scene 1.

=2142.= To add to golden numbers, golden numbers.--DECKER, THOMAS.

_Patient Grissell, Act 1, scene 1._

=2143.=

I’ve read that things inanimate have moved, And, as with living souls, have been inform’d, By magic numbers and persuasive sound. --CONGREVE, RICHARD.

_The Morning Bride, Act 1, scene 1._

=2144.= ... the Yancos on the Amazon, whose name for three is

Poettarrarorincoaroac,

of a length sufficiently formidable to justify the remark of La Condamine: Heureusement pour ceux qui ont à faire avec eux, leur Arithmetique ne va pas plus loin.--PEACOCK, GEORGE.

_Encyclopedia of Pure Mathematics (London, 1847); Article “Arithmetic,” sect. 32._

=2145.= There are three principal sins, avarice, luxury, and pride; three sorts of satisfaction for sin, fasting, almsgiving, and prayer; three persons offended by sin, God, the sinner himself, and his neighbour; three witnesses in heaven, _Pater_, _verbum_, and _spiritus sanctus_; three degrees of penitence, contrition, confession, and satisfaction, which Dante has represented as the three steps of the ladder that lead to purgatory, the first marble, the second black and rugged stone, and the third red porphyry. There are three sacred orders in the church militant, _sub-diaconati_, _diaconiti_, and _presbyterati_; there are three parts, not without mystery, of the most sacred body made by the priest in the mass; and three times he says _Agnus Dei_, and three times, _Sanctus_; and if we well consider all the devout acts of Christian worship, they are found in a ternary combination; if we wish rightly to partake of the holy communion, we must three times express our contrition, _Domine non sum dignus_; but who can say more of the ternary number in a shorter compass, than what the prophet says, _tu signaculum sanctae trinitatis_. There are three Furies in the infernal regions; three Fates, Atropos, Lachesis, and Clotho. There are three theological virtues: _Fides_, _spes_, and _charitas_. _Tria sunt pericula mundi: Equum currere; navigare, et sub tyranno vivere._ There are three enemies of the soul: the Devil, the world, and the flesh. There are three things which are of no esteem: the strength of a porter, the advice of a poor man, and the beauty of a beautiful woman. There are three vows of the Minorite Friars: poverty, obedience, and chastity. There are three terms in a continued proportion. There are three ways in which we may commit sin: _corde_, _ore_, _ope_. Three principal things in Paradise: glory, riches, and justice. There are three things which are especially displeasing to God: an avaricious rich man, a proud poor man, and a luxurious old man. And all things, in short, are founded in three; that is, in number, in weight, and in measure.

--PACIOLI, _Author of the first printed treatise on arithmetic._

_Quoted in Encyclopedia of Pure Mathematics (London, 1847); Article “Arithmetic,” sect. 90._

=2146.= Ah! why, ye Gods, should two and two make four?

--POPE, ALEXANDER.

_The Dunciad, Bk. 2, line 285._

=2147.=

By him who stampt _The Four_ upon the mind,-- _The Four_, the fount of nature’s endless stream. --_Ascribed to_ PYTHAGORAS.

_Quoted in Whewell’s History of the Inductive Sciences, Bk. 4, chap. 3._

=2148.=

Along the skiey arch the goddess trode, And sought Harmonia’s august abode; The universal plan, the mystic Four, Defines the figure of the palace floor. Solid and square the ancient fabric stands, Raised by the labors of unnumbered hands. --NONNUS.

_Dionysiac, 41, 275-280. [Whewell]._

=2149.= The number seventy-seven figures the abolition of all sins by baptism.... The number ten signifies justice and beatitude, resulting from the creature, which makes seven with the Trinity, which is three: therefore it is that God’s commandments are ten in number. The number eleven denotes sin, because it _transgresses_ ten.... This number seventy-seven is the product of eleven, figuring sin, multiplied by seven, and not by ten, for seven is the number of the creature. Three represents the soul, which is in some sort an image of Divinity; and four represents the body, on account of its four qualities....

--ST. AUGUSTINE.

_Sermon 41, art. 23._

=2150.= Heliodorus says that the Nile is nothing else than the year, founding his opinion on the fact that the numbers expressed by the letters Νειλος, Nile, are in Greek arithmetic, Ν = 50; Ε = 5; Ι = 10; Λ = 30; Ο = 70; Σ = 200; and these figures make up together 365, the number of days in the year.

_Littell’s Living Age, Vol. 117, p. 380._

=2151.= In treating 666, Bungus [Petri Bungi Bergomatis Numerorum mysteria, Bergamo, 1591] a good Catholic, could not compliment the Pope with it, but he fixes it on Martin Luther with a little forcing. If from A to I represent 1-9, from K to S 10-90, and from T to Z 100-500, we see--

M A R T I N L U T E R A 30 1 80 100 9 40 20 200 100 5 80 1

which gives 666. Again in Hebrew, _Lulter_ [Hebraized form of Luther] does the same:--

ר ת ל י ל 200 400 30 6 30 --DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 37._

=2152.= Stifel, the most acute and original of the early mathematicians of Germany, ... relates ... that whilst a monk at Esslingen in 1520, and when infected by the writings of Luther, he was reading in the library of his convent the 13th Chapter of _Revelations_, it struck his mind that the _Beast_ must signify the Pope, Leo X.; He then proceeded in pious hope to make the calculation of the sum of the numeral letters in _Leo decimus_, which he found to be M, D, C, L, V, I; the sum which these formed was too great by M, and too little by X; but he bethought him again, that he has seen the name written Leo X., and that there were ten letters in _Leo decimus_, from either of which he could obtain the deficient number, and by interpreting the M to mean _mysterium_, he found the number required, a discovery which gave him such unspeakable comfort, that he believed that his interpretation must have been an immediate inspiration of God.

--PEACOCK, GEORGE.

_Encyclopedia of Pure Mathematics (London, 1847); Article “Arithmetic,” sect. 89._

=2153.= Perhaps the best anagram ever made is that by Dr. Burney on Horatio Nelson, so happily transformed into the Latin sentence so truthful of the great admiral, _Honor est a Nilo_. Reading this, one is almost persuaded that the hit contained in it has a meaning provided by providence or fate.

This is also amusingly illustrated in the case of the Frenchman André Pujom, who, using j as i, found in his name the anagram, Pendu à Riom. Riom being the seat of justice for the province of Auvergne, the poor fellow, impelled by a sort of infatuation, actually committed a capital offence in that province, and was hanged at Riom, that the anagram might be fulfilled.

_New American Cyclopedia, Vol. 1; Article “Anagram.”_

=2154.= The most remarkable pseudonym [of transposed names adopted by authors] is the name of “Voltaire,” which the celebrated philosopher assumed instead of his family name, “François Marie Arouet,” and which is now generally allowed to be an anagram of “Arouet, l. j.,” that is, Arouet the younger.

_Encyclopedia Britannica, 11th Edition; Article “Anagram.”_

=2155.= Perhaps the most beautiful anagram that has ever been composed is by Jablonsky, a former rector of the school at Lissa. The occasion was the following: When while a young man king Stanislaus of Poland returned from a journey, the whole house of Lescinsky assembled to welcome the family heir. On this occasion Jablonsky arranged for a school program, the closing number of which consisted of a ballet by thirteen pupils impersonating youthful heroes. Each of them carried a shield on which appeared in gold one of the letters of the words _Domus Lescinia_. At the end of the first dance the children were so arranged that the letters on their shields spelled the words _Domus Lescinia_. At the end of the second dance they read: _ades incolumis_ (sound thou art here). After the third: _omnis es lucida_ (wholly brilliant art thou); after the fourth: _lucida sis omen_ (bright be the omen). Then: _mane sidus loci_ (remain our country’s star); and again: _sis columna Dei_ (be a column of God); and finally: _I! scande solium_ (Proceed, ascend the throne). This last was the more beautiful since it proved a true prophecy.

Even more artificial are the anagrams which transform one verse into another. Thus an Italian scholar beheld in a dream the line from Horace: _Grata superveniet, quae non sperabitur, hora_. This a friend changed to the anagram: _Est ventura Rhosina parataque nubere pigro._ This induced the scholar, though an old man, to marry an unknown lady by the name of Rosina.--HEIS, EDUARD.

_Algebraische Aufgaben (Köln, 1898), p. 331._

=2156.= The following verses read the same whether read forward or backward:--

Aspice! nam raro mittit timor arma, nec ipsa Si se mente reget, non tegeret Nemesis;[13]

also,

Sator Arepo tenet opera rotas. --HEIS, EDUARD.

_Algebraische Aufgaben (Köln, 1898), p. 328._

[13] The beginning of a poem which Johannes a Lasco wrote on the count Karl von Südermanland.

=2157.= There is a certain spiral of a peculiar form on which a point may have been approaching for centuries the center, and have nearly reached it, before we discover that its rate of approach is accelerated. The first thought of the observer, on seeing the acceleration, would be to say that it would reach the center sooner than he had before supposed. But as the point comes near the center it suddenly, although still moving under the same simple law as from the beginning, makes a very short turn upon its path and flies off rapidly almost in a straight line, out to an infinite distance. This illustrates that apparent breach of continuity which we sometimes find in a natural law; that apparently sudden change of character which we sometimes see in man.--HILL, THOMAS.

_Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 521._

=2158.= One of the most remarkable of Babbage’s illustrations of miracles has never had the consideration in the popular mind which it deserves; the illustration drawn from the existence of isolated points fulfilling the equation of a curve.... There are definitions of curves which describe not only the positions of every point in a certain curve, but also of one or more perfectly isolated points; and if we should attempt to get by induction the definition, from the observation of the points on the curve, we might fail altogether to include these isolated points; which, nevertheless, although standing alone, as miracles to the observer of the course of the points in the curve, are nevertheless rigorously included in the law of the curve.--HILL, THOMAS.

_Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 516._

=2159.= Pure mathematics is the magician’s real wand.--NOVALIS.

_Schriften, Zweiter Teil (Berlin, 1901), p. 223._

=2160.= Miracles, considered as antinatural facts, are amathematical, but there are no miracles in this sense, and those so called may be comprehended by means of mathematics, for to mathematics nothing is miraculous.--NOVALIS.

_Schriften, Zweiter Teil (Berlin, 1911), p. 222._

INDEX

=Black-faced numbers refer to authors=

Abbreviations:--m. = mathematics, math. = mathematical, math’n. = mathematician.

Abbott, =1001=.

Abstract method, Development of, 729.

Abstract nature of m., Reason for, 638.

Abstract reasoning, Objection to, 1941.

Abstractness, math., Compared with logical, 1304.

Adams, Henry, M. and history, =1599=. Math’ns practice freedom, =208=, =805=.

Adams, John, Method in m., =226=.

Aeneid, Euler’s knowledge of, 859.

Aeschylus. On number, 1606.

Aim in teaching m., 501-508, 517, 844.

Airy, Pythagorean theorem, =2126=.

Akenside, =1532=.

Alexander, 901, 902.

Algebra, Chapter XVII. Definitions of, 110, 1714, 1715. Problems in, 320, 530, 1738. Of use to grown men, 425. And geometry, 525-527, 1610, 1707. Advantages of, 1701, 1703, 1705. Laws of, 1708-1710. As an art, 1711. Review of, 1713. Designations of, 1717. Origin of, 1736. Burlesque on modern, 1741. Hume on, 1863.

Algebraic notation, value of, 1213, 1214.

Algebraic treatises, How to read, 601.

Amusements in m., 904, 905.

Anagrams, On De Morgan, 947. On Domus Lescinia, 2155. On Flamsteed, 968. On Macaulay, 996. On Nelson, 2153. On Newton, 1028. On Voltaire, 2154.

Analysis, Invigorates the faculty of resolution, 416. Relation of geometry to, 1931.

Analytical geometry, 1889, 1890, 1893. Method of, 310. Importance of, 949. Burlesque on, 2040.

Ancient geometry, Characteristics of, 712, 714. Compared with modern, 1711-1716. Method of, 1425, 1873-1875.

Ancients, M. among the, 321.

Anecdotes, Chapters, IX, X.

Anger, M. destroys predisposition to, 458.

Angling like m., 739.

Anglo-Danes, Aptitude for m., 836.

Anglo-Saxons, Aptitude for m., 837. Newton as representative of, 1014.

Anonymous, Song of the screw, 1894.

Appolonius, 712, 714.

Approximate m., Why not sufficient, 1518.

Aptitude for m., 509, 510, 520, 836-838, 976, 1617.

Arabic notation, 1614.

Arago, M. the enemy of scientific romances, =267=. Euler, “analysis incarnate,” =961=. Euler as a computer, =962=. On Kepler’s discovery, =982=. Newton’s efforts superhuman, =1006=. On probabilities, =1591=. Geometry as an instrument, 1868.

Arbuthnot, M. frees from prejudice, credulity and superstition, =449=. M. the friend of religion, =458=. M. compared to music, =1112=. On math, reasoning, =1503=.

Archimedes, His machines, 903, 904. Estimate of math, appliances, 904-906, 908. Wordsworth on, 906. Schiller on, 907. And engineering, 908. Death of, 909. His tomb, 910. Compared with Newton, 911. Character of his work, 912, 913. Applied m., 1312.

Architecture and m., 276.

Archytas, 904. And Plato, 1427.

Aristippus the Cyrenaic, 845.

Aristotle, 914. On relation of m. to esthetics, =318=.

Arithmetic, Chapter XVI. Definitions of, 106, 110, 1611, 1612, 1714. Emerson on advantage of study of, 408. Problems in, 528. A master-key, 1571. Based on concept of time, 1613. Method of teaching, 1618. Purpose of teaching, 454, 1624. As logic, 1624, 1625. The queen of m., 1642. Higher, 1755. Hume on, 1863.

Arithmetical theorems, 1639.

Art, M. as a fine, Chapter XI

Arts, M. and the, 1568-1570, 1573.

Astronomy and m., 1554, 1559, 1562-1567.

“Auge et impera.,” 631.

Authority in science, 1528.

Axioms, 518, 2015. In geometry, 1812, 2004, 2006. Def. in disguise, 2005. Euclid’s, 2007-2010, 2014. Nature of, 2012. Proofs of, 2013. And the idea of space, 2004.

Babbage, 923.

Bacon, Lord, Classification of m., =106=. M. makes men subtile, =248=. View of m., 316, 915, 916. M. held in high esteem by ancients, =321=. On the generalizing power of m., =327=. On the value of math, studies, =410=. M. develops concentration of mind, =411=. M. cures distraction of mind, =412=. M. essential to study of nature, =436=. His view of m., =915=, =916=. His knowledge of m., 917, 918. M. and logic, =1310=. Growth of m., =1511=.

Bacon, Roger, Neglect of m. works injury to all science, =310=. On the value of m., =1547=.

Bain, Importance of m. in education, =442=. On the charm of the study of m., =453=. M. and science teaching, =522=. Teaching of arithmetic, =1618=.

Ball, R. S., =2010=.

Ball, W. W. R., On Babbage, =923=. On Demoivre’s death, =944=. De Morgan and the actuary, =945=. Gauss as astronomer, =971=. Laplace’s “It is easy to see.” =986=. Lagrange, Laplace and Gauss contrasted, =993=. Newton’s interest in chemistry and theology, =1015=. On Newton’s method of work, =1026=. On Newton’s discovery of the calculus, =1027=. Gauss’s estimate of Newton, =1029=. M. and philosophy, =1417=. Advance in physics, =1530=. Plato on geometry, 1804. Notation of the calculus, =1904=.

Barnett, M. the type of perfect reasoning, =307=.

Barrow, On the method of m., =213=, =227=. Eulogy of m., =330=. M. as a discipline of the mind, =402=. M. and eloquence, =830=. Philosophy and m., =1430=. Uses of m., =1572=. On surd numbers, =1728=. Euclid’s definition of proportion, =1835=.

Beattie, =1431=.

Beauty of m., 453, 824, 1208. Consists in simplicity, 242, 315. Sylvester on, 1101. Russell on, 1104. Young on, 1110. Kummer on, 1111. White on, 1119. And truth, 1114. Boltzmann on, 1116.

Beltrami, On reading of the masters, =614=.

Berkeley, On geometry as logic, =428=. On math. symbols, =1214=. On fluxions, =1915=, =1942-1944=. On infinite divisibility, =1945=.

Bernoulli, Daniel, 919.

Bernoulli, James, Legend for his tomb, 920, 922. Computation of sum of tenth powers of numbers, 921. Discussion of logarithmic spiral, 922.

Berthelot, M. inspires respect for truth, =438=.

Bija Ganita, Solution of problems, =1739=.

Billingsley, M. beautifies the mind, =319=.

Binary arithmetic, 991.

Biology and m., 1579-1581.

Biot, Laplace’s “It is easy to see,” 986.

Bôcher, M. likened to painting, =1103=. Interrelation of m. and logic, =1313=. Geometry as a natural science, =1866=.

Boerne, On Pythagoras, =1855=.

Bois-Reymond, On the analytic method, =1893=. Natural selection and the calculus, =1921=.

Boltzmann, On beauty in m., =1116=.

Bolyai, Janos, Duel with officers, 924. Universal language, 925. Science absolute of space, 926.

Bolyai, Wolfgang, 927. On Gauss, =972=.

Bolzano, 928. Cured by Euclid, =929=. Parallel axiom, =2110=.

Book-keeping, Importance of the art of, 1571.

Boole, M. E. =719=.

Boole’s Laws of Thought, 1318.

Borda-Demoulins, Philosophy and m., =1405=.

Boswell, =981=.

Bowditch, On Laplace’s “Thus it plainly appears,” =985=.

Boyle, Usefulness of m. to physics, =437=. M. and science, =1513=, =1533=. Ignorance of m., =1577=. M. and physiology, =1582=. Wings of m., =1626=. Advantages of algebra, =1703=.

Brahmagupta, Estimate of m., =320=.

Brewster, On Euler’s knowledge of the Aeneid, =959=. On Euler as a computer, =963=. On Newton’s fame, =1002=.

Brougham, =1202=.

Buckle, On geometry, =1810=, =1837=.

Burke, On the value of m., =447=.

Burkhardt, On discovery in m., =618=. On universal symbolism, =1221=.

Butler, N. M., M. demonstrates the supremacy of the human reason, =309=. M. the most astounding intellectual creation, =707=. Geometry before algebra, =1871=.

Butler, Samuel, =2118=.

Byerly, On hyperbolic functions, =1929=.

Cajori, On the value of the history of m., =615=. On Bolyai, =927=. Cayley’s view of Euclid, =936=. On the extent of Euler’s work, =960=. On Euler’s math. power, =964=. On the Darmstaetter prize, =967=. On Sylvester’s first class at Johns Hopkins, =1031=. On music and m. among the Pythagoreans, =1130=. On the greatest achievement of the Hindoos, =1615=. On modern calculation, =1614=. On review in arithmetic, =1713=. On Indian m., =1737=. On the characteristics of ancient geometry, =1873=. On Napier’s rule, =1888=.

Calculating machines, =1641=.

Calculation, Importance of, 602. Not the sole object of m., 268.

Calculus, Chapter XIX. Foundation of 253. As a method, 309. May be taught at an early age, 519, 1917, 1918.

Cambridge m., 836, 1210.

Cantor, On freedom in m., =205=, =207=. On the character of Gauss’s writing, =975=. Zeno’s problem, 1938. On the infinite, =1952=.

Carlisle life tables, 946.

Carnot, On limiting ratios, =1908=. On the infinitesimal method, =1907=.

Carson, Value of geometrical training, =1841=.

Cartesian method, 1889, 1890.

Carus, Estimate of m., =326=. M. reveals supernatural God, =460=. Number and nature, =1603=. Zero and infinity, =1948=. Non-euclidean geometry, =2016=.

Cathedral, “Petrified mathematics,” 1110.

Causation in m., 251, 254.

Cayley, Advantage of modern geometry over ancient, =711=. On the imaginary, =722=. Sylvester on, 930. Noether on, 931. His style, 932. Forsyth on, 932-934. His method, 933. Compared with Euler, 934. Hermite on, 935. His view of Euclid, 936. His estimate of quaternions, 937. M. and philosophy, =1420=.

Certainty of m., 222, 1440-1442, 1628, 1863.

Chamisso, Pythagorean theorem, =1856=.

Chancellor, M. develops observation, imagination and reason, =433=.

Chapman, Different aspects of m., =265=.

Characteristics of m., 225, 229, 247, 263.

Characteristics of modern m., 720, 724-729.

Charm in m., 1115, 1640, 1848.

Chasles, Advantage of modern geometry over ancient, =712=.

Checks in m., 230.

Chemistry and m., 1520, 1560, 1561, 1750.

Chess, M. like, 840.

Chrystal, Definition of m., =113=. Definition of quantity, =115=. On problem solving, =531=. On modern text-books, =533=. How to read m., =607=. His algebra, 635. On Bernoulli’s numbers, =921=. On math. versus logical abstractness, =1304=. Rules of algebra, =1710=. On universal arithmetic, =1717=. On Horner’s method, =1744=. On probabilities, =1967=.

Cicero, Decadence of geometry among Romans, =1807=.

Circle, Properties of, 1852, 1857.

Circle-squarers, 2108, 2109.

Clarke, Descriptive geometry, =1882=.

Classic problems, Hilbert on, 627.

Clebsch, On math. research, =644=.

Clifford, On direct usefulness of math. results, =652=. Correspondence the central idea of modern m., =726=. His vision, 938. His method, 939. His knowledge of languages, 940. His physical strength, 941. On Helmholtz, =979=. On m. and mineralogy, =1558=. On algebra and good English, =1712=. Euclid the encouragement and guide of scientific thought, =1820=. Euclid the inspiration and aspiration of scientific thought, =1821=. On geometry for girls, =1842=. On Euclid’s axioms, =2015=. On non-Euclidean geometry, =2022=

Colburn, 967.

Coleridge, On problems in m., =534=. Proposition, gentle maid, =1419=. M. the quintessence of truth, =2019=.

Colton, On the effect of math. training, =417=.

Commensurable numbers, 1966.

Commerce and m., 1571.

Committee of Ten, On figures in geometry, =524=. On projective geometry, =1876=.

Common sense, M. the etherealization of, 312.

Computation, Not m., 515. And m., 810. Not concerned with significance of numbers, 1641.

Comte, On the object of m., =103=. On the business of concrete m., =104=. M. the indispensable basis of all education, =334=. Mill on, 942. Hamilton on, 943. M. and logic, =1308=, =1314=, =1325=. On Kant’s view of m., =1437=. Estimate of m., =1504=. M. essential to scientific education, =1505=. M. and natural philosophy, =1506=. M. and physics, =1535=, =1551=. M. and science, =1536=. M. and biology, =1578=, =1580=, =1581=. M. and social science, =1587=. Every inquiry reducible to a question of number, =1602=. Definition of algebra and arithmetic, =1714=. Geometry a natural science, =1813=. Ancient and modern methods, =1875=. On the graphic method, =1881=. On descriptive geometry, =1883=. Mill’s estimate of, 1903.

Congreve, =2143=.

Congruence, Symbol of, 1646.

Conic sections, 658, 660, 1541, 1542.

Conjecture, M. free from, 234.

Contingent truths, 1966.

Controversies in m., 215, 243, 1859.

Correlation in m., 525-527, 1707, 1710.

Correspondence, Concept of, 725, 726.

Coulomb, 1516.

Counting, Every problem can be solved by, 1601.

Cournot, On the object of m., =268=. On algebraic notation, =1213=. Advantage of math, notation, =1220=.

Craig, On the origin of a new science, =646=.

Credulity, M. frees mind from, 450.

Cremona, On English text-books, =609=.

Crofton, On value of probabilities, =1590=. On probabilities, =1952=, =1970=,=1972=.

Cromwell, On m. and public service, =328=.

Curiosities, Chapter XXI.

Curtius, M. and philosophy, =1409=.

Curve, Definition of, 1927.

Cyclometers, Notions of, 2108.

Cyclotomy depends on number theory, 1647.

D’Alembert, On rigor in m., =536=. Geometry as logic, =1311=. Algebra is generous, =1702=. Geometrical versus physical truths, =1809=. Standards in m., =1851=.

Dante, =1858=, =2117=.

Darmstaetter prize, 2129.

Davis, On Sylvester’s method, =1035=. M. and science, =1510=. On probability, =1968=.

Decimal fractions, 1217, 1614.

Decker, =2142=.

Dedekind, Zeno’s Problem, 1938.

Deduction, Why necessary, 219. M. based on, 224. And Intuition, 1413.

Dee, On the nature of m., =261=.

Definitions of m., Chapter I. Also 2005.

Democritus, 321.

Demoivre, His death, 944.

Demonstrations, Locke on, 236. Outside of m., 1312. In m., 1423.

De Morgan, Imagination in m., =258=. M. as an exercise in reasoning, =430=. On difficulties in m., =521=. On correlation in m., =525=. On extempore lectures, =540=. On reading algebraic works, =601=. On numerical calculations, =602=. On practice problems, =603=. On the value of the history of m., =615=, =616=. On math’ns., =812=. On Bacon’s knowledge of m., =918=. And the actuary, 945. On life tables, =946=. Anagrams’ on his name, =947=. On translations of Euclid, =953=. Euclid’s elements compared with Newton’s Principia, =954=. Euler and Diderot, =966=. Lagrange and the parallel axiom, =984=. Anagram on Macaulay’s name, =996=. Anagrams on Newton’s name, =1028=. On math, notation, =1216=. Antagonism of m. and logic, =1315=. On German metaphysics, =1416=. On m. and science, =1537=. On m. and physics, =1538=. On the advantages of algebra, =1701=. On algebra as an art, =1711=. On double algebra and quaternions, =1720=. On assumptions in geometry, =1812=. On Euclid in schools, =1819=. Euclid not faultless, =1823=. On Euclid’s rigor, =1831=. Geometry before algebra, =1872=. On trigonometry, =1885=. On the calculus in elementary instruction, =1916.= On integration, =1919=. On divergent series, =1935=, =1936=. Ad infinitum, =1949=. On the fourth dimension, =2032=. Pseudomath and graphomath, =2101=. On proof, =2102=. On paradoxers, =2105=. Budget of paradoxes, =2106=. On D’Israeli’s six follies of science, =2107=. On notions of cyclometers, =2108=. On St. Vincent, =2109=. Where Euclid failed, =2114=. On the number of the beast, =2151=.

Descartes, On the use of the term m., =102=. On intuition and deduction, =219=, =1413=. Math’ns alone arrive at proofs, =817=. The most completely math. type of mind, 948. Hankel on, 949. Mill on, 950. Hankel on, 1404. On m. and philosophy, =1425=, =1434=. Estimate of m., =1426=. Unpopularity of, =1501=. On the certainty of m., =1628=. On the method of the ancients, =1874=. On probable truth, =1964=. Descriptive geometry, 1882, 1883.

Dessoir, M. and medicine, =1585=.

Determinants, 1740, 1741.

Diderot and Euler, 966.

Differential calculus, Chapter XIX. And scientific physics, 1549.

Differential equations, 1549-1552, 1924, 1926.

Difficulties in m., 240, 521, 605-607, 634, 734, 735.

Dillmann, M. a royal science, =204=. On m. as a high school subject, =401=. Ancient and modern geometry compared, =715=. On ignorance of, =807=. On m. as a language, =1204=. Number regulates all things, =1505=.

Dirichlet, On math, discovery, =625=. As a student of Gauss, 977.

Discovery in m., 617-622, 625.

_D_-ism versus _dot_-age, 923.

Disquisitiones Arithmeticae, 975, 977, 1637, 1638.

D’Israeli, 2007.

Divergent series, 1935-1937.

“Divide et impera,” 631.

Divine character of m., 325, 329.

“Divinez avant de demontrer,” 630.

Division of labor in m., 631, 632.

Dodgson, On the charm of, =302=. Pythagorean theorem, =1854=. Ignes fatui in m., =2103=.

Dolbear, On experiment in math. research, =613=.

Domus Lescinia, Anagram on, 2155.

Donne, =1816=.

_Dot_-age versus _d_-ism, 923.

Durfee, On Sylvester’s forgetfulness, =1038=.

Dutton, On the ethical value of m., =446=.

“Eadem mutata resurgo.” 920, 922.

Echols, On the ethical value of m., =455=.

Economics and m., 1593, 1594.

Edinburgh Review, M. and astronomy, =1565=, =1566=.

Education, Place of m. in, 334, 408. Study of arithmetic better than rhetoric, 408. M. as an instrument in, 413, 414. M. in primary, 431. M. as a common school subject, 432. Bain on m. in, 442. Calculus in elementary, 1916, 1917.

Electricity, M. and the theory of, 1554.

Elegance in m., 640, 728.

Ellis, On precocity in m., =835=. On aptitude of Anglo-Danes for m., =836=. On Newton’s genius, =1014=.

Emerson, On Newton and Laplace, =1003=. On poetry and m., =1124=.

Endowment of math’ns, 818.

Enthusiasm, 801.

Equality, Grassmann’s definition of, =105=.

Equations, 104, 526, 1891, 1892.

Errors, Theory of, 1973, 1974.

Esthetic element in m., 453-455, 640, 1102, 1105, 1852, 1853.

Esthetic tact, 622.

Esthetic value of m., 1848, 1850.

Esthetics, Relation of m. to, 318, 319, 439.

Estimates of m., Chapter III. See also 1317, 1324, 1325, 1427, 1504, 1508.

Ethical value of m., 402, 438, 446, 449, 455-457.

Euclid, Bolzano cured by, 929. And Ptolemy, 951, 1878. And the student, 952.

Euclid’s Elements, Translations of, 953. Compared with the Principia, 954. Greatness of, 955. Greatest of human productions, 1817. Performance in, 1818. In English schools, 1819. Encouragement and guide, 1820. Inspiration and aspiration, 1821. The only perfect model, 1822. Not altogether faultless, 1823. Only a small part of m., 1824. Not fitted for boys, 1825. Early study of, 1826. Newton and, 1827. Its place, 1828. Unexceptional in rigor, 1829. Origin of, 1831. Doctrine of proportion, 1834. Definition of proportion, 1835. Steps in demonstration, 1839. Parallel axiom, 2007.

Euclidean geometry, 711, 713, 715.

Eudoxus, 904.

Euler, the myriad-minded, 255. Pencil outruns intelligence, 626. On theoretical investigations, 657. Merit of his work, 956. The creator of modern math. thought, 957. His general knowledge, 958. His knowledge of the Aeneid, 959. Extent of his work, 960. “Analysis incarnate,” 961. As a computer, 962, 963. His math. power, 964. His _Tentamen novae theorae musicae_, 965. And Diderot, 966. Error in Fermat’s law of prime numbers, 967.

Eureka, 911, 917.

Euripedes, 1568.

Everett, Estimate of m., =325=. Value of math. training, =443=. Theoretical investigations, =656=. Arithmetic a master-key, =1571=. On m. and law, =1598=.

Exactness, See precision.

Examinations, 407.

Examples, 422.

Experiment in m., 612, 613, 1530, 1531.

Extent of m., 737, 738.

Fairbairn, 528.

Fallacies, 610.

Faraday, M. and physics, 1554.

Fermat, 255, 967, 1902.

Fermat’s theorem, 2129.

Figures, Committee of Ten on, 524. Democritus view of, 321. Battalions of, 1631.

Fine, Definition of number, =1610=. On the imaginary, =1732=.

Fine Art, M. as a, Chapter XI.

Fisher, M. and economics, =1594=.

Fiske, Imagination in m., =256=. Advantage of m. as logic, =1324=.

Fitch, Definition of m., =125=. M. in education, =429=. Purpose of teaching arithmetic, =1624=, =1625=.

Fizi, Origin of the Liliwati, =995=.

Flamsteed, Anagram on, =968=.

Fluxions, 1911, 1915, 1942-1944.

Fontenelle, Bernoulli’s tomb, =920=.

Formulas, Compared to focus of a lens, 1515.

Forsyth, On direct usefulness of math. results, =654=. On theoretical investigations, =664=. Progress of m. =704=. On Cayley, =932-934=. On m. and physics, =1539=. On m. and applications, =1540=. On invariants, =1747=. On function theory, =1754=, =1755=.

Foster, On m. and physics, =1516=, =1522=. On experiment in m., =1531=.

Foundations of m., 717.

Four, The number, 2147, 2148.

Fourier, Math, analysis co-extensive with nature, =218=. On math. research, =612=. Hamilton on, 969. On m. and physics, =1552=, =1553=. On the advantage of the Cartesian method, =1889=.

Fourier’s theorem, 1928.

Fourth dimension, 2032, 2039.

Frankland, A., M. and chemistry, =1560=.

Frankland, W. B., Motto of Pythagorean brotherhood, =1833=. The most beautiful truth in geometry, =1857=.

Franklin, B., Estimate of m., =322=. On the value of the study of m., =323=. On the excellence of m., =324=. On m. as a logical exercise, =1303=.

Franklin, F., On Sylvester’s weakness, =1033=.

Frederick the Great, On geometry, =1860=.

Freedom in m., 205-208, 805.

French m., 1210.

Fresnel, 662.

Frischlinus, =1801=.

Froebel, M. a mediator between man and nature, =262=.

Function theory, 709, 1732, 1754, 1755.

Functional exponent, 1210.

Functionality, The central idea of modern m., 254. Correlated to life, 272.

Functions, 1932, 1933. Concept not used by Sylvester, 1034.

Fundamental concepts, Chapter XX.

Fuss, On Euler’s _Tentamen novae theorae musicae_, =965=.

Galileo, On authority in science, =1528=.

Galton, 838.

Gauss, His motto, 649. Mere math’ns, =820=. And Newton compared, 827. His power, 964. His favorite pursuits, 970. The first of theoretical astronomers, 971. The greatest of arithmeticians, 971. The math. giant, 972. Greatness of, 973. Lectures to three students, =974=. His style and method, 983. His estimate of Newton, 1029. On the advantage of new calculi, =1215=. M. and experiment, 1531. His _Disquisitiones Arithmeticae_, 1639, 1640. M. the queen of the sciences, =1642=. On number theory, =1644=. On imaginaries, =1730=. On the notation sin²φ, =1886=. On infinite magnitude, =1950=. On non-euclidean geometry, =2023-2028=. On the nature of space, 2034.

Generalization in m., 245, 246, 252, 253, 327, 728.

Genius, 819.

Geometrical investigations, 642, 643.

Geometrical training, Value of, 1841, 1842, 1844-1846.

Geometry, Chapter XVIII. Bacon’s definition of, 106. Sylvester’s definition of, 110. Value to mankind, 332, 449. And patriotism, 332. An excellent logic, 428. Plato’s view of, 429. The fountain of all thought, 451. And algebra, 525-527. Lack of concreteness, 710. Advantage of modern over ancient, 711, 712. And music, 965. And arithmetic, 1604. Is figured algebra, 1706. Name inapt, 1801. And experience, 1814. Halsted’s definition of, 1815. And observation, 1830. Controversy in, 1859. A mechanical science, 1865. A natural science, 1866. Not an experimental science, 1867. Should come before algebra, 1767, 1871, 1872. And analysis, 1931.

Germain, Algebra is written geometry, =1706=.

Gilman, Enlist a great math’n, =808=.

Glaisher, On the importance of broad training, =623=. On the importance of a well-chosen notation, =634=. On the expansion of the field of m., =634=. On the need of text-books on higher m., =635=. On the perfection of math. productions, =649=. On the invention of logarithms, =1616=. On the theory of numbers, =1640=.

Goethe, On the exactness of m., =228=. M. an organ of the higher sense, =273=. Estimate of m., =311=. M. opens the fountain of all thought, =451=. Math’ns must perceive beauty of truth, =803=. Math’ns bear semblance of divinity, =804=. Math’ns like Frenchmen, =813=. His aptitude for m., =976=. M. like dialectics, =1307=. On the infinite, 1957.

Golden age of m., 701, 702. Of art and m. coincident, 1134.

Gordan, When a math. subject is complete, =636=.

Gow, Origin of Euclid, =1832=.

Gower, =1808=.

Grammar and m. compared, 441.

Grandeur of m., 325.

Grassmann, Definition of m., =105=. Definition of magnitude, =105=. Definition of equality, =105=. On rigor in m., =538=. On the value of m., =1512=.

Greek view of science, 1429.

Graphic method, 1881.

Graphomath, 2101.

Group, Notion of, 1751.

Growth of m., 209, 211, 703.

Hall, G. S., M. the ideal and norm of all careful thinking, =304=.

Hall and Stevens, On the parallel axiom, =2008=.

Haller, On the infinite, =1958=.

Halley, On Cartesian geometry, 716.

Halsted, On Bolyai, =924-926=. On Sylvester, =1030=, =1039=. And Sylvester, =1031=, =1032=. On m. as logic, =1305=. Definition of geometry, =1815=.

Hamilton, Sir William, His ignorance of m., 978.

Hamilton, W. R., Importance of his quaternions, 333. Estimate of Comte’s ability, =943=. To the memory of Fourier, =969=. Discovery in light, 1558. On algebra as the science of time, =1715=, =1716=. On quaternions, =1718=. On trisection of an angle, =2112=.

Hankel, Definition of m., =114=. On freedom in m., =206=. On the permanency of math. knowledge, =216=. On aim in m., =508=. On isolated theorems, =621=. On tact in m., =622=. On geometry, 714. Ancient and modern m. compared, =718=, =720=. Variability the central idea in modern m., =720=. Characteristics of modern m., =728=. On Descartes, =949=. On Euler’s work, =956=. On philosophy and m., =1404=. On the origin of m., =1412=. On irrationals and imaginaries, =1729=. On the origin of algebra, =1736=. Euclid the only perfect model, =1822=. Modern geometry a royal road, =1878=.

Harmony, 326, 1208.

Harris, M. gives command over nature, =434=.

Hathaway, On Sylvester, =1036=.

Heat, M. and the theory of, 1552, 1553.

Heath, Character of Archimedes’ work, =913=.

Heaviside, The place of Euclid, =1828=.

Hebrew and Latin races, Aptitude for m., 838.

Hegel, =1417=.

Heiss, Famous anagrams, =2055=. Reversible verses, =2056=.

Helmholtz, M. the purest form of logical activity, =231=. M. requires perseverance and great caution, =240=. M. should take more important place in education, =441=. Clifford on, =979=. M. the purest logic, =1302=. M. and applications, =1445=. On geometry, =1836=. On the importance of the calculus, =1939=. A non-euclidean world, =2029=.

Herbart, Definition of m., =117=. M. the predominant science, =209=. On the method of m., =212=, =1576=. M. the priestess of definiteness and clearness, =217=. On the importance of checks, =230=. On imagination in m, =257=. M. and invention, =406=. M. the chief subject for common schools, =432=. On aptitude for m., =509=. On the teaching of m., =516=. M. the greatest blessing, =1401=. M. and philosophy, =1408=. If philosophers understood m., =1415=. M. indispensable to science, =1502=. M. and psychology, =1583=, =1684=. On trigonometry, =1884=.

Hermite, On Cayley, =935=.

Herschel, M. and astronomy, =1564=. On probabilities, =1592=.

Hiero, 903, 904.

Higher m., Mellor’s definition of, =108=.

Hilbert, On the nature of m., =266=. On rigor in m., =537=. On the importance of problems, =624=, =628=. On the solvability of problems, =627=. Problems should be difficult, =629=. On the abstract character of m., =638=. On arithmetical symbols, =1627=. On non-euclidean geometry, =2019=.

Hill, Aaron, On Newton, =1009=.

Hill, Thomas, On the spirit of mathesis, =274=. M. expresses thoughts of God, =275=. Value of m., =332=. Estimate of Newton’s work, =333=. Math’ns difficult to judge, =841=. Math’ns indifferent to ordinary interests of life, =842=. A geometer must be tried by his peers, =843=. On Bernoulli’s spiral, =922=. On mathesis and poetry, =1125=. On poesy and m., =1126=. On m. as a language, =1209=. Math, language untranslatable, =1210=. On quaternions, =1719=. On the imaginary, =1734=. On geometry and literature, =1847=. M. and miracles, =2157=, =2158=.

Hindoos, Grandest achievement of, 1615.

History and m., 1599.

History of m., 615, 616, 625, 635.

Hobson, Definition of m., =118=. On the nature of m., =252=. Functionality the central idea of m., =264=. On theoretical investigations, =663=. On the growth of m., =703=. A great math’n a great artist, =1109=. On m. and science, =1508=. Hoffman, Science and poetry not antagonistic, =1122=.

Holzmüller, On the teaching of m., =518=.

Hooker, =1432=.

Hopkinson, M. a mill, =239=.

Horner’s method, 1744.

Howison, Definition of m., =134=, =135=. Definition of arithmetic, =1612=.

Hudson, On the teaching of m., =612=.

Hughes, On science for its own sake, =1546=.

Humboldt, M. and astronomy, =1567=.

Hume, On the advantage of math, science, =1438=. On geometry, =1862=. On certainty in m., =1863=. Objection to abstract reasoning, =1941=.

Humor in m., 539.

Hutton, On Bernoulli, =919=. On Euler’s knowledge, =958=. On the method of fluxions, =1911=.

Huxley, Negative qualities of m., =250=.

Hyper-space, 2030, 2031, 2033, 2036-2038.

Hyperbolic functions, 1929, 1930.

Ignes fatui in m., 2103.

Ignorabimus, None in m., 627.

Ignorance of m., 310, 331, 807, 1537, 1577.

Imaginaries, 722, 1729-1735.

Imagination in m., 246, 251, 253, 256-258, 433, 1883.

Improvement of elementary m., 617.

Incommensurable numbers, contingent truths like, 1966.

Indian m., 1736, 1737.

Induction in m., 220-223, 244. And analogy, 724.

Infinite collection, Definition of, 1959, 1960.

Infinite divisibility, 1945.

Infinitesimal analysis, 1914.

Infinitesimals, 1905-1907, 1940, 1946, 1954.

Infinitum, Ad, 1949.

Infinity and infinite magnitude, 723, 928, 1947, 1948, 1950-1958.

Integers, Kronecker on, 1634, 1635.

Integral numbers, Minkowsky on, 1636.

Integrals, Invention of, 1922.

Integration, 1919-1921, 1923, 1925.

International Commission on m., =501=, =502=, =938=.

Intuition and deduction, 1413.

Invariance, Correlated to life, 272. MacMahon on, 1746. Keyser on, 1749.

Invariants, Changeless in the midst of change, 276. Importance of concept of, 727. Sylvester on, 1742. Forsyth on, 1747. Keyser on, 1748. Lie on, 1752.

Invention in m., 251, 260.

Inverse process, 1207.

Investigations, See research.

Irrationals, 1729.

Isolated theorems in m., 620, 621.

“It is easy to see,” 985, 986, 1045.

Jacobi, His talent for philology, 980. Aphorism, =1635=. Die “Ewige Zahl,” =1643=.

Jefferson, On m. and law, =1597=.

Johnson, His recourse to m., 981. Aptitude for numbers, =1617=. On round numbers, =2137=.

Journals and transactions, 635.

Jowett, M. as an instrument in education, =413=.

Judgment, M. requires, 823.

Jupiter’s eclipses, 1544.

Justitia, The goddess, 824.

Juvenal, Nemo mathematicus etc., =831=.

Kant, On the a priori nature of m., =130=. M. follows the safe way of science, =201=. On the origin of scientific m., 201. On m. in primary education, =431=. M. the embarrassment of metaphysics, =1402=. His view of m., =1436=, =1437=. On the difference between m. and philosophy, =1436=. On m. and science, =1508=. Esthetic elements in m., =1852=, =1853=. Doctrine of time, =2001=. Doctrine of space, =2003=.

Karpinsky, M. and efficiency, =1673=.

Kasner, “Divinez avant de demontrer,” =630=. On modern geometry, =710=.

Kelland, On Euclid’s elements, =1817=.

Kelvin, Lord, See William Thomson.

Kepler, His method, 982. Planetary orbits and the regular solids, =2134=.

Keyser, Definition of m., =132=. Three characteristics of m., =225=. On the method of m., =244=. On ratiocination, =246=. M. not detached from life, =273=. On the spirit of mathesis, =276=. Computation not m., =515=. Math, output of present day, =702=. Modern theory of functions, =709=. M. and journalism, =731=. Difficulty of m., =735=. M. appeals to whole mind, =815=. Endowment of math’ns, =818=. Math’ns in public service, =823=. The aim of the math’n, =844=. On Bolzano, =929=. On Lie, =992=. On symbolic logic, =1321=. On the emancipation of logic, =1322=. On the Principia Mathematica, =1326=. On invariants, =1728=. On invariance, =1729=. On the notion of group, =1751=. On the elements of Euclid, =1824=. On protective geometry, =1880=. Definition of infinite assemblage, =1960=. On the infinite, =1961=. On non-euclidean geometry, =2035=. On hyper-space, =2037=, =2038=.

Khulasat-al-Hisab, Problems, =1738=.

Kipling, =1633=.

Kirchhoff, Artistic nature of his works, 1116.

Klein, Definition of m., =123=. M. a versatile science, =264=. Aim in teaching, =507=, =517=. Analysts versus synthesists, =651=. On theory and practice, =661=. Math, aptitudes of various races, =838=. Lie’s final aim, =993=. Lie’s genius, =994=. On m. and science, =1520=. Famous aphorisms, =1635=. Calculating machines, =1641=. Calculus for high schools, =1918=. On differential equations, =1926=. Definition of a curve, =1927=. On axioms of geometry, =2006=. On the parallel axiom, =2009=. On non-euclidean geometry, =2017=, =2021=. On hyper-space, =2030=.

Kronecker, On the greatness of Gauss, =973=. God made integers etc., =1634=.

Kummer, On Dirichlet, =977=. On beauty in m., =1111=.

LaFaille, Mathesis few know, =1870=.

Lagrange, On correlation of algebra and geometry, =527=. His style and method, 983. And the parallel axiom, 984. On Newton, =1011=. Wings of m., =1604=. Union of algebra and geometry, =1707=. On the infinitesimal method, =1906=.

Lalande, M. in French army, =314=.

Langley, M. in Prussia, 513.

Lampe, On division of labor in m., =632=. On Weierstrass, =1049=. Weierstrass and Sylvester, =1050=. Qualities common to math’ns and artists, =1113=. Charm of m., =1115=. Golden age of art and m. coincident, =1134=.

Language, Chapter XII. See also 311, 419, 443, 1523, 1804, 1889.

Laplace, On instruction in m., =220=. His style and method, =983=. “Thus it plainly appears,” 985, 986. Emerson on, 1003. On Leibnitz, =991=. On the language of analysis, =1222=. On m. and nature, =1525=. On the origin of the calculus, =1902=. On the exactitude of the differential calculus, =1910=. The universe in a single formula, =1920=. On probability, =1963=, =1969=, =1971=.

Laputa, Math’ns of, 2120-2122, Math. school of, 2123.

Lasswitz, On modern algebra, =1741=. On function theory, =1934=. On non-euclidean geometry, =2040=.

Latin squares, 252.

Latta, On Leibnitz’s logical calculus, =1317=.

Law and m., 1597, 1598.

Laws of thought, 719, 1318.

Leadership, M. as training for, 317.

Lecture, Preparation of, 540.

Lefevre, M. hateful to weak minds, =733=. Logic and m., =1309=.

Leibnitz, On difficulties in m., =241=. His greatness, 987. His influence, 988. The nature of his work, 989. His math. tendencies, 990. His binary arithmetic, 991. On Newton, =1010=. On demonstrations outside of m., =1312=. Ars characteristica, =1316=. His logical calculus, 1317. Union of philosophical and m. productivity, 1404. M. and philosophy, =1435=. On the certainty of math. knowledge, =1442=. On controversy in geometry, =1859=. His differential calculus, 1902. His notation of the calculus, 1904. On necessary and contingent truth, =1966=.

Leverrier, Discovery of Neptune, 1559.

Lewes, On the infinite, =1953=.

Lie, On central conceptions in modern m., =727=. Endowment of math’ns, =818=. The comparative anatomist, 992. Aim of his work, 993. His genius, 994. On groups, =1752=. On the origin of the calculus, =1901=. On differential equations, =1924=.

Liliwati, Origin of, 995.

Limitations of math. science, 1437.

Limits, Method of, 1905, 1908, 1909, 1940.

Lindeman, On m. and science, =1523=.

Liouville, 822.

Lobatchewsky, =2022=.

Locke, On the method of m., =214=, =235=. On proofs and demonstrations, =236=. On the unpopularity of m., =271=. On m. as a logical exercise, =423=, =424=. M. cures presumption, =425=. Math, reasoning of universal application, =426=. On reading of classic authors, =604=. On Aristotle, =914=. On m. and philosophy, =1433=. On m. and moral science, =1439=, =1440=. On the certainty of math. knowledge, =1440=, =1441=. On unity, =1607=. On number, =1608=. On demonstrations in numbers, =1630=. On the advantages of algebra, =1705=. On infinity, =1955=, =1957=. On probability, =1965=.

Logarithmic spiral, 922.

Logarithmic tables, 602.

Logarithms, 1526, 1614, 1616.

Logic and m., Chapter XIII. See also 423-430, 442.

Logical calculus, 1316, 1317.

Longevity of math’ns, 839.

Lovelace, Why are wise few etc., =1629=.

Lover, =2140=.

Macaulay, Plato and Bacon, =316=. On Archimedes, =905=. Bacon’s view of m., =915=, =916=. Anagram on his name, =996=. Plato and Archytas, =1427=. On the power of m., =1527=.

Macfarlane, On Tait, Maxwell, Thomson, =1042=. On Tait and Hamilton’s quaternions, =1044=.

Mach, On thought-economy in m., =203=. M. seems possessed of intelligence, =626=. On aim of research, =647=. On m. and counting, =1601=. On the space of experience, =2011=.

MacMahon, Latin squares, 252. On Sylvester’s bend of mind, =645=. On Sylvester’s style, =1040=. On the idea of invariance, =1746=.

Magnitude, Grassmann’s definition, 105.

Magnus, On the aim in teaching m., =505=.

Manhattan Island, Cost of, 2130.

Marcellus, Estimate of Archimedes, =909=.

Maschke, Man above method, =650=.

Masters, On the reading of the, 614.

Mathematic, Sylvester on use of term, 101. Bacon’s use of term, 106.

Mathematical faculty, Frequency of, 832.

Mathematical mill, The, 239, 1891.

Mathematical productions, 648, 649.

Mathematical theory, When complete, 636, 637.

Mathematical training, 443, 444. Maxims of math’ns, 630, 631, 649. Not a computer, 1211. Intellectual habits of math’ns, 1428. The place of the, 1529. Characteristics of the mind of a, 1534.

Mathematician, The, Chapter VIII.

Mathematics, Definitions of, Chapter I. Objects of, Chapter I. Nature of, Chapter II. Estimates of, Chapter III. Value of, Chapter IV. Teaching of, Chapter V. Study of, Chapter VI. Research in, Chapter VI. Modern, Chapter VII. As a fine art, Chapter XI. As a language, Chapter XII. Also 445, 1814. And logic, Chapter XIII. And philosophy, Chapter XIV. And science, Chapter XV. And applications, Chapter XV. Knowledge most in, 214. Suppl. brevity of life, 218. The range of, 269. Compared to French language, 311. The care of great men, 322. And professional education, 429. And science teaching, 522. The queen of the sciences, 975. Advantage over philosophy, 1436, 1438. As an instrument, 1506. For its own sake, 1540, 1541, 1545, 1546. The wings of, 1604.

Mathesis, 274, 276, 1870, 2015.

Mathews, On Disqu. Arith. =1638=. On number theory, =1639=. The symbol ≡, =1646=. On Cyclotomy, =1647=. Laws of algebra, =1709=. On infinite, zero, infinitesimal, =1954=.

Maxims of great math’ns, 630, 631, 649.

Maxwell, 1043, 1116.

McCormack, On the unpopularity of m., =270=. On function, =1933=.

Méchanique céleste, 985, 986.

Medicine, M. and the study of, 1585, 1918.

Mellor, Definition of higher m., =108=. Conclusions involved in premises, =238=. On m. and science, =1561=. On the calculus, =1912=. On integration, =1923=, =1925=.

Memory in m., 253.

Menæchmus, 901.

Mere math’ns, 820, 821.

Merz, On the transforming power of m., =303=. On the dominant ideas in m., =725=. On extreme views in m., =827=. On Leibnitz’s work, =989=. On the math. tendency of Leibnitz, =990=. On m. as a lens, =1515=. M. extends knowledge, =1524=. Disquisitiones Arithmeticae, =1637=. On functions, =1932=. On hyper-space, =2036=.

Metaphysics, M. the only true, 305.

Meteorology and m., 1557.

Method of m. 212-215, 226, 227, 230, 235, 244, 806, 1576.

Metric system, 1725.

Military training, M. in, 314, 418, 1574.

Mill, On induction in m., =221=, =222=. On generalization in m., =245=. On math. studies, =409=. On m. in a scientific education, =444=. Math’ns hard to convince, =811=. Math’ns require genius, =819=. On Comte, =942=. On Descartes, =942=, =948=. On Sir William Hamilton’s ignorance of m., =978=. On Leibnitz, =987=. On m. and philosophy, =1421=. On m. as training for philosophers, =1422=. M. indispensable to science, =1519=. M. and social science, =1595=. On the nature of geometry, =1838=. On geometrical method, =1861=. On the calculus, =1903=.

Miller, On the Darmstaetter prize, =2129=.

Milner, Geometry and poetry, =1118=.

Minchin, On English text-books, =539=.

Mineralogy and m., 1558.

Minkowski, On integral numbers, =1636=.

Miracles and m., 2157, 2158, 2160.

Mixed m., Bacon’s definition of, 106. Whewell’s definition of, 107.

Modern algebra, 1031, 1032, 1638, 1741.

Modern geometry, 1710-1713, 715, 716, 1878.

Modern m., Chapter VII.

Moebius, Math’ns constitute a favorite class, =809=. M. a fine art, =1107=.

Moral science and m., 1438-1440.

Moral value of m., See ethical value.

Mottoes, Of math’ns, 630, 631, 649. Of Pythagoreans, 1833.

Murray, Definition of m., =116=.

Music and m., 101, 276, 965, 1107, 1112, 1116, 1127, 1128, 1130-1133, 1135, 1136.

Myers, On m. as a school subject, =403=. On pleasure in m., =454=. On the ethical value of m., =457=. On the value of arithmetic, =1622=.

Mysticism and numbers, 2136-2141, 2143.

Napier’s rule, 1888.

Napoleon, M. and the welfare of the state, =313=. His interest in m., 314, 1001.

Natural science and m., Chapter XV. Also 244, 444, 445, 501.

Natural selection, 1921.

Nature of m., Chapter II. See also 815, 1215, 1308, 1426, 1525, 1628.

Nature, Study of, 433-436, 514, 516, 612.

Navigation and m., 1543, 1544.

Nelson, Anagram on, 2153.

Neptune, Discovery of, 1554, 1559.

Newcomb, On geometrical paradoxers, =2113=.

Newton, Importance of his work, 333. On correlation in m., =526=. On problems in algebra, =530=. And Gauss compared, 827. His fame, 1002. Emerson on, 1003. Whewell on, 1004, 1005. Arago on, 1006. Pope on, 1007. Southey on, 1008. Hill on, 1009. Leibnitz on, 1010. Lagrange on, 1011. No monument to, 1012. Wilson on, 1012, 1013. His genius, 1014. His interest in chemistry and theology, 1015. And alchemy, 1016, 1017. His first experiment, 1018. As a lecturer, 1019. As an accountant, 1020. His memorandum-book, 1021. His absent-mindedness, 1022. Estimate of himself, =1023-1025=. His method of work, 1026. Discovery of the calculus, 1027. Anagrams on, 1028. Gauss’s estimate of, 1029. On geometry, =1811=. Compared with Euclid, 1827. Geometry a mechanical science, =1865=. Test of simplicity, =1892=. Method of fluxions, 1902.

Newton’s rule, 1743.

Nile, Origin of name, 2150.

Noether, On Cayley, =931=. On Sylvester, =1034=, =1041=.

Non-euclidean geometry, 1322, 2016-2029, 2033, 2035, 2040.

Nonnus, On the mystic four, =2148=.

Northrup, On Lord Kelvin, =1048=.

Notation, Importance of, 634, 1222, 1646. Value of algebraic, 1213, 1214. Criterion of good, 1216. On Arabic, 1217, 1614. Advantage of math., 1220. See also symbolism.

Notions, Cardinal of m., 110. Indefinable, 1219.

Novalis, Definition of pure m., =112=. M. the life supreme, =329=. Without enthusiasm no m., =801=. Method is the essence of m., =806=. Math’ns not good computers, =810=. Music and algebra, =1128=. Philosophy and m., =1406=. M. and science, =1507=, =1526=. M. and historic science, =1599=. M. and magic, =2159=. M. and miracles, =2160=.

Number, Every inquiry reducible to a question of, 1602. And nature, 1603. Regulates all things, 1605. Aeschylus on, 1606. Definition of, 1609, 1610. And superstition, 1632. Distinctness of, 1707. Of the beast, 2151, 2152.

Number-theory, The queen of m., 975. Nature of, 1639. Gauss on, 1644. Smith on, 1645. Notation in, 1646. Aid to geometry, 1647. Mystery in, 1648.

Number-work, Purpose of, 1623.

Numbers, Pythagoras’ view of, 321. Mighty are, 1568. Aptitude for, 1617. Demonstrations in, 1630. Prime, 1648. Necessary truths like, 1966. Round, 2137. Odd, 2138-2141. Golden, 2142. Magic, 2143.

Obscurity in m. and philosophy, 1407.

Observation in m., 251-253, 255, 433, 1830.

Obviousness in m., 985, 986, 1045.

Olney, On the nature of m., =253=.

Oratory and m., 829, 830.

Order and arrangement, 725.

Origin of m., 1412.

Orr, Memory verse for π, =2127=.

Osgood, On the calculus, =1913=.

Ostwald, On four-dimensional space, 2039.

π. In actuarial formula, 945. Memory verse for, 2127.

Pacioli, On the number three, =2145=.

Painting and m., 1103, 1107.

Papperitz, On the object of pure m., 111.

Paradoxes, Chapter XXI.

Parallel axiom, Proof of, 984, 2110, 2111. See also non-euclidean geometry.

Parker, Definition of arithmetic, =1611=. Number born in superstition, =1632=. On geometry, =1805=.

Parton, On Newton, =1917-1919=, =1021=, =1022=, =1827=.

Pascal, Logic and m., =1306=.

Peacock, On the mysticism of Greek philosophers, =2136=. The Yankos word for three, =2144=. The number of the beast, =2152=.

Pearson, M. and natural selection, =834=.

Peirce, Benjamin, Definition of m., =120=. M. as an arbiter, =210=. Logic dependent on m., =1301=. On the symbol √-1, =1733=.

Peirce, C. S. Definition of m., =133=. On accidental relations, =2128=.

Perry, On the teaching of m., =510=, =511=, =619=, =837=.

Persons and anecdotes, Chapters IX and X.

Philosophy and m., Chapter XIV. Also 332, 401, 414, 444, 445, 452.

Physics and m., 129, 437, 1516, 1530, 1535, 1538, 1539, 1548, 1549, 1550, 1555, 1556.

Physiology and m., 1578, 1581, 1582.

Picard, On the use of equations, =1891=.

Pierce, On infinitesimals, =1940=.

Pierpont, Golden age of m., =701=. On the progress of m., =708=. Characteristics of modern m., =717=. On variability, =721=. On divergent series, =1937=.

Plato, His view of m., 316, 429. M. a study suitable for freemen, =317=. His conic sections, 332. And Archimedes, 904. Union of math. and philosophical productivity, 1404. Diagonal of square, =1411=. And Archytas, 1427. M. and the arts, =1567=. On the value of m., =1574=. On arithmetic, =1620=, =1621=. God geometrizes, 1635, 1636. 1702. On geometry, 429, 1803, 1804, =1806=, =1844=, =1845=.

Pleasure, Element of in m., 1622, 1629, 1848, 1850, 1851.

Pliny, =2039=.

Plus and minus signs, 1727.

Plutarch, On Archimedes, =903=, =904=, =908-910=, =912=. God geometrizes, =1802=.

Poe, =417=.

Poetry and m., Weierstrass on, 802. Pringsheim on, 1108. Wordsworth on, 1117. Milner on, 1118. Workman on, 1120. Pollock on, 1121. Hoffman on, 1122. Thoreau on, 1123. Emerson on, 1124. Hill on, 1125, 1126. Shakespeare on, 1127.

Poincaré, On elegance in m., =640=. M. has a triple end, =1102=. M. as a language, =1208=. Geometry not an experimental science, =1867=. On geometrical axioms, =2005=.

Point, 1816.

Political science, M. and, 1201, 1324.

Politics, Math’ns and, 814.

Pollock, On Clifford, =938-941=, =1121=.

Pope, =907=, =2015=, =2031=, =2046=.

Precision in m., 228, 639, 728.

Precocity in m., 835.

Predicabilia a priori, 2003.

Press, M. ignored by daily, 731, 732.

Price, Characteristics of m., =247=. On m. and physics, =1550=.

Prime numbers, Sylvester on, 1648.

Principia Mathematica, 1326.

Pringsheim, M. the science of the self-evident, =232=. M. should be studied for its own sake, =439=. On the indirect value of m., =448=. On rigor in m., =535=. On m. and journalism, =732=. On math’ns in public service, =824=. Math’n somewhat of a poet, =1108=. On music and m., =1132=. On the language of m., =1211=. On m. and physics, =1548=.

Probabilities, 442, 823, 1589, 1590-1592, 1962-1972, 1975.

Problem solving, 531, 532.

Problems, In m., 523, 534. In arithmetic, 528. In algebra, 530. Should be simple, 603. In Cambridge texts, 608. On solution of, 611. On importance of, 624, 628. What constitutes good, 629. Aid to research, 644. Of modern m., 1926.

Proclus, Ptolemy and Euclid, =951=. On characteristics of geometry, =1869=.

Progress in m., 209, 211, 212, 216, 218, 702-705, 708.

Projective geometry, 1876, 1877, 1879, 1880.

Proportion, Euclid’s doctrine of, 1834. Euclid’s definition of, 1835.

Proposition, 1219, 1419.

Prussia, M. in, 513.

Pseudomath, Defined, 2101.

Psychology and m., 1576, 1583, 1584.

Ptolemy and Euclid, 951.

Public service, M. and, 823, 824, 1303, 1574.

Public speaking, M. and, 420, 829, 830.

Publications, Math. of present day, 702, 703.

Pure M., Bacon’s definition of, 106. Whewell’s definition of, 107. On the object of, 111, 129. Novalis’ conception of, 112. Hobson’s definition of, 118. Russell’s definition of, 127, 128.

Pursuit of m., 842.

Pythagoras, Number the nature of things, 321. Union of math, and philosophical productivity, 1404. The number four, =2147=.

Pythagorean brotherhood, Motto of, 1833.

Pythagorean theorem, 1854-1856, 2026.

Pythagoreans, Music and M., 1130.

Quadrature, See Squaring of the circle.

Quantity, Chrystal’s definition of, =115=.

Quarles, On quadrature, =2116=.

Quaternions, 333, 841, 937, 1044, 1210, 1718-1726.

Quetelet, Growth of m., =1514=.

Railway-making, 1570.

Reading of m., 601, 604-606.

Reason, M. most solid fabric of human, 308. M. demonstrates supremacy of human, 309.

Reasoning, M. a type of perfect, 307. M. as an exercise in, 423-427, 429, 430, 1503.

Recorde, Value of arithmetic, 1619.

Regiomontanus, 1543.

Regular solids, 2132-2135.

Reid, M. frees from sophistry, =215=. Conjecture has no place in m., =234=. M. the most solid fabric, =308=. On Euclid’s elements, =955=. M. manifests what is impossible =1414=. On m. and philosophy, =1423=. Probability and Christianity, =1975=. On Pythagoras and the regular solids, =2132=.

Reidt, M, as an exercise in language, =419=. On the ethical value of m., =456=. On aim in math. instruction, =506=.

Religion and m., 274-276, 459, 460, 1013.

Research in m., Chapter VI.

Reversible verses, 2156.

Reye, Advantages of modern over ancient geometry, =714=.

Rhetoric and m., 1599.

Riemann, On m. and physics, 1549.

Rigor in m., 535-538.

Rosanes, On the unpopularity of m., =730=.

Royal road, 201, 901, 951, 1774.

Royal science, M. a, 204.

Rudio, On Euler, =957=. M. and great artists, =1105=. On m. and navigation, =1543=.

Rush, M. cures predisposition to anger, =458=.

Russell, Definition of m., =127=, =128=. On nineteenth century m., =705=. Chief triumph of modern m., =706=. On the infinite, =723=. On beauty in m., =1104=. On the value of symbols, =1219=. On Boole’s Laws of Thought, =1318=. Principia Mathematica, 1326. On geometry and philosophy, =1410=. Definition of number, =1609=. Fruitful uses of imaginaries, =1735=. Geometrical reasoning circular, =1864=. On projective geometry, =1879=. Zeno’s problems, =1938=. Definition of infinite collection, =1959=. On proofs of axioms, =2013=. On non-euclidean geometry, =2018=.

Safford, On aptitude for m., =520=. On m. and science, =1509=.

Sage, Battalions of figures, =1631=.

Sartorius, Gauss on the nature of space, =2034=.

Scepticism, 452, 811.

Schellbach, Estimate of m., =306=. On truth, =1114=.

Schiller, Archimedes and the youth, =907=.

Schopenhauer, Arithmetic rests on the concept of time, =1613=. Predicabilia a priori, =2003=.

Schröder, M. as a branch of logic, =1323=.

Schubert, Three characteristics of m., =229=. On controversies in m., =243=. Characteristics of m., =263=. M. an exclusive science, =734=.

Science and m., Chapter XV. M. an indispensible tool of, 309. Neglect of m. works injury to, 310. Craig on origin of new, 646. Greek view of, 1429. Six follies of, 2107. See also 433, 436, 437, 461, 725.

Scientific education, Math. training indispensable basis of, =444=.

Screw, The song of the, 1894. As an instrument in geometry, 2114.

Sedgwick, Quaternion of maladies, =1723=.

Segre, On research in m., =619=. What kind of investigations are important, =641=. On the worthlessness of certain investigations, =642, 643=. On hyper-space, =2031=.

Seneca, Alexander and geometry, =902=.

Seventy-seven, The number, 2149.

Shakespeare, 1127, 1129, 2141.

Shaw, J. B., M. like game of chess, =840=.

Shaw, W. H., M. and professional life, =1596=.

Sherman, M. and rhetoric, =1599=.

Smith, Adam, 1324.

Smith, D. E., On problem solving, =532=. Value of geometrical training, =1846=. Reason for studying geometry, =1850=.

Smith, H. J. S., When a math. theory is completed, =637=. On the growth of m., =1521=. On m. and science, =1542=. On m. and physics, =1556=. On m. and meteorology, =1557=. On number theory, =1645=. Rigor in Euclid, =1829=. On Euclid’s doctrine of proportion, =1834=.

Smith, W. B., Definition of m., =121=. On infinitesimal analysis, =1914=. On non-euclidean and hyperspaces, =2033=.

Simon, On beauty and truth, =1114=.

Simplicity in m., 315, 526.

Sin²φ, On the notation of, 1886.

Six hundred sixty-six, The number, 2151, 2152.

Social science and m., 1201, 1586, 1587.

Social service, M. as an aid to, 313, 314, 328.

Social value of m., 456, 1588.

Solitude and m., 1849, 1851.

Sophistry, M. free from, 215.

Sound, M. and the theory of, 1551.

Southey, On Newton, =1008=.

Space, Of experience, 2011. Kant’s doctrine of, 2003. Schopenhauer’s predicabilia, 2004. Whewell, On the idea of, 2004. Non-euclidean, 2015, 2016, 2018. Hyper-, 2030, 2031, 2033, 2036-2038.

Spedding, On Bacon’s knowledge of m., =917=.

Speer, On m. and nature-study, =514=.

Spence, On Newton, =1016=, =1020=.

Spencer, On m. in the arts, =1570=.

Spherical trigonometry, 1887.

Spira mirabilis, 922.

Spottiswoode, On the kingdom of m., =269=.

Squaring the circle, 1537, 1858, 1934, 1948, 2115-2117.

St. Augustine, The number seventy seven, =2149=.

St. Vincent, As a circle-squarer, 2109.

Steiner, On projective geometry, =1877=.

Stewart, M. and facts, =237=. On beauty in m., =242=. What we most admire in m., =315=. M. for its own sake, =440=. M. the noblest instance of force of the human mind, =452=. Math’ns and applause, =816=. Mere math’ns, =821=. Shortcomings of math’ns, =828=. On the influence of Leibnitz, =988=. Reason supreme, =1424=. M. and philosophy compared, =1428=. M. and natural philosophy, =1555=.

Stifel, The number of the beast, =2152=.

Stobæus, Alexander and Menæchmus, =901=. Euclid and the student, =952=.

Study of m., Chapter VI.

Substitution, Concept of, 727.

Superstition, M. frees mind from, 450. Number was born in, 1632.

Surd numbers, 1728.

Surprises, M. rich in, 202.

Swift, On m. and politics, =814=. The math’ns of Laputa, =2120-2122=. The math. school of Laputa, =2123=. His ignorance of m., 2124, 2125.

Sylvester, On the use of the terms mathematic and mathematics, =101=. Order and arrangement the basic ideas of m., =109=, =110=. Definition of algebra, =110=. Definition of arithmetic, =110=. Definition of geometry, =110=. On the object of pure m., =129=. M. requires harmonious action of all the faculties, =202=. Answer to Huxley, =251=. On the nature of m., =251=. On observation in m., =255=. Invention in m., =260=. M. entitled to human regard, =301=. On the ethical value of m., =449=. On isolated theorems, =620=. “Auge _et impera._” =631=. His bent of mind, =645=. Apology for imperfections, =648=. On theoretical investigations, =658=. Characteristics of modern m., =724=. Invested m. with halo of glory, 740. M. and eloquence, =829=. On longevity of math’ns, =839=. On Cayley, =930=. His view of Euclid, 936. Jacobi’s talent for philology, =980=. His eloquence, 1030. Researches in quantics, =1032=. His weakness, 1033, 1036, 1037. One-sided character of his work, 1034. His method, 1035, 1036, 1041. His forgetfulness, 1037, 1038. Relations with students, 1039. His style, 1040, 1041. His characteristics, 1041. His enthusiasm, 1041. The math. Adam, =1042=. And Weierstrass, 1050. On divine beauty and order in m., =1101=. M. among the fine arts, =1106=. On music and m., =1131=. M. the quintessence of language, =1205=. M. the language of the universe, =1206=. On prime numbers, =1648=. On determinants, =1740=. On invariants, =1742=. Contribution to theory of equations, 1743. To a missing member etc., =1745=. Invariants and isomerism, =1750=. His dislike for Euclid, =1826=. On the invention of integrals, =1922=. On geometry and analysis, =1931=. On paradoxes, =2104=.

Symbolic language, M. as a, 1207, 1212. Use of, 1573.

Symbolic logic, 1316-1321.

Symbolism, On the nature of math., 1210. Difficulty of math., 1218. Universal impossible, 1221. See also notation.

Symbols, Burlesque on, 1741.

Symbols, M. leads to mastery of, 421. Value of math., 1209, 1212, 1219. Essential to demonstration, 1316. Arithmetical, 1627.

Tact in m., 622, 623.

Tait, On the unpopularity of m., =740=. And Thomson, 1043. And Hamilton, 1044. On quaternions, =1724-1726=. On spherical trigonometry, =1887=.

Talent, Math’ns men of, 825.

Teaching of m., Chapter V.

Tennyson, 1843.

Teutonic race, Aptitude for m., 838.

Text-books, Chrystal on, 533. Minchin on, 539. Cremona on English, 609. Glaisher on need of, 635.

Thales, 201.

Theoretical investigations, 652-664.

Theory and practice, 661.

Thompson, Sylvanus, Lord Kelvin’s definition of a math’n, =822=. Cayley’s estimate of quaternions, =937=. Thomson’s “It is obvious that,” =1045=. Anecdote of Lord Kelvin, =1046=, =1047=. On the calculus for beginners, =1917=.

Thomson, Sir William, M. the only true metaphysics, =305=. M. not repulsive to common sense, =312=. What is a math’n? =822=. And Tait, 1043. “It is obvious that,” 1045. Anecdotes concerning, 1046, 1047, 1048. On m. and astronomy, =1562=. On quaternions, =1721, 1722=.

Thomson and Tait, 1043. On Fourier’s theorem, =1928=.

Thoreau, On poetry and m., =1123=.

Thought-economy in m., 203, 1209, 1704.

Three, The Yankos word for, 2144. Pacioli on the number, 2145.

Time, Arithmetic rests on notion of 1613. As a concept in algebra, 1715, 1716, 1717. Kant’s doctrine of, 2001. Schopenhauer’s predicabilia, 2003.

Todhunter, On m. as a university subject, =405=. On m. as a test of performance, =408=. On m. as an instrument in education, =414=. M. requires voluntary exertion, =415=. On exercises, =422=. On problems, =523=, =608=. How to read m., =605=, =606=. On discovery in elementary m., =617=. On Sylvester’s theorem, =1743=. On performance in Euclid, =1818=.

Transformation, Concept of, 727.

Trigonometry, 1881, 1884-1889.

Trilinear co-ordinates, 611.

Trisection of angle, 2112.

Truth, and m., 306. Math’ns must perceive beauty of, 803. And beauty, 1114.

Tzetzes, Plato on geom., =1803=.

Unity, Locke on the idea of, 1607.

Universal algebra, 1753.

Universal arithmetic, 1717.

Universal language, 925.

Unpopularity of m., 270, 271, 730-736, 738, 740, 1501, 1628.

Usefulness, As a principle in research, 652-655, 659, 664.

Uses of m., See value of m.

Value of m., Chapter IV. See also 330, 333, 1414, 1422, 1505, 1506, 1512, 1523, 1526, 1527, 1533, 1541, 1542, 1543, 1547-1576, 1619-1626, 1841, 1844-1851.

Variability, The central idea of modern m., 720, 721.

Venn, On m. as a symbolic language, =1207=. M. the only gate, =1517=.

Viola, On the use of fallacies, =610=.

Virgil, =2138=.

Voltaire, Archimedes more imaginative than Homer, =259=. M. the staff of the blind, =461=. On direct usefulness of results, =653=. On infinite magnitudes, =1947=. On the symbol, =1950=. Anagram on, 2154.

Walcott, On hyperbolic functions, =1930=.

Walker, On problems in arithmetic, =528=. On the teaching of geometry, =529=.

Wallace, On the frequency of the math. faculty, =832=. On m. and natural selection, =833, 834=. Parallel growth of m. and music, =1135=.

Walton, Angling like m., =739=.

Weber, On m. and physics, =1549=.

Webster, Estimate of m., =331=.

Weierstrass, Math’ns are poets, =802=. Anecdote concerning, 1049. And Sylvester, 1050. Problem of infinitesimals, 1938.

Weismann, On the origin of the math. faculty, =1136=.

Wells, On m. as a world language, =1201=.

Whately, On m. as an exercise, =427=. On m. and navigation, =1544=. On geometrical demonstrations, =1839=. On Swift’s ignorance of m., =2124=.

Whetham, On symbolic logic, =1319=.

Whewell, On mixed and pure math., =107=. M. not an inductive science, =223=. Nature of m., 224. Value of geometry, 445. On theoretical investigations, =660=, =662=. Math’ns men of talent, =825=. Fame of math’ns, =826=. On Newton’s greatness, =1004=. On Newton’s theory, =1005=. On Newton’s humility, =1025=. On symbols, 1212. On philosophy and m., =1429=. On m. and science, =1534=. Quotation from R. Bacon, =1547=. On m. and applications, =1541=. Geometry and experience, 1814. Geometry not an inductive science, 1830. On limits, 1909. On the idea of space, 2004. On Plato and the regular solids, =2133=, =2135=.

White, H. S., On the growth of m., =211=.

White, W. F., Definition of m., =131=, =1203=. M. as a prerequisite for public speaking, =420=. On beauty in m., =1119=. The place of the math’n, =1529=. On m. and social science, =1586=. The cost of Manhattan island, =2130=.

Whitehead, On the ideal of m., =119=. Definition of m., =122=. On the scope of m., =126=. On the nature of m., =233=. Precision necessary in m., =639=. On practical applications, =655=. On theoretical investigations, =659=. Characteristics of ancient geometry, =713=. On the extent of m., =737=. Archimedes compared with Newton, =911=. On the Arabic notation, =1217=. Difficulty of math. notation, =1218=. On symbolic logic, =1320=. Principia Mathematica, 1326. On philosophy and m., =1403=. On obscurity in m. and philosophy, =1407=. On the laws of algebra, =1708=. On + and − signs, =1727=. On universal algebra, =1753=. On the Cartesian method, =1890=. On Swift’s ignorance of m., =2125=.

Whitworth, On the solution of problems, =611=.

Williamson, On the value of m., =1575=. Infinitesimals and limits, =1905=. On infinitesimals, =1946=.

Wilson, E. B., On the social value of m., =1588=. On m. and economics, =1593=. On the nature of axioms, =2012=.

Wilson, John, On Newton and Shakespeare, =1012=. Newton and Linnæus, =1013=.

Woodward, On probabilities, =1589=. On the theory of errors, =1973=, =1974=.

Wordsworth, W., On Archimedes, =906=. On poetry and geometric truth, =1117=. On geometric rules, =1418=. On geometry, =1840=, =1848=. M. and solitude, =1859=.

Workman, On the poetic nature of m., =1120=.

Young, C. A., On the discovery of Neptune, =1559=.

Young, C. W., Definition of m., =124=.

Young, J. W. A., On m. as type a of thought, =404=. M. as preparation for science study, =421=. M. essential to comprehension of nature, =435=. Development of abstract methods, =729=. Beauty in m., =1110=. On Euclid’s axiom, =2014=.

Zeno, His problems, 1938.

Zero, 1948, 1954.

* * * * *

Transcriber’s Notes

Punctuation has been standardised.

Characters in small caps have been replaced by all caps.

Italic text has been denoted by _underscores_ and bold text by =equal signs=.

Em-dash added before all attribution names for consistency.

The two omitted illustrations have been identified by an [Illustration:] tag with a short description.

Mis-alphabetized entries in the Index have been corrected

Non-printable superscripts are represented by a caret followed by the character, i.e. x^n.

Non-printable subscripts have been represented by an underscore followed by the subscript in braces i.e. _{a}.

Book was written in a period when many words had not become standarized in their spelling. Numerous words have multiple spelling variations in the text. These have been left unchanged unless noted below:

§230 - “elmenetary” corrected to “elementary” (the most elementary use of)

§437 - “Mathematiks” corrected to “Mathematicks” (The Usefulness of Mathematicks) as in the quoted text.

§511 - Block number shown as 517

§517 - “hoheren” corrected to “höheren” (höheren Schulen) for consistency

§540 - duplicate word “the” removed (let the mind)

§657 - “anaylsis” corrected to “analysis” (field of analysis.)

§729 - “Geomtry” corrected to “Geometry” (Algebra and Geometry)

§822 - end of quote not identified; placement unclear.

§823 - “heros” corrected to “heroes” (many of the major heroes)

§917 - “εὓυρηκα” corrected to “εὔυρηκα” (speaks of the εὔυρηκα)

§986 - added missing end quote

§1132 - “Vereiningung” corrected to “Vereinigung” ( Deutschen Mathematiker Vereinigung)

§1325 - “Philosphy” corrected to “Philosophy” (Positive Philosophy)

§1421 - “1427” corrected to block “1421” (=1421.=)

§1503 - “Todhunder’s” corrected to “Todhunter’s” (Todhunter’s History of)

§1535 - “uses” corrected to “use” (the use of analysis)

§1803 - “τὴυ” corrected to “τὴν” (μοῦ τὴν στέγην)

§1874 - “anaylsis” corrected to “analysis” (a kind of analysis)

§1930 - “Hyberbolic” corrected to “Hyperbolic” (Mathematical Tables, Hyperbolic Functions)

§2009 - “Stanfpunkte” corrected to “Standpunkte” (höheren Standpunkte aus)

§2126 - Omitted block number added

§2135 - “astromomy” corrected to “astronomy” (history of astronomy)

§2151 - “10” corrected to “9” (A to I represent 1-9)

Appolonius - also spelled “Apollonius” but not referenced at §523 and §917

Bôcher - “Bocher” corrected to “Bôcher” as given in text

Halsted - “Slyvester” corrected to “Sylvester” (And Sylvester)

Jefferson - “Om” corrected to “On” (On m. and law)

Peacock - “philosphers” corrected to “philosophers” (Greek philosophers)

End of Project Gutenberg's Memorabilia Mathematica, by Robert Edouard Moritz