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