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Isaac Newton on Mathematical Certainty and Method Isaac Newton on Mathematical Certainty and Method Niccol`oGuicciardini The MIT Press Cambridge, Massachusetts London, England c 2009 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and re- trieval) without permission in writing from the publisher. For information about special quantity discounts, please email special [email protected] This book was set in Computer Modern by Compomat s.r.l., Configni (RI), Italy. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Guicciardini, Niccol`o. Isaac Newton on mathematical certainty and method / Niccol`o Guicciardini. p. cm. - (Transformations : studies in the history of science and technology) Includes bibliographical references and index. isbn 978-0-262-01317-8 (hardcover : alk. paper) 1. Newton, Isaac, Sir, 1642–1727—Knowledge-Mathematics. 2. Mathematical analy- sis. 3. Mathematics-History. I. Title. QA29.N4 G85 2009 510-dc22 2008053211 10987654321 Index Abbreviations, xxi Andersen, Kirsti, 9, 140 Abel, Niels H., 40–41 Apagogical proofs (in Barrow’s sense), 177 Accountants, 5, 351 Apollonius, xviii, 15, 56, 63, 80, 81, 91, 105, Acerbi, Fabio, xviii, 83, 86, 88, 102 118, 145, 253, 342, 385 Adams, John C., 248, 307, 340, 348 Archimedes, xiii, 65–66, 145, 342 Affected equations, 136, 154, 156, 158, 162, Aristaeus, 81 166, 167, 179, 193, 194, 212, 231, Aristotelian conception of pure and mixed 345, 355, 356, 376 mathematics, 146, 172 Alchemy, 3, 238, 313, 342 Aristotelian inertia, 235 Algebra speciosa, 339 Aristotelian logic, 23 Algebraic curves, 6, 15, 42, 104, 157, 188 Aristotelian substantial forms and occult Algebraic equations qualities, 297 Newton’s method of resolution, 158–164, Aristotelian textbook tradition, 323 179, 355 Aristotle (pseudo) Problemata Mechanica, to be neglected, 256, 266, 289, 311, 344 4 used in common analysis, 5 Arithmetica speciosa, 298 used in the Principia, 259 Arthur, Richard T. W., 171 Algebraic operations (in Descartes), 38 Astronomy, 12, 57, 251, 255, 371, 372 Analogy (in Wallis’s sense), 143, 150, 212, Attraction of extended bodies, 251 308 Analysis Bacon, Francis, 26 and synthesis in natural philosophy, 315– Baconian inductivism, 23, 343 327 Barrow, Isaac, xiv, xv, 3, 5, 12, 14, 19, 21, by means of organic descriptions, 102, 26–29, 59, 66, 81, 82, 146, 154, 169– 309 180, 183, 189, 212, 302, 304, 313, by means of porisms, 79, 81–84, 95, 102 324–325, 340, 342–344, 346, 347, 351, common, xiv, xviii, 5, 11, 29, 117–121, 353, 362, 377–379, 381, 385 164–167 Bartolazzi, Margherita, 62 defined by Descartes, 38–40 Beal, Peter, 349 defined by Newton, 76–78, 309–313 Bennett, Jim, 27 defined by Pappus, 33–35, 38–40 Bentley, Richard, 283 defined in the Aristotelian tradition, 323– Berkeley, George, 230, 313 324 Bernoulli, Jacob, 334, 366 new, xiv, xviii, 5, 130–212, 308 Bernoulli, Johann, 240, 253, 334, 335, 346, of the ancients, xv, 14, 15, 37, 79–81, 308 362, 366, 370, 375 problematic, 33–35 Bernoulli, Nicolaus, 111 theorematic, 33–35 Bertoloni Meli, Domenico, 27, 95, 332 used in the Principia, 255 Bessel, Friedrich Wilhelm, 252 Analysis Veterum. See Analysis (of the an- Biot, Jean-Baptiste, 373 cients) Blay, Michel, 140, 257, 289 Analytic geometry, 32–33 Boas, Mary, 25–26 Analytical parallelogram, 130, 159–163, 356, Bonelli, Maddalena, xviii 357, 376 Borelli, Giovanni Alfonso, 81, 347 414 Index Bos, Henk, xix, 33, 38, 40–41, 44–45, 49, Newton’s letter (August 20, 1672), 94– 65 96, 106 Bottani, Andrea, xviii Newton’s letter (December 10, 1672), 352– Bottazzini, Umberto, xviii 353 Boutroux, Pierre, 32 Newton’s letter (November 8, 1676), 353– Boyle, Robert, xiv, xvii, 23, 25–26, 28–29 354 Brachistochrone problem, 335 Colson, John, 78, 169, 347, 350, 368 Brackenridge, Bruce, 237, 246, 250, 271, Comets, 251 274, 294 Commandino, Federico, 33, 296 Brigaglia, Aldo, 76, 312 Composition. See Synthesis Briggs, Henry, 144 Conchoid, 7–9, 67, 68 Brouncker, William, 12, 25, 347, 352 and neusis, 42–43, 68 Brunschvigc, L´eon, 32 its simplicity according to Newton, 65 Buchwald,JedZ.,xix,3,23 mechanical construction of, 69 tangent, 191 Campbell, Colin, 347 Concrete mathematics, 27–28 Cantor, Moritz, 164 Conic sections, 42–43 Cardano, Girolamo, 40–41 anharmonic property, 87 Carpentry, 372 in Wallis, 139 Cartesian parabola, 112 organic description, 88, 95, 300 Cartographers, 5 Conservation of mechanical energy, 270 Carus, Andr´e, xix Constant of integration, xvii Casey, John, 85 Construction of equations Cauchy, Augustin Louis, xvii, 220 according to Descartes, 41, 51–52 Cavalieri, Bonaventura, 57, 140 according to Newton, 64–78 Central forces Convergence, xvii, 153, 154, 231 direct problem, 244–246 Copernicus, Nicolaus, 37, 297 inverse problem, 246–249, 268–278, 322 Corpuscular models, 22, 25–26 Chandrasekhar, Subrahmanyan, 260, 294 Costabel,Pierre,32 Characteristica Universalis, 329 Cotes, Roger, 20, 246, 254, 268, 283, 317, Chasles, Michel, 82–83 318 Cheyne, George, 366 Newton-Cotes formula, 210 Child, James M., 170, 177 Craig, John, 202, 334, 345–348, 350, 361, Circle 362, 366 arc length, 155–156, 376 Cramer, Gabriel, 116 quadrature of, 143, 150–152 Crooke, John, 342 Cissoid, 7–9 Cross-ratio, 83–85 and two mean proportionals, 42–43 Cubic curves, xvii, 6, 15 equation of, 112 as shadows of divergent parabolas, 112, Newton’s organic description, 75 121–130 quadrature of, 10, 200–202, 214–216 divergent parabolas, 122 Clairaut, Alexis Claude, 123 genus, 110 Clarke, Samuel, 316, 384 Newton’s classification in 72 species, 111 Clavius, Christoph, 48 order, 110 Cohen, Bernard I., xviii, xxii, 98, 237, 249, parabolic hyperbola, 130–132 252, 288, 293, 298, 303, 326 reduction to four canonical forms, 110– Collins, John, 12, 14, 16, 62, 154, 158, 164, 112 166, 332, 334, 336, 337, 339–343, redundant hyperbola, 132 346–348, 350–353, 357, 360, 369, 373, use of series in the study of, 130–136 377 Cuomo, Serafina, 296 Index 415 Curvature in the study of trajectories, 236 Earth’s shape, 251 Cycloid, 6–9, 42, 66–67 Edleston, Joseph, 350 quadrature of, 10, 156–157 Edwards, Charles H., 154 Elegance D’Alembert, Jean Le Rond, 257 according to Fermat, 145 Damiani, Sara, xix according to Newton, 64–80, 231, 311– Dates, old style (O.S.) and new style (N.S), 312 xxi according to Wallis, 146 Davies, Richard, xviii Eratosthenes, 81 De Gandt, Fran¸cois, 140, 219, 235, 294 Erlichson, Herman, 250, 271, 274 De Moivre, Abraham, 254, 366 Ether, 237 De Morgan, Augustus, 373 Euclid, xviii, 4, 15, 23, 33, 37, 80, 103, 144, De Witt, Jan, 94, 112, 140 145, 253, 344, 345, 385 Dear, Peter, 294 Data, 81, 83 Dedekind-Peano Axioms, 142 Elements, 81, 154, 218, 254, 304 Densmore, Dana, 246, 294, 298 Porisms, 15, 81–83 Desargues, G´erard, 83 Euler, Leonhard, 240, 257, 288, 346 theorem of, 86 Exactness Descartes, Ren´e, xiv–xvii, 1, 4–6, 10, 15, according to Descartes, 43–44 21–24, 26, 27, 29, 31–60, 63–70, 73, according to Newton, 297–298, 300–302 74, 76, 79–81, 89, 90, 93, 97, 101, 106, 112, 117, 129, 130, 140, 154, Fatio de Duillier, Nicolas, 335, 345, 346, 156, 165, 167, 235–237, 261, 262, 348, 350, 363–365, 375 265, 266, 294, 299–301, 307, 308, Feingold, Mordechai, xix, 3, 25–26, 153, 311, 315, 316, 329, 344–346, 362, 170 363, 385, 386 Fermat, Pierre de, 41–42, 81, 144–146, 174, Discours de la M´ethode,29 344, 362, 381, 385 G´eom´etrie, xvii, 5, 6, 15, 31–60, 76, 112, Ferrari, Ludovico, 40–41 385 Ferrier, Jean, 97 Meditationes de Prima Philosophia, 15, Figala, Karin, 313 81, 385 Finite equations. See Algebraic equations Principia Philosophiae, xiii, 21–23, 235 First and ultimate ratios, 13, 147, 191, 219– Regulae ad Directionem Ingenii, 310 223, 241–242 Deschales, Milliet, 147 Flamsteed, John, 237, 346, 350 Describing and describend curves, 94–97 Fleckenstein, Joachim O., 336 Di Sieno, Simonetta, 92 Fluents Differential calculus, 253, 329, 332, 360 according to Barrow, 171–173 Differential equations, 57, 194, 247, 255, according to Newton, 180, 226 257, 270, 277, 288, 359, 360, 375 as geneses that exist in rerum natura, Digby, Kenelm, 145 226, 315, 382 Diophantus, 37 definition, 180 DiSalle, Robert, 239 sensible finite magnitudes, 314 Ditton, Humphry, 366, 367 Fluxions Domski, Mary, 294 as finite and real quantities, 226, 382 Ducheyne, Steffen, 323 definition, 180 Duplication of cube, 42–43 visible to the eye, 226 Dupont, Pascal, 206 Folium of Descartes, 112 Dur´an Guarde˜no, Antonio J., 117, 135 Folkes, Martin, 25–26 Fontenelle, Bernard Le Bovier de, 288 Eagles, Christina M., 276, 366 Fortification, 372 416 Index Franci, Raffaella, 62 Heron of Alexandria, 296 Fundamental theorem of calculus, 170, 177, Hiscock, Walter George, xv, 17, 77, 341, 182–185, 205–206, 358, 360 369 Funkenstein, Amos, 325 Hobbes, Thomas, 80, 144–146, 302, 313, 324–325, 342, 344, 385 Gabbey, Alan E., 294 Hoff Kjeldsen, Tinne, xix Galilei, Galileo, 20, 57, 235, 242, 254 Hofmann, Joseph E., 170, 336 Galois, Evariste,´ 41 Hooke, Robert, xiv, xvii, 12, 23–27, 244, Galuzzi, Massimo, xix, 55, 90, 92 342 Garber, Daniel, 23 Micrographia, 26–27 Garrison, James W., 294, 313, 321 and mathematics, 27 Gassendi, Pierre, 313 planetary theory, 242 Gaukroger, Stephen, 23 Horizon line, 125–127 Gauss, Carl Friedrich, xiii, 252 Horrocks, Jeremiah, 342 Genitum, 223 Horsley, Samuel, 347 Geometrical curves, 42, 47 Hudde, Johan van Waveren, 186–188, 190, Geometrically rational and irrational curves, 334, 353 306 Huygens, Christiaan, xiv, 17, 81, 146, 254, Ghetaldi, Marino
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