chapter xix, this connection does exist, for all the molecules

rest on one and the same body. This body is capable of trans- mitting 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
•i76 MATTER AND ETHER THEORIES [CH. XXI
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 treating the atoms as indivisible aggregates of sub-atoms which are in all respects equal and similar, and can be combined in only a limited number of forms which are permanent. But most of the old difficulties connected with the atoms arise again in connection with the sub-atoms.
(7) I am not aware that Clerk Maxwell discussed any other hypotheses in connection with this point, but it has been sug- gested 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 pendulum 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 properties of its successive members connected by a regular periodic law. This conjecture,
* See Nature, Sept. 2, 1886, vol. xxxiv, pp. 423—432.
CH. XXl] MAITER AND ETHER THEORIES 477
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 combination. 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 some- thing 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 precursors 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 withiQ the domain of mathematics. It is however very re- markable that J. J. Thomson's conclusions on the stability of the orbital systems he devised should agree so closely with Mendelejev's periodic law.
On the whole Clerk Maxwell thought that the phenomena poiat 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 beyond 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 development of mathematical physics in the course of the nineteenth century.
Sir William Thomson suggested* four distinct methods of
* See iVat«?-e, March 31, 1870, vol. i, pp. 551 — 5;">3; and T nil's Recent Advances, pp. 303 — 318. The fourth method hud been proposed by Loschmidt in 18G3.
478 MATTER AND ETHER THEORIES [CH. XXI
attacking the problem. They lead to results which are not very different.
The first of these rests on an assertion of Cauchy that the phenomena of prismatic colours show that the distance between consecutive molecules of matter is comparable with the wave- lengths of light. Taking the most unfavourable case this would seem to indicate that in a transparent homogeneous solid or liquid medium there are not more than 64 x 10^ molecules in a cubic inch, that is, that the distance between consecutive molecules is greater than 1/(4 x lO^jth 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/lO^th of an inch there would be but a few molecules in its thickness. This method is not quantitative.
Thirdly, Thomson proved that the contact phenomena of electricity require that in an alloy of brass the distance be- tween two molecules, one of zinc and one of copper, shall be greater than 1/(7 x 10^)th of an inch ; hence the number of molecules in a cubic inch of zinc or copper is not greater than 35 x 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 tem- perature. This leads to the conclusion that the distance between two consecutive molecules of a gas at normal pressure and temperature is greater than 1/(6 x 10^)th of an inch, and is less than l/10''th of an inch ; while the actual size of the molecule is a trifle gTeater than 1/(3 x 102'*)th of a cubic inch ; and the number of molecules in a cubic inch is about 3 X 10-".
Thus it would seem that a cubic inch of gas at ordinary
CH. XXl] MATTER AND ETHER THEORIES 479
pressure and temperature contains about 3 x 10^° molecules, all similar and equal, and each molecule has a volume of about 1/(3 X 10-^)th of a cubic inch ; while a cubic inch of the simplest solid or liquid contains rather less than 10-'' molecules, and perhaps each molecule has a volume of about 1/(3 x 10^)th of a cubic inch. For instance, if a pea or a drop of water whose radius is 1/1 6th 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 w^ould be about the same as that of a fives-ball. The average size of the minute drops of w^ater 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 about l/30000th of an inch. Such a drop therefore would contain about 2 x 10^^ separate molecules. In gases and vapours, the number of atoms required to make 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 is l/4000th of a millimetre 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 elementary molecules of the so-called elements are required to constitute one molecule of organic matter. At least half of every living organism consists of water, and we may for the moment suppose that the remainder consists of organic matter. Hence the smallest living being which is visible 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. Clerk Maxwell did not consider that this was sufficient to justify the current con- clusions of physiologists, and said that they must not suppose that structural details of infinitely small dimensions can furnish by themselves an explanation of the variety known to exist
480 MATTER AND ETHER THEORIES [CH. XXI
in the properties and functions of the most minute organisms ; hut physiologists have replied that whether their conjectures be right or wrong Clerk 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.
Throughout this chapter I have written as if the mass of a body were independent of whether it is or is not in motion relative to the hypothetical ether. This is assumed in the usual, or Newtonian, system of dynamics, but it has been recently called in question, notably by H. A. Lorentz, A. Einstein, and H. Minkowski.
The ultimate reason for this scepticism is the absence of any recognizable phenomena arising from the earth's motion relative to the ether : a question which was the subject of a series of experiments made in 1882 by A. A. Michelson and E. W. Morley. To account for this, Einstein propounded a theory of Relativity* in which he assumed certain relations between the measures of space and time employed by two observers who have a mutual relative velocity v. If the origin of coordinates be the same for both observers at the instant at which they both commence to reckon time, and if the axis of X be taken in the direction of v, he assumes that the relations between the coordinates of a point and the times T, t which have apparently elapsed at any subsequent instant are
X = ^{x-vt\ Y=y, Z = z, T = 13 {t - va;/c%
where X, Y, Z, T refer to the observations of the first observer, and X, y, z, t to those of the other ; c is the velocity of light ;
and /S = (1 — ■yY^O ~ • If ^ be negligible compared with c, these relations are the same as in the Newtonian system.
The theory leads to the result that moving bodies contract in the direction of their advance, and the greater the velocity the greater the contraction ; thus, since the earth rotates from
* For an account of the theory, see N. R. Campbell, Philosophical Magazine^ London, April, 1911, pp. 502—517.
CH. XXl] MATTER AND ETHER THEORIES 481
west to east, the bulk of a man walking eastwards will be somewhat smaller than his bulk when he walks westwards. Again, on this theory the mass of an electron may be taken to increase with its velocity, and it would become indefinitely great if its velocity were equal to that of light. At present the theory is beyond the range of direct verification, but it is not inconsistent with known facts, and possibly it may explain some phenomena connected with the motion of atoms and ions.
B. R. 31
483
INDEX.
Abbot, W., 255. Abbott, E. A., 424. Abel, N. H., on Quintics, 329. Achilles and the Tortoise, 84. Acts or Disputations, chap. xi. Agrippa. Cornelius, 138. Ahrens, W., 113, 139. Airy, Sir Geo., 108, 273. Aix, Labyrinth at, 186. Albohazen on Astrology, 381. Alcuin, 2, 71. Alfred the Great, 454. Algebraic Equations, chap. xiv. Alkarismi on tt, 299. Alkborough, Labyrinth at, 186. Allman, G. J., 308. Alphabet, Morse, 416. Amicable Numbers, 38.
AXALLAGMATIC ARRANGEMENTS, 65-66.
Anaxagoras, 297.
Anderson, T., 419.
Angle-Sum Theorem, chap. xiii.
Angular Motion, 86.
Anstice, E. K., 194.
Antipho, 297.
Apollonius, 286, 289, 298.
Ai-chimedes, 90, 293, 297, 299, 301.
Archytas on Delian Problem, 287.
Argyie, Earl of, 402.
Arithmetic, Higher, 36-43.
Arithmetical Fallacies, 28-31.
Arithmetical Puzzles, 4-33.
— Recreations, chap. i.
Arya-Bhata on tt, 298.
Asenby, Labyrinth at, 186.
Astrological Planets, 138, 384, 442.
Astrology, chap. xvii.
Atomic Theories, chap. xxi.
Atoms, Size of, 478.
Attraction, Law of, 470-474.
Augustine, St, on Astrology, 389.
Augustus, 418, 444.
Ayrton, W. E., on Magic Mirrors, 108.
Babbage, C, 271.
Baby's-Cot, String Figure, 354.
Bachet's Prohlemcs, 2-25, 34-30, 71,
138, 142, 238, 240. Bacon, Francis, 418. Bailey, J. E., 395. Baker, H. F., 235. Ball, W. W. R., 248, 250, 285, 333,
430. Bardesan on Fate, 380. Barrow, I., 249. Bats, String Figure, 366. Baudhayana on ir, 298. Bazeries, 414. Beautfort Cipher, 411. Becquerel Bays, 467, 468. Bedwell, T., 249. Beltrami, E., 325, 424. Benham, C. E., 108. Bennett, G. T., 118. Bentley, R., Newton to, 470. Bergholt, E. G. B., 33. Bernoulli, John, 29. Berosus, 450. Berri, Duchesse de, 413. Bertrand on Parallel Postulate, 315. Bertrand, J. L. F., 28. Bertrand, L. (of Geneva), 124. Besant on Hauksbee's Law, 101. Bewley, E. D., 33. Bezout, E., on Parallels, 322. Bhaskara on tt, 299. Bickmore, C. E., 340, 341, 347. Biering, C. H., 285. Bilguer, von, on Chess Pieces, 112. Billingsley, H., 249. Bills, S., on Kirkman's Problem, 223. Binary Povveus, Fermat on, 39-40. Biquadrate Theorem, 319. Birch, J. G., 343. Birds, Flight of, 106-107. Bishop's Re-entrant Path, 134. Bjerknes, 466. Blackburn, J., 264. Blunderville, T., 249. Board of Studies, 277-279. Boat-racing with a Rope, 101. Boddicker, 0., on Knots, 379.
484
INDEX
Bolyai, J., on Hyper-Space, 324, 424.
Bolyai, W., on Hyper-Space, 324.
Bonnycastle, J., 268.
Bonnycastle, J., on Parallels, 322.
Bonola, E., on Geometry, 323.
Bordered Magic Squares, 152-154.
Boscovich, K., 322, 461, 474, 476.
Boughton Green, Labyrinth at, 186.
Bouniakowski, V., on Shuffling, 235.
Bourget, M. J., on Shuffling, 235.
Bourlet, 22, 23.
Boussinesq on Ether, 462.
Brackets, in Tripos, 260, 267, 276, 277.
Brahmagupta on tt, 299.
Bray, A. , on Kirkman's Prob. , 194, 203.
Breton on Mosaics, 185.
Brewster. Sir David, 394.
Bkidge Problem, 221.
Briggs, H., 249.
Bristed, C. A., 271.
Bromton, 184.
Brouncker, Viscount, on tt, 302.
Brown, J. (Saint), 269.
Bryan on Bird Flight, 106, 107.
Bryso, 297.
Bubble Theory of Matter, 469.
Buckley, W., 249.
Butcher, S., on the Calendar, 449.
Butterfly, String Figure, 370.
Button-Hole, String Trick, 377.
Caesar, Julius, 388, 418, 443. Calendar, the Civil, 443-445.
— the Ecclesiastical, 445-448.
— the Gregorian, 445-448.
— the Julian, 443-444. Calendars, University, 265. Callet, J. F., on tt, 304. Cambridge lyiATHEMATics, chap. xi.
— Studies at, chap. xi. Campbell, N. E., 480.
Candles, Pound-of-, String Figure, 354.
Cantor, M., on tt, 296.
Cardan, G., 2. 229, 232, 380, 391-
393 395. Cards! Problems with, 16-18, 32, 33,
71, 235-246. Carnot, L. N. M., 321. Caroline Catch, String Trick, 372. Cartwright, W., 419. Cat's-Cradles, chap. xvi. Cat's-Eye, String Figure, 352. Cauchy, 61, 462, 478. Cayley, A., 55, 59.188,245, 302, 317, 435.
Cellini, 394.
Cells of a Chess-board, 122. Centrifugal Force, 89. Ceulen, van, on tt, 301, 302. Challis MSS., 261. Charles I, 419, 430, 449.
/
Charles V of Germany, 455. Chartres, Labyrinth at, 186. Chartres, E., 28, 51, 306. Chasles on Trisection of Angle, 293. Chaucer on the Sun-cylinder, 450. Cheke, Sir John, 392. Chess-board, Games on, 74-80, chap. vi.
— Problems, 74-80, chap. vi.
— knights' moves on. 111. Chess-board, Notation of, 109. Chess-board Eecreations, chap. vi. Chess, Maximum Pieces Prob., 119. Chess, Minimum Pieces Prob,, 119. Chess, Number of Initial Moves, 110. Chess Pieces, Value of, 110-113. Chifu-Chemulpo Puzzle, 70, 82. Chilcombe, Labyrinth at, 186. Chinese on tt, 299.
Chinese Eings, 229-234. Church Window, String Figure, 352. Ciccolini, T., on Chess, 129. Cicero on Astrology, 388. Ciphers, chap, xviii.
— Definition of, 396.
— Four types of, 403-413. Circle, Quadrature of, 293-306. Cissoid, the, 287, 289, 295. Clairaut on Trisection of Angle, 292. Clairaut, A. C, 321.
Clarke, S., 249.
Classical Tripos, 275.
Claus, 228.
Clausen on v, 304.
Clavius on Calendar, 444, 445, 446, 449.
Clavius, C, 321, 322.
Clepsydras, 453.
Clerk Maxwell, J., 59, 108, 459, 466,
471, 473, 475, 476, 479. Clerke, G., 249. Clifford, 87.
Climbing a Tree, String Figure, 359. Clocks, 96, 453-456. Cnossus, Coins of, 184, 185. Coat and Waistcoat Trick, 378. Coccoz, 46, 163. Code-Book Ciphers, 421. Code, Morse, 416.
Cole, F. N., 334, 336, 339, 342, 344. Colebrooke, H. T. , Indian Algebra, 299. Collini on Chess, 128. Collins, Letter from J. Gregory, 303. Colour-cube Problem, 67-69. Colouring Maps, 54-59. Columbus, 449. Columbus's Egg Puzzle, 93. Comberton, Labyrinth at, 186. Compasses, Watches as, 457-458. Competition, in Tripos, 267, 270, 271. Composite Magic Squares, 152. Conchoid, the, 287, 291, 295.
INDEX
485
Cones moving uphill, 93.
Congruent Figures, 484.
Conrad's Tables, 405.
Continuity of Matter, 4G0.
Contour-lines, 59-60.
Cotes, R., 249.
Counters, Games with, 62-64, 74-80.
Cox, James, on Clocks, 96.
Cradle, String Figure, 351.
Craig, J., 249.
Crassus, 388,
Cretan Labyrinth, 182, 184-185.
Cricket-ball, Spin on, 105.
Cross-Fours, 67.
Cryptographs, chap, xviii.
— Definition of, 396.
— Three types of, 397-403. Cryptograpuy, chap, xviii. Cube, Duplication of, 285-291. Cubes, Coloured, 67-69.
— Skeleton, 32.
Cubic Equation, Solution of, 328. Cudworth, W., on Sharp, 303. Cumberland, R., 254-255. Cumulative Vote, 33. Cunningham, A. J. C, 40, 334, 336,
339, 340, 341, 342, 344, 345. Cureton, W., on Astrology, 380. Curie on Eadio-Activity, 467. Curiosa Physica, 107-108. Curl on a Cricket-ball, 105. Cursor, Papirius, 450. Cusa on w, 300. Cusps, Astrological, 382. Cut on a Tennis-ball, 103-105. Cutting Cards, Problems on, 17. Cylinders, Sun-, 450.
D acres. A., 249.
D'Alembert, 28, 30.
D'Alembert, J., 323.
Dase on tt, 304, 305.
Dasypodius, 455.
Davis, E. P., on Kirkman'sProb., 218.
Day, Def. of, 440.
— Commencement of, 442.
— Sidereal and Solar, 441,
Days of Week, Names of, 442, 443. Days of Week from Date, 449. Dealtry, W,, 268, 269, De Berri, Duchesse, 414. Decimation, 24-27. Dee, J., 249.
De Fonteney on Ferry Problem, 72. De Fouqui^res, 63. De Haan, B., on tt, 296, 301. Dehn, M,, 436. De Lagny on tt, 303. De la Hire on Magic Squares, 138, 139, 142-144, 149-152, 155, 156.
De la Loubere on Magic Squares, 140-
142, 157, Delambre on Calendar, 448, 449. Delannoy, 72, De la Pryme, 252. Delastelle, F., 395. Delbceuf, J., on Parallels, 321. Delian Problem, 285-291. De Longchamps, G., 345. De Moivre, A., 122, 123. De Montmort, 1, 123. De Morgan, A., 55, 84, 247, 251, 268,
269, 293, 295, 296, 306, 320, 391, 446. Denary Scale of Notation, 10-11. De Parville on Tower of Hanoi, 229. De Polignac on Knight's Move, 133. De Rohan, 421.
Derrington, on Queens' Problem, 118. De St Laurent, 235. Descartes, 290, 292, 303. Des Ourmes, 138, 139. Diabolic Magic Squares, 156-162. Dials, Sun-, 450-452. Diamonds, String Figure, 361-363. Dickson, L. E,, 42, 244. Diego Palomino, 23. Digby, Lord, 419. Digges, T., 249.
Diodes on Delian Problem, 289. Diodorus on Lake Moeris, 184. Dircks, H., 421.
Dircks, H., on Pei-petual Motion, 94. Dirichlet, Lejeune, 42. Dissection, Proofs by, 52-54. Dodecahedron Game, 189-192. Dodgson, C. L., on Parallels, 45, 321. Dominical Letter, 448, Dominoes, 22-23, 168-169. Dominoes, Arrangements of, 178-181. D'Ons-en-bray, 138, 139. Door, Apache, String Figure, 358. Double-Crowns, String Figure, 355. Doubly Magic Squares, 163. Douglas, S., 270, 272. Drayton, 184. Dudeney, H. E., 26, 33, 47, 119, 168,
194, 203. Duplication of Cube, 285-291. Durations, see Time. Diirer, A., 138, 321, 322. Dynamical Games, 69-80.
Earnshaw, S., 108, 276. Easter, Date of, 445-449. Eckenstein, 0.,onKirkman's Problem,
193, 199, 203, 209, 217, 220. Edward VI, 383, 391-394. — Horoscope of, 393, Eight Queens Problem, 113-118. Einstein, A., 480.
3i— 3
486
INDEX
Eisenlohr, A., on Ahmes, 297. Eisenstein, 40. Electrons, 465-467. Elliptic Geometries, 324-326. Elliptic Geometry, 433-436. Elusive Loop, String Trick, 376. Enestrdm, G., on tt, 296. Engel, F., on Parallels, 307. Epicm'us on Gravitation, 471. Equilibrium, Puzzles on, 90-93. Eratosthenes, 287. Escott, E. B., 305, 346. Ether-Squirts, 465. Ether Theories, 462-466. Etten, van, 11.
Euclid, 38, 44-45, 297, 310, 321.
Euclid on Parallels, chap, xni, 433.
Euclid's Axioms, &c., 433.
Euc. I. 32, 52.
Euc. I. 47, 52.
Euclidean Geometry, 325, 433-435.
Euclidean Space, 326, 433-436.
Eudemus, 310.
Euler, 38-41, 61, 122-127, 139, 156, 166, 303, 335, 336, 337.
Euler's Unicursal Prob., 170-182.
Examination, Printed, 262, 274.
Exploration IProblems, 23.
Fairfax, 419.
Fallacies, Arithimetical, 28-31.
— Geometrical, 44-52.
_ Mechanical, 84-87, 93-GS.
Faraday on Matter, 461.
Fauquembergue, E., 339.
Fenn, J., 255.
Fermat, P., 36-43, 138, 139, 334, 335,
336, 337, 346. Fermat on Binary Powers, 39-40. Fermat's Last Theorem, 40-43. Ferry-boat Problems, 71-73. Fifteen Girls Problem, chap. ix. Fifteen Puzzle, 224-228. Figulus on Astrology, 388. Firmicus on Astrology, 381. Firth, W., 145.
Fish-in-a-Dish, String Figure, 353. Fish-Pond, String Figure, 352. Fitzpatrick, J., 75. Flamsteed, J., 249.' Flamsteed on Astrology, 390. Flat-land, 426-431. Fluid Motion, 101-107. Fluxions, 268, 271, 272. Fly-on-the-Nose, String Trick, 375. Fonteney on Ferry Problem, 72. Force, Definition of, 87. Foster, S., 249.
Fouqui^res, Becq de, on Games, 63. Four- Colour Map Theorem, 54-59.
Four "3's" Problem, 14. Four "4's" Problem, 14. Four "9's" Problem, 14. Four Digits Problem, 13. Fours, 1?roblem of, 14. Fox, Captain, on tt, 306. Frankenstein, G., 163. Franklin, B., 468. Frederick II of Germany, 454. Frenicle, 138, 139, 152. Frere, J., 256. Fresnel on Ether, 462. Friedlein, G., 310, 313. Frost, A. H., 156, 194.
Galileo on Pendulum, 455.
Galois, E., on Quiutic Equation, 329.
Galton, 28.
Games, Dynamical, 69-80.
— Statical, 62-69.
— with Counters, 74-80. Gases, Theory of, 478-479. Gauss, K. F., 43, 323, 341. Geminus, 308.
Geodesic Problems, 73-74. Geography, Physical, 59-61. Geojietrical Fallacies, 44-52. Geoseetrical Problems, Three Clas- sical, chap. XII. Geometrical Eecreations, chaps.
III-IV.
Geometry, Non-Euclidean, 323-328,
chap. XIX. George I of England, 253. Gerard, M. L., 435. Gerbert, 298.
Gergonne's Problem, 240-244. Germain, S., 42.
Gill, T. H., on Kirkman's Prob,, 194. Glaisher, J. W. L., 114, 248, 296. Glamorgan, Earl of, 420. Gnomons, 450. Goldbach's Theorem, 39. Golden Number, 448. Golf-balls, Flight of, 105. Gooch, W., 263.
Gravity, Hypotheses on, 470-474. Great Northern Puzzle, 69, 82. Green on Ether, 462. Greenwich, Labyrinth at, 186. Gregorian Calendar, 444, 445. Gregory XIII, 444-446. Gregory, Jas.. 294, 302. Gregory of St Vincent, 290. Gregory's Series, 303. Grienberger on tt, 302. Grille, The, 401.
Grimthoi^pe, Lord, on Clocks, 454. Gronfeld's Method in Ciphers, 409. Gros, L., on Chinese Eings, 232, 234.
INDEX
487
GuAKiNi's Problem, 135-136. Gun, Report of, 108. Gunning, H., 262, 264. Giinther, S., 113, 139. Guthrie on Colouring Maps, 54.
Haan, B. de, on tt, 296, 301. Haddon, A. C, on String Figures,
348, 349, 360, 365, 367, 368, 369,
372, 373. Haddon, K., on Cat's Cradles, 348. Halley on tt, 303. Halter, String Trick, 374. Hamilton, Archbishop, 391. Hamilton, Sir Wm., 189-192. Hamiltokian Game, 189-192, Hammock, String Figure, 354, Hampton Court, Maze at, 182, 186. Handcuffs, String Trick, 375. Hanoi, Tower of, 228-229. Harris on Pendulum Clock, 455. Harvey, J., 249. Harvey, E., 249. Harzer, P., on tt, 299. Hauksbee's Law, 101-106. Hayward, Sir J., 405. Head-Hunters, String Figure, 3GS. Heawood, P. J., on Maps, 56. Hegesippus on Decimation, 24. Hele, P., 455. Helmholtz, H. L. F. von, 97, 424,
462, 463. Henry VHI of England, 454. Henry, Ch., on Euler's Problem, 170. Hei-mann, A., 65. Hermary, 192.
Herodotus on Lake Moeris, 184. Hero of Alexandria on v, 287, 298. Herschel, Sir John, 271, 444. Hezekiah, 451.
Hicks, W. M., on Matter, 464, 466. Hiero of Syracuse, 90. Higher Arithmetic, 36-43. Hilbert, D., 435, 436. Hill, M. J. M., 464. Hill, T., 249. Hills and Dales, 59-60. Hinton, C. H., 424, 428, 430. Hipparchus on Hours of Day, 442. Hippias, 297.
Hippocrates of Chios, 287, 297. Hodson, W., 261. Homaloidal Geometries, 325-320. Honorary Optimes, 251, 254, 275. Hood, T., 249. Hooke on Timepieces, 455, Horary Astrology, 381. Hornbuckle, T. W., 269, 270. Horoscopes, chap. xvu. — Example of, 393.
Horoscopes, Rules to cast, 381-333. — Rules to read, 383-387. Horrox, J., 249. Hour-glasses, 453. Hours, Def. of, 440, 442. Houses, Astrological, 381, 382. Huddling, 250. Hudson, C. T., 244. Hudson, W. H. H., 236. Hustler, J. D., 268, 269. Hutton, C, 3, 303. Huygens, 289, 293, 302, 455. Hyperbolic Geometries, 323-326. Hyperbolic Geometry, 433-435. Hyper-magic Squares, 156-163. Hyper- Space, chap. xix.
IcosiAN Game, 189-192.
Ideler, J. L., on the Calendar, 449.
Inertia, 88, 89.
Inwards on the Cretan Maze, 184.
Isaiah, 451.
Jacob, E., 269, 270.
Jacobi, 341, 345.
Jaenisch, C. F. de, 120, 122, 128, 132.
James II of England, 402.
Japanese Magic Mirrors, 108.
Jayne, C. F., on String Figures, 348, 359, 361, 362, 363, 365, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376.
Jebb, J., 255, 257-2-59.
Johnson, W., on Fifteen Puzzle, 224,
Jones, W., on ir, 296, 303.
Josephus Problem, 23-27,
Julian Calendar, 444.
Julian's Bowers, 186.
Julius Caesar, 388, 415, 443.
Junior Optimes, 251-252, 255.
Jurin, J., 249.
Kelvin, Lord, 461, 462, 464, 471, 473,
477, 478. Kempe, A. B., on Colouring Maps, 56. Ketteler on Ether, 462. Killing, W., on Parallels, 322. Kinetic Theory of Gases, 478-479. King's Re-entrant Path, 133. Kirchhoff on Ether, 462. Kirkman, T. P., 193, 222. Kirkman's Problem, chap. ix. Klein, F. C, 284, 325, 426, 435. Kliiber, J. L., 395. Knight, Re-entrant Path, 122-132. Knights of the Round Table, 33. Knots, 379, 426. Knyghton, 184. Konigsberg Problem, 170-183. Kummer on Format's Theorem, 42.
488
INDEX
Labile Ether, 462.
Labosne on Magic Squares, 149.
Labyrinths, 182-187.
Lacroix, P. L., 293.
Lacroix, S. F., 315, 322.
Lagny on tt, 303.
Lagrange, J. L., 271, 320, 327, 337.
Lagrange's Theorem, 39.
La Hire, 138, 139, 142-144, 149-152.
Laisant, C. A., 12, 347.
La Loub^re, 140-142.
Lambert on tt, 293, 294.
Lambert, J. H., 322.
Lam6, 42, 462.
Landry, F., 334, 336, 338, 339, 342.
Langley on Bird Flight, 106.
Laplace on Velocity of Sound, 461.
Laplace, P. S., 271, 321.
Laqui^re on Knight's Path, 131.
Larmor, J., on Electrons, 459, 465.
Latruaumont, 421.
Lattice Work, String Figure, 355.
Laughton, R., 249.
Lawrence, F. W., 37, 347.
Lax, W., 263.
Lea, W., on Kirkman's Problem, 223.
Leake, 11, 14, 18, 22.
Leap-year, 443-445.
Lebesgue on Fermat's Theorem, 42.
Le Bon, G., on Matter, 468.
Legendre, A. M., 42, 124, 132, 271,
293, 294, 318, 319, 320, 321, 335,
337, 340, 341. Legros, L. A., 194, 195. Leibnitz on Games, 1. Lejeune Dirichlet on Fermat, 42. Le Lasseur, 334, 336, 338, 339. Leonardo of Pisa on tt, 300. Le Sage on Gravity, 472, 473. Leslie, J., 287, 291, 319. Leurechon, J., 2, 11. Lie, S., 435.
Lightning, String Figure, 369. Lilius on the Calendar, 444, 445. Lilly, W., on Astrology, 390. Linde, A. van der, 122. Lindemann on tt, 294. Line-land, 426. Lines of Slope, 60. Lippeus, 455.
Listing, J. B., 81, 172, 379. Liveing on the Spectrum Top, 108. Lizard Twist, String Trick, 372. Lobatschewsky, N. I., 324, 325, 424. Locke, J., 260, 262. Lommel on Ether, 462. London and Wise, 186. Longchamps, G. de, 346. Loop Trick, String Trick, 378. Lorentz, F., 321.
Lorentz, H. A., 462, 480.
Loschmidt on Molecules, 477, 479.
Loubere, de la, 140-142.
Louis XI of France, 389-390.
Louis XIV of France, 140.
Loyd, S., 19.
Lucas, E., 34, 67, 72, 77, 78, 80, 170,
178, 183, 218, 228, 232, 338-340. Lucas di Burgo, 2. Lucca, Labyrinth at, 186. Ludlam, W., 321. Lydgate on the Sun-cylinder, 450.
McClintock, E., 156.
MacCullagh on Ether, 462, 465.
Machin's Series for tt, 303, 304, 305.
Maclaurin on Newton, 472.
MacMahon, 35-36, 67.
Magic Bottles, 90, 91.
Magic Mirrors, 108.
Magic Pencils, 163-165.
Magic Squares, chap. vn.
Magic Square Puzzles, 166-169.
Magic Stars, 154-155.
Magnus on Hauksbee's Law, 103.
Manger, String Figure, 354.
Map Colour Theorem, 54-59.
Marie Antoinette, 421.
Marsden, E., on Kirkman's Prob., 194.
Mathematics, Cambridge, chap. xi.
Mathews, G. B., 344.
Matter, Constitution of, chap. xxi.
— Hypotheses on, 460-470.
— Kinds of, limited, 475-477.
— Size of Molecules, 477-480. Maxim on Bird Flight, 106. Maxwell, J. Clerk, 59, 108, 459, 466,
471, 473, 475, 476, 479. Maxwell's Demon, 108. Mazes, 182-187. Mean Time, 441, 442. Mechanical Recreations, chap. v. Medieval Problems, 18-25. Menaechmus, 290. Menage Problem, 34. Mendelejev, D. L, 468, 469, 477. Mersenne on Primes, 37. Mersenne's Numbers, chap, xv, 37—
38, 333, 334, 335. Mesolabum, 287. Metius, A., on tt, 300. Meton, 450. Meziriac, see Bachet. Michelson-Morley Experiments, 480. Milner, L, 261, 269. Minding on Knight's Path, 132. Minos, 182, 286. Minotaur, 184. Minskowski, H., 480. Minutes, Def. of, 440, 441.
INDEX
489
Mirrors, Magic, 108. Miscellaneous Problems, 224. Models, 97-98. Moderators, chap. xi. Mohammed's Sign-Manual, 176. Moivre, A. de, 122, 123. Molecules, Size of, 477-480. Money, Question on, 9-10, Monge on Shuffling Cards, 235-237. Months, 443. Montmort, de, 1, 123. Montucla, 3, 90, 91, 123, 139, 149,
151, 293, 294, 302. Moon, E., 132. Moore, E. H., 221. Morcom, R. K., 21. Morehead, J. C, 40. Morgan, A. de, $ee De Morgan. Morland, S., 249. Morley on Cardan, 392. Morse Code, 418-9. Mosaic Pavements, 64, 185. Moschopulus, 138, 142. Motion in Fluids, 101-107. Motion, Laws of, 83, 87-93.
— Paradoxes on, 84-87.
— Perpetual, 93-96. Mousetrap, Game of, 245-24G. Mouse Trick, 374.
Movements A, B, and T, in String
Figures, 357, 358. Miiller (Eegiomontanus), 300. Mullinger, J. B., 282. Mydorge, 2.
Nasik Maoic Squares, 156-162.
Natal Astrology, 381.
Nauclc, F., 113.
Neale, C. M., 254.
Needle Threading, String Trick, 373.
Neumann on Ether, 462.
Newton, Isaac, 94, 103, 249, 268, 269,
270, 290, 292, 294, 295, 461, 468,
470, 471, 472. Newtonian Laws of Motion, 83-93. Nicene Council on Easter, 446. Niceron, J. F., 395. Nicomedes, 287. Nigidius on Astrology, 388. Non-Archimedian Geometry, 436. Non-Euclidean Geometries, 433-435.
NON-EUCLIDEAN GEOMETRY, 312, 322-
326. Nonez on Sun-dials, 450, Non-Legendrian Geometry, 436. Notation, Denary Scale of, 10-11. Noughts and Crosses, 62. Numa on the Year, 443. Numbers, Perfect, 334.
— Puzzles with, 4-27.
Numbers, Theory of, 36-43.
Oliver, General, on Sun-dials, 452.
Ons-en-bray, 138, 139.
Oppert on tt, 297.
Optimes, ch. xi, 251-252, 255, 275.
Oram on Eight Queens, 117.
Oughtred, W., 249.
Oughtred's Recreations, 11, 14, 18,
22, 91, 92. Ourmes on Magic Squares, 138, 139. Ovid, 184.
Owls, String Figure, 365. Ozanam, A. F., on Labyrinths, 185. Ozanam's Recreations, 2, 3, 11, 18,
25, 54, 71, 90, 91, 92, 93, 96, 98,
123, 138, 139, 149, 166, 229, 450,
452, 456.
TT, 293-306; see Table of Contents.
Pacificus on Clocks, 454.
Pacioli di Burgo, 2.
Pairs-of-Cards Trick, 238-240.
Paley, W., 256, 262, 264, 265, 269, 274.
Palomino, 23.
Pandiagonal Magic Squares, 156-162.
Pappus, 287, 291, 292.
Parabolic Geometries, 325.
Parabolic Geometry, 433, 436.
Paradromic Rings, 80-81.
Parallel Postulate, chap. xm.
Parallels, Definitions of, 322, 323.
— Theory of, 433.
Parkinson, J., on String Figures, 368.
Parmentier, on Knight's Path, 122.
Parrot Cage, String Figure, 368.
Parry on Sound, 108.
Parville, de, 229.
Pascal on Angle-Sum Theorem, 308.
Pawns, Games with, 74-80.
Paynell, N., 249.
Pearson, K., on Ether-Squirts, 465.
Pein on Ten Queens, 118.
Peirce, B., on Kirkman's Problem, 194.
Peirce's Problem of n^ Girls, 219.
Pencils, Magic, 163-165.
Pepys, S., 420.
Perfect Magic Squares, 156-102.
Perfect Numbers, 38, 334.
Permutation Problems, 32.
Perpetual Motion, 93-96.
Perrin, 12.
Perry, J., on Magic Mirrors, 108.
Peterson on Maps, 57.
Peyrard, F., 310.
Philo, 287.
Philoponus on Dclian Problem, 285.
I'hysical Geography, 59-61.
Pile Problems, 240-245.
Piuetti, 378.
490
INDEX
Pirie, G., on tt, 302.
Pitatus on the Calendar, 445.
Pittenger, 89.
Plana, G. A. A., 334, 336, 338, 342.
Planck C. 145.
Planets (Astrological), 138, 384, 442.
— Signification of, 384-386. Plato on Delian Problem, 285, 286. Play fair Cipher, 411.
Playfair, J., 307, 316, 317, 320, 322.
Pliny, 184, 388.
Pocock, W. I., 348, 373, 377.
Poe, E. A., 395, 405.
Poignard, 138, 139.
Poincare, J. H., 437.
Poitiers, Labyrinth at, 186.
Polignac on Knight's Path, 133.
Poll Examinations, 275.
Poll-men, 252.
Pollock, Sir F., 268-270.
Pompey, 388.
Porta, G., 395.
Portier, B., on Magic Squares, 163.
Pound-of-Candles, String Figure, 354.
Powers, K. E., 334, 336, 340.
Pratt on Knight's Path, 128.
Pretender, The Young, 402.
Primes, 37.
Probabilities and tt, 305.
Probabilities, Fallacies in, 30-32, 52.
Problem Papers, 261, 262, 265.
Proclus, 310, 313, 314.
Ptolemy, 298, 380, 381, 442.
Ptolemy on Parallel Postulate, 313.
Purbach on tt, 300.
Puzzles, Arithmetical, 4-36.
— Geometrical, 62-81.
— Mechanical, 84-93. Pythagorean Symbol, 176. Pythagoreans on Angle-Sum Theorem,
310.
Quadratic Equation, Solution of, 328. Quadrature of Circle, 293-306. Quartic Equation, Solution of, 328. Queen, Paths on Chess-board, 133,
134, 135. Queens Problem, Eight, 113-118. Queens, Problems with, 113-118. QuiNTic Equations, Algebraic, ch. xiv.
Racquet-ball, Cut on, 103-105.
Railway Puzzles (Shunting), 69-71.
Ramesam, 339.
Ramification, 188.
Raphael on Astrology, chap. xvii.
Ravenna, Labyrinth at, 186.
Rayleigh, Lord, 103, 105, 106, 257,
462. Record, R., 249. ,
Re-entrant Paths on Chess-board,
122-134. Regiomontanus on tt, 300. Reimer, N. T., 285. Reiss, 80.
Reiss on Dominoes, 181. Relative Motion, 87. Relativity, Theory of, 480. Reneu, W., 252. Renton, W., 52. Resolvants, 327. Reuschle, C. G., 334, 336, 333. Reversible Magic Squares, 167. Reynolds, O., 469, 470. Rhind Papyrus, 297. Riccioli on the Calendar, 445. Rich, J., 419. Richard, J., 87, 307, 311. Richards, W. H., 457. Richelieu, 401. Richter on tt, 304. Riemann, G. F. B., 324, 325, 424. Rigaud, S. P., 471. Rilly, A., 163. Ring-Dial, 452, 453. Rivers, W. H. R., on String Figures,
349, 367, 368, 372, 373. Rockliff Marshes, Lalsyrinth at, 186. Rodet, L., on Arya-Bhata, 298. Rodwell, G. F., on Hyper-Space, 424. Roget, P. M., 122, 127-132. Romanus on tt, 300. Rome, Labyrinth at, 186. Rontgen Rays, 467. Rook, Re-entrant Path, 133-134. Rooke, L., 249. Rosamund's Bower, 184. Rosen, F., on Arab values of tt, 299. Rothschild, F., 389. Round Table, Knights of, 33. Route Method in Ciphers, 399. Routes on Chess-board, 122-135. Row, Counters in a, 62-64, 74-78. Rudio, F., on tt, 293. Ruffini, P., on Algebraic Quintic, 329. Russell, B. A. W., 85. Rutherford on tt, 304.
Saccheri, J,, 424. Saccheri, J., on Parallels, 323. Saffron Walden, Labyrinth at, 186. Sailing, Theory of, 98-101. Sand-clocks, 453. Sarrau on Ether, 462. Saunderson, N., 249. Sauveur, J., 138, 139, 156. Scale of Notation, Denary, 10-11. — Puzzles dependent on, 11-14. Schlegel, V., 424. School-girls, Fifteen, 193-223.
INDEX
491
Schooling, J. 11., 10. Schotten, H., on Parallels, 307. Schubert, H., on tt, 293. Schumacher, 424. Scott, Sir Walter, 390. Scytale, The, 403. Seconds, Def. of, 440, 441. Secuet Commuxications, chap, xviii. Seelhoff, P. H. H., 334, 336, 339. See-saw, String Figure, 356, 369. Selander, K. E. I., on tt, 296. Senate-House Exajiination, chap. xi. Seneca on Astrology, 389. Senior Optimes, 251-252, 255. Setting-Sun, String Figure, 367. Seventy-seven Puzzle, 54. Shanks, W., on tt, 304. Sharp, A., on tt, 303. Shelton, T., 419. Sherwin's Tables, 303. Shuffling Cards, 235-237. Shunting Problems, 69-71, 82. Sidereal Time, 440, 441. Simon, M., on Parallels, 307. Simpson, R., on Parallels, 321. Simpson, T, , on Parallels, 321. Simpson's Euclid, 268. Simultaneity, 438. Sixteen Counter Problem, 64, 82. Sixty-five Puzzle, 52-53. Skeleton Cubes, 32. Smith, A., on tt, 306.
— Hen., on Numbers, 42.
— R., 249, 257.
— R. C, see Raphael. Snell on tt, 301, 302. Snuffer-Trays, String Figure, 852. Solar Time, 441-442.
Soldier's Bed, String Figure, 352. Solitaire, 80.
Somerv'ille, D. M. Y., 323, 424, 426. Sosigenes on Calendar, 444. Sound, Problem in, 107-108.
— Velocity of, 400. Southey on Astrology, 394. Southwark, Labyrinth at, 186. Sovereign, Change for, 32. Space, IProperties of, chap. xix. Spear, Throwing, String Figure, 360. Spectrum Analysis, 474, 475. Spectrum Top, 108.
Spin on a Cricket-ball, 105. Spirits, Raising, 394. Sporus on Delian Problem, 289. Sprague on Eleven Queens, 118. Squaring the Circle, 293-306. Stability of Equilibrium, 90-93. Stachel, P., on Parallels, 307. Stars, String Figures, 304, 366. Statical Games, 62-69.
St Cyr Method in Ciphers, 410.
Steen on the Mousetrap, 246.
Steiner'sCoinbinatorischeAufgabe,223.
St Laurent on Cards, 235.
StolBer on the Calendar, 445.
Stokes on Ether, 462.
St Omer, Labyrinth at, 186.
Storey on the Fifteen Puzzle, 224.
Strabo on Lake Moeris, 184.
String Figures, chap. xvi.
String Tricks, 371-378.
Stringham on Hypcr-Space, 432.
Sturm, A., 285.
St Vincent, Gregory of, 290.
Styles, 450.
Suetonius, 418.
Sun-cylinders, 450.
Sun-dials, 450-452.
Sun-rings, 452-453.
Sun-setting, String Figure, 367.
Sun, the Mean, 441.
Suspension Bridge, String Figure, 355.
Svastika, 185.
Swift, 84.
Sylvester, J. J., 63, 65, 222.
Tacitus on Astrology, 389.
Tait, P. G., 25, 56, 57, 58, 75, 172,
176, 379, 459, 463, 473. Tangrams, 69.
Tanner, L., on Shuffling Cards, 235. Tarry, G., 72, 163, 166, 177, 178. Tartaglia, 2, 18, 24, 34, 71. Tate, 417. Tavel, G. F., 270. Taylor, B., 123, 249. Taylor, Ch., on Trisection Prob., 292. Taylor, H. M., 110. Tennis-ball, Cut on, 103-105. Tesselation, 64-67. Thales on Angle-Sum Theorem, 308,
321. Theon of Alexandria, 310. Theory of Numbers, 36-43, chap. xv. Thibaut, G., on Baudhavana, 298. Thompson, T. P., on Parallels, 307. Thomson, J. J., 105, 459, 464, 467,
477. Thomson, Sir Wm., see Kelvin. Thrasyllus on Astrology, 389, 390. Threading Needle, String Trick, 373. Three-in-a-row, 62-64. Three-pile Problem, 240-245. Three-Things Problem, 19-23. Throwing Spear, String Figure, 300. Tiberius on Astrology, 389. Time, chap. xx.
— Equation of, 442.
— Measurement of, 438-441.
— Units of, 438-443.
492
INDEX
Todhunter, J., 278.
Tonstall, C, 249.
Tower or Hanoi, 228-229.
Trastevere, Labyrinth at, 186.
Tbebly Magic Squakes, 163.
Tree, Climbing, String Figure, 359.
Trees, Geometrical, 188.
Treize, Game of, 245-246.
Trellis-Bridge, String Figure, 355.
Tremaux on Mazes, 183.
Triangle, Sum of Angles of, ch. siu.
Tricks, String, 371-378.
Tricks with Numbers, 3-34.
Tridents, String Figure, 355.
Tripos, Mathematical, chap. xi.
Tripos, Origin of Term, 281-283.
Trisection of Angle, 291-293.
Tritheim, J., 395.
Trollope, E., on Mazes, 184.
Troy-towns, 186.
Turton, W. H., 49, 54, 121.
Uhlemann on Astrology, 380. Unicursal Problems, chap. viii.
Van Ceulen on tt, 301, 302.
Vandermonde, 80, 122, 127.
Van Etten, 11.
Varignon, P., on Parallels, 322.
Vase Problem, 18.
Vega on ir, 304.
Vick on Clocks, 455.
Vieta, 290, 300.
Vince, S., 268, 269.
Violle, B., Magic Squares, 139.
Virgil, 184.
Voigt on Ether, 462.
VolpiceUi, P., on Knight's Path, 122.
Von Bilguer on Chess Pieces, 112.
Von Helmholtz, H. F. L., 97, 424,
462, 463. Vortex Eings, 463, 464.
— Spheres, 464.
— Sponges, 464, 465. Voting, Question on, 33.
Waistcoat Puzzle, 378.
Walecki on Kirkman'sProb., 218, 219.
Walker, G. T., 28.
Wallis, J., 229, 232, 249, 302.
Wallis, J., on Parallels, 314, 320.
Wantzell, P. L., 284.
Ward, S., 249.
Waring, E., 255, 263, 269.
Warnsdorff, Knight's Path, 128.
Watch Problem, 14-15.
Watches, 96, 455.
— as Compasses, 456-458.
Water-clocks, 453, 456.
Waterloo, Battle of, 449.
Watersheds and Watercourses, 60-;j51.
Watson, G. N., vi.
Watson, R., 256, 265.
Waves, Superposition of, 108.
Weber- Wellstein, 339.
Week, Days of, from date, 449.
Week, Names of Days, 442-443.
Weights Problem, The, 34-36.
Western, A. E., on Binary Powers, 40.
Wheatstone, C, on Ciphers, 414, 419,
420. Whewell, W., 248, 269, 270, 272, 273,
380, 389. Whist, Number of Hands at, 33. Whiston, W., 249. Whitehead, A. N., 435. Whittaker, E. T., 459. Wiedemann, A., on Lake Moeris, 184. Wiles, J. P., 450. Wilkins, J., on Ciphers, 395, 403, 410,
412. William HI of England, 186. Willis on Hauksbee's Law, 101. Wilson, J,, on Ptolemy, 381. Wilson's Theorem, 269. Wing, Labyrinth at, 186. Withers, J. W., on Parallels, 307. Wolfe on Parallels, 322. Wood, J., 263, 268. Woodall, H. J., 334, 336, 339, 342. Woodhouse, R., 269, 271. Worcester, Marquis of, 420. Wordsworth, C, 248, 255, 263, 265,
283. Work, 89-93.
Wostrowitz, E. B. von, 395. Wranglers, 251-252, chap. xi. Wright, E., 249. Wright, J. M. F., 273.
Yam Thief, String Trick, 374. Year, Civil, 443-445. Year, Mohammedan, 445.
Zach, Baron, on tt, 304. Zech, R., 455. Zeller, C, 449. Zeno on Motion, 84-85. Zodiac Signs in Astrology, 383, 386- 387.
CAMBRIDGE; PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS
493 A SHORT ACCOUNT OF THE
HISTORY OF MATHEMATICS
By W. W. rouse BALL. [Fifth Edition, 1911. Pp. xxiv + 522. Price 10s. net]
MACMILLAN AND CO. Ltd., LONDON AND NEW YORK.
This book gives an account of the lives and discoveries of those mathematicians to whom the development of the subject is mainly due. The use of technicalities has been avoided and the work is intelligible to any one acquainted with the elements of mathe- matics.
It commences with an account of the origin and progress of Greek mathematics, from which the Alexandrian, the Indian, and the Arab schools may be said to have arisen. Next the mathematics of medieval Europe and the renaissance are described. The latter part of the book is devoted to the history of modern mathematics, beginning with the invention of analytical geometry and the in- finitesimal calculus. The history is brought down to the present time.
This excellent snmmary of the history of mathematics supplies a want wbicli has long been felt in this country. The extremely difficult question, how far such a work should be technical, has been solved with great tact. . . The work contams many valuable hints, and is thoroughly readable. The biographies, which include those of most of the men who jdayed important parts in the development of culture, are full and general enough to interest the ordinary reader as well as the specialist. Its value to the latter is much increased by the numerous references to authorities, a good table of contents, and a full and accurate index. — The Saturday Review.
Mr. Ball's book should meet with a hearty welcome, for though we possess other histories of special branches of mathematics, this is the first serious attempt that has been made in the English language to give a systematic account of the origin and development of the science as a whole. It is \\Titten too i)i an attractive style. Technicalities are not too numerous or obtnisive, and the work is inter.spersed with biographical sketches and
494
anecdotes likely to interest the general rearler. Thus the tyro and the advanced mathematician alike may read it with pleasure and profit. — The Athenceum.
A wealth of authorities, often far from accordant with each other, renders a work such as this extremely formidable ; and students of mathematics have reason to be grateful for the vast amount of information which has been condensed into this short account. ... In a survey of so wide extent it is of course impossible to give anything but a bare sketch of the various lines of research, and this circumstance tends to render a narrative scrappy. It says much for Mr. Ball's descriptive skill that his history reads more like a con- tinuous story than a series of merely consecutive summaries. — The Academy.
We can heartily recommend to our mathematical readers, and to others also, Mr. Ball's History of Mathematics. The history of what might be supposed a dry subject is told in the pleasantest and most readable style, and at the same time there is evidence of the most careful research. — The Observatory.
All the salient points of mathematical history are given, and many of the results of recent antiquarian research ; but it must not be imagined that the