Recently the geometry of quantum states has been under considerable development. Every good geometry deserves, if possible, an accompanying trigonometry. I will here introduce such a trigonometry to accompany the geometry of quantum states.

En 1676, alors qu'il sejourne encore a Paris, Leibniz entreprend de composer un volumineux traite qui restera sans doute l'un de ses ecrits mathematiques les plus fortement charpentes: La quadrature arithmetique du cercle, de l'ellipse et de l'hyperbole et la trigonometrie sans tables qui en est le corollaire. Ce traite se presente comme un abrege exhaustif de la geometrie infinitesimale, dont Leibniz avait pu esperer qu'elle lui ouvrirait les portes de l'Academie des Sciences. Cependant, contraint de quitter la capitale avant (...) sa publication, il abandonnera derriere lui un ouvrage qui n'allait voir le jour qu'en 1993. Le probleme de la quadrature est pretexte a soulever plusieurs enjeux capitaux tant pour les mathematiques que pour la philosophie. Il permet notamment a Leibniz d'examiner avec une grande precision la question de la methode, des fondements, ainsi que des notions cruciales de rigueur et d'infini, tout en mettant en evidence leurs applications pratiques. Si la presente edition de ce texte foisonnant ne reproduit pas l'integralite des variantes, elle en propose une version corrigee et annotee, qui s'accompagne d'une traduction francaise en regard. (shrink)

Reviews Wildberger's account of his rational trigonometry project, which argues for a simpler way of doing trigonometry that avoids irrationals.

With mathematical reconstructions and philosophical arguments I show that Toomer's 1973 paper never contained any conclusive evidence for his claims that Hipparchus had a 3438'-based chord table, and that the Indians used that table to compute their sine tables. Recalculating Toomer's reconstructions with a 3600' radius -- i.e. the radius of the chord table in Ptolemy's Almagest, expressed in 'minutes' instead of 'degrees' -- generates Hipparchan-like ratios similar to those produced by a 3438' radius. It is therefore possible that the (...) radius of Hipparchus's chord table was 3600', and that the Indians independently constructed their 3438'-based sine table. (shrink)

After having reformulated optics, Ibn al-Haytham conceived of an analogous project for astronomy. This has just been revealed by an important book by the mathematician which has never been studied until now. Ibn al-Haytham's reform consists in excluding all cosmology, and in developing a systematic study of a celestial kinematics that has been completely geometrized. In turn, the realization of such a reform demanded innovative research in infinitesimal geometry. In this article, an attempt is made to present this new geometry, (...) as well as the mathematical means invented to elaborate it. (shrink)

Suppose you are stranded on an island and you want to get over to the nearby mainland. Your only option is to swim. But is the other shore close enough? If you embark and it isn’t, you drown. So you prefer to know before taking off. Happily, you are well equipped. You have not only a yardstick, but also a theodolite for measuring angles, and a good knowledge of trigonometry. You then determine the distance to the other shore by (...) means of triangulation. You fix two points, A and B, on your own shore, and identify a third point, C, in sight on the opposite shore. You then measure the distance between A and B with your yardstick, and the two angles ABC and CAB with your theodolite. With this information you can calculate the distances AC and BC to the other shore. (shrink)

The book of Aristarchus of Samos, On the distances and sizes of the sun and moon, is one of the few pre-Ptolemaic astronomical works that have come down to us in complete or nearly complete form. The simplicity and cleverness of the basic ideas behind the calculations are often obscured in the reading of the treatise by the complexity of the calculations and reasoning. Part of the complexity could be explained by the lack of trigonometry and part by the (...) fact that Aristarchus appears unwilling to make some simplifications that could be simply taken for granted. But an important part of the complexities is due to some unnecessary inconsistencies, as recently discovered by Berggren and Sidoli (Arch Hist Exact Sci 61:213–254, 2007). In the first part of this paper, I will try to show that some of these inconsistencies are just apparent. But the complexity of the calculations and reasoning is not the only reason that could disturb a reader of the treatise. The great inaccuracy—even for the measurement methods and instruments available at those times—of one of the three input values of the treatise is really astonishing. In the sixth and last hypothesis, Aristarchus states that the moon’s apparent size is equal to 2∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\circ }$$\end{document}, while the correct value is one-fourth of that. Some attempts have been made in order to explain such a big value, but all of them have problems. In the second part of this paper, I will propose a new speculative but plausible explanation of the origin of this value. (shrink)

In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. (...) The article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids. (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)

The twelfth century was a period of transmission and absorption of Arabic learning though it filtered outside of the Arabic world as early as the second half of the tenth century. In general, the lure of Spain began to act only in the twelfth century, and the active impulse toward the spread of Arabic mathematics came from beyond the Pyrenees and from men of diverse origins. The chief names are Adelard of Bath, Robert of Chester, Hermann of Carinthia and Gerard (...) of Cremona. In this time the Latin world became acquainted with the Hindu numerals, the Arabic Algebra and Euclid'sElements. However, not only Spain, but also the Norman kingdom of southern Italy and Sicily occupies a position of peculiar importance, though the works of the translators did not become very influential. There were made direct translations from Greek into Latin. One had to wait a century more to obtain a translation from Greek into Latin of the chief Archimedean scientific and mathematical treatises by William of Moerbeke. In the thirteenth century Fibonacci and Jordanus Nemorarius stand at the threshold of European mathematics. Not only was Fibonacci the first to explain Arabic arithmetic, but his works, especially his later ones, contain many original ideas. Jordanus continued the Greco-Roman tradition rather than the Greco-Arabic one, but he did so with much independence. To Nicole Oresme (fourteenth century) was due a broadened view of proportionality, a geometric proof to determine the summation of convergent infinite series and the proof, evidently the first in the history of mathematics, that the harmonic series is divergent. The Configuration Doctrine was treated by Merton College authors and by Oresme. In the fifteenth century theDe triangulis omnimodis of Regiomontan, a systematic account of the methods for solving triangles, marked the rebirth of trigonometry. (shrink)

In The Sea–Mans Kalendar, Henry Bond predicted that magnetic declination would be 0° in 1657, and would then increase westerly for 30 years. Based on these predictions, Bond went on to claim in The Longitude Found that, by using his model of magnetism, he can offer a technique for determining longitude. This paper offers an assessment of Bond’s method for longitude determination and critically evaluates Thomas Hobbes’s so–far neglected response to Bond’s proposal in Decameron physiologicum, in which Hobbes complains about (...) what he takes to be Bond’s implicit natural philosophy and about his use of spherical trigonometry. (shrink)

China was the first country beyond Europe with an important civilization to receive scientific theory from the West in the modern era. Neither in India nor in Japan (where the first Western works arrived from China and were quickly banned) nor a fortiori in other missionary countries was there an early acquaintance with European sciences. In China the first handbook of Western geometry was printed in 1607, the first treatise of astronomy in 1614. After 1584 a map of the world, (...) inspired no doubt by that of Ortelius of 1570, was engraved in stone. Printed versions spread quickly in China and continued into Korea and Japan. The algebraic notation of Viète, plane and solid trigonometry, logarithms (called then the rules of Neper), the cosmology of Aristotle and the methods of astronomical calculation of Tycho Brahe were all introduced into China during the course of the 17th century. (shrink)

Diderot s’est passionné toute sa vie pour les questions musicales. C’est à travers elles que cet article traite du thème de la « perception des rapports » dans l’esthétique diderotienne. Le trait saillant des « Principes généraux d’acoustique » (1748) est le refus de distinguer un point de vue strictement scientifique et esthétiquement neutre, et un point de vue proprement esthétique. Dans les pas du Descartes de l’Abrégé de musique, et dans une discussion serrée avec le père André, Diderot soutient (...) une thèse que l’on pourrait qualifier d’empirico-malebranchienne. Si, sur le modèle de l’optique cartésienne, Diderot soutient que l’oreille perçoit les consonances et le beau musical selon une sorte de trigonométrie naturelle, il refuse de l’appuyer sur l’action directe d’une puissance supérieure à l’esprit, comme chez l’auteur de De la recherche de la vérité. La « perception des rapports » est au contraire fondée sur une conception empiriste du nombre : c’est dans l’expérience du dénombrement, et par le biais des sens, que nous connaissons le nombre. La conception des « corps harmoniques », développée dans la « Lettre à Mlle *** », permet d’associer la réalité des rapports des nombres, leur rationalité et le plaisir des corps – sans faire appel à une raison universelle sur le modèle malebranchien. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 27.2 (2003) 284-303 [Access article in PDF] Accidental Art:Tolstoy's Poetics of Unintentionality Michael A. Denner I ART'S ABILITY TO INFECT another with an emotion, the concept that has come to be probably the most readily identified catchphrase in What Is Art? (though it crops up in his earlier writings on art), derives from L. N. Tolstoy's dynamic identity claim about art: we know an artist has (...) created a genuine work of art when he "hands on to others feelings he has lived through, and that others are infected by these feelings and also experience them." 1 "What is art?" is then a taxonomical question, and, according to Tolstoy, art is identifiable from non-art not by some set of external characteristics—the way, say, Aristotle categorized animals in his natural history—rather art is recognizable by its function, by what it does. In this sense art is like a tool: A given thing is art because it performs a certain task, much as one might claim a screwdriver is anything that sets and removes screws, regardless of its external characteristics. Tolstoy, in fact, repeatedly refers to art as "orudie," "a tool," or even "a weapon."It is worth noting from the outset that this metaphor of contagiousness is one of sublimated violence. For its passive "perceiver" (Tolstoy's characteristic term for the consumer of art), art amounts to irresistible, surreptitious emotional manipulation by an external entity. Tolstoy is ever attentive to the underlying meanings of the words he chooses, and I think he means to emphasize the latent, etymological meaning of the zarazit', to infect, the root of which is razit', to strike, a military term that likely resonated with Tolstoy, a war veteran. 2 Etymological arguments aside, Tolstoy makes explicit the psychological violation that goes hand-in-hand [End Page 284] with the experience of a work of art in a diary entry from a period of work on an earlier variant of What Is Art?: "One must hone the artistic work so that it penetrates. To hone a work means to make it perfect artistically, in which case it will pierce through the indifference...." 3 In an earlier version of the essay, Tolstoy offers "hypnotism" as a synonym for infection (p. 53).With its ability to "penetrate" through indifference and colonize, willy-nilly, the other, art accomplishes directly and immediately what no amount of rational argument could accomplish: "Art is differentiated from the activity of rational activity, which demands preparation and a certain sequence of knowledge (so that one cannot learn trigonometry before knowing geometry), by the fact that it acts on people independently of their state of development and education, that the charm of a picture, of sounds, or of forms, infects any man whatever his plane of development" (p. 178).The effect of art does not depend on some graduated, sequential process as it does for rational activity: art's infectiousness is immediate and irresistible. According to What Is Art?, the summum bonum of art's infectiousness, its best use, but notably not its only, is bringing about the Kingdom of God by effecting a change in the psychology of all the individuals who experience it. "Religious art," Tolstoy's supreme category of art, surmounts the (ultimately illusory for Tolstoy) individuality of the perceiver by inducing in him or her an emotional union with the artist and all others who have experienced the work of art, thereby inducting all perceivers into the universal collective of humankind: "In this freeing of our personality from its separation from other people, from its isolation, in this uniting of personality with others, lies the chief characteristic and the great attractive force of art" (p. 228). Art is a sacrament and the artist a priest of sorts.It is often said, even by those who have read the work, that Tolstoy "rejected" art in the last decades of his life, and What Is Art? is often cited as evidence. Granted, much of the book excoriates much of the art of the ages, but one must not... (shrink)

Aesthetics is regarded, traditionally, as an aspect of philosophy. Arthur Edwards' approach is different. Ignoring philosophy, he points out that any work of art is devised in the mind of the artist and interpreted through the mind of the beholder and the object must therefore constitute a device of communication between these two minds. In this agreeably written, fully illustrated and constantly fascinating study he explores the implications of this idea, remembering that both artist and experiencer may be of any (...) culture. (We can respond, granted a degree of familiarity, to works produced by artists from cultures as different from our own as those of a Palaeolithic cave man, an Aztec priest, a West African wood-carver or a medieval monk.) The result is an astonishingly wide-ranging quest, one which extends from an investigation of communication theory to an account of the ecological concepts held by certain Amazonian tribes, from a demolition of Freud's model of the psyche to a study of the trigonometry of the Great Pyramid and from an account of the evolution of the human mind to a theory, albeit partial, concerning the nature of beauty. All are held within two complementary themes: the relevance to his enquiry of Jungs theory of the collective unconscious together with the archetypes of which it is composed, and the confirmation of these concepts by independent evidence. (shrink)

The true form of the seasonal-hour sundial, in both vertical and horizontal planes, has been calculated by spherical trigonometry and displayed with the aid of computer graphics. These grids are presented, and show that the hour lines are in fact shallow curves at sites not on the equator. The curvature becomes very apparent at latitudes exceeding 50°. The true seasonal-hour pattern for a vertical dial at a latitude of 52·6°N is compared with the equiangular scratch dial.

This user-friendly guide offers pragmatic recommendations on teaching various elements of algebra, including trigonometry, finite mathematics, and statistics to nontraditional students.

Les Sphériques ont été écrites par Ménélaos sous la forme d’un traité d’apparence purement mathématique. Cependant, la matière qui y est développée dans les deuxième et troisième livres est étroitement liée à des problèmes rencontrés en astronomie: calcul des coordonnées équatoriales du Soleil, établissement de tables d’ascensions, étude du mouvement du Soleil dans la sphère oblique, levers simultanés. Ce lien, qui demeure implicite dans le texte, a été mis en évidence par deux mathématiciens et astronomes arabo-musulmans, qui ont exposé la (...) signification astronomique de certains théorèmes des Sphériques. Nous présenterons, commenterons et complèterons cette explicitation en classant les théorèmes concernés en trois groupes: sens de variation de certaines grandeurs sur la sphère, propositions de trigonométrie sphérique et applications, sens de variation de rapports entre certaines grandeurs sur la sphère. The Spherics were written by Menelaos in the form of a purely mathematical treatise. However, the material developed in the second and third book is closely linked to problems met in astronomy: computation of equatorial coordinates of the Sun, setting up of rising-time tables, study of the motion of the Sun in the sphaera obliqua, simultaneous risings. This link, which remains implicit in the text, was clearly displayed by two arabo-islamic mathematicians and astronomers, who expounded the astronomical meaning of some theorems of the Spherics. We describe, comment and complement their explanations, by classifying the implications of the theorems in three groups: direction of variation of some quantities on the sphere, spherical trigonometry and applications, direction of variation of ratios of some quantities on the sphere. (shrink)

What does quilting have to do with electric circuit theory? The answer is just one of the fascinating ways that best-selling popular math writer Paul Nahin illustrates the deep interplay of math and physics in the world around us in his latest book of challenging mathematical puzzles, Mrs. Perkins's Electric Quilt. With his trademark combination of intriguing mathematical problems and the historical anecdotes surrounding them, Nahin invites readers on an exciting and informative exploration of some of the many ways math (...) and physics combine to create something vastly more powerful, useful, and interesting than either is by itself. In a series of brief and largely self-contained chapters, Nahin discusses a wide range of topics in which math and physics are mutually dependent and mutually illuminating, from Newtonian gravity and Newton's laws of mechanics to ballistics, air drag, and electricity. The mathematical subjects range from algebra, trigonometry, geometry, and calculus to differential equations, Fourier series, and theoretical and Monte Carlo probability. Each chapter includes problems--some three dozen in all--that challenge readers to try their hand at applying what they have learned. Just as in his other books of mathematical puzzles, Nahin discusses the historical background of each problem, gives many examples, includes MATLAB codes, and provides complete and detailed solutions at the end. Mrs. Perkins's Electric Quilt will appeal to students interested in new math and physics applications, teachers looking for unusual examples to use in class--and anyone who enjoys popular math books. (shrink)

Thomas Urquhart (1611–1660), celebrated for his English translation of Rabelais’ Gargantua et Pantagruel, has earned some notoriety for his eccentric, putatively incomprehensible early book on trigonometry The Trissotetras (1645). The Trissotetras was too impractical to succeed in its own day as a textbook, since it lacked both trigonometric tables and sample calculations. But its current bad reputation is based on literary authors’ amplifications of the verdict prefaced to its 19th century reprinting by one mathematician, William Wallace, who lacked the (...) background to appreciate the book’s historical context. Considering that context (including seventeenth century ‘copious’ prose, and medieval logic and ‘art of memory’), the bad reputation is undeserved: the book is mathematically clear, clever (e.g. in superimposing 16 problems into one diagram), and complete. The Trissotetras may thus be viewed as simply one more of Urquhart’s polymathic projects and involvements – which included education, rise of the middle class, religious and class conflicts, development of science and mathematics, search for patronage, universal language construction, and development of English prose – which serve to make him a lively and instructive intellectual Everyman for his time. (shrink)

Mathematics is a struggle for many. To make it more accessible, behavioral and educational scientists are redesigning how it is taught. To a similar end, a few rogue mathematicians and computer scientists are doing something more radical: they are redesigning mathematics itself, improving its ergonomic features. Charles Peirce, an important contributor to ordinary symbolic logic, also introduced a rigorous but non-symbolic, graphical alternative to it that is easier to picture. In the spirit of this iconic logic, George Spencer-Brown founded iconic (...) mathematics. Performing iconic arithmetic, algebra, and even trigonometry, resembles doing calculations on an abacus, which is still popular in education today, has aided humanity for millennia, helps even when it is merely imagined, and ameliorates severe disability in basic computation. Interestingly, whereas some intellectually disabled individuals excel in very complex numerical tasks, others of normal intelligence fail even in very simple ones. A comparison of their wider psychological profiles suggests that iconic mathematics ought to suit the very people traditional mathematics leaves behind. (shrink)

This is the first book in a two-volume series. The present volume introduces the basics of the conceptual foundations of quantum physics. It appeared first as a series of video lectures on the online learning platform Udemy.]There is probably no science that is as confusing as quantum theory. There's so much misleading information on the subject that for most people it is very difficult to separate science facts from pseudoscience. The goal of this book is to make you able to (...) separate facts from fiction with a comprehensive introduction to the scientific principles of a complex topic in which meaning and interpretation never cease to puzzle and surprise. An A-Z guide which is neither too advanced nor oversimplified to the weirdness and paradoxes of quantum physics explained from the first principles to modern state-of-the-art experiments and which is complete with figures and graphs that illustrate the deeper meaning of the concepts you are unlikely to find elsewhere. A guide for the autodidact or philosopher of science who is looking for general knowledge about quantum physics at intermediate level furnishing the most rigorous account that an exposition can provide and which only occasionally, in few special chapters, resorts to a mathematical level that goes no further than that of high school. It will save you a ton of time that you would have spent searching elsewhere, trying to piece together a variety of information. The author tried to span an 'arch of knowledge' without giving in to the temptation of taking an excessively one-sided account of the subject. What is this strange thing called quantum physics? What is its impact on our understanding of the world? What is ‘reality’ according to quantum physics? This book addresses these and many other questions through a step-by-step journey. The central mystery of the double-slit experiment and the wave-particle duality, the fuzzy world of Heisenberg's uncertainty principle, the weird Schrödinger's cat paradox, the 'spooky action at a distance' of quantum entanglement, the EPR paradox and much more are explained, without neglecting such main contributors as Planck, Einstein, Bohr, Feynman and others who struggled themselves to come up with the mysterious quantum realm. We also take a look at the experiments conducted in recent decades, such as the surprising "which-way" and "quantum-erasure" experiments. Some considerations on why and how quantum physics suggests a worldview based on philosophical idealism conclude this first volume. This treatise goes, at times, into technical details that demand some effort and therefore requires some basics of high school math (calculus, algebra, trigonometry, elementary statistics). However, the final payoff will be invaluable: Your knowledge of, and grasp on, the subject of the conceptual foundations of quantum physics will be deep, wide, and outstanding. Additionally, because schools, colleges, and universities teach quantum physics using a dry, mostly technical approach which furnishes only superficial insight into its foundations, this manual is recommended for all those students, physicists or philosophers of science who would like to look beyond the mere formal aspect and delve deeper into the meaning and essence of quantum mechanics. The manual is a primer that the public deserves. (shrink)

RésuméLe problème de la détermination de la Qibla est l'une des questions cruciales qui se posent à la culture scientifique de l'Islam médiéval; le résoudre correctement nécessite tant des théories mathématiques que des observations. Les mathématiques relèvent de deux chapitres: la trigonométrie plane et la trigonométrie sphérique. L'observation et les instruments d'observation sont indispensables à la détermination des coordonnées géographiques de La Mecque et du lieu donné; ces coordonnées sont en effet les données que l'on entre dans les formules donnant (...) la Qibla. Dans son Almageste, Abū al-Wafāʾ résout brillamment ce problème. Son travail porte tant sur les mathématiques que sur l'observation; il obtient des solutions modernes, adéquates et faciles. En trigonométrie plane, il donne de nouvelles définitions des fonctions trigonométriques, démontre les formules des sinus, donne une approximation de sin 1° grâce à laquelle il construit des tables de sinus et de tangentes d'une grande précision. En trigonométrie sphérique, il démontre quatre nouveaux théorèmes dont la règle des tangentes ; cette règle lui permet de trouver, comme nous le montrerons, des solutions simples au problème de la détermination de la Qibla. En ce qui concerne les observations, il décrit trois instruments utilisés par lui plusieurs années de suite à Bagdad. Cet article, nourri de nombreux textes arabes originaux traduits et commentés, donne une description détaillée, technique et analytique, des méthodes mathématiques d'Abū al-Wafāʾ pour la détermination de la Qibla. (shrink)

Combining historical fact with a retelling of ancient myths and legends, Alistair Wilson shows how mathematics arose out of the problems of everyday life. He introduces concepts such as geometry, prime numbers, and trigonometry in a way that will totally disarm the reader who fears mathematics.

1. A Road to Philosophy of Mathematics l became interested in philosophy and mathematics at more or less the same time, rather late in high school; and my interest in the former certainly influenced my attitude towards the latter, leading me to ask what mathematics is really about at a fairly early stage. I don ’t really remember how it was that I got interested in either subject. A very good math teacher came to my school when I was in (...) 9th grade and I got caught up in his course on solid geometry; but he soon left and math then lost its luster again in the hands of teachers who neither liked nor understood it. Calculus wasn’t taught in high school in those days, or at least not in mine: besides geometry we learned some algebra and trigonometry. I doubt that even the word “ philosophy ” passed the lips of any of my teachers. My mother, who worked for a publishing house, brought home for me copies of, among other works, the Jowett translations of Plato’s Dialogues, Will Durant’s Story of Philosophy and Courant and Robbins’ What Is Mathematics?; but I can’t remember why she did that: She wasn’t at all intellectual and, as far as I recall, my interests at the time were mostly confined to sports and girls—in some order. Maybe she just thought it was time for me to develop new interests. After high school, I went in 1948 to Lehigh University, then at least primarily an engineering school, on an athletic scholarship. There I had the good fortune in my first year to have an introduction to philosophy course with Lewis White Beck. He had just moved there from the University of Delaware and shortly thereafter moved on to the University of Rochester, where he became one of the leading lights of American Kant studies. My good luck was compounded when, in my second year, Adolph Gr¨ unbaum arrived at Lehigh, fresh from graduate school at Yale, and stayed at least long enough for me to graduate, before moving to the University of Pittsburgh as Andrew Mellon Professor of Philosophy of Science.. (shrink)

This paper presents an edition of al-Hasan ibn al-asan ibn al-Haytham's treatise, Qawl fi samt al-qibla bi-al-isāb with translation and commentary. In it Ibn al-Haytham provides a universal method for finding the direction of the qibla at any location on the surface of the earth by using spherical trigonometry and accurate calculation. Ibn al-Haytham's computational solution has not been studied before, and it has often been confused with another work of his in which he uses an analemma construction to (...) solve the problem of the qibla graphically. As a result of this confusion, contemporary scholars have mistakenly attributed the first universal solution of the qibla problem to Jamshīd al-Kāshī, some four centuries after the introduction of this method by Ibn al-Haytham. The present treatise represents an important juncture in the history of the development of mathematics of the qibla, and sheds more light on the contributions of one of the most important scientists of medieval Islam. (shrink)

An ingenious study in the interplay between mereology and set theory, this book is launched innocuously enough with the thesis that a class just is the mereological sum or "fusion" of its sub-classes. The sub-classes of a class are parts of a class in the literal sense of the word "part," as trigonometry is literally a part of mathematics. We are thus urged to resist the suggestion that the word "part" applies first and foremost to the spatial parts of (...) a material object. The atomic parts of a class being seen to be its singleton or "unit" sub-classes, a class is now to be understood on the most basic level as a fusion of singletons but for the distressing fact that how a class might have only one member was never explained to us when we first learned that a class was supposed to be "a many combined into one.". (shrink)