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Published: 25 July 2024

Last updated: 31 July 2024

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1

Tripepi, Alessandro. "International Perspectives on the Florentine Edition of Apollonius’ Conics." Nuncius 38, no. 3 (November 23, 2023): 690–710. http://dx.doi.org/10.1163/18253911-bja10085.

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Abstract In 1661 Giovanni Alfonso Borelli published his edition of the three hitherto lost books belonging to the treatise on Conics by the Hellenistic mathematician Apollonius of Perga. The long and complex editorial operation is here reconstructed drawing on an unpublished document which had not been redacted within the Florentine circles that promoted the editorial initiative, but rather in the Roman circles which provided indispensable support to the venture. The examined letter, written by the Roman intellectual Michelangelo Ricci to prince Leopoldo de’ Medici, allows us to assess the significance of the effort made by a large a team involving numerous scholars experts in geometry and philology; and it allows us also to emphasise the important international dimension of a work that—from its genesis to its dissemination—has been able to connect the whole Continent.

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2

Stavek, Jiri. "Newton’s Hyperbola Observed from Newton’s Evolute (1687), Gudermann’s Circle (1833), the Auxiliary Circle (Pedal Curve and Inversion Curve), the Lemniscate of Bernoulli (1694) (Pedal Curve and Inversion Curve) (09.01.2019)." Applied Physics Research 11, no. 1 (January 29, 2019): 65. http://dx.doi.org/10.5539/apr.v11n1p65.

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Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets orbiting around the Sun. After the intensive study of those conic sections during the last four hundred years it is believed that this topic is practically closed and the 21st Century cannot bring anything new to this subject. Can we add to those visible orbits from the Aristotelian World some curves from the Plato’s Realm that might bring to us new information about those conic sections? Isaac Newton in 1687 discovered one such curve - the evolute of the hyperbola - behind his famous gravitation law. In our model we have been working with Newton’s Hyperbola in a more complex way. We have found that the interplay of the empty focus M (= Menaechmus - the discoverer of hyperbola), the center of the hyperbola A (= Apollonius of Perga - the Great Geometer), and the occupied focus N (= Isaac Newton - the Great Mathematician) together form the MAN Hyperbola with several interesting hidden properties of those hyperbolic paths. We have found that the auxiliary circle of the MAN Hyperbola could be used as a new hodograph and we will get the tangent velocity of planets around the Sun and their moment of tangent momentum. We can use the lemniscate of Bernoulli as the pedal curve of that hyperbola and we will get the normal velocities of those orbiting planets and their moment of normal momentum. The first derivation of this moment of normal momentum will reveal the torque of that hyperbola and we can estimate the precession of hyperbolic paths and to test this model for the case of the flyby anomalies. The auxiliary circle might be used as the inversion curve of that hyperbola and the Lemniscate of Bernoulli could help us to describe the Kepler’s Equation (KE) for the hyperbolic paths. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

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3

Stavek, Jiri. "Galileo’s Parabola Observed from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Galileo’s Empty Focus, Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Ptolemy’s Circle (Hodograph), and Dürer-Simon Parabola (16.03.2019)." Applied Physics Research 11, no. 2 (March 30, 2019): 56. http://dx.doi.org/10.5539/apr.v11n2p56.

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Galileo’s Parabola describing the projectile motion passed through hands of all scholars of the classical mechanics. Therefore, it seems to be impossible to bring to this topic anything new. In our approach we will observe the Galileo’s Parabola from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Galileo’s Empty Focus, Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Ptolemy’s Circle (Hodograph), and Dürer-Simon Parabola. For the description of events on this Galileo’s Parabola (this conic section parabola was discovered by Menaechmus) we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the Galileo’s empty focus that plays an important function, too. We will study properties of this MAG Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. For the visible Galileo’s Parabola in the Aristotelian World, there might be hidden curves in the Plato’s Realm behind the mechanism of that Parabola. The analysis of these curves could reveal to us hidden properties describing properties of that projectile motion. The parabolic path of the projectile motion can be described by six expressions of projectile speeds. In the Dürer-Simon’s Parabola we have determined tangential and normal accelerations with resulting acceleration g = 9.81 msec-2 directing towards to Galileo’s empty focus for the projectile moving to the vertex of that Parabola. When the projectile moves away from the vertex the resulting acceleration g = 9.81 msec-2 directs to the center of the Earth (the second focus of Galileo’s Parabola in the “infinity”). We have extracted some additional properties of Galileo’s Parabola. E.g., the Newtonian school correctly used the expression for “kinetic energy E = ½ mv2 for parabolic orbits and paths, while the Leibnizian school correctly used the expression for “vis viva” E = mv2 for hyperbolic orbits and paths. If we will insert the “vis viva” expression into the Soldner’s formula (1801) (e.g., Fengyi Huang in 2017), then we will get the right experimental value for the deflection of light on hyperbolic orbits. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

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4

Florio, Emilia. "Claude Mydorge Reader and Interpreter of Apollonius’ Conics." Mathematics 9, no. 3 (January 28, 2021): 261. http://dx.doi.org/10.3390/math9030261.

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In 1639, the treatise Prodromi catoptricorum et dioptricorum sive Conicorum operis ad abdita radii reflexi et refracti mysteria praevij et facem praeferentis. Libri quatuor priores by Claude Mydorge was printed in Paris. This volume, which followed the printing of his first two books in 1631, has resonance especially in the writings of those who, after him, addressed the conics. This fact raises the question of who Mydorge was and what his knowledge of the “doctrine” of the conics was, what is the most appropriate cultural context in which to properly read this writing, and finally, what is the place of its content in the development of thought placed between the Veteres and the Recentiores. In this paper, I attempt to elaborate an answer to these different questions, with the aim of emphasizing how the author reads and interprets the first books of Apollonius’ Conics. Neither the treatise, nor the figure of Mydorge, have received much attention in the current literature, although he was estimated as a savant in Paris and he was believed by Descartes to be one of the greatest mathematicians of his time.

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5

Hogendijk, Jan P. "Desargues' Brouillon Project and the Conics of Apollonius." Centaurus 34, no. 1 (March 1991): 1–43. http://dx.doi.org/10.1111/j.1600-0498.1991.tb00687.x.

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6

Bello-Chávez, Jhon Helver. "Elementa Curvarum Linearum more Apollonius that Descartes." Visión electrónica 2, no. 2 (December 6, 2019): 435–38. http://dx.doi.org/10.14483/22484728.18442.

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This document shows an analysis of the second book Elementa Curvarum Linearum written by Jan De Witt, published for the first time in the second edition of Geometry [1]. This writing is considered the first analytical geometry textbook. The influence of the work carried out by Apollonius in his conics book is studied, the use and interpretation of diagrams is debated. The development of the analytical method and the generation of curves by means of movement are also studied. Some propositions were renewed versions in terms of eighteenth-century mathematics, they used symbology, algebraic techniques and curves were classified by means of their symbolic representations, in these propositions a work closer to Apollonius is seen, the conic is not generated, it is assumed its existence, its nature is geometric. The study concludes that, although the textbook was published in the second edition of Geometry, the genesis of the curves remains geometric. The conics appear as objects of study in action immersed in the symbolic and algebraic practice characteristic of the time.

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7

Olmstead, Eugene A., and Arne Engebretsen. "Technology Tips: Exploring the Locus Definitions of the Conic Sections." Mathematics Teacher 91, no. 5 (May 1998): 428–34. http://dx.doi.org/10.5951/mt.91.5.0428.

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Conic sections were first studied in 350 B.C. by Menaechmus, who cut a circular conical surface at various angles. Early mathematicians who added to the study of conics include Apollonius, who named them in 220 B.C., and Archimedes, who studied their fascinating properties around 212 B.C. In previous articles in this journal, conic sections have been shown both as an algebraic, or parametric, representation (Vonder Embse 1997) and as a geometric, that is, a paper-folding, model (Scher 1996). Both articles offer important insights into the mathematical nature of the conic sections and into teaching methods that can integrate conics into our curriculum. Even though many textbooks discuss conic equations and their graphs, they do not fully develop locus definitions of conic sections.

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8

Hogendijk, Jan P. ""Apollonius Saxonicus": Die Restitution eines verlorenen Werkes des Apollonius von Perga durch Joachim Jungius, Woldeck Weland und Johannes Müller. Bernd Elsner." Isis 83, no. 4 (December 1992): 665–66. http://dx.doi.org/10.1086/356327.

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9

Jones, Alexander. "Book Review: On Isagogical Questions: Prolegomena Mathematica: From Apollonius of Perga to the Late Neoplatonists." Journal for the History of Astronomy 30, no. 3 (August 1999): 315–16. http://dx.doi.org/10.1177/002182869903000309.

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10

Bellosta, Hélène. "DE L'USAGE DES CONIQUES CHEZ IBRĀHĪM IBN SINĀN." Arabic Sciences and Philosophy 22, no. 1 (February 27, 2012): 119–36. http://dx.doi.org/10.1017/s0957423911000129.

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AbstractOnce Apollonius' Conics had been translated from Greek into Arabic, they became a main reference and the principal tool in studying solid problems, algebraic equations of 3rd and 4th degrees, infinitesimal mathematics, etc. Mathematicians of the 9th–10th centuries also studied the conic sections' constructions, as well as their continuous drawing and their drawing by points. Ibrāhīm ibn Sinān (909–946), as his grandfather Thābit ibn Qurra (826–901), was one of the most active and inventive mathematicians in these fields. Late Hélène Bellosta examined in this article Ibn Sinān's contribution.

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11

Sasaki, Chikara. "D'al-Khwārizmī à Descartes." Arabic Sciences and Philosophy 23, no. 2 (July 24, 2013): 319–25. http://dx.doi.org/10.1017/s0957423913000052.

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The volume D'al-Khwārizmī à Descartes is a monumental contribution to the world history of mathematical sciences, showing clearly that Arabic mathematics was an indispensable predecessor of early modern European mathematics. Roshdi Rashed is known, first of all, as an editor of classical mathematical writings in Arabic by such authors as al-Khwārizmī, Thābit ibn Qurra, Ibrāhīm ibn Sinān, Ibn al-Haytham, al-Khayyām, Sharaf al-Dīn al-Ṭūsī, as well as of the Arabic versions of Apollonius' Conics, Diophantus' Arithmetica, and Diocles' Burning Mirrors. As the volume under review shows, he is also a historian of mathematics of the first class who has transformed historiography. This book is, in a sense, a manifesto of Prof. Rashed's entire œuvre.

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12

Brigaglia, Aldo. "Remarks on the Historiography of Mathematics." Aestimatio: Sources and Studies in the History of Science 1 (April 30, 2021): 205–22. http://dx.doi.org/10.33137/aestimatio.v1i1.37627.

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In this paper, I examine aspects of the methodological debate that originated in 2010, when the distinguished historian of mathematics Sabetai Unguru reviewed Roshdi Rashed’s edition of the Arabic translation of Apollonius’ Conics. In his review, Unguru criticized what Rashed calls “l’usage instrumental d’une autre mathématique pour commenter une oeuvre ancienne”. I consider this debate very important and will try to place it within in the discussion of the so-called “geometric algebra” that goes back to the seventies, by tracing the contributions of the main figures who took part in it. Published Online (2021-04-30)Copyright © 2021 by Aldo Brigaglia Article PDF Link: https://jps.library.utoronto.ca/index.php/aestimatio/article/view/37627/28622 Corresponding Author: Aldo Brigaglia,University of PalermoE-Mail: aldo.brigaglia@gmail.com

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13

Sidoli, Nathan. "Apollonius's Conics: The Greek and Arabic TraditionsApollonius de Perge. Coniques. Volume 1, Part 1: Livre I: Commentaire historique et mathématique, edition et tradition du texte arabe. Volume 1, Part 2: Livre I: Édition et traduction du text grec. Edited by, Roshdi Rashed, Micheline Decorps-Foulquier, and Michel Federspiel. xiv + 666 pp., lxxxiv + 275 pp. Berlin/New York: Walter de Gruyter, 2008. $207 (cloth).Apollonius de Perge. Coniques. Volume 2, Part 1: Livres II et III: Commentaire historique et mathématique, edition et tradition du texte arabe. Edited by, Roshdi Rashed. xiv + 682 pp. Berlin/New York: Walter de Gruyter, 2010. $140 (cloth).Apollonius de Perge. Coniques. Volume 2, Part 2: Livre IV: Commentaire historique et mathématique, edition et tradition du texte arabe. Edited by, Roshdi Rashed. x + 319 pp. Berlin/New York: Walter de Gruyter, 2009. $140 (cloth).Apollonius de Perge. Coniques. Volume 2, Part 3: Livres II–IV: Édition et traduction du text grec. Edited by, Micheline Decorps-Foulquier and Michel Federspiel. xxx + 506 pp. Berlin/New York: Walter de Gruyter, 2010. $140 (cloth).Apollonius de Perge. Coniques. Volume 3: Livre V: Commentaire historique et mathématique, edition et tradition du texte arabe. Edited by, Roshdi Rashed. xi + 550 pp. Berlin/New York: Walter de Gruyter, 2008. $140 (cloth).Apollonius de Perge. Coniques. Volume 4: Livres VI et VII: Commentaire historique et mathématique, edition et tradition du texte arabe. Edited by, Roshdi Rashed. x + 572 pp. Berlin/New York: Walter de Gruyter, 2009. $137 (cloth)." Isis 102, no. 3 (September 2011): 537–42. http://dx.doi.org/10.1086/661629.

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14

Auffret, Thomas. "SERENUS D'ANTINOË DANS LA TRADITION GRÉCO-ARABE DESCONIQUES." Arabic Sciences and Philosophy 24, no. 2 (August 5, 2014): 181–209. http://dx.doi.org/10.1017/s0957423914000010.

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AbstractMainly based on the study of a hitherto neglected epigraphic document from Antinoë, the present article aims at showing that the geometer Serenus – the author of two treatisesOn the Section of a CylinderandOn the Section of a Cone– lived at the beginning of the 3rdcentury AD. On the ground of a renewed study of various elements taken both from the treatises and the indirect tradition, it also suggests that Serenus must be placed among a scientific tradition closely linking geometry of conics and catoptrics that can be traced back to the works of Conon of Samos and Pythion of Thasos. This hypothesis raises the problem of the nature of his alleged Platonism, which is examined in relation to Menaechmus' heterodox constructivism. Finally, the study of an element in the Arabic transmission of the treatise onConicsby Apollonius enables us to clarify some point regarding the textual tradition of the treatisesOn the Section of a CylinderandOn the Section of a Cone.

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15

GUERRINI, LUIGI. "PER UN ARCHIVIO DELLA CORRISPONDENZA DEGLI SCIENZIATI ITALIANI MATEMATICA ED ERUDIZIONE. GIOVANNI ALFONSO BORELLI E L'EDIZIONE FIORENTINA DEI LIBRI V, VI E VII DELLE CONICHE DI APOLLONIO DI PERGA." Nuncius 14, no. 2 (1999): 505–68. http://dx.doi.org/10.1163/182539199x00067.

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Abstracttitle SUMMARY /title As a consequence of the recovery of thirty-four of Giovanni Alfonso Borelli's letters to Carlo Roberto Dati relevant to the Florentine edition of the V, VI and VII books of Apollonius of Perga's CONICHE (1661), the article completes the known reconstruction of the various stages of the preparation of the edition with previously unknown events and gives some information about the meaning of some philological and mathematical controversy to which it gave rise. The letters show to the reader the lofty cultural quality of the edition of 1661 and enlighten the seventeenth-century debates on the genuineness of the codicological tradition as the means of transmission of the mathematical books of ancient times. The intellectual connexion between the letterato Carlo Roberto Dati and the mathematician Giovanni Alfonso Borelli gives furthermore a proof of the very close links binding erudition and science in seventeenth-century Italy.

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16

Tybjerg, Karin. "Jaap Mansfeld. Prolegomena Mathematica: From Apollonius of Perga to Late Neoplatonism. (Philosophia Antiqua, 80.) viii+178 pp., bibl., indexes. Leiden/Boston: E. J. Brill, 1998." Isis 94, no. 4 (December 2003): 705–6. http://dx.doi.org/10.1086/386420.

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17

Rigby, John, and G. J. Toomer. "Apollonius: Conics Books V to VII; The Arabic Translation of the Lost Greek Original in the Version of the Banu Musa." Mathematical Gazette 76, no. 477 (November 1992): 427. http://dx.doi.org/10.2307/3618416.

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18

Stavek, Jiri. "Newton’s Parabola Observed from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Castillon’s Cardioid, and Ptolemy’s Circle (Hodograph) (09.02.2019)." Applied Physics Research 11, no. 2 (February 25, 2019): 30. http://dx.doi.org/10.5539/apr.v11n2p30.

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Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets and other astronomical objects orbiting around the Sun. The books of these two Old Masters “Astronomia Nova” and “Principia…” were originally written in the geometrical language. However, the following generations of researchers translated the geometrical language of these Old Masters into the infinitesimal calculus independently discovered by Newton and Leibniz. In our attempt we will try to return back to the original geometrical language and to present several figures with possible hidden properties of parabolic orbits. For the description of events on parabolic orbits we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the focus occupied by our Sun discovered in several stages by Aristarchus, Copernicus, Kepler and Isaac Newton (The Great Mathematician). We will study properties of this PAN Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. In the Plato’s Realm some curves carrying hidden information might be waiting for our research. One such curve - the evolute of parabola - discovered Newton behind his famous gravitational law. We have used the Castillon’s cardioid as the curve describing the tangent velocity of objects on the parabolic orbit. In the PAN Parabola we have newly used six parameters introduced by Gottfried Wilhelm Leibniz - abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal. We have obtained formulae both for the tangent and normal velocities for objects on the parabolic orbit. We have also obtained the moment of tangent momentum and the moment of normal momentum. Both moments are constant on the whole parabolic orbit and that is why we should not observe the precession of parabolic orbit. We have discovered the Ptolemy’s Circle with the diameter a (distance between the vertex of parabola and its focus) where we see both the tangent and normal velocities of orbiting objects. In this case the Ptolemy’s Circle plays a role of the hodograph rotating on the parabolic orbit without sliding. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

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19

Wallace, Richard. "(R.) Netz The Shaping of Deduction in Greek Mathematics. Cambridge UP, 1999. Pp. xvii + 327. £40. 0521622794. - (J.) Mansfeld Prolegomena mathematica: From Apollonius of Perga to Late Neoplatonism. Leiden: Brill, 1998. Pp. vii + 178. 9004112677. - (D.) Fowler The Mathematics of Plato's Academy (2nd edn). Oxford UP, 1999. Pp. xxiii + 441, ill. £60. 0198502583. - (G.) Wöhrle Ed.Biologie. (Geschichte der Mathematik und der Naturwissenschaften in der Antike, 1). Stuttgart: Franz Steiner Verlag, 1999. Pp. 284, ill. DM 88. 3515073892." Journal of Hellenic Studies 123 (November 2003): 259–61. http://dx.doi.org/10.2307/3246317.

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20

Gisch, David, and Jason Ribando. "Apollonius’ Problem: A Study of Solutions and Their Connections." Volume 3, Issue 1 3, no. 1 (June 17, 2004). http://dx.doi.org/10.33697/ajur.2004.010.

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In Tangencies Apollonius of Perga showed how to construct a circle that is tangent to three given circles. More generally, Apollonius' problem asks to construct the circle which is tangent to any three objects that may be any combination of points, lines, and circles. The case when all three objects are circles is the most complicated case since up to eight solution circles are possible depending on the arrangement of the given circles. Within the last two centuries, solutions have been given by J. D. Gergonne in 1816, by Frederick Soddy in 1936, and most recently by David Eppstein in 2001. In this report, we illustrate the solution using the geometry software Cinderella™, survey some connections among the three solutions, and provide a framework for further study.

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21

Duffee, Linden Anne. "The Instructive Function of Mathematical Proof: A Case Study of the Analysis cum Synthesis method in Apollonius of Perga’s Conics." Axiomathes, April 12, 2021. http://dx.doi.org/10.1007/s10516-021-09551-w.

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