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[PDF page 334] 318 MATTER AND ETHER THEORIES. [CH. XIV four dimensions as suggested above in chapterxii, this connection does exist, for all the molecules rest on one and the same body. This body is capable of transmitting vibrations, hence, no matter how the molecules were set vibrating originally, they would fall into certain groups, and all the members of each group would vibrate at the same rate. It was the possibility of obtaining thus a physical connection between the various particles in our universe that first suggested to me the idea of a supporting medium in a fourth dimension. (4) If we accept Boscovich’s hypothesis or that of an elastic solid ether, and if we may lay it down as axiomatic that the mass of every sub-atom is the same, we may conceive that the number of ways of combining the sub-atoms into a permanent system is limited, and that the period of vibration depends on the form in which the sub-atoms are combined into an atom. This view is not inconsistent with any known facts. I may add that it is probable that the chemical atom is the essential vibrating system, for the sodium spectrum, to take one instance, is the same as that of all its compounds. (5) In the same way we may suppose that the vortex rings are formed so that they can have only a definite number of stable forms produced by interlinking or kinking. (6) Similarly we may modify the popular hypothesis by treat- ing the atoms as indivisible aggregates of sub-atoms which are in all respects equal and similar, and can be combined in only a limited num- ber of forms which are permanent. But most of the old difficulties con- nected with the atoms arise again in connection with the sub-atoms. (7) I am not aware that Maxwell discussed any other hypotheses in connection with this point, but it has been suggested recently that, if the various forms of matter were evolved originally out of some one primitive material, then there may have been periodic disturbances in this matter when the atoms were being formed, such that they were produced only at some definite phase in the period*. Thus, if the disturbance is represented by the swinging of a pendu- lum in a resisting medium, it might be supposed that the atoms were formed at the points of maximum amplitude, and we should expect that the atoms successively thrown off would form a series having the prop- erties of its successive members connected by a regular periodic law. * See Crookes on Mendeléeff’s periodic law,Nature, Sept. 2, 1886, vol. xxxiv, pp. 423–432.

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[PDF page 335] CH. XIV] THE SIZE OF MOLECULES OF MATTER. 319 This conjecture, when worked out in some detail, led to the conclusion that some elements which ought to have appeared in the series were missing, but it was possible to predict their properties and to suggest the substances with which they were most likely to be found in com- bination. Guided by these theoretical conclusions a careful chemical analysis revealed the fact that such elements did exist. That this hypothesis has led to new discoveries is something in its favour, but I do not wish to be understood to say that it is a theory which leads to results that have been verified subsequently. I should say rather that we have obtained an analogy which is sufficiently like the truth to suggest new discoveries. Such analogies are often the pre- cursors of laws, so that it is not unreasonable to hope that ere long our knowledge of this border-land of chemistry and physics may be more definite, and thus that molecular physics may be brought within the domain of mathematics. It is however very remarkable that J.J. Thom- son’s conclusions on the stability of the orbital systems he devised should agree so closely with Mendeléeff’s periodic law. On the whole Maxwell thought that the phenomena point to a common origin of all molecules of the same kind, that this was an event not belonging to that order of nature under which we live, but must have originated when or before the existing order was established, and that so long as the present order exists it is immutable. This is equivalent to saying that we have arrived at a point be- yond which our limited experience does not enable us to carry the explanation. That we should be able to form an approximate idea of the size of the molecules of matter is a testimony to the extraordinary advance which mathematical physics has made recently. Sir William Thomson, now Lord Kelvin, whose ingenuity seems to know no limits—has suggested* four distinct methods of attacking the problem. They lead to results which are not very different. The first of these rests on an assertion of Cauchy that the phenom- ena of prismatic colours show that the distance between consecutive molecules of matter is comparable with the wave-lengths of light. Tak- ing the most unfavourable case this would seem to indicate that in a * See Nature, March 31, 1870, vol.i, pp. 531–553; and Tait’sRecent Advances, pp. 303–318. The fourth method had been proposed by Loschmidt in 1863.

PG26839.B00337
[PDF page 336] 320 MATTER AND ETHER THEORIES. [CH. XIV transparent homogeneous solid or liquid medium there are not more than 64 × 1024 molecules in a cubic inch, that is, that the distance between consecutive molecules is greater than1/(4 × 108)th of an inch. The second method is founded on the amount of work required to draw out a film of liquid, such as a soap-bubble, to a given thickness. This can be calculated from experiments in a capillary tube, and it is found that, if a soap-bubble could be drawn out to a thickness of 1/108th of an inch there would be but a few molecules in its thickness. This method is not quantitative. Thirdly, Sir William Thomson proved that the contact phenomena of electricity require that in an alloy of brass the distance between two molecules, one of zinc and one of copper, shall be greater than 1/(7 × 108)th of an inch; hence the number of molecules in a cubic inch of zinc or copper is not greater than35 × 1025. Lastly, the kinetic theory of gases leads to the conclusion that certain phenomena of temperature and viscosity depend, inter alia, on inter-molecular collisions, and so on the sizes and velocities of the molecules, while the average velocity with which the molecules move increases with the temperature. This leads to the conclusion that the distance between two consecutive molecules of a gas at normal pres- sure and temperature is greater than1/(6 × 106)th of an inch, and is less than 1/107th of an inch; while the actual size of the molecule is a trifle greater than 1/(3 × 1020)th of a cubic inch; and the number of molecules in a cubic inch is about3 × 1020. Thusitwouldseemthatacubicinchofgasatordinarypressureand temperature contains about 3 × 1020 molecules, all similar and equal, and each molecule has a volume of about1/(3 ×1025)th of a cubic inch; while a cubic inch of the simplest solid or liquid contains rather less than 1027 molecules, and perhaps each molecule has a volume of about 1/(3 × 1026)th of a cubic inch. For instance, if a pea or a drop of water whose radius is1/16th inch was magnified to the size of the earth, then there would be about thirty molecules in every cubic foot of it, and probably the size of a molecule would be about the same as that of a fives-ball. The average size of the minute drops of water in a very light cloud can be calculated from the coloured rings produced when the sun or moon shines through it. The radius of a drop is about1/30000th of an inch. Such a drop therefore would contain about2 × 1013 separate molecules. In gases and vapours, the number of atoms required to make

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[PDF page 337] CH. XIV] THE SIZE OF MOLECULES OF MATTER. 321 up one of these molecules can be estimated, but in liquids the number is not as yet known. Loschmidt asserted that a cube whose side is1/4000th of a mil- limetre is the smallest object which can be made visible at the present time. Such a cube of oxygen or nitrogen would contain from 60 to 100 millions of molecules of the gas. Also on an average about 50 elemen- tary molecules of the so-called elements are required to constitute one molecule of organic matter. At least half of every living organism con- sists of water, and we may for the moment suppose that the remainder consists of organic matter. Hence the smallest living being which is vis- ible under the microscope contains from 30 to 50 millions of elementary molecules which are combined in the form of water, and from 30 to 50 millions of elementary molecules which are combined so as to make not more than one million organic molecules. Hence a very simple organism might be built up out of as few as a million similar organic molecules. Maxwell did not consider that this wassufficienttojustifythecurrentconclusionsofphysiologists, andsaid that they must not suppose that structural details of infinitely small di- mensionscanfurnishbythemselvesanexplanationofthevarietyknown to exist in the properties and functions of the most minute organisms; but physiologists have replied that whether their conjectures be right or wrong Maxwell’s argument is vitiated by his non-consideration of differences due to the physical (as opposed to the chemical) structure of the organism and the consequent motions of the component parts.

PG26839.B00339
[PDF page 338] INDEX Abbot, W., 176 Abbott, E.A., 276 Achilles and the Tortoise, 67 Acts or Disputations, chap.vii Agrippa, Cornelius, 122 Ahrens, 97, 103, 123 Airy, Sir Geo., 87, 193 Aix, Labyrinth at, 152 Albohazen on Astrology, 237 Alcuin, 2, 55 Alfred the Great, 300 Alkarisimi on π, 216 Alkborough, Labyrinth at, 152 Anallagmatic Squares, 51 Anaxagoras, 214 Anderson, T., 271 Angular Motion, 69 Anstice, 103, 105–106 Antipho, 214 Apollonius, 205, 207, 215 Archimedes, 72, 211, 214, 218 Archytas on Delian Problem, 206 Argyle, 257 Arithmetic, Higher, 29–34 Arithmetical Fallacies, 20–24 Arithmetical Puzzles, 4–26 — Recreations, chap. i Arya Bhata onπ, 215 Asenby, Labyrinth at, 152 Astrological Planets, 122, 239, 290 Astrology, chap. x Atomic Theories, chap. xiv Atoms, Size of, 319 Attraction, Law of, 313–317 Augustine on Astrology, 243 Augustus, 268, 291, 292 Ayrton on Magic Mirrors, 87 Babbage, C., 191 Bachet’sProblèmes, 2–20, 27–29, 55, 113, 115, 122, 125, 132 Bacon, Francis, 268 Bailey, J.E., 250 Ball, 170, 172, 204, 207, 223, 281 Bardesan on Fate, 236 Barrow, I., 171 Baudhayana on π, 215 Becquerel Rays, 311, 312 Bedwell, T., 171 Beltrami on Space, 276 Benham on Spectrum Top, 87 Bentley, Newton to, 313 Bentley, R., 175 Bernoulli, John, 22 Berosus, 297 Berri, de, 266 Bertrand, J.L.F., 21 Bertrand, L. (of Geneva), 160 Besant on Hauksbee’s Law, 82 Bhaskara on π, 216 Bickmore, C.E., 229, 230, 235 322

PG26839.B00340
[PDF page 339] INDEX. 323 Biering on Delian Problem, 204 Billingsley, H., 171 Bills on Kirkman’s Problem, 109 Binary Powers, Fermat on, 31–32 Birch, J.G., 232 Birds, Flight of, 85–86 Bjerknes, 310 Blackburn, J., 184 Blundeville, T., 171 Board, Mathematical, 197, 198 Boat-racing with a rope, 81–82 Bolyai, J., 284 Bonnycastle, J., 188 Bordered Magic Squares, 135–136 Boscovich on Matter, 306, 316, 318 Boughton Green, Labyrinth at, 152 Bouniakowski, V., on shuffling, 110 Bourget, M.J., on shuffling, 110 Bourlet, 18, 19 Boussinesq on Ether, 307 Brackets, in Tripos, 181, 187, 196, 197 Brahmagupta on π, 216 Breton on Mosaics, 152 Brewster, Sir David, 248 Briggs, H., 171 Bristed, C.A., 191 Bromton, 151 Brouncker on π, 219 Brown, J. (Saint), 189 Bryan on Bird Flight, 86 Bryso, 214 Bubble Theory of Matter, 312 Buckley, W., 171 Burnside, Kirkman’s Problem, 103 Butcher on the Calendar, 296 Caesar, Julius, 243, 268, 291 Calendar, the Civil, 291–293 — the Ecclesiastical, 293–296 — the Gregorian, 293–295 — the Julian, 291–292 Calendars, University, 185 Cambridge, Mathematics, chap. vii — Studies at, chap. vii Cantor on π, 214 Cardan, 2, 93, 95, 236, 246–248, 274 Cards, Problems with, 14–15, 26, 55, 109–120 Carpmael, Kirkman’s Problem, 103 Cartwright, W., 270 Cauchy, 48, 306, 319 Cayley, 44, 46, 103, 119, 154, 219 Cellini, 248 Cells of a Chess-board, 158 Centrifugal Force, 71–72 Ceulen, van, onπ, 217, 218 Challis MSS, 182 Charles I, 268–270, 271, 296 Charles V of Germany, 301 Chartres, Labyrinth at, 152 Chartres, R., 21, 41, 222 Chasles on Trisection of Angle, 210 Chaucer on the Sun-cylinder, 297 Cheke, Sir John, 246 Chess-board, Games on , 60–64, 103, 158–168 — knights’ moves on, 63, 158–168 — problems, 25, 60–64, 103, 158–168 Chilcombe, Labyrinth at, 152 Chinese on π, 216 Chinese rings, 93–97 Ciccolini on Chess, 164 Cicero on Astrology, 243

PG26839.B00341
[PDF page 340] 324 INDEX. Ciphers, chap. xi — Definition of, 251 — Four types of, 258–266 — Historical, 268–275 — Requisites for good, 267–268 Circle, Quadrature of, 211–222 Cissoid, the, 205, 207, 212 Clairaut on Trisection of Angle, 210 Clarke, S., 171 Classical Tripos, 195 Claus, 91 Clausen on π, 220 Clavius on Calendar, 292, 293, 296 Clepsydras, 300 Clerk Maxwell,see Maxwell Clerke, G., 171 Clifford, 70 Clocks, 77, 300–302 Cnossus, Coins of, 151 Coat and Waistcoat Trick, 64 Coccoz, 37, 137 Code-Book Ciphers, 273 Cole, F.N., 224, 226, 229, 231, 233 Colebrooke on Indian Algebra, 216 Collini on Chess, 163 Collins, Letter from J. Gregory, 219 Colour-cube Problem, 51–52 Colouring Maps, 44–46 Columbus, 296 Columbus’s Egg Puzzle, 74 Comberton, Labyrinth at, 152 Compasses, Watches as, 302–303 Competition, in Tripos, 187, 191 Composite Magic Squares, 134 Conchoid, the, 205, 209, 212 Cones moving uphill, 75 Conradus, D.A., 260 Continuity of Matter, 305 Contour-lines, 47 Cotes, R., 171 Counters, Games with, 48–50, 58–63 Craig, J., 171 Crassus, 243 Cretan Labyrinth, 151, 152 Cricket-Ball, Spin on, 85 Crookes on Mendeléeff’s Laws, 318 Cross-fours, 51 Cryptographs, Definition of, 250 — Three types of, 252–258 Cryptography, chap. xi Cube, Duplication of , 204–208 Cubes, Coloured, 51–52 Cubes, Skeleton, 26 Cudworth on Sharp, 219 Cumberland, R., 175–176 Cumulative Vote, 26 Cunningham, A.J.C., 32, 224, 227, 228, 229, 230, 233, 234 Cureton on Syriac Astrology, 236 Curie on radio-activity, 311 Curiosa Physica, 86–87 Cursor, 297 Cusa on π, 216 Cusps, Astrological, 237 Cut on a Tennis-ball, 83–85 Cutting Cards, Problems on, 15 Cylinders, Sun-, 297 Dacres, A., 171 Daedalus, Labyrinth of, 151 D’Alembert, 21, 24 Darwin, G.H., 42 Dase on π, 220 Dasypodius, 301 Day, Definition of, 288 — Commencement of, 290 — Sidereal and Solar, 289 Days of Week from date, 296 Days of Week, Names of, 290–291 Dealtry, W., 188, 189, 190 De Berri, 266

PG26839.B00342
[PDF page 341] INDEX. 325 Decimation, 19–20 Dee, J., 171 De Fonteney on Ferry Problem, 55 De Fouquières, 49 De Haan onπ, 214, 217 De Lagny onπ, 220 De la Hire on Magic Squares, 122, 123, 126–127, 128, 132–134 De la Loubère on Magic Squares, 124–125, 137 Delambre on Calendar, 295, 296 Delannoy, 55, 60 De la Pryme, 174 Delian Problem, 204–208 De Longchamps, G., 234 De Moivre, on Knight’s Move, 159 De Montmort, 1, 159 De Morgan, A., 44, 67, 169, 172, 188, 189, 211, 213, 214, 222, 245, 293 Denary scale of notation, 10, 11 De Parville on Tower of Hanoï, 92 De Polignac on Knight’s Move, 166 De Rohan, 271 De St Laurent, 110 Descartes, 208, 210, 219, 305 Des Ourmes on Magic Squares, 122 Diabolic Squares, 136–137 Dials, Sun-, 296–298 Dickson, L.E., 118 Diego Palomino, 19 Digby, Lord, 270 Digges, T., 171 Diocles on Delian Problem, 207 Diodorus on Lake Moeris, 150 Dircks, H., 271 Dirichlet on Fermat’s Theorem, 33 Dissection, Proofs by, 42–44 Dixon, A.C., 46, 103 Dodecahedron Game, 155–158 Dominical Letter, 295 Dominoes, 48, 141 D’Ons-en-bray, Magic Squares, 122 Doubly Magic Squares, 137 Douglas, S., 191, 193 Drach on Magic Squares, 123 Drayton, 151 Duplication of Cube , 204–208 Durations, see Time Dürer, A., 122 Dynamical Games, 52–64 Earnshaw, S., 87, 196 Easter, Date of, 293–296 Edward VI, 239, 246–248, 259 — Horoscope of, 247 Eight Queens Problem , 97–103 Eisenlohr on Ahmes, 214 Eisenstein, 32 Electrons, 310–312 Elliptic Geometry, 284–285 Eneström on π, 213 Epicurus on Gravitation, 313 Equilibrium, Puzzles on, 72–74 Eratosthenes, 204 Escott, E.B., 235 Ether Theories, 306–310 Ether-Squirts, 309 Etten, van, 2, 10 Euclid, 30, 36, 214 Euclid’s Axioms &c., 284 — Parallel Postulate, 284 Euclidean Geometry, 283–286 Euclidean Space, 283–286 Euclid i. 32, 42 Euclid i. 47, 42 Euler, 29, 30, 32, 48, 122, 123, 136, 140, 159, 160–162, 213, 219, 220, 224, 225, 226, 227 Euler’s Unicursal Problem, 143–149 Examination, Printed, 183, 195

PG26839.B00343
[PDF page 342] 326 INDEX. Exploration Problems, 26 F allacies, Arithmetical, 20–24 — Geometrical, 35–41 — Mechanical, 67–69, 75–78 Faraday on Matter, 306 Fauquembergue, E., 229 Fenn, J., 176 Fermat on Binary Powers, 31–32 Fermat, P., 29, 31–34, 123, 224, 225, 226, 227, 234 Fermat’s Last Theorem, 32–34 Ferry-boat Problems, 55–57 Fifteen Puzzle, 88–91 Fifteen School-girls, 103–109 Figulus on Astrology, 243 Firmicus on Astrology, 237 Fitzpatrick, J., 59 Flamsteed on Astrology, 245 Flamsteed, J., 171 Flat-land, 277–282 Fluid Motion, 82–86 Fluxions, 179, 182, 186, 188, 189, 191, 192 Fonteney on Ferry Problem, 55 Force, Definition of, 70 Foster, S., 171 Fouquières on Ancient Games, 49 Four-Colour Theorem, 44–46 Fox onπ, 222 Frankenstein on Magic Pencils, 138 Franklin, B., 312 Frederick II of Germany, 300 Frénicle, Magic Squares, 122, 135 Frere, J., 177 Fresnel on Ether, 306 Frolow on Magic Squares, 136 Frost, A.H., 103, 104–105, 123, 136 Galileo on Pendulum, 301 Galton, 21 Games, Dynamical, 52–64 — Statical, 48–52 — with Counters, 58–63 Gases, Theory of, 320 Gauss, 29, 34, 230, 276, 294 Geodesic Problems, 57–58 Geography, Physical, 46–48 Geometrical F allacies, 35–41 Geometrical Recreations, chap. ii Geometry, Non-Euclidean, chap. xii George I of England, 174 Gerbert, 215, 300 Gergonne’s Problem, 115–118 Gill, Kirkman’s Problem, 106–107 Glaisher, J.W.L., 98, 99, 170, 214 Glamorgan, Earl of, 271 Gnomons, 296 Goldbach’s Theorem, 31 Golden Number, 295 Gooch, W., 183 Gra vity, Hypotheses on, 313–317 Green on Ether, 306 Greenwich, Labyrinth at, 153 Gregorian Calendar, 292, 293 Gregory XIII, 292–293 Gregory, Jas., 212, 218 Gregory of St Vincent, 208 Gregory’s Series, 219 Grienberger on π, 218 Grille, The, 255 Grimthorpe on Clocks, 300 Gros on Chinese Rings, 95, 96 Guarini’s Problem, 63 Gun, Report of, 87 Gunning, H., 183, 184 Günther, S., 97, 99, 123 Guthrie on colouring maps, 44

PG26839.B00344
[PDF page 343] INDEX. 327 Haan, de, onπ, 214, 217 Halley on π, 219 Halsted on Hyper-space, 276 Hamilton, Archbishop, 246 Hamilton, Sir Wm., 155–158 Hamiltonian Game, 155–158 Hampton Court, Maze at, 149, 153 Hanoï, Tower of , 91–93 Harris on pendulum clock, 301 Harvey, J., 171 Harvey, R., 171 Hauksbee’s La w, 82–85 Hayward, J., 259 Heawood on Colouring Maps, 46 Hegesippus on Decimation, 19 Hele, P., 301 Helmholtz, 78, 276, 308 Helmholz, 307 Henrietta Maria, 269 Henry on Unicursal Problems, 143 Henry VIII of England, 247, 300 Hermary, 158 Hero of Alexandria onπ, 205, 215 Herodotus on Lake Moeris, 150 Herschel, Sir John, 191, 291 Hezekiah, 298 Hicks on Matter, 304, 308, 310 Hiero of Syracuse, 72 Higher Arithmetic, 29–34 Hill, C., 272 Hill, M.J.M., 308 Hill, T., 171 Hills and Dales , 46–48 Hinton on Space, 276, 279, 281 Hipparchus on hours of day, 290 Hippias, 214 Hippocrates of Chios, 205, 214 Hodson, W., 182 Holditch on Magic Squares, 123 Homaloidal Geometry, 285 Honorary Optimes, 173, 175, 195 Hood, T., 171 Hooke on Timepieces, 301 Horary Astrology, 237 Hornbuckle, T.W., 189, 190 Horner on Magic Squares, 123 — Rules to cast, 238 Horoscopes, chap. x — Example of, 247 — Rules to cast, 237 — Rules to read, 239–242 Horrox, J., 171 Hour-glasses, 300 Hours, definition of, 289, 290 Houses, Astrological, 237 Huddling, 171, 172 Hudson, C.T., on cards, 118 Hudson, W.H.H., on cards, 111 Hustler, J.D., 189 Hutton, C., 3, 220 Huygens, 208, 211, 212, 218, 219, 301 Hyper-magic Squares, 136–137 Hyper-space, chap. xii Hyperbolic Geometry, 284–285 Icosian Game, 155–158 Ideler on the Calendar, 296 Inertia, 70, 71 Inwards on the Cretan Maze, 151 Isaiah, 298 Jacob, E., 189, 190, 191 Jacobi, 230, 234 Jaenisch, 158, 164, 167, 168 James II of England, 257 Japanese Magic Mirrors, 87 Jebb, J., 176, 178–180 Johnson on Fifteen Puzzle, 88 Jones on π, 220 Jones on π, 213 Josephus on Decimation, 19 Julian Calendar, 291–292

PG26839.B00345
[PDF page 344] 328 INDEX. Julian’s Bowers, 152 Julius Caesar, 243, 268 Junior Optimes, 173, 180 Jurin, J., 171 Kelvin, 306, 307, 308, 314, 315, 319 Kempe on Colouring Maps, 45 Ketteler on Ether, 307 Kinetic Theory of Gases, 320 Kirchhoff on Ether, 307 Kirkman, T.P., 103, 108 Kirkman’s Problem, 103–109 Klein, 203 Knight’s Path Problem, 158–168 Knyghton, 151 Königsberg Problem, 143–149 Kummer on Fermat’s Theorem, 33 Labile Ether, 307 Labosne on Magic Squares, 132 Labyrinths, 149–153 Lacroix, 211 Lagny on π, 220 Lagrange, 191, 227 Lagrange’s Theorem, 31 La Hire, 122, 123, 126–127, 128, 132–134 Laisant, C.A., 11, 235 La Loubère, 124–125 Lambert on π, 211 Lambert on π, 211 Lamé, 33, 307 Landry, 224, 226, 228, 229, 231 Langley on Bird Flight, 86 Laplace, 191 Laplace on velocity of sound, 306 Laquière on Knight’s Path, 166 Larmor on Electrons, 304, 310 Latruaumont, 271 Laughton, R., 171 Lawrence, F.W., 235 Lax, W., 183, 184 Lea on Kirkman’s Problem, 109 Leap-year, 291–292 Lebesgue on Fermat’s Theorem, 33 Legendre, 33, 160, 168, 191, 211, 227, 230 Leibnitz on Games, 1 Lejeune Dirichlet on Fermat, 33 Le Lasseur, 224, 226, 227, 228 Leonardo of Pisa onπ, 216 Le Sage on Gravity, 314, 315 Leslie, J., 205, 209 Leurechon, 2 Lilius on the Calendar, 292 Lilly on Astrology, 245 Linde on Knight’s Path, 158 Lindemann on π, 211 Line-land, 278 Lines of Slope, 47 Lippeus, 301 Listing’sTopologie, 65, 145 Liveing on the Spectrum Top, 87 Lobatschewsky, N.I., 276, 284 Locke, J., 181, 183 Lommel on Ether, 307 London and Wise, 153 Lorentz on Ether, 307 Loschmidt on Molecules, 319, 321 Loubère, de la, 124–125 Louis XI of France, 244 Louis XIV of France, 124 Loyd, S., 17 Lucas di Burgo, 2 Lucas, E., 50, 55, 60, 61, 63, 91, 95, 103, 107, 143, 155, 228, 229 Lucca, Labyrinth at, 152 Lydgate on the Sun-cylinder, 297

PG26839.B00346
[PDF page 345] INDEX. 329 MacCullagh on Ether, 306 Machin’s series forπ, 220, 221 Maclaurin on Newton, 314 MacMahon, 28–29, 51 Magic Bottles, 73 Magic Mirrors, 87 Magic Pencils, 137–140 Magic Squares, chap. v Magic Square Puzzles, 140–142 Magnus on Hauksbee’s Law, 83 Map Colour Theorem , 44–46 Marie Antoinette, 272 Mathematics, Cambridge, chap. vii Mathews, G.B., 233 Matter, Constitution of, chap.xiv — Hypotheses on, 305–313 — Kinds of, limited, 317–319 — Size of Molecules, 319–321 Maxim on Bird Flight, 86 Maxwell’s Demon, 87 Maxwell, J. Clerk, 46, 87, 304, 310, 314, 315, 317, 318, 321 Mazes, 149–153 Mean Time, 289, 290 Mechanical Recreations, chap. iii Medieval Problems, 16–20 Menaechmus, 206 Mendeléeff, 312, 318, 319 Mersenne on Primes, 29 Mersenne’s Numbers, chap. ix, 30, 223, 224 Mesolabum, 206 Metius on π, 217 Meton, 297 Méziriac, see Bachet Milner, I., 181, 189 Minding on Knight’s Path, 168 Minos, 149, 204 Minotaur, 151 Minutes, def. of, 289, 290 Mirrors, Magic, 87 Models, 78 Moderators, chap. vii Mohammed’s sign-manual, 148 Moivre, A. De, 159 Molecules, Size of , 319–321 Money, Question on, 8–9 Monge on Shuffling Cards, 109–111 Months, 291 Montmort, De, 1, 159 Montucla, 3, 50, 72, 73, 122, 123, 132, 133, 159, 211, 212 Moon, R., 123, 167 Morgan, A. De,see De Morgan Morland, S., 171 Morley on Cardan, 246 Mosaic Pavements, 50, 152 Moschopulus, 122, 126 Motion in Fluids, 82–86 Motion, Laws of, 66, 70–75 — Paradoxes on, 67–69 — Perpetual, 75–78 Mousetrap, Game of , 119–120 Müller (Regiomontanus), 216 Mullinger, J.B., 201 Mydorge, 2 Nasik Squares, 136–137 Natal Astrology, 237 Nauck, 97 Neumann on Ether, 306 Newton, 75, 171, 188, 189, 190, 208, 210, 212, 305, 312, 313, 314 Newtonian Laws of Motion, 66–75 Nicene Council on Easter, 293 Niceron, 274 Nicomedes, 205 Nigidius on Astrology, 243 Non-Euclidean Geometry, 283–286

PG26839.B00347
[PDF page 346] 330 INDEX. Nonez on Sun-dials, 297 Notation, Denary scale of, 10 Noughts and Crosses, 48 Numa on the Year, 291 Numbers, Perfect, 224 — Puzzles with, 4–20 — Theory of, 29–34 Oliver on Sun-dials, 298 Ons-en-bray on Magic Squares, 122 Oppert on π, 214 Optimes, chap. vii, 173, 176, 195 Oram on Eight Queens, 101 Oughtred, W., 171 Oughtred’sRecreations, 2, 10, 13, 16, 18, 73, 74 Ourmes on Magic Squares, 122 Ovid, 150 Ozanam’sRécréations, 2, 10, 16, 20, 43, 50, 55, 64, 72, 73, 74, 75, 77, 79, 93, 122, 123, 132, 140, 159, 297, 298, 302 Ozanam, A.F., on Labyrinths, 152 π, 211–222,see table of contents Pacificus on Clocks, 300 Pacioli di Burgo, 2 Pairs of Cards Trick, 113–115 Paley, W., 177, 183, 185, 189, 194 Palomino, 19 Pappus, 205, 209, 210 Parabolic Geometry, 285 Paradromic Rings, 64–65 Parallels, Theory of, 284 Parmentier on Knight’s Path, 158 Parry on Sound, 87 Parville on Tower of Hanoï, 92 Pa wns, Games with, 58–63 Paynell, N., 171 Pearson on Ether-Squirts, 309 Pein on Ten Queens, 101 Pencils, Magic, 137–140 Pepys, S., 270–271 Perfect Numbers, 30, 224 Permutation Problems, 26 Perpetual Motion, 75–78 Perrin, 11 Perry on Magic Mirrors, 87 Peterson on maps, 46 Philo, 205 Philoponus on Delian Problem, 204 Physical Geography, 46–48 Pierce on Kirkman’s Problem, 103 Pile Problems, 115–119 Pirie on π, 219 Pitatus on the Calendar, 292 Pittenger, 71 Plana, G.A.A., 224, 226, 228, 231 Planets (astrological), 122, 239, 290 — Signification of, 239–241 Plato on Delian Problem, 204, 205, 206 Pliny, 150, 243 Poe, E.A., 260, 263, 274 Poignard, Magic Squares, 122, 123, 127 Poincaré, H., 285 Poitiers, Labyrinth at, 152 Polignac on Knight’s Path, 166 Poll Examinations, 195, 196 Poll-Men, 173 Pollock, F., 188–190 Pompey, 243 Porta, G., 274 Power, Kirkman’s Problem, 103, 105 Pratt on Knight’s Path, 163 Pretender, The Young, 257 Primes, 29 Probabilities and π, 221 Probabilities, Fallacies in, 24, 42 Problem Papers, 182, 183, 185 Ptolemy, 215, 236, 237, 290
