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## Solar and lunar observations at Istanbul in the 1570s

### Archive for History of Exact Sciences (2015-07-01) 69: 343-362 , July 01, 2015

From the early ninth century until about eight centuries later, the Middle East witnessed a series of both simple and systematic astronomical observations for the purpose of testing contemporary astronomical tables and deriving the fundamental solar, lunar, and planetary parameters. Of them, the extensive observations of lunar eclipses available before 1000 AD for testing the ephemeredes computed from the astronomical tables are in a relatively sharp contrast to the twelve lunar observations that are pertained to the four extant accounts of the measurements of the basic parameters of Ptolemaic lunar model. The last of them are Taqī al-Dīn Muḥammad b. Ma‘rūf’s (1526–1585) trio of lunar eclipses observed from Istanbul, Cairo, and Thessalonica in 1576–1577 and documented in chapter 2 of book 5 of his famous work, *Sidrat muntaha al-afkar fī malakūt al-falak al-dawwār (The Lotus Tree in the Seventh Heaven of Reflection)*. In this article, we provide a detailed analysis of the accuracy of his solar (1577–1579) and lunar observations.

## The Conical Sundial from Thyrrheion – Reconstruction and Error Analysis of a Displaced Antique Sundial

### Archive for History of Exact Sciences (2000-12-01) 55: 163-176 , December 01, 2000

### Summary

The conical sundial from the museum Thyrrheion is found to be designed with cardinal parameters

geographical latitude ϕ = arc tan(3/5) = 30°57′50″

half cone angle α = arc tan(4/9) = 23°57′45″

radius at equinox r_{0} = 4 unciae = 98.7mm (pes monetalis)

position of the cone tip h = 18 unciae = 444.3 mm

The half cone angle is equal to the angle of the ecliptic which leads to the special case of a conical sundial with the associated sphere being tangent at the day line of the winter solstice. The ratio of sides in the generating triangle 4:9 may thus be interpreted as an approximation for the angle of ecliptic.

The place of finding (operation) is 10° North of the intended latitude (design). The error of the sundial's indications due to the displacement are analyzed.

## Einstein Equations and Hilbert Action: What is missing on page 8 of the proofs for Hilbert's First Communication on the Foundations of Physics?

### Archive for History of Exact Sciences (2005-10-01) 59: 577-590 , October 01, 2005

The history of the publication of the gravitational field equations of general relativity in November 1915 by Einstein and Hilbert is briefly reviewed. An analysis of the internal structure and logic of Hilbert's theory as expounded in extant proofs and in the published version of his relevant paper is given with respect to the specific question what information would have been found on a missing piece of Hilbert's proofs. The existing texts suggest that the missing piece contained the explicit form of the Riemann curvature scalar in terms of the Ricci tensor as a specification of the axiomatically underdetermined Lagrangian in Hilbert's action integral. An alternative reading that the missing piece of the proofs already may have contained the Einstein tensor, i.e. an explicit calculation of the gravitational part of Hilbert's Lagrangian is argued to be highly implausible.

## Wie erfolgreich waren die im 19. Jahrhundert betriebenen Versuche einer mechanischen Grundlegung des zweiten Hauptsatzes der Thermodynamik?

### Archive for History of Exact Sciences (1987-03-01) 37: 77-99 , March 01, 1987

## Construction of Magic Squares Using the Knight’s Move in Islamic Mathematics

### Archive for History of Exact Sciences (2003-11-01) 58: 1-20 , November 01, 2003

## The lunar theories of al-Baghdādī

### Archive for History of Exact Sciences (1972-01-01) 8: 321-328 , January 01, 1972

### Conclusions

We have seen that the first two of *al-Baghdādī's* three methods originate from his predecessors
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{H}$$
*abash* and *Ya*
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{H}$$
*yā*. On the other hand, the ideas underlying the third method, which may well be due to *al-Baghdādī* himself, may have inspired *al-Kāshī* to construct similar two-argument tables in his *zīj*. This hypothesis is supported by the fact that the *zījes* of *al-Kāshī* and of *al-Baghdādī* exhibit another similarity: they contain the same rule for performing parabolic interpolation in planetary velocities.

One important contribution due to Muslim scientists to the development of the exact sciences in the Middle Ages consists in inventions of improved and more expedient computational methods. *Al-Baghdādī's* lunar theories therefore represent a stage on this road beginning with *Ya*
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{H}$$
*yā*/
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{H}$$
*abash* and terminating with *al-Kāshī's* highly sophisticated methods.

## Cyclotomie et formes quadratiques dans l’œuvre arithmétique d’Augustin-Louis Cauchy (1829–1840)

### Archive for History of Exact Sciences (2013-07-01) 67: 349-414 , July 01, 2013

Augustin-Louis Cauchy publie une majorité de ses recherches arithmétiques entre 1829 et 1840. Celles-ci ne sont pourtant qu’évoquées dans certaines histoires de la théorie des nombres centrées sur les lois de réciprocité ou sur la théorie des nombres algébriques. Elles y sont décrites comme contenant quelques résultats similaires à ceux de Gauss, Jacobi ou Dirichlet mais de manière incomplète et désordonnée. L’objectif de cet article est de présenter une analyse des textes arithmétiques de Cauchy publiés entre 1829 et 1840 pour montrer qu’ils contiennent au contraire un ensemble cohérent de résultats en lien avec les formes quadratiques $$4p^{\mu }=x^2+ny^2$$ , où $$p$$ est un nombre premier et $$n$$ un diviseur de $$p-1$$ . Nous discuterons également la forme particulière de ce corpus et la stratégie utilisée pour retrouver les lignes directrices du travail de Cauchy. Augustin-Louis Cauchy published most of his arithmetical research between 1829 and 1840. These are however only mentioned in some number theory history centered on reciprocity laws or on theory of algebraic numbers. They are described as containing some results similar to those of Gauss, Jacobi and Dirichlet but in a incomplete and disorganized way. The objective of this paper is to present an analysis of Cauchy’s arithmetical texts published between 1829 and 1840 to show that they contain a rather consistent set of results related to quadratic forms $$4p^{\mu } = x ^2 + ny ^2 $$ , where $$p$$ is a prime and $$n$$ a divisor of $$ p-1 $$ . We will also discuss the particular form of this body of texts and the strategy we used to find the guidelines of the work of Cauchy.

## Mean Motions in Ptolemy’s Planetary Hypotheses

### Archive for History of Exact Sciences (2009-07-04) 63: 635-654 , July 04, 2009

In the *Planetary Hypotheses*, Ptolemy summarizes the planetary models that he discusses in great detail in the *Almagest*, but he changes the mean motions to account for more prolonged comparison of observations. He gives the mean motions in two different forms: first, in terms of ‘simple, unmixed’ periods and next, in terms of ‘particular, complex’ periods, which are approximations to linear combinations of the simple periods. As a consequence, all of the epoch values for the Moon and the planets are different at era Philip. This is in part a consequence of the changes in the mean motions and in part due to changes in Ptolemy’s time in the anomaly, but not the longitude or latitude, of the Moon, the mean longitude of Saturn and Jupiter, but not Mars, and the anomaly of Venus and Mercury, the former a large change, the latter a small one. The pattern of parameter changes we see suggests that the analyses that yielded the *Planetary Hypotheses* parameters were not the elegant trio analyses of the *Almagest* but some sort of serial determinations of the parameters based on sequences of independent observations.