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Sources of illustrations

Pictures on pages vi and x

All these pictures are freely available on the internet. Jan de Witt Painting by Jan de Baen (1633–1702)

Source: Rijksmuseum Amsterdam http://commons.wikimedia.org/wiki/File:Grand_Pensionary_Johan_de_Witt.jpg

René Descartes

Painting by Frans Hals (ca 1583−1666) Source: Louvre, Museum, Paris http://en.wikipedia.org/wiki/File:Frans_Hals_-_Portret_van_René_Descartes.jpg

Pierre de Fermat

http://commons.wikimedia.org/wiki/File:Pierre_de_Fermat.jpg

Frans van Schooten, jr.

http://commons.wikimedia.org/wiki/File:Frans_van_schooten_jr.jpg

François Viète

http://commons.wikimedia.org/wiki/File:Francois_Viete.jpg

Other pictures in Chapters 1−5

Most figures were drawn by Tobias Baanders (CWI, Amsterdam), with the following exceptions • Figure 5.8 was inspired by Heath [30] • Figure 5.9 was derived from Descartes [16] • Figure 5.13 (drawn by Jan Aarts) was derived from J.D. LAWRENCE, A catalog

of Special Curves, New York, Dover publications, 1972

Index

abscissa-axis 10, 17, 18, 27, 38, 49, 54, 259, 267, 270, 282, 286, 302

algebraic curve 301, 304 analysis 3−6, 257 analytic geometry 2−4, 7, 9, 14, 16, 25,

26 apex 26 Apollonius 2, 3, 5, 13, 14, 17, 18, 25,

38, 261, 270, 271, 277, 293, 295 Apollonius Batavus 3 Apollonius Gallus 3, 5 Apollonius Redivivus 3 Archimedes 2, 22, 261 Aristaeus the Elder 3 Aristotle 6, 258 Arithmetica Infinitorum by Wallis 25 Artis analyticae praxis by Harriot 25 axis/axes 14−18, 21, 28, 34, 37, 44, 46,

48, 122, 138, 158, 160, 178, 180, 236, 261, 262, 271, 276, 277, 281–285, 292–295, 302, 305, 308

Barrow, Isaac (1630–1677) 25 Bombelli, Rafael (1526–1572) 6 center(s) 9, 13, 17, 18, 20, 28, 49–51,

53, 54, 56–58, 126, 130, 132, 138, 144, 148, 152, 154, 158, 160, 170, 176, 178, 212, 218, 222, 226, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 268, 283, 284, 289, 290, 292, 294, 295, 297, 302, 308

circle(s) 7, 9, 13–15, 17, 18, 21–24, 38, 39. 41, 44, 50, 55–57, 60, 132, 134, 168, 172, 186, 210, 238, 240, 242, 244, 246, 250, 254, 257, 262, 264, 265, 277, 282, 286, 295–297. 300–303

coefficient(s) 6, 7, 13, 14, 17, 26, 28, 31, 265, 267, 269, 281, 282, 304

Collectio (Συναγωγη) 2–5, 298, 300 Collège de France 25

compositio, composition 1, 3, 17, 18, 28, 160, 272, 299

conic(s) 2, 5, 7, 12–14, 17, 21, 25–28, 43, 118, 202, 232, 252, 257, 293, 296, 297, 300, 304, 308

conical wedge 26 Conica 13, 261, 270, 271, 277, 293,

295, 296 conjugate 234, 252, 291 conjugate axis/axes 17, 46, 47, 51, 56,

57, 277, 280, 282, 284, 294 conjugate axis, associated 20, 45, 49 conjugate diameter 49, 52, 53, 56, 192,

263, 268, 272, 275, 276 conjugate direction 10, 17, 260, 289 constructive resolution (of equations) 9,

14, 23, 25, 26 conversim 32 coordinate axis/axes 38 coordinate system 7, 15, 26, 28 corollary (corollarium) 29, 31, 37, 41,

42, 122, 124, 160, 162, 164, 176, 180, 182, 262, 271, 277, 280

cubus 6 curve(s) 2, 7, 9, 10, 13–18, 21, 24–28,

31, 32, 34–36, 38–42, 44, 45, 49, 50, 51, 53, 54, 56, 60, 62, 74, 76, 82, 86, 88, 92, 84, 98, 100, 104, 114, 116, 122, 126, 130, 132, 134, 138, 144, 148, 154, 160, 162, 170, 172, 178, 180, 186, 238, 240, 244, 246, 250, 257–259, 264, 265, 267, 268, 276. 277, 282–284, 286, 290, 291, 298, 300– 308

Debeaune, Florimond (1601–1652) 25 definition 2–4, 9, 12, 29, 38, 40, 42, 47,

49, 50, 52, 53, 56, 62, 257, 276, 281, 282, 285, 289, 294, 299, 305

degenerations 28, 270 demonstratio 28, 33–35, 40, 41, 45,

257–259, 261, 266, 282, 283, 285, 291, 292

316 Index

demonstration 10, 62, 104, 108, 110, 160, 188, 254, 257

Descartes, René (1596–1650) 2, 6–14, 16, 24, 25, 257, 281, 282, 300–308

descriptio 28, 33, 35, 40, 41, 257–259, 261, 265, 288, 290–292

description 2, 6, 16, 34−36, 39, 49, 53, 100, 136, 148, 168, 236, 265, 282

determinatio 28, 33, 35, 40, 257, 260, 265, 282, 283

determination 6, 7, 36, 62, 84, 88, 90, 96, 106, 108, 112, 128, 136, 142, 148, 152, 168, 236, 257, 301

Diogenes Laertius (3rd century) 3 Diophantus (200?–284?) 6 division(s) 5, 7, 66, 68, 92, 98, 106,

110, 140, 144, 148, 150, 152, 156, 158, 172, 176, 178, 222, 224, 230, 246, 250, 267, 270

Elements 2, 3, 29, 258, 262, 279, 299 ellipse 15–17, 20, 21, 26, 38, 39, 41, 42,

44, 50, 55–58, 60, 118, 132, 134, 168, 170, 176, 178, 180, 182, 186, 210, 238, 240, 242, 244, 246, 248, 250, 252, 254, 257, 261–265, 270, 271, 277, 280, 282, 283, 286, 291–295, 297, 301

Elsevier 24 equality sign 6 equation(s) 2, 6, 7, 9, 10, 12–18, 21–28,

31–51, 53–56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 78, 80, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 112, 114, 116, 118, 120, 124, 126, 128, 132, 134, 136, 140, 142, 144, 148, 152, 154, 158, 162, 166, 168, 170, 174, 176, 182, 186, 188, 190, 192, 194, 196, 198, 202, 204, 206, 210, 216, 218, 226, 230, 232, 234, 236, 238, 240, 242, 250, 254, 257–260, 262, 264, 266–271, 275, 276, 280, 282, 284, 286, 288–291, 300–306, 308

algebraic 6, 21, 25, 26, 301, 304

linear 16, 28, 31

Eratosthenes (276–194 BC) 24 Euclid (325?–265? BC) 2, 3, 29, 262,

272, 273, 279, 293, 295, 299 Eudoxus (408–355 BC) 3 exegetic analysis 4, 6 Exempla 28 Fermat, Clément-Samuel de 24 Fermat, Pierre de (1601–1665) 2, 3, 7,

10, 14–19, 21–25, 28, 273, 304 first quadrant 15, 27, 31 focal properties 25 focus/foci 37, 40, 42, 122, 160, 176,

261, 270–271, 277 Geometria à Renato Descartes 25 Géométrie by Descartes 2, 8–10, 12, 13,

16, 24, 25, 257, 281, 300, 302, 308 Ghetaldi, Marino (1566–1626) 3 Golius, Jacobus (1596?–1667) 9 half-line 15, 258, 259 Harriot, Thomas (1560–1621) 25 Heath, Thomas L. (1861–1940) 3, 295,

297, 298 Heiberg, Johan Ludvig (1854 –1928) 3 Hero, Heron (10–70) 3, 299 Hierius (early 5th century) 2 Hire, Philippe de la (1640–1718) 25, 26 homogenea 6, 18 Hultsch, F. 4, 298 hyperbola(s) 16, 17, 22, 23, 26, 28, 29,

38–41, 44, 48–54, 60, 80, 118, 124, 126, 128, 130, 134, 136, 138, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 176. 186, 210, 212, 214, 216, 218, 220, 222, 226, 230, 232, 234, 236, 238, 257, 258, 261–271, 275, 277, 280, 282, 283, 286, 288–291, 295, 301, 302

immutable initial point 28, 31, 62, 70, 72, 74, 80, 84, 88, 90, 96, 100, 108, 112, 114, 124, 126, 128, 132, 136, 142, 152, 168

indefinite(ly) 28, 31, 62, 70, 72, 74, 80, 84, 86, 88, 90, 92, 96, 98, 100, 106, 108, 112, 122, 124, 126, 128, 136, 142, 144, 152, 154, 168, 170, 186, 196, 257, 258

Index 317

indeterminate 15, 28, 60, 62, 114, 116, 120, 132, 156, 170, 174, 240, 257

infinite(ly) 15, 28, 64, 118, 258, 307 infinitesimal methods 25 Isagoge 5, 14, 16, 21, 24 latitudo 6 latus 6 latus rectum 13, 18, 28, 36, 37, 46, 48,

49, 52, 53, 56–58, 72, 74, 76, 82, 86, 88, 92, 98, 102, 104, 106, 108, 110, 112, 116, 118, 126, 132, 144, 154, 156, 170, 214, 218, 222, 224, 226, 228, 230, 232, 234, 236, 240, 242, 244, 246, 248, 250, 252, 254, 258–264, 268, 270, 271, 275–277, 282, 284–286, 289, 292, 305

latus transversum 122, 132, 144, 154, 156, 170, 212, 214, 218, 222, 224, 226, 228, 230, 232, 234, 236, 240, 242, 244, 246, 248, 250, 252, 254

Leibniz, Gottfried Wilhelm (1646–1716) 26, 301

Leiden 9, 24 Liber Primus 2, 7, 24–26. 28, 29, 31,

32, 38, 49, 54, 56, 124, 126, 257, 258, 262, 263, 270–272, 277, 289, 292, 294, 302

Liber Secundus 1, 2, 25, 26, 28, 31, 263 (line) segment(s) 2, 4, 5, 7–9, 12, 15,

19, 21, 24–26, 28, 31, 37, 40–43, 54, 60, 64, 66, 70, 78, 90, 92, 96, 98. 102, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 142, 144, 152, 154, 156, 160, 164, 170, 174, 176, 178, 180, 182, 186, 192, 196, 198, 202, 204, 206, 208, 210, 212, 216, 226, 236, 238, 240, 242, 244, 257–259, 267, 270, 274, 277, 279–282, 286, 287, 298–300, 308

Lieux Géométriques 25, 26 Loci plani 3, 14, 300 locus/loci 5, 7, 12–16, 18, 21, 24–26,

29, 31–33, 37–40, 42, 54, 56, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 98, 100, 104, 106, 108, 110, 112, 114, 116, 118,

120, 124, 126, 128, 130, 132, 134, 136, 138, 142, 144, 148, 152, 154, 156, 158, 168, 170, 174, 176, 182, 186, 188, 190, 204, 210, 216, 230, 232, 234, 236, 238, 240, 242, 250, 254, 257, 262, 270, 277, 281, 291, 293, 300, 302, 305

logistica numerosa 6 logistica speciosa 6 longitudo 6 Maire, Jan 24 major axis 17, 178, 281 Menaechmus 7 Mersenne, Père Marin Étienne (1568–

1648) 24 mesolabium 24 method 2, 3, 7, 9, 10, 12, 16, 17, 21–28,

31, 33, 39, 43, 84, 182, 257, 258, 266, 270, 298, 302, 306–308

~ of Descartes 2, 10, 12, 24 minor axis 178, 281 multiplication 7, 78, 80, 82, 92, 98, 104,

128, 140, 144, 148, 150, 152, 156, 172, 267, 299

negative abscissa 25 Notae Breves 24 ordinate-axis 10, 17, 34, 303 Oresme, Nicole (ca 1320–1382) 2 origin 10, 15, 17, 18, 38, 267, 282, 303,

306 Ozonam, Jacques (1640–1717) 26 Pappus of Alexandria (ca 290 – ca 350)

2–5, 9, 10, 13, 14, 21, 257, 293, 298–300, 303, 304, 306–308

parabola(s) 7, 13, 15–18, 22, 23, 26, 27, 32–37, 43, 45–48, 60, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 136, 186, 190, 192, 194, 196, 204, 206, 257–262, 264, 265, 270, 271, 275, 282–285, 301, 303–306, 308

Parabola of Descartes 13, 308 parallel translation(s) 26, 54, 282 Parent, Antoine (1666–1716) 25 Pascal, Blaise 24

318 Index

Pascal, Étienne 24 plane curves 2, 10, 13, 21, 24, 25, 300,

304 planimetric 25 planum 6 Plato 3 poristic analysis 4, 6 power taking 7 Prade 3 Problema(ta) 2, 21, 28, 29 product(s) 6–9, 12, 21, 60, 78, 84, 170,

182, 184, 254, 262, 286, 288, 290, 291, 299, 300

propositio 21, 29 Pythagoras, school of 3 quadratic equation(s) 9, 12, 16, 25, 27,

28, 43 quadrato-quadratum 6 quadratum 6, 16 quotient 8, 9, 98, 118, 144 Rabuel, C. (1669–1728) 16, 31, 306 radix 6 Recorde, R. (ca 1510–1558) 6 Regula Universalis 27, 28, 258, 270 Regula Universalis, second 27, 262 Regula Universalis, third 27, 28 rhetic analysis 4, 6 ridge 26 Roberval, G.P. de (1602–1675) 24 root 7–9, 12, 13, 22, 26 rotation 16, 18, 27, 282, 284 Snellius, W. (1580–1626) 3 solidum 6 straight line 4, 7, 15, 16, 26–28, 31, 32,

41–44, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 104, 112, 114, 118, 122, 124, 136, 138, 142, 144, 152, 162, 164, 166, 168, 170, 180, 182, 186, 192, 194, 196, 198, 200, 202, 204, 206, 210, 214, 216, 218, 220, 222, 226, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 254, 257, 258, 262, 282, 300, 301, 304, 308

surface 25

symmetry axis/axes 14, 17, 20, 28, 37, 261, 277, 278, 283, 284–286

symptomata of Apollonius 25 synthesis (συνθεσις) 3–5, 17, 28 terminus alterius 15 terminus localis 16 Theaitetus 3 Theorema(ta) 2, 28, 29 Thomas, I. 4, 298 Tractatus de sectionibus conicis of

Wallis 25 transverse axis 41, 42, 46, 47, 49–51,

53, 54, 56, 160, 162, 176, 180, 182, 271, 275, 277, 280, 285, 286, 294

Treatise of Algebra, both Historical and Practical of Wallis 26

trident 13, 303, 305, 306 trisection of the angle 24 umbilical point 37, 122, 160 unit element 8 v-axis 269 Van Schooten, Jr., Frans (1615–1660)

1, 3, 9, 24, 25, 281 vertex/vertices 14, 17, 28, 34, 37, 45–

48, 53–55, 74, 76, 82, 86, 88, 92, 98, 102, 104, 110, 112, 114, 122, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 236, 258–262, 270, 283–286, 291, 305

Viète, François (1540–1603) 2–7, 9, 14, 16, 21, 23, 261

Wallis conocuneus 26 Wallis, John (1616–1703) 25–26 Wantzel, P.L. (1814–1848) 12 Witt, Jan de (1625–1672) 1, 10, 16, 17,

25–29, 31, 33, 34, 38–40, 42–45, 48–50, 55, 56, 60, 257–264, 266, 267, 269–274, 277–284, 286–291, 294, 302

Wolff, Christian von (1679–1754) 2 x-axis 16, 31, 32, 34, 40, 282 y-axis 15, 31, 282 zetetic analysis 4, 5