This paper tries to make sense of Kant's scattered remarks about conic sections to see what light they shed on his philosophy of mathematics. It proceeds by confronting his remarks with the source that seems to have informed his thinking about conic sections: the Conica of Apollonius. The paper raises questions about Kant's attitude towards mathematics and the way he understood the cognitive resources available to us to do mathematics.

Due to the complexity of the cavity/vehicle and oscillation characteristics, streamlined shape integrated design of conventional fully wetted vehicles is not suitable for supercavitating vehicles. In this paper, a set of design criteria is highlighted to optimize the length and streamlined shape of a conical section subjected to realistic design constraints, which integrate the complex characteristics of the cavity/vehicle system under the condition of auto-oscillation of supercavitating vehicles. The auto-oscillation and its time-domain characteristics are determined. By deriving the equation (...) describing the cavity/vehicle relationship and identifying the maximum amplitude of the Euler angle, the cavity/vehicle tangent point criterion is proposed to determine the theoretical optimum value of the length of the conical section. A method of equal cross-sectional area for gas flow is proposed to design the streamlined shape of the conical section. Water tunnel and autonomous flight experiments were carried out to validate the feasibility of the design methodology developed in this work. (shrink)

Through an in-depth analysis of the notes that Leibniz made while reading Pascal’s manuscript treatise on conic sections, we aim to show the real extension of what he called “hexagrammum mysticum”, and to highlight the main results he achieved in this field, as well as proposing plausible proofs of them according to the methods he seems to have developed.

From the second half of the 10th century, mathematicians developed a new chapter in the geometry of conic sections, dealing with the theory and practice of their continuous drawing. In this article, we propose to sketch the history of this chapter in the writings of al-Qūhī and al-Sijzī. A hitherto unknown treatise by al-Sijzī - established, translated, and commented - has enabled us better to situate and understand the themes of this new research, and how it eventually approached the (...) problem of the classification of curves in previously unknown terms. (shrink)

In this paper, we describe the genesis of Boscovich’s Sectionum Conicarum Elementa and discuss the motivations which led him to write this work. Moreover, by analysing the structure of this treatise in some depth, we show how he developed the completely new idea of “eccentric circle” and derived the whole theory of conic sections by starting from it. We also comment on the reception of this treatise in Italy, and abroad, especially in England, where—since the late eighteenth century—several authors (...) found inspiration in Boscovich’s work to write their treatises on conic sections. (shrink)

In this paper, we describe the genesis of Boscovich’s Sectionum Conicarum Elementa and discuss the motivations which led him to write this work. Moreover, by analysing the structure of this treatise in some depth, we show how he developed the completely new idea of “eccentric circle” and derived the whole theory of conic sections by starting from it. We also comment on the reception of this treatise in Italy, and abroad, especially in England, where—since the late eighteenth century—several authors (...) found inspiration in Boscovich’s work to write their treatises on conic sections. (shrink)

In this study, we test the security of a crucial plank in the Principia’s mathematical foundation, namely Newton’s path leading to his solution of the famous Inverse Kepler Problem: a body attracted toward an immovable center by a centripetal force inversely proportional to the square of the distance from the center must move on a conic having a focus in that center. This path begins with his definitions of centripetal and motive force, moves through the second law of motion, (...) then traverses Propositions I, II, and VI, before coming to an end with Propositions XI, XII, XIII and this trio’s first corollary. To test the security of this path, we answer the following questions. How far is Newton’s path from being truly rigorous? What would it take to clarify his ambiguous definitions and laws, supply missing details, and close logical gaps? In short, what would it take to make Newton’s route to the Inverse Kepler Problem completely convincing? The answer is very surprising: it takes far less than one might have expected, given that Newton carved this path in 1687.Keywords: Isaac Newton; Principia; Centripetal force; Second law of motion; Conic orbits; Inverse Kepler Problem. (shrink)

The perfect compass, used by al-Qūhī, al-Sijzī and his successors for the continuous drawing of conic sections, reappeared after a long eclipse in the works of Renaissance mathematicians like Francesco Barozzi in Venice. The resurgence of this instrument seems to have depended on its interest to solve new optico-perspective problems. Having reviewed the various instruments designed for the drawing of conic sections, the article is focused on the sole conic compass. Theoretical and empirical applications are detailed. Contrarily (...) to the common thesis of an independant discovery, various elements suggest a direct descent between the birkār al-tāmm of the Arabic mathematical tradition and the Italian conic compass. Then we present the most probable transmission hypothesis involving: 1° Ibn Yūnus and his disciples of Mosul, 2° Sultan Malik al-Kāmil in Damas, 3° Master Theodore and Frederick II at the court of Sicily, 4° Andalò di Negro and Robert of Anjou in Naples, 5° Lorenzo della Volpaia, Vinci, Sangallo and Michelangelo in Florence, 6° Ausonio, Contarini, Thiene and Francesco Barozzi in Venice. (shrink)

From the second half of the 10th century, mathematicians developed a new chapter in the geometry of conic sections, dealing with the theory and practice of their continuous drawing. In this article, we propose to sketch the history of this chapter in the writings of al-Qūhī and al-Sijzī. A hitherto unknown treatise by al-Sijzī - established, translated, and commented - has enabled us better to situate and understand the themes of this new research, and how it eventually approached the (...) problem of the classification of curves in previously unknown terms. (shrink)

In the Tractatus, Wittgenstein conceives of tautology as 'saying nothing'. More precisely, he holds -- or so this essay contends -- that it says nothing in virtue of possessing a zero quantity of sense. Insofar as it is the limit of a series of propositions of diminishing quantity of sense, tautology resembles a degenerate conic section. But it also resembles the result of a summing together of equal and opposite linear vector quantities. Both of these models shape Wittgenstein's (...) conception of tautology in the Tractatus, though they are not fully reconciled. Many of the puzzling features of the Tractatus's view of logic arise from this failure to affect the needed reconciliation. These points are argued for by working through the development of Wittgenstein's views on these matters in the Wartime Notebooks and other pre-Tractatus writings. (shrink)

This article studies a fragment on the conic sections that appear in the Codex Atlanticus, fols. 611rb/915ra. Arguments are put forward to assemble these two folios. Their comparison with the Latin texts available before 1500 shows that they derive from the De speculis comburentibus of Alhacen and the De speculis comburentibus of Regiomontanus, joined together in his autograph manuscript. Having identified the sources, and discussed their mathematics, the issue of their transmission is targeted. It is shown that these notes (...) were written by Paolo dal Pozzo Toscanelli, through whom they reached the notebooks of Leonardo da Vinci. (shrink)

A textbook designed for the mandatory semester ethics course at the United States Naval Academy by USNA Ethics Section, with contributions by the Distinguished Chair in Ethics.

In Résolution des quatre principaux problèmes d’architecture (1673) then in Cours d’architecture (1683), the architect–mathematician Nicolas-François Blondel addresses one of the most famous architectural problems of all times, that of the reduction in columns (entasis). The interest of the text lies in the variety of subjects that are linked to this issue. (1) The text is a response to the challenge launched by Curabelle in 1664 under the name Étrenne à tous les architectes; (2) Blondel mathematicizes the problem in the (...) “style of the Ancients”; (3) The problem is reformulated and solved through the continuous drawing of the curve; (4) Blondel refutes the uniqueness of the curve by enumerating a variety of solutions (conchoid, spiral, parabola, ellipse, circle, hyperbola). This exuberance responds to an intention that does not coincide with the state of the art of mathematics at the end of the seventeenth century, nor with the taste for geometry of the Ancients, nor with any pedagogical project. This feature is explained by Blondel’s plan to found architecture on scientific bases. The reasons for his failure are analysed. (shrink)

The Greek mathematician Diophantos of Alexandria lived during the third century CE. Apart from his age, very little else is known about his life. Even the exact form of his name is uncertain, and only a few incomplete manuscripts of his greatest work, Arithmetica, have survived. In this impressive scholarly investigation, first published in 1885, Thomas Little Heath meticulously presents what can be gleaned from Greek, Latin and Arabic sources, and guides the reader through the algebraist's idiosyncratic style of mathematics, (...) discussing his notation and originality. This was the first thorough survey of Diophantos' work to appear in English. Also reissued in this series are Heath's two-volume History of Greek Mathematics, his treatment of Greek astronomy through the work of Aristarchus of Samos, and his edition in modern notation of the Treatise on Conic Sections by Apollonius of Perga. (shrink)

In his Brouillon Project on conic sections, Girard Desargues studies the notion of traversale, which generalizes that of diameter introduced by Apollonius. One often reads that it is equivalent to the notion of polar, a concept that emerged in the beginning of 19th century. In this article we shall study in great detail the developments around that notion in the middle part of the Brouillon project. We shall in particular show, using the notes added by Desargues after the first (...) draft was written, how here arises a new theory, that of the polarity associated to a conic section. We shall show that besides a natural correspondance between points and lines, Desargues has understood that a conic induces on every line in its plane an involution, and how he progressively handles the considerable conceptual difficulties he has to deal with when trying to explain clearly his novative ideas. (shrink)

The treatise on conic sections by the Hellenistic mathematician Apollonius from Perga is regarded as a supreme achievement of Greek mathematics and maintained its authority right up to the 18th century. This new edition is the first to consider all Greek and Arabic sources, with the Arabic texts being presented in the first ever critical edition. Both versions of the text are accompanied by a French translation, an extensive mathematical commentary, numerous philological notes and a complete glossary. The four (...) volumes of the edition are intended to be completed in 2009. (shrink)

In this paper we discuss in some depth the main theorems pertaining to Carnot’s theory of transversals, their initial reception by Servois, and the applications that Brianchon made of them to the theory of conic sections. The contributions of these authors brought the long-forgotten theorems of Desargues and Pascal fully to light, renewed the interest in synthetic geometry in France, and prepared the ground from which projective geometry later developed.

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such (...) as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Authorship ofthe Abstract Revisited David Raynor In a recent issue ofHume Studies, J. 0¿ Nelson challenges the received view that Hume himself composed the Abstract, and argues instead that we know that Adam Smith wrote it.1 But his main argument is so blatantly fallacious that charity requires that we interpret his intervention as ajeu d'esprit. I have no idea why he wishes to tease Hume scholars so mercilessly. (...) Most probably he wishes to keep the issue of the authorship of the Abstract alive until either someone disproves his belief that Smith wrote the pamphlet, or the scholarly community comes round to accepting Smith as the real author. Whatever his motive, I intend here to squelch his mischievous conjecture once and for all. Nelson first sought to cast doubt on Hume's authorship of the Abstract in an article published in The Philosophical Quarterly for 1976. In the fifteen years since that paper appeared, he has become convinced that Smith wrote the pamphlet. It was not always so. Here is how he judiciously concluded his earlier article: "According to the external evidence it is most improbable that Hume was the author of the Abstract and it is plausible to suppose that Adam Smith was; but according to the internal evidence, it is most improbable that Adam Smith was its author and almost certain that Hume was. How these incompatible conclusions are to be resolved I have no idea." 2 Nelson now has no doubt that any such discomfort should be resolved by allowing the external evidence tooutweigh the internal, becausehehas come to regard the external evidence as showing that it is not simply improbable that Hume wrote the work, but downright impossible. Nelson'sentire case rests on theidentityofthe "Mr Smith"referred to in a letter of 4 March, 1740 from Hume at Ninewells to Professor Francis Hutcheson at Glasgow. The crucial passage reads: My Bookseller has sent toMr Smith aCopy ofmy Book, which I hope he has received, as well as your Letter. I have not yet heard whathe has done with the Abstract. Perhaps you have. I have got it printed in London; but not in the WOrAe ofthe Learned; therehavingbeen an article with regard to my Book, somewhat abusive, printed in that Work, before I sent up the Abstract.3 Volume XTX Number 1 213 DAVID RAYNOR Until Keynes and Sraffa argued otherwise in their 1938 edition of the Abstract, some scholars believed that Adam Smith was the "Mr Smith" referred to in H, and that he wrote theAbstract at Hutcheson's suggestion. Keynes and Sraffainstead argued that the "MrSmith"was John Smith, Hutcheson's Dublin pubUsher, and that Hume himself wrote the Abstract (as all the internal evidence suggests). Norman Kemp Smith, in a review ofKeynes and Sraffa's edition, accepted their interpretation.4 Now Nelson seeks to prove all of them wrong. He argues thus: the "Mr Smith" in His eitherJohn Smith or Adam Smith; itcan'tbeJohn Smith;thereforeitmustbeAdamSmith.We canreadily agree with Nelson that the "Mr Smith" was notJohn Smith the Dublin publisher. But it will not follow—as Nelson supposes—that the man referred to must have been Adam Smith who, at the time, was still an undergraduate at Glasgow. "But ifthe Mr Smith ofH was not John Smith... who could he be excepttheAdam Smithofthe traditional theory?" asks Nelson.6 1have elsewhere suggested that the man in question was William Smith, one ofthepublishers oftheAmsterdam periodicalBibliothèqueraisonnée.6 Hutcheson'sInquiry had been pubUshed in Dublin by John Smith andWilliam Smith. The latter subsequently moved to Amsterdam and married into the Wetstein pubhshing family. William Smith was responsible for directing the firm's periodical, the Bibliothèque raisonnée, which employed Pierre Desmaizeaux as its London correspondent. The issue of this periodical for April-June, 1735, published aletterfrom Hutcheson tohis "ancien & intimeAmi"William Smith concerning Robert Simson's book on conic sections. In light ofH, it is probable that copies of the first two books of the Treatise, a manuscript copy oftheAbstract, and aletterfrom Hutcheson were sent to WilUam Smith in early 1740. The April-May-June issue of the Bibliothèqueraisonnée for 1740 published a largely favourable review of the first... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Two Sonnets DANIEL GALEF Thales to Thratta (spoken by “The Astrologer Who Fell into a Well”) All things are full of spirits. So said I, who plumbed the well of science, saw the sun made black and tracked its course across the sky, who, armed with muscle, wrote the river’s run. Where is my spirit? Thratta! I feel cold and wet. This well is deep. The world is wet (...) and cold. The world I knew was filled with gold and fire.... How quick the Olympic sun has set, how quick my words rewritten, and my land made palimpsest. The sea has left Miletus, the shore on which I walked and lectured—and the wise men who walked with me, whom I taught. The boundless sea, whose tides once rose to meet us has changed its shape, moved on. But I cannot. arion 27.2 fall 2019 Eudemus to Apollonius (spoken by the dedicatee of Apollonius of Perga’s first books of the Conic Sections) I write this note in the sand because I know the waves will wash it clean, and you won’t see it. Like the circles we traced here so long ago, our footprints fade from Pergamum. So be it. Your book was well-received. As was your son; his seems the echo of your dimming face. How did my thoughts, which like an hour-glass run, once hold the eternal laws of boundless space? The beach bends on forever. How might a man reckon every mote, each grit and grain? The universe of truths seems greater than could ever fit within the human brain— Yet love, which binds over such a greater span, all mortals its completeness can contain. 104 two sonnets... (shrink)

Johann Bernoulli in 1710 affirmed that Newton had not proved that conic sections, having a focus in the force centre, were necessary orbits for a body accelerated by an inverse square force. He also criticized Newton's mathematical procedures applied to central forces in Principia mathematica, since, in his opinion, they lacked generality and could be used only if one knew the solution in advance. The development of eighteenth-century dynamics was mainly due to Continental mathematicians who followed Bernoulli's approach rather (...) than Newton's. The ways of thinking of the British Newtonians have, therefore, been somewhat forgotten. This paper is an attempt to assess what Bernoulli was criticizing and what were the immediate reactions of the Newtonians. In particular, I will concentrate on two papers by John Keill, submitted to the Royal Society in 1708 and 1714, in which the results on central forces achieved by the British were summarized and their methods defended. (shrink)

In order to find positive solutions for third-degree equations, which he did not know how to solve for roots, m proceeds by the intersections of conic sections. The representation of an algebraic equation by a geometrical curve is made possible by the choices of units of measure for lengths, surfaces, and volumes. These units allow a numerical quantity to be associated with a geometrical magnitude. Is there a trace of this unit in the mathematicians to whom al-Khayyām refers directly (...) in his Algebra? How does this unit enable the measurement of quantities and rational and irrational relations? We find answers to these questions in the commentaries to Books V and X of the Elements. (shrink)

This book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high school concepts, such as numbers, polynomials, functions, and trigonometry. It retells math as a story, where the next chapter follows from the previous one. Readers will see how various mathematical concepts are tied, will see mathematics is (...) not a pile of formulas and facts, but has an orderly and beautiful edifice. The author begins with basic rules of logic, and then progresses through the topics already familiar to the students: numbers, inequalities, functions, polynomials, exponents, and trigonometric functions. There are also beautiful proofs for conic sections, sequences, and Fibonacci numbers. Each chapter has exercises for the reader. (shrink)

Excerpt from Selections From Pascal Blaise pascal was born at Clermont - Ferrand, in the center of F rance, on June 19, 1623. Three years later his mother died, and his father, taking the family duties most seriously, decided to be his son's own educator. At this time the father occupied a judicial position of considerable importance, but in 1630 he retired from it, moved the household to Paris, and gave himself up entirely to his work of preceptor. He taught (...) Blaise Latin and Greek, Mathematics and Physics, Logic and Philoso phy. For scientific studies the; boy showed unusual aptitude. At the age of twelve he was so far advanced as to write an essay on acoustics, which was followed at fifteen by treatise on conic sections. At eighteen he invented machine for count ing numbers. So great was his zeal for learning that he over taxed his strength by too steady application. Disorders ap peared, headaches and neuralgia, and his sister, Mme Perier, is authority for the statement that, from this time on, he rarely passed day without pain. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Over the course of the nineteenth century mathematicians became vividly aware that great advances in intuitive “understanding” could be obtained if novel definitions were devised for old notions such as “conic section”, for one thereby often gained a deeper appreciation for why old theorems in the subject had to be true. From a naïve philosophical standpoint, such definitional alterations look as if they must properly displace the “propositional contents” of the very theorems they seek to illuminate. Haven’t our (...) reformers merely “changed the subject”, rather than truly provided. The conceptual enlightenment they claim? Many practitioners of the time claimed that “Science” enjoys a special prerogative to ignore “surface content” in its search for truth, a sentiment with which Frege often concurs, at least in his early writings. Yet it is hard to render these opinions consistent with his official views on sense andreference, as this essay details. It also surveys Russell’s views on such topics, although he was generally less aware than Frege of the revolutionary mathematical work pursued within the “search for fruitful definitions” program. (shrink)

In the 1687 Principia, Newton gave a solution to the direct problem (given the orbit and center of force, find the central force) for a conic-section with a focal center of force (answer: a reciprocal square force) and for a spiral orbit with a polar center of force (answer: a reciprocal cube force). He did not, however, give solutions for the two corresponding inverse problems (given the force and center of force, find the orbit). He gave a cryptic (...) solution to the inverse problem of a reciprocal cube force, but offered no solution for the reciprocal square force. Some take this omission as an indication that Newton could not solve the reciprocal square, for, they ask, why else would he not select this important problem? Others claim that ``it is child's play'' for him, as evidenced by his 1671 catalogue of quadratures (tables of integrals). The answer to that question is obscured for all who attempt to work through Newton's published solution of the reciprocal cube force because it is done in the synthetic geometric style of the 1687 Principia rather than in the analytic algebraic style that Newton employed until 1671. In response to a request from David Gregory in 1694, however, Newton produced an analytic version of the body of the proof, but one which still had a geometric conclusion. Newton's charge is to find both ``the orbit'' and ``the time in orbit.'' In the determination of the dependence of the time on orbital position, t(r), Newton evaluated an integral of the form ∫dx/xn to calculate a finite algebraic equation for the area swept out as a function of the radius, but he did not write out the analytic expression for time t = t(r), even though he knew that the time t is proportional to that area. In the determination of the orbit, θ (r), Newton obtained an integral of the form ∫dx/√(1−x2) for the area that is proportional to the angle θ, an integral he had shown in his 1669 On Analysis by Infinite Equations to be equal to the arcsin(x). Since the solution must therefore contain a transcendental function, he knew that a finite algebraic solution for θ=θ(r) did not exist for ``the orbit'' as it had for ``the time in orbit.'' In contrast to these two solutions for the inverse cube force, however, it is not possible in the inverse square solution to generate a finite algebraic expression for either ``the orbit'' or ``the time in orbit.'' In fact, in Lemma 28, Newton offers a demonstration that the area of an ellipse cannot be given by a finite equation. I claim that the limitation of Lemma 28 forces Newton to reject the inverse square force as an example and to choose instead the reciprocal cube force as his example in Proposition 41. (shrink)

The Greek astronomer and geometrician Apollonius of Perga produced pioneering written work on conic sections in which he demonstrated mathematically the generation of curves and their fundamental properties. His innovative terminology gave us the terms 'ellipse', 'hyperbola' and 'parabola'. The Danish scholar Johan Ludvig Heiberg, a professor of classical philology at the University of Copenhagen, prepared important editions of works by Euclid, Archimedes and Ptolemy, among others. Published between 1891 and 1893, this two-volume work contains the definitive Greek text (...) of the first four books of Apollonius' treatise together with a facing-page Latin translation. Volume 1 contains the first three books, with the editor's introductory matter in Latin. (shrink)

The Greek astronomer and geometrician Apollonius of Perga produced pioneering written work on conic sections in which he demonstrated mathematically the generation of curves and their fundamental properties. His innovative terminology gave us the terms 'ellipse', 'hyperbola' and 'parabola'. The Danish scholar Johan Ludvig Heiberg, a professor of classical philology at the University of Copenhagen, prepared important editions of works by Euclid, Archimedes and Ptolemy, among others. Published between 1891 and 1893, this two-volume work contains the definitive Greek text (...) of the first four books of Apollonius' treatise together with a facing-page Latin translation. Volume 2 contains the fourth book in addition to other Greek fragments and ancient commentaries, notably that of Eutocius, as well as the editor's Latin prolegomena comparing the various manuscript sources. (shrink)

The Greek astronomer and geometrician Apollonius of Perga produced pioneering written work on conic sections in which he demonstrated mathematically the generation of curves and their fundamental properties. His innovative terminology gave us the terms 'ellipse', 'hyperbola' and 'parabola'. The Danish scholar Johan Ludvig Heiberg, a professor of classical philology at the University of Copenhagen, prepared important editions of works by Euclid, Archimedes and Ptolemy, among others. Published between 1891 and 1893, this two-volume work contains the definitive Greek text (...) of the first four books of Apollonius' treatise together with a facing-page Latin translation. Volume 1 contains the first three books, with the editor's Latin introductory matter. Volume 2 contains the fourth book in addition to other Greek fragments and ancient commentaries, notably that of Eutocius. (shrink)

In this paper, we give a thorough account of Brianchon and Poncelet’s joint memoir on equilateral hyperbolas subject to four given conditions, focusing on the most significant theorems expounded therein, and the determination of the “nine-point circle”. We also discuss about the origin of this very rare example of collaborative work for the time, and the general challenge of finding the nature of the loci described by the centres of the conic sections required to pass through m points and (...) to be tangent to n straight lines given in position, m + n = 4, which was posed at the end of their work. In the case m = 4, i.e. when the conic sections have to pass through the vertices of a quadrilateral, the locus of centres is another conic section passing through the intersection points of the opposite sides and the two diagonals of the quadrilateral, respectively, and, as Gergonne showed analytically shortly after, through other significant points connected with the quadrilateral; this curve was later given the name of the “nine-point conic”, being a natural generalization of the above mentioned circle. (shrink)

Once 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, as his grandfather Thābit ibn Qurra, was one of the most active and inventive mathematicians in these fields. Late Hélène Bellosta examined (...) in this article Ibn Sinān's contribution.Une fois les Coniques d'Apollonius traduits en arabe, ces livres sont devenus la référence indispensable et le principal instrument dans l'étude des problèmes solides, des équations algébriques des 3e et 4e degrés, des problèmes des mathématiques infinitésimales, etc. Les mathématiciens des ixe-xe siècles ont également étudié les constructions des sections coniques ainsi que leur tracé par points et leur tracé continu. Ibrāhīm ibn Sinān, comme son grand-père Thābit ibn Qurra, était l'un des mathématiciens les plus actifs et les plus inventifs en ce domaine. Dans cet article, la regrettée Hélène Bellosta a examiné la contribution d'Ibn Sinān.Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.DE L'USAGE DES CONIQUES CHEZ IBRĀHĪM IBN SINĀNVolume 22, Issue 1Hélène Bellosta DOI: https://doi.org/10.1017/S0957423911000129Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. DE L'USAGE DES CONIQUES CHEZ IBRĀHĪM IBN SINĀNVolume 22, Issue 1Hélène Bellosta DOI: https://doi.org/10.1017/S0957423911000129Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. DE L'USAGE DES CONIQUES CHEZ IBRĀHĪM IBN SINĀNVolume 22, Issue 1Hélène Bellosta DOI: https://doi.org/10.1017/S0957423911000129Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

This article makes two claims. The first is that Archimedes’ On Floating Bodies included a punning reference, in its key diagrammatic figure AΠΟΛ: the precise purpose of the pun may not be recovered by us, but even so it remains a powerful example of the playful in Archimedes’ writing. The second is that Apollonius could have been Archimedes’ younger contemporary. The outcome could be that we find Archimedes addressing a playful, hidden message to Apollonius, providing us with a unique insight (...) into the communication of Greek mathematics. (shrink)

The constraint-handling methods using multiobjective techniques in evolutionary algorithms have drawn increasing attention from researchers. This paper proposes an efficient conical area differential evolution algorithm, which employs biased decomposition and dual populations for constrained optimization by borrowing the idea of cone decomposition for multiobjective optimization. In this approach, a conical subpopulation and a feasible subpopulation are designed to search for the global feasible optimum, along the Pareto front and the feasible segment, respectively, in a cooperative way. In particular, the conical (...) subpopulation aims to efficiently construct and utilize the Pareto front through a biased cone decomposition strategy and conical area indicator. Neighbors in the conical subpopulation are fully exploited to assist each other to find the global feasible optimum. Afterwards, the feasible subpopulation is ranked and updated according to a tolerance-based rule to heighten its diversity in the early stage of evolution. Experimental results on 24 benchmark test cases reveal that CADE is capable of resolving the constrained optimization problems more efficiently as well as producing solutions that are significantly competitive with other popular approaches. (shrink)

It is well known that there is a categorical equivalence between lattice-ordered Abelian groups (or l-groups) and conical BCK-algebras (see [COR 80]). The aim of this paper is to study this equivalence from the perspective of logic, in particular, to study the relationship between two deductive systems: conical logic Co and a logic of l-groups, Balo. In [GAL 04] the authors introduce a system Bal which models the logic of balance of opposing forces with a single distinguished truth value, that (...) represents equilibrium. Its equivalent algebraic semantics BAL (via Blok-Pigozzi's construction) is definitionally equivalent to the variety of l-groups. In this paper we define the system Balo which is equivalent to Bal and whose Lindenbaum-Tarski algebra is an l-group. On the other hand, we define the conic logic Co and we prove that it can be naturally merged in the system Balo. Also, we prove that every formula of Balo has a normal form that depends on translations of formulas of Co. (shrink)

Summary The conical sundial from the museum Thyrrheion is found to be designed with cardinal parametersgeographical latitude ϕ = arc tan(3/5) = 30°57′50″half cone angle α = arc tan(4/9) = 23°57′45″radius at equinox r0 = 4 unciae = 98.7mm (pes monetalis)position of the cone tip h = 18 unciae = 444.3 mmThe 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. (shrink)

This document describes the implementation of a conical tank control system using an adaptive neurofuzzy system. For implementation, an indirect approach is used where the controller is optimized using the model obtained during the plant identification carried out using data obtained during the system operation. Furthermore, implementation includes training of neuro fuzzy-systems and application to control a conical tank. Regarding plant identification, preliminary training takes place using data obtained for different input values. The controller configuration is established considering the analogy (...) with a discrete-time linear system. The simulation shows that the control system manages to approach the desired response given by the considered reference model. (shrink)