe than hitherto realized and that it frequently came as the result of empirical considerations. This critical attitude towards trepidation reached an early culmination point with the work of Agostino Ricci (De motu octavae spherae, 1513), who demonstrated the theory’s redundancy with a penetrating analysis of the role of observational error in Ptolemy’s Almagest.     

Der Eigenartige Einfluß Witelos auf die Entwicklung der Dioptrik

Summary Witelo's Perspectiva, which was printed three times in the sixteenth century, profoundly influenced the science of dioptrics until the Age of Newton. Above all, the optical authors were interested in the so-called Vitellian tables, which Witelo must have copied from the nearly forgotten optical Sermones of Claudius Ptolemy. Research work was often based on these tables. Thus Kepler relied on the Vitellian tables when he invented his law of refraction. Several later authors adopted Kepler's law, not always because they believed it to be true, but because they did not know of any better law. Also Harriot used the Vitellian tables until his own experiments convinced him that Witelo's angles were grossly inaccurate. Unfortunately Harriot kept his results and his sine law for himself and for a few friends. The sine law was not published until 1637, by Descartes, who gave an indirect proof of it. Although this proof consisted in the first correct calculation of both rainbows, accomplished by means of the sine law, the Jesuits Kircher (Ars Magna, 1646) and Schott (Magia Optica, 1656) did not mention the sine law. Marci (Thaumantias, 1648) did not know of it, and Fabri (Synopsis Opticæ, 1667) rejected it. It is true that the sine law was accepted by authors like Maignan (Perspectiva Horaria, 1648) and Grimaldi (Physico-Mathesis, 1665), but since they used the erroneous Vitellian angles for computing the refractive index, they discredited the sine law by inaccurate and even ludicrous results.That even experimental determinations might be unduly biased by the Vitellian angles is evident from the author's graphs of seventeenth century refractive angles. These graphs also show how difficult it was to measure such angles accurately, and how the Jesuit authors of the 1640's adapted their experimental angles to the traditional Vitellian ones. Witelo's famous angles, instead of furthering the progress of dioptrics, delayed it. Their disastrous influence may be traced for nearly thirty years after Descartes had published the correct law of refraction.

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Vorgelegt von C. Truesdell     

Henry Dresser and Victorian Ornithology: Birds,Books and Business

The metaphysical commitment to the circle as the essential element in the analysis of celestial motion has long been recognized as the hallmark of classical astronomy. Part I of this paper contains a discussion of how, for Kepler, the circle also functions in geometry to select the basic polygons, in music to select the basic harmonies, and in astrology to select the basic aspects. In Part II, the discussion centres on the question of how the replacement of circular planetary orbits by elliptical orbits in the Astronomia Nova of 1609 affected Kepler's metaphysical commitment to celestial circularity that was made manifest in the derivation of planetary radii in the Mysterium Cosmographicum of 1596. The answer is found in the new and much more accurate derivation of both the planetary radii and their eccentricities in the Harmonice Mundi of 1619. It is the relationship of the diurnal movements of single planets at aphelion and perihelion to specific musical consonances that provides the first step. Then, in the second step, these ratios are ‘tempered’ so that all six planets can provide a heavenly choir. The third and final step employs the ‘mean period’, which is obtained directly from the tempered ratios given by musical theory and diurnal (not annual) motion, in the 3/2 power law to calculate the planetary radii and eccentricities with amazing accuracy. Thus the ellipse is necessary to supply the variation in angular velocities that contain the Creator's archetypal celestial circularity.     

The planetary increase of brightness during retrograde motion: An explanandum constructed ad explanantem

In Ancient Greek two models were proposed for explaining the planetary motion: the homocentric spheres of Eudoxus and the Epicycle and Deferent System. At least in a qualitative way, both models could explain the retrograde motion, the most challenging phenomenon to be explained using circular motions. Nevertheless, there is another explanandum: during retrograde motion the planets increase their brightness. It is natural to interpret a change of brightness, i.e., of apparent size, as a change in distance. Now, while according to the Eudoxian model the planet is always equidistant from the earth, according to the epicycle and deferent system, the planet changes its distance from the earth, approaching to it during retrograde motion, just as observed. So, it is usually affirmed that the main reason for the rejection of Eudoxus' homocentric spheres in favor of the epicycle and deferent system was that the first cannot explain the manifest planetary increase of brightness during retrograde motion, while the second can. In this paper I will show that this historical hypothesis is not as firmly founded as it is usually believed to be.     

Sauerstoff-Heteroringe. Die Konfiguration und Synthese desd,l-Homopterocarpins

Summary The racemic form of homopterocarpin (V), a constituent of red sandal-wood, has been synthesized by a method which has previously been proposed by the authors. N. M. R. spectral studies in conjunction with the examination of models have enabled the assignment of thecis configuration (V) to homopterocarpin.

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V. Mitteilung. IV. Mitteilung sieheH. Suginome, Exper.18, 161 (1962).     

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.     

Arginine-vasopressin,lysine-vasopressin,and oxytocin,C14-labeled in the glycine residue

Zusammenfassung Die Synthese von Arginin-Vasopressin, Lysin-Vasopressin und Oxytocin, deren Glycinrest eine¹⁴C-Markierung trägt, wird mit Hilfe der Festkörpermethode nachMerrifield beschrieben.

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Supported by National Institutes of Health grants No. AM-13567 and No. AM-10080 and the Atomic Energy Commission.

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Abbreviations follow the rules of the IUPAC-IUB Commission on Biochemical Nomenclature in Biochemistry5, 2485 (1966). All optically-active amino acids are ofl-configuration. The following additional abbreviations were used: N-hydroxysuccinimide ester (OSu), ethanol (EtOH), methanol (MeOH), acetic acid (AcOH),n-butanol (n-BuOH), pyridine (Pyr) and N,N-dicyclohexylcarbodiimide (DCCI). Protected peptides and hormones were visualized on thinlayer plates according to the procedure byH. Zahn andE. Rexroth, Z. analyt. Chem.148, 181 (1955). The biological activities of the hormones were measured against the U.S.P. Posterior Pituitary Reference Standard; the four-point design was used for these bioassays and standard errors were calculated according to the method ofC. I. Bliss,The Statistics of Bioassay (Academic Press, New York, N.Y. 1952).

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Acknowledgments. The authors wish to thank Mr.D. Schlesinger for some of the amino acid derivatives used, and MissM. Wahrenburg and Mrs.A. Silverman for bioassays.     

Encephalitogenic protein: Aβ-pleated sheet conformation (102–120) yields a possible molecular form of a serotonin receptor

Zusammenfassung Es besteht die Annahme (Carnegie 1970), dass das basische Gehirnprotein als Rezeptor für Serotonin wirkt, und es wird der Vorschlag für molekularbiologische Reaktionen von Serotonin-Rezeptoren gemacht.

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Acknowledgments. We are most grateful to Dr.D. Urry for advice on protein conformation, to Dr.P. R. Carnegie for helpful discussions and the Ealing Corporation for the generous loan of molecular models.     

Die Welt der kleinen Planeten

Summary After a historical survey on the discovery of the small planets and the international organization of the calculation of orbits the importance of the divergences from the ordinary orbits between Mars and Jupiter is discussed. On the one side those small planets which approach Mars, Jupiter, even Earth and Saturn, in strongly excentric orbits offer new methods for the mass determination of the large planets; e.g. the perigee of Eros made possible a new determination of the parallax of the sun (8,79). On the other side the group of the Trojans e.g. gives new aspects for the problem of the three bodies. In the breaches of commensurability the statistical distribution of the orbits of the small planets hints to the problem of the stability of the planetary system.     

Bromothricin

Summary Streptomyces antibioticus (Waksman andWoodruff)Waksman andHenrici 1948, strain Tü 99, produced Bromothricin instead of Chlorothricin, when it was grown in a nutrient medium containing 0.5% potassium bromide. The properties of bromothricin are very close to those of chlorothricin. Its nature was elucidated by degradation to methyl 5-bromo-2-methoxy-6-methylbenzoate.     

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.     

Cystinurie undl -cystinsteinbildung beim hund

Summary Cystinuria andl-Cystine lithiasis, is a rather rare hereditary metabolic disease in human beings. Due to the high purity ofl-Cystine stones formed in the urinary tract, they can serve as ideal models for studies on the chemical dissolution of kidney stones in general. By systematic examinations, it is possible to find for these purposes, dogs in which a cystinuria is present.

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Herrn Prof.Forenbacher, Vet. Med. Fakultät der Universität Zagreb, danken wir für die Übersendung von Steinen, und FrauM. Stoffers für die technische Mitarbeit.     

Geminus and the concept of mean motion in Greco-Latin astronomy

Conclusion Geminus account of lunar motion in chapter 18 of hisIntroductio astronomiae is, in our view, an important contribution to Greco-Latin astronomy because, in attempting to reconstruct arithmetically (the parameters of) the Moon's motion in longitude, he undermines the task astronomers had hitherto set for themselves. This undermining of a commonly acknowledged view of the purpose of astronomy is articulated in a whole new set of questions concerning the nature and place of both observation and mathematical reasoning in the science of the heavens. Yet, one must not overlook the fact thatGeminus reconstruction also indicates resources for addressing these questions. Of these resources, the most powerful proved to be the idea that irregular motion could be quantified as a systematic departure from a mean motion, and the idea that observational data could be organized and structured by means of genetic arithmetical reconstructions.But, since we limit our attention to extant treatises and decline to speculate about works or parts of works that have not survived, we must say that it would takePtolemy to discern the new direction for astronomy thatGeminus opened up and to pursue it. In part, this involved straightening out the conflated conception of mean motion in chapter 18 — the qua arithmetic mean daily displacement can only be anapparent lunar motion in longitude and not one the Moonreally makes, but the same need not be true of the qua periodic mean daily displacement — and determining its proper relation to real and apparent planetary motion. Indeed,Ptolemy's genius lay, we think, in seeing that even though, in assimilating Babylonian astronomy, earlier and contemporary Greco-Latin writers betrayed a confused, inconsistent, and unsophisticated grasp of the proper role of arithmetic, geometry, and observation in astronomical argument [seeBowen 1994], the solution lay in a mathematical reconstruction of the observed celestial motions, in which mean motion played an essential role.