Isaac Newton on Mathematical Certainty and Method

Home , Arithmetica Universalis, General Scholium, Method of Fluxions

Niccol`oGuicciardini

The MIT Press Cambridge, Massachusetts London, England c 2009 Massachusetts Institute of Technology

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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 analysis. 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, Léon, 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, André, 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, Gérard, 83 Euler, Leonhard, 240, 257, 288, 346 theorem of, 86 Exactness Descartes, René, 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 Méthode,29 344, 362, 381, 385 Géométrie, 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 Durán Guardeño, 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 (Getaldić), 41–42 326, 331, 332, 346, 362, 363, 385 Giudice, Franco, xviii Horologium Oscillatorium, 16, 59, 236, Giusti, Enrico, 31, 140 344 Glanvill, Joseph, 25 law for circular uniform motion, 236 Godbid, William, 342 Huygens, Constantijn, 97 Granger, Gilles-Gaston, 32 Hyperbola Grattan-Guinness, Ivor, xix quadrature of, 11, 143, 153 Gregorie, James Gregory, James12 tangent, 8 Gregory, David, 13, 111, 202, 241, 250, 254, 267, 268, 273–277, 283, 334, 343, 345–348, 350, 351, 361–363, 366, 377, Ihmig, Karl-Norbert, 317 380 Inchoative or inceptive quantities, 147, 148, Gregory, James, 12, 333, 341, 343, 347, 352, 224 353, 357, 361, 378, 381, 383 Indivisibles Gresham College, 28 arithmetic of, 139–148 Grosholz, Emily, 57 geometry of, 140 Gua de Malves, Jean Paul, 116 Guerlac, Henry, 315, 320, 321 Induction (in Wallis’s sense), 140–148, 153, 209, 212, 308 Inertia, 235 Hall, A. Rupert, 170, 331, 335, 336, 379 Infinite equations. See Series and the “fable of fluxions”, 289 Halley, Edmond, 26, 82, 238, 293, 345, 346, Infinite products, 144 349–351, 361 Infinitesimal triangle Hanoverian succession, xvi, 329 according to Barrow, 173 Hardy, Godfrey H., 14 according to Newton, 183, 191 Harper, William L., 246, 252 Infinitesimals, 16, 137, 140, 180, 314, 382 Harrison, John, 23, 82, 316 as heuristic tools, 219, 230 Haugen, Kristine, xix higher-order, 190, 232 Hayes, Charles, 348 Instrument makers, 5, 97 Hercules, 311 Integral calculus, 253, 329, 332 Herivel, John, 237 Integration, 10, 204 Hermann, Jacob, 375 Riemann’s definition, 221 Index 417

Interpolation Linguiti, Alessandro, xviii according to Wallis, 143 Liveing, George D., 348 Newton’s method, 210 Loettgers, Andrea, xix Logarithmic curve, 46–47 Jacquier, Fran¸cois, 367 Love, Harold, 348, 350 Janiak, Andrew, 326 Luard, Henry R., 340, 348 Jardine, Nicholas, 4 Lucasian Chair, 3, 14, 19, 59, 61, 169, 346 Jesseph, Douglas M., 142, 145, 324, 342 Lützen, Jesper, xix Jones, Alexander, 83, 85–87 Jones, William, 78, 115, 210, 336, 337, 341, Macclesfield Collection, 337, 348, 350, 373 347, 350, 351, 367, 369–372, 379 Maclaurin, Colin, 110, 313, 320 Jullien, Vincent, 32 Mahoney, Michael S., 33, 172, 174, 177 Maierù, Luigi, 148 Keill, John, 336, 348, 375, 378, 381 Malet, Antoni, 140, 179, 325 Kepler’s area law, 242, 326 Mamiani, Maurizio, 298 Kepler problem, 248, 305 Mancosu, Paolo, xiv, 140 Kepler, Johannes, 41–42, 305 Mandelbrote, Scott, xix, 335 Kersey, John, 164, 259, 342, 352 Maschio, Silvia, xix Keylway, Robert, 164 McGuire, James E., 32, 239, 315, 325 Kinckhuysen, Gerard, xvii, 14, 62, 76, 340, McMullin, Ernan, 293 341, 344, 352 Mechanical curves, 6, 7, 42, 44, 66–67, 299, Knobloch, Eberhard, xix, 332 301 Knoespel, Kenneth J., 313 and infinite series, 154 Knorr, Wilbur, 83 Mechanical lines. See Mechanical curves Kollerstrom, Nicholas, 158, 164, 238, 252, Mechanics 307 according to Descartes, 43–44, 299, 301 Kormos-Buchwald, Diana, xix according to Newton, 72–73, 293–308 according to Wallis, 294 La Hire, Philippe de, 83, 124 practical, 294, 296, 297 Lacroix, Sylvestre F., 259 rational, 169, 294, 296, 297 Lagrange, Joseph-Louis, 164, 288 universal, 293, 298 Lalande, Joseph Jérôme L. de, 259 use of mechanical tools in mathematics, Land surveyors, 5, 97 27, 95–97 Landsman, Klaas, xix Mellon Foundation, xix Le Seur, Thomas, 367 Mendell, Henry, 36 L’Echelle, Abraham de, 82 Mercator, Nicolaus (Niklaus Kauffman), 11, Lefort, Felix, 373 14, 62, 153, 165, 166, 340, 352 Leibniz, Gottfried Wilhelm, xiv–xvi, 17, 81, Mesolabum, 44–45, 104 115, 151, 155, 159, 165, 170, 177, Method of exhaustion, 16, 145, 147, 154, 187, 253, 254, 288, 314, 318, 320, 230 329–333, 335, 336, 341, 347, 351, Method of series and fluxions, xviii, 9 353–362, 364–366, 369, 372–386 analytical, xviii, 137, 304 L’Hospital, Guillaume F. A. de, 57, 255, and the second law of motion, 239, 288 288, 346, 368, 375 determination of tangents, 190–193, 227– Limits 229 according to Cauchy, 220 direct, 11, 13, 185–193 according to Leibniz, 332 employed in the Principia, 16, 248–257, according to Newton, 217–229, 241–242 267–290, 303 according to Wallis, 142, 147 inverse, 11, 13, 193–210 existence and uniqueness, 222 notation, 181–182 418 Index

rules of direct method, 186–189, 224–225 deduction from phenomena, 21, 23, 315– synthetic, xviii, 13, 137, 213–232, 304 327 Methuselah, 311 “De Gravitatione et Aequipondio Flu- Miller, Maximilian, 104, 105, 118 idorum”, 314 Mixed mathematics, 27–28, 171 De Methodis Serierum et Fluxionum, 13, Molland, Andrew George, 31 159–161, 179–202, 341, 347 Moment “De Motu Corporum in Gyrum”, 238, definition, 180 326 in Lemma 2, Book 2, of the Principia, De Quadratura, xviii, 16, 202–210, 226– 223 229, 335, 367–368 Montucla, Jean E., 259, 373, 378, 380 didactic style, 189, 204, 211 contribution of Lalande and Lacroix to early mathematical studies, 3–11 the Histoire, 259 early thoughts on planetary motion, 236 Moon’s motion, 251 Enumeratio Linearum Tertii Ordinis, 16, Motte, Andrew, 284, 335, 348, 362 96, 109–136, 367–368 Mugnai, Massimo, xviii epistola posterior for Leibniz, 202–203, Murdoch, Patrick, 110, 123 333, 334, 356–360, 376–377 epistola prior for Leibniz, 333, 354–356 “Errores Cartesij Geometriae”, 89–90 Nauenberg, Michael, xviii, 27, 237, 244, 251, experimentum crucis, 24–26, 344 252 first and ultimate ratios, 137, 219–223, Navigation, 12, 351 241–242, 382 Neal, Katherine, 146 “Geometria Curvilinea”, xviii, 13, 16, 77, Neil’s semicubic parabola, 112, 122 217–219, 254, 333 Neusis, 42–43, 68–71 “Geometriae Libri Duo”, 17, 103, 109, Newman, William R., 313 125, 127, 300–303, 309–313 Newton, Isaac “Geometriae Libri Tres”, 84, 86 “Account” of Commercium Epistolicum, “Inventio Porismatum”, 84 329, 381–384 Lucasian Lectures on Algebra, 14, 59, 61– Addendum to De Methodis, 13, 191, 217 78, 104, 106, 113 Analysis per Quantitatum, Series, Flux- “Matheseos Universalis Specimina”, 361 iones, ac Differentias, 115, 369–372 Methodus Differentialis, 210 anni mirabiles, xvii, 6–11, 204, 290, 307 “New Theory about Light and Colors” anti-Cartesianism, 15, 21, 59, 78, 93, (1672), 12, 24–25 103, 212, 294, 343, 344 Newton-Cotes formula, 210 Arithmetica Universalis, xv, 15, 61–78, Newton-Raphson method, 162, 307, 355 96, 104–106, 369 October 1666 Tract on Fluxions, 11, 13, binomial series, 11, 148–154, 227, 355 167, 169, 179, 203, 211, 340, 348 certainty in natural philosophy, 19–21, “Of Quadrature by Ordinates”, 210 343–344 on mathematical elegance, 64–80, 344 Commercium Epistolicum, xv, 155, 329, on mathematical ingenuity, 102 330, 336–337, 372–381 on postulates in geometry, 103–104 convergence of series, 153, 154, 164 on the ancients, 77, 93, 212, 218, 227, criticisms against Leibniz’s calculus, 80– 231, 301–303, 311–313, 344, 382 81, 302, 381–384 Optical Lectures, 19–29, 343–344 De Analysi per Aequationes Numero Ter- Opticks, 14, 116, 315–318, 321, 322, 324, minorum Infinitas, 12, 13, 136, 154– 365 167, 185, 333, 340–341, 347, 376 organic description of conics, 97–103 “De Compositione et Resolutione Vete- policy of publication, 12, 110, 113–117, rum Geometrarum”, 84 136, 230–232, 249, 256, 312, 339– Index 419

351, 361–369 Pappus problem, 15, 16 Principia Mathematica. See Philosophiae Descartes’ solution, 53–56 Naturalis Principia Mathematica for n lines, 80, 93 priority dispute with Leibniz, 203–204, Newton’s solution, 90–93 223, 314, 329–337, 365–367, 369–377, Pascal triangle, 150 384 Pascal, Blaise, 83 Questiones [sic] Qaedam Philosophicae, Pell, John, 164, 352 235, 325 Pellet, Thomas, 347 relationship with acolytes, 17, 238, 283, Pelseneer, Jean, 312 288, 340 Pemberton, Henry, 344, 369 rule of signs, 62, 113 Perspective transformation, 84 “Solutio Problematis Veterum de Loco Perturbation theory, 251, 307 Solido”, 15, 90–94, 113 Pesic, Peter, 305 Theory of the Moon, 252 Piccolomini, Alessandro, 3, 302, 324, 325 “Tractatus de Compositione Locorum Soli- Picot, Claude, 23, 316 dorum”, 90–94 Pitcairn, Archibald, 202, 362, 366 unproven mathematical statements, 113, Point-wise construction of curves, 45–46, 267 48, 103 “Veterum Loca Solida Restituta”, 80, Polynomial equations, 40–41 90–94, 136 ﬂuxion of, 186–187 Nicole, Fran¸cois, 111, 123 Porisms, 15, 79, 81–89, 344 Notation according to Newton, 81–84 Leibniz’s, xvii hyptios porism, 85–87 Newton’s, xvii, 181–182, 204, 379 main porism, 87–89, 97 Wallis’s, 153 Portsmouth Collection, 340 Pourciau, Bruce, 219, 244, 246, 306 Oldenburg, Henry, 12, 24–25, 139, 332, 336, Precession of equinoxes, 251 339, 341, 353, 355, 360, 373, 378 Precession of orbits, 251 Omerique, Antonio Hugo de, 77, 312, 345 Principe, Lawrence M., 313 Opticians, 97 Philosophiae Naturalis Principia Mathemat- Organic description of curves, 6–7, 9, 27, ica 45–46, 93–103, 112, 300, 371 Book 1, Deﬁnitions, 238 analysis and synthesis by means of, 102 Book 1, Laws of motion, 239 use in natural philosophy, 319–320, 323– Book 1, Lemma 1, 219–220 326 Book 1, Lemma 2, 221 Oughtred, William, 4, 139, 164, 189, 212, Book 1, Lemma 7, 220–221 259 Book 1, Lemma 9, 241 Ovals Book 1, Lemma 10, 242 algebraic nonintegrability, 248, 305–307 Book 1, Lemma 17, 92 Cartesian, 48, 97 Book 1, Lemma 19 (Corollary 2), 90 Ozanam, Jacques, 38 Book 1, Lemma 20, 99 Book 1, Lemma 21, 94, 97–99 Painting, 372 Book 1, Lemma 22, 123–127 Panza, Marco, 11, 153 Book 1, Lemma 28, 248, 305–307 Pappus, xvii, 15, 17, 33, 41–43, 65–66, 68, Book 1, Proposition 1, 242–243 79, 81, 83, 94, 297, 310, 344 Book 1, Proposition 2, 243, 318 Collectio, xv, 15, 33–35, 37, 56, 68–70, Book 1, Proposition 4, 318 81–84, 296, 310 Book 1, Proposition 6, 244–245, 320 Proposition 48 of the Collectio, 35–37 Book 1, Proposition 9, 245 theorem of, 83 Book 1, Proposition 10, 245 420 Index

Book 1, Proposition 11, 245 solid, 42–43, 65 Book 1, Proposition 12, 245 Proclus, 3 Book 1, Proposition 13, 245 Projective geometry, 15, 17, 83, 121–130, Book 1, Proposition 13 (Corollary 1), 247 246–247, 322 horizon line, 125–127 Book 1, Proposition 22, 94, 95, 97–101, perspective and projective transforma- 299–300 tions, 84 Book 1, Proposition 30, 259–266 vanishing line, 126 Book 1, Proposition 31, 307 Puiseux, Victor-Alexandre, 11, 158 Book 1, Proposition 41, 247–250, 268– Pure mathematics, 27–28 271, 348, 363 Pycior, Helena M., 62, 63, 164, 171, 313, Book 1, Proposition 41 (Corollary 3), 342, 351 250, 271–278, 346 Book 1, Proposition 45, 252, 318, 362 Quadratrix, 6–9, 42 Book 1, Proposition 90, 279–280 Quadratures, 10, 154–158, 329 Book 1, Proposition 91, 280–281, 363 as solutions to all problems, 203 Book 1, Proposition 91 (Corollary 2), by substitution of variable, 196–202, 383 282–288 definition of, 140, 193 Book 1, Proposition 93, 362 in closed form, 194–196, 206 Book 1, Section 1, Scholium, 222–223 not worthy of public utterance, 214, 256 Book 1, Section 5, 90–95 prime theorem, 202–203, 357, 358, 362, Book 1, Section 9, 251 364 Book 1, Section 11, 251 reduction to inverse method of fluxions, Book 1, Section 12, 251 182 Book 1, Section 13, 251 synthetic, 213–217 Book 2, Lemma 2, 223–226, 333–334, via infinite series, 154–158, 193–194 362 Quaestio de certitudine mathematicarum, Book 2, Proposition 10, 252, 362 4, 324 Book 2, Proposition 24, 239 Quartic curve, 96 Book 2, Proposition 35 (=34), 251, 268, 348 Rahn, Johann Heinrich, 259, 352 Book 3, Introduction, 322 Raphson, Joseph, 158, 164, 349, 381 Book 3, Lemma 5, 210, 251 Newton-Raphson method, 307 Book 3, Proposition 28, 252 Rashed, Roshdi, 32, 42 General Scholium, 22, 318 Rau, Christian, 82 Preface by Cotes (1713), 20, 317, 318, Resolution. See Analysis 322 Riccati, Jacopo, 375 Preface by Newton (1687), 293–299, 317, Riccati, Vincenzo, 115 322 Rigaud, Stephen P., 267 use of fluxions, xv, 16, 248–257, 267–290, Roberval, Gilles Personne de, 174, 304 303, 370 Rouse Ball, William W., 111, 116–117, 123– Prisca sapientia, xviii, 17, 78, 312–313 130 Probabilism, 14 Rovelli, Daniela, 55 Problems Royal Society, 331, 333, 335, 336, 351, 353, determinate, 38–42, 49–53 369, 372–381 indeterminate, 38–40, 47–49, 53–56, 79– Ruler and compass, 42–43, 103 107 linear, 42–43, 65 Sageng, Erik, 313 locus problems, 47–48, 83 Saint Vincent, Grégoire de, 153 planar, 42–43, 65 Saladini, Girolamo, 115 Index 421

Salmon, George, 85, 100 Lineae Tertii Ordinis Neutonianae, 91, Sarasa, Alphonse Antonio de, 153 114, 118, 121, 132 Sargent, Rose-Mary, xix Stokes, George G., 348 Sauer, Tilman, xix Subtangent, 172, 174, 190, 191, 228 Savilian Chair of Geometry, 139 Sylvester, James J., 62 Schooten, Frans van, 4–6, 31, 42, 46, 81, Synthesis, 5 94, 97, 363 according to Descartes, 40–42, 47–56 Scriba, Christioph, 139 according to Newton, 74–76, 309–313 Scribano, Emanuela, xviii by means of organic descriptions, 102 Sellés, Manuel A., 148 defined by Pappus, 33–35 Sepkoski, David, 313 Serfati, Michel, 32, 154 Talbot, Christopher R. M., 116–117, 123– Series, xviii, 5, 9, 130–136, 139–167 130 arcsin, 156 Tamny, Martin, 32, 325 binomial series, 148–154 Tangents convergence, 153, 154, 164 Barrow’s method, 173–174 ln(1 + x), 153 Newton’s method, 190–193, 227–229 long division, 11, 179 Taylor, Brook, 254 quadratures by means of, 154–158, 193– Theodosius, 342 194 Three-body problem, 251 reversion of, 162 Tides, 251 root extraction, 11, 179 Time sin, 156, 162 absolute, 315, 320 Shapin, Stephen, 25–26 in the method of fluxions, 180 Shapiro, Alan E., 19, 20, 24, 315, 316, 322 Torricelli, Evangelista, 57, 140 Shapiro, Barbara, 19, 23 Transcendental curves, 10, 42 Shkolenok, Galina, 96 lines of infinitesimal order, 110 Simplicity Trascendental functions, 346 according to Barrow, 174 Trisection of angle, 42–43, 65–66, 68 according to Descartes, 47, 64–65 Descartes’ solution, 49–53 according to Newton, 64–67, 102, 106, Newton’s solution, 71–72 112, 218, 231 Trochoid. See Cycloid Simpson, Thomas, 158, 164 Tschirnhaus, Ehrenfried Walther von, 362, Simson, Robert, 82–83 377 Skepticism, 23, 25–26 Turnbull, Herbert W., 96 Sloane, Hans, 25–26, 335 Sluse, René-Fran¸cois de, 334, 347, 353, 360, 362, 363, 377, 378 Valluri, Sree Ram, 252 Smeenk, Christopher, xix Vanishing line, 125, 126 Smith, George, xix, 246, 293, 326 Vanishing magnitudes, 137, 218, 219 Snobelen, Stephen, 313 Van Maanen, Jan A., 364 Solid of least resistance, 251 Varignon, Pierre, 254, 257, 334, 375 Spiral, 6–9, 42, 44, 172 Vermij, Rienk H., 364 Sprat, Thomas, 25 Viète, Fran¸cois, 4, 34, 37, 41–42, 68–70, 144 Stedall, Jacqueline, 62, 140, 143, 145, 147, Vortex theory, 237, 307 148 Voss, Gerardus Joannes, 147 Steiner’s Theorem, 88, 99, 101 Vuillemin, Jules, 32 Stewart, Ian, 4 Stewart, John, 208, 367, 368 Wallis, John, 4, 6, 11, 12, 26–28, 37, 57, Stirling, James, 110, 111, 252, 254 80, 81, 112, 155, 165, 166, 212, 230, 422 Index

294, 331, 334, 335, 342, 344, 347, 351, 352, 356, 359, 361–365, 377 Algebra, 158, 364 Arithmetica Inﬁnitorum, 11, 139–148, 153, 166, 331, 356 De Sectionibus Conicis, 6, 112, 140 Westfall, Richard S., 3, 28, 109, 185, 237 Whewell, William, 235 Whiston, William, 15, 63, 66 Whiteside, Derek T., xiii, xviii, 3, 13, 62, 66, 80–82, 84, 86, 89, 90, 92, 94, 96, 101, 102, 109–111, 113, 117, 123, 124, 127, 128, 132, 136, 139, 153, 166, 167, 170, 171, 202, 203, 210, 214, 217, 223, 224, 237, 240, 250, 252, 255, 283, 303, 310, 333, 335– 337, 339, 340, 342, 343, 347, 348, 350, 351, 366 Wilson, Curtis, 251, 252, 257, 293 Wilson, James, 347, 349, 350 Wren, Christopher, 26–29, 238

Yoder, Joella G., 16

Zabarella, Jacopo, 34