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PrefaceList of AbbreviationsChapter One: The Mathematical Career of the Monster of MalmesburyChapter Two: The Reform of Mathematics and of the UniversitiesIdeological Origins of the DisputeChapter Three: De Corpore and the Mathematics of MaterialismChapter Four: Disputed FoundationsHobbes vs. Wallis on the Philosophy of MathematicsChapter Five: The "Modern Analytics" and the Nature of DemonstrationChapter Six: The Demise of Hobbesian GeometryChapter Seven: The Religion, Rhetoric, and Politics of Mr. Hobbes and Dr. WallisChapter Eight: Persistence in ErrorWhy Was Hobbes So Resolutely Wrong?Appendix: Selections from Hobbes's Mathematical WritingsReferencesIndex Copyright © Libri GmbH. All rights reserved.
Quick Overview of the Primal Symbols of Cosmogenesis. How do the Two Ones form the Intelligible World? The First One is a Ray from Parabrahman (Absoluteness). The Second One, Logos Demiourgos (Creative), is a mere reflection of the First. The Virgin Matrix of the Universe is cold Fire — cool Radiance, colourless, formless, devoid of every quality. Only the eye of the Seer can follow and behold the basic line of the Pythagorean Triangle in all its pregenetic glory. How did the Heavenly Snails clothed themselves in the Fabric of Darkness? How does the Triangle becomes Square, and the Square a Six-faced Cube? What is the key to the septenary significance of the Primordial Circle? What are the Four Cardinal Points upon which the rock-cut temples of India were built? The Four Maharajahs, or Great Kings, are the Divine Instructors of nascent humanity and agents of Karma on Earth, whereas the Lipikas are concerned with humanity’s hereafter. The Pythagorean Monas is a Solitary Ray. It strides through the Regions of the Universe in steps, the steps of Vishnu. Unmanifested Logos is The First One. The meaning of the Two Ones explained. Second Logos is The One made Three, concreting into the Third Logos, where the Three live within The One, thus making up the Perfect Square in heaven and a Cube on earth. The meaning of the dotted line explained. Light drops one Solitary Ray. Then number strides in the regions of the universe. Eastern and Kabbalistic Cosmogonies are Identical (Diagram). Cosmogenesis and Anthropogenesis numerically and geometrically expressed. The One attracts within itself the Divine Unity of the One Circle and forms out of It the Perfect Square, thus “squaring the circle.” The Great Circle or Ring Pass-Not is the “Rope of the Angels” hedging off the phenomenal from the noumenal Kosmos. That Circle is the Universal Principle which, from any given point, expands to embrace all things, while embodying the potentiality of every action in Kosmos. How does the Triangle become the Square? How does the Square become the Six-faced Cube? How does the Triad becomes a Tetrad, the Pythagorean Perfect Square in heaven, and a Cube on earth? Mathematically expressed, Logos become Tetragrammaton, i.e., the Three become Four. When the Spiritual Ego, the holy number , germ and matrix of the , enters the animal body, the faces of the cube unfold, thus forming the cross of passions upon which material man (Chrēstos) crucifies himself and disappoints his glorious Spirit (Christos). Kabbalistic interpretation of the “crucifixion nails” reveals their sexual meaning. The occult meaning of the Svastika, emblem of the activity of Fohat, symbolised by the figure and the Sacred Four, explained. The true Word may only be found by tracing the mystery of the passage inward and outward of Life Eternal, through the states typified in the Unmanifested Circle, the Triangle, and the Perfect Square. “The true Word may only be found by tracing the mystery of the passage inward and outward of the Eternal Life, through the states typified in these three geometric figures,” says a Master of Wisdom. The Divine Heptad is the key to squaring the circle and to the philosophers’ stone, which is no stone. The Ineffable Word, being composed of Seven Letters, represents the First Hebdomad. The Seventh Letter is the highest in initiations; the remaining six are substitutes. The Sacred Word is the Central Mathematical Point, around which the Six-pointed star, emblem of the Theosophical Society, revolves and evolves. The meaning of two Interlaced Triangles explained by a Master of Wisdom. The Unmanifested Circle or Absolute Life, is non-existent outside the Ideal Triangle and Perfect Square. It manifests through them as the “Son,” i.e., Kosmos and Man. On the outward Path of Action, the Second One (Atma), in order to manifest itself as Logos, Its concealed duality (Atma-Buddhi) has to become three (Atma-Buddhi-Manas). On the inward Path of Renunciation, Logos Revealed attracts within Itself the Circle and reverts to Its original state of Absolute Unity by forming out of it the Perfect Square, and inscribing within it the Ineffable Name. The Rope of the Angels expressed numerically.
This book is about James Gregory’s attempt to prove that the quadrature of the circle, the ellipse and the hyperbola cannot be found algebraically. Additonally, the subsequent debates that ensued between Gregory, Christiaan Huygens and G.W. Leibniz are presented and analyzed. These debates eventually culminated with the impossibility result that Leibniz appended to his unpublished treatise on the arithmetical quadrature of the circle. The author shows how the controversy around the possibility of solving the quadrature of the circle by certain means (algebraic curves) pointed to metamathematical issues, particularly to the completeness of algebra with respect to geometry. In other words, the question underlying the debate on the solvability of the circle-squaring problem may be thus phrased: can finite polynomial equations describe any geometrical quantity? As the study reveals, this question was central in the early days of calculus, when transcendental quantities and operations entered the stage. Undergraduate and graduate students in the history of science, in philosophy and in mathematics will find this book appealing as well as mathematicians and historians with broad interests in the history of mathematics.
Geometry: The Line and the Circle is an undergraduate text with a strong narrative that is written at the appropriate level of rigor for an upper-level survey or axiomatic course in geometry. Starting with Euclid's Elements, the book connects topics in Euclidean and non-Euclidean geometry in an intentional and meaningful way, with historical context. The line and the circle are the principal characters driving the narrative. In every geometry considered—which include spherical, hyperbolic, and taxicab, as well as finite affine and projective geometries—these two objects are analyzed and highlighted. Along the way, the reader contemplates fundamental questions such as: What is a straight line? What does parallel mean? What is distance? What is area? There is a strong focus on axiomatic structures throughout the text. While Euclid is a constant inspiration and the Elements is repeatedly revisited with substantial coverage of Books I, II, III, IV, and VI, non-Euclidean geometries are introduced very early to give the reader perspective on questions of axiomatics. Rounding out the thorough coverage of axiomatics are concluding chapters on transformations and constructibility. The book is compulsively readable with great attention paid to the historical narrative and hundreds of attractive problems.