Notes

¹The foot-notes give only a few of the authorities which might easily be cited. They are thought to include those from which considerable extracts have been made, the necessary condensation of these extracts making any other form of acknowledgment impossible.

²For a list of current mathematical journals see the Jahrbuch über die Fortschritte der Mathematik. A small but convenient list of standard periodicals is given in Carr’s Synopsis of Pure Mathematics, p. 843; Mackay, J. S., Notice sur le journalisme mathématique en Angleterre, Association française pour l’Avancement des Sciences, 1893, II, 303; Cajori, F., Teaching and History of Mathematics in the United States, pp. 94, 277; Hart, D. S., History of American Mathematical Periodicals, The Analyst, Vol. II, p. 131.

³For a list of such societies consult any recent number of the Philosophical Transactions of Royal Society of London. Dyck, W., Einleitung zu dem für den mathematischen Teil der deutschen Universitätsausstellung ausgegebenen Specialkatalog, Mathematical Papers Chicago Congress (New York, 1896), p. 41.

⁴Klein, F., The Present State of Mathematics, Mathematical Papers of Chicago Congress (New York, 1896), p. 133.

⁵Cantor, M., Geschichte der Mathematik, Vol. III, p. 94; Smith, H. J. S., Report on the theory of numbers; Collected Papers, Vol. I; Stolz, O., Grössen und Zahien, Leipzig. 1891.

⁶Brocard, H., Sur la fréquence et la totalité des nombres premiers; Nouvelle Correspondence de Mathématiques, Vols. V and VI; gives recent history to 1879.

⁷But see Favaro, A., Notizie storiche sulle frazioni continue dal secolo decimoterzo al decimosettimo, Boncompagni’s Bulletino, Vol. VII, 1874, pp. 451, 533.

⁸Klein, F., Vorträge über ausgewählte Fragen der Elementargeometrie, 1895, p. 38; Bachmann, P., Vorlesungen über die Natur der Irrationalzahlen, 1892.

⁹Riecke, F., Die Rechnung mit Richtungszahlen, 1856, p. 161; Hankel, H., Theorie der komplexen Zahlensysteme, Leipzig, 1867; Holzmüller, G., Theorie der isogonalen Verwandtschaften, 1882, p. 21; Macfarlane, A., The Imaginary of Algebra, Proceedings of American Association 1892, p. 33; Baltzer, R., Einführung der komplexen Zahlen, Crelle, 1882; Stolz, O., Vorlesungen über allgemeine Arithmetik, 2. Theil, Leipzig, 1886.

¹⁰Chapman, C. H., Weierstrass and Dedekind on General Complex Numbers, in Bulletin New York Mathematical Society, Vol. I, p. 150; Study, E., Aeltere und neuere Untersuchungen über Systeme complexer Zahlen, Mathematical Papers Chicago Congress, p. 367; bibliography, p. 381.

¹¹Klein, F., Evanston Lectures, Lect. VIII.

¹²Tait, P. G., on Quaternions, Encyclopædia Britannica; Schlegel, V., Die Grassmann’sche Ausdehnungslehre, Schlömilch’s Zeitschrift, Vol. XLI.

¹³These are set forth in a paper by J. W. Gibbs, Nature, Vol. XLIV, p. 79.

¹⁴For bibliography see Schlegel, V., Die Grassmann’sche Ausdehnungslehre, Schlömilch’s Zeitschrift, Vol. XLI.

¹⁵For Macfarlane’s Digest of views of English and American writers, see Proceedings American Association for Advancement of Science, 1891.

¹⁶Cayley, A., Equations, and Kelland, P., Algebra, in Encyclopædia Britannica; Favaro, A., Notizie storico-critiche sulla costruzione delle equazioni. Modena, 1878; Cantor, M., Geschichte der Mathematik, Vol. III, p. 375.

¹⁷Loria, Gino, Esame di alcune ricerche concernenti l’esistenza di radici nelle equazioni algebriche; Bibliotheca Mathematica, 1891, p. 99; bibliography on p. 107. Pierpont, J., On the Ruffini-Abelian theorem, Bulletin of American Mathematical Society, Vol. II, p. 200.

¹⁸Harley, R., A contribution of the history …of the general equation of the fifth degree, Quarterly Journal of Mathematics, Vol. VI, p. 38.

¹⁹See Art. 7.

²⁰Klein, F., Vorlesungen über das Ikosaeder, 1884.

²¹Netto, E., Theory of Substitutions, translated by Cole; Cayley, A., Equations, Encyclopædia Britannica, 9th edition.

²²Pierpont, James, Lagrange’s Place in the Theory of Substitutions, Bulletin of American Mathematical Society, Vol. I, p. 196.

²³Matthiessen, L. Grundzüge der antiken und modernen Algebra der litteralen Gleichungen, Leipzig, 1878.

²⁴Burkhardt, H., Die Anfänge der Gruppentheorie und Paolo Ruffini, Abhandlungen zur Geschichte der Mathematik, VI, 1892, p. 119. Italian by E. Pascal, Brioschi’s Annali di Matematica, 1894.

²⁵Muir, T., Theory of Determinants in the Historical Order of its Development, Part I, 1890; Baltzer, R., Theorie und Anwendung der Determinanten. 1881. The writer is under obligations to Professor Weld, who contributes Chap. II, for valuable assistance in compiling this article.

²⁶“Numerum $bb-ac$, cuius indole proprietates formæ imprimis pendere in sequentibus docebimus, determinantem huius uocabimus.”

²⁷Meyer, W. F., Bericht über den gegenwärtigen Stand der Invariantentheorie. Jahresbericht der deutschen Mathematiker-Vereinigung, Vol. I, 1890-91; Berlin 1892, p. 97. See also the review by Franklin in Bulletin New York Mathematical Society, Vol. III, p. 187; Biography of Cayley, Collected Papers, VIII, p. ix, and Proceedings of Royal Society, 1895.

²⁸See Art. 2.

²⁹Klein’s Evanston Lectures, Lect. I.

³⁰Williamson, B., Infinitesimal Calculus, Encyclopædia Britannica, 9th edition; Cantor, M., Geschichte der Mathematik, Vol. III, pp. 150-316; Vivanti, G., Note sur l’histoire de l’infiniment petit, Bibliotheca Mathematica, 1894, p. 1; Mansion, P., Esquisse de l’histoire du calcul infinitésimal, Ghent, 1887. Le deux centième anniversaire de l’invention du calcul différentiel; Mathesis, Vol. IV, p. 163.

³¹Carll, L. B., Calculus of Variations, New York, 1885, Chap. V; Todhunter, I., History of the Progress of the Calculus of Variations, London, 1861; Reiff, R., Die Anfänge der Variationsrechnung, Mathematisch-naturwissenschaftliche Mittheilungen, Tübingen, 1887, p. 90.

³²Bacharach, M., Abriss der Geschichte der Potentialtheorie, 1883. This contains an extensive bibliography.

³³Cantor, M., Geschichte der Mathematik, Vol. III, p. 429; Schlesinger, L., Handbuch der Theorie der linearen Differentialgleichungen, Vol. I, 1895, an excellent historical view; review by Mathews in Nature, Vol. LII, p. 313; Lie, S., Zur allgemeinen Theorie der partiellen Differentialgleichungen, Berichte über die Verhandlungen der Gesellschaft der Wissenschaften zu Leipzig, 1895; Mansion, P., Theorie der partiellen Differentialgleichungen ter Ordnung, German by Maser, Leipzig, 1892, excellent on history; Craig, T., Some of the Developments in the Theory of Ordinary Differential Equations, 1878-1893, Bulletin New York Mathematical Society, Vol. II, p. 119 ; Goursat, E., Leçons sur l’intégration des équations aux dérivées partielles du premier ordre, Paris, 1895; Burkhardt, H., and Heffier, L., in Mathematical Papers of Chicago Congress, p.13 and p. 96.

³⁴“In der ganzen modernen Mathematik ist die Theorie der Differentialgleichungen die wichtigste Disciplin.”

³⁵Grunert’s Archiv für Mathematik, Vol. LIV.

³⁶Klein’s Evanston Lectures, Lect. I.

³⁷Klein’s Evanston Lectures, Lect. II, III.

³⁸Riquier, C., Mémoire sur l’existence des intégrales dans un système differentiel quelconque, etc. Mémoires des Savants étrangers, Vol. XXXII, No. 3.

³⁹Cantor, M., Geschichte der Mathematik, Vol. III, pp. 53, 71; Reiff, R., Geschichte der unendlichen Reihen, Tübingen, 1889; Cajori, F., Bulletin New York Mathematical Society, Vol. I, p. 184; History of Teaching of Mathematics in United States, p. 361.

⁴⁰Bibliotheca Mathematica, 1892-94; historical.

⁴¹Historical Summary by Bôcher, Chap. IX of Byerly’s Fourier’s Series and Spherical Harmonics, Boston, 1893; Sachse, A., Essai historique sur la représentation d’une fonction …par une série trigonométrique. Bulletin des Sciences mathématiques, Part I, 1880, pp. 43, 83.

⁴²Brill, A., and Noether, M., Die Entwickelung der Theorie der algebraischen Functionen in alterer und neuerer Zeit, Bericht erstattet der Deutschen Mathematiker-Vereinigung, Jahresbericht, Vol. II, pp. 107-566, Berlin, 1894; Königsberger, L., Zur Geschichte der Theorie der elliptischen Transcendenten in den Jahren 1826-29, Leipzig, 1879; Williamson, B., Infinitesimal Calculus, Encyclopædia Britannica; Schlesinger, L., Differentialgleichungen, Vol. I, 1895; Casorati, F., Teorica delle funzioni di variabili complesse, Vol. I, 1868; Klein’s Evanston Lectures. For bibliography and historical notes, see Harkness and Morley’s Theory of Functions, 1893, and Forsyth’s Theory of Functions, 1893; Eneström, G., Note historique sur les symboles …Bibliotheca Mathematica, 1891, p. 89.

⁴³Enneper, A., Elliptische Funktionen, Theorie und Geschichte, Halle, 1890; Königsberger, L., Zur Geschichte der Theorie der elliptischen Transcendenten in den Jahren 1826-29, Leipzig, 1879.

⁴⁴Klein, Evanston Lectures, p. 3; Riemann and Modern Mathematics, translated by Ziwet, Bulletin of American Mathematical Society, Vol. I, p. 165; Burkhardt, H., Vortrag uber Riemann, Göttingen, 1892.

⁴⁵Klein, F., Riemann and Modern Mathematics, translated by Ziwet, Bulletin of American Mathematical Society, Vol. I, p. 165.

⁴⁶Bôcher, M., A bit of mathematical history, Bulletin of New York Mathematical Society, Vol. II, p. 107.

⁴⁷Merriman, M., Method of Least Squares, New York, 1884, p. 182; Transactions of Connecticut Academy, 1877, Vol. IV, p. 151, with complete bibliography; Todhunter, I., History of the Mathematical Theory of Probability, 1865; Cantor, M., Geschichte der Mathematik, Vol. III, p. 316.

⁴⁸Eneström, G., Review of Cantor, Bibliotheca Mathematica, 1896, p. 20.

⁴⁹Bulletin of New York Mathematical Society, Vol. II, p. 57.

⁵⁰Loria, G., Il passato e il presente delle principali teorie geometriche. Memorie Accademia Torino, 1887; translated into German by F. Schutte under the title Die hauptsächlichsten Theorien der Geometrie in ihrer früheren und heutigen Entwickelung, Leipzig, 1888; Chasles, M., Aperçu historique sur l’origine et le développement des méthodes en Géométrie, 1889; Chasles, M., Rapport sur les Progrès de la Géométrie, Paris, 1870; Cayley, A., Curves, Encyclopædia Britannica; Klein, F., Evanston Lectures on Mathematics, New York, 1894; A. V. Braunmühl, Historische Studie über die organische Erzeugung ebener Curven, Dyck’s Katalog mathematischer Modelle, 1892.

⁵¹Ball, W. W. R., On Newton’s classification of cubic curves. Transactions of London Mathematical Society, 1891, p. 104.

⁵²For details see Loria, Il passato e il presente, etc.

⁵³Loria, G., Notizie storiche sulla Geometria numerativa. Bibliotheca Mathematica, 1888, pp. 39, 67; 1889, p. 23.

⁵⁴Biographical Notice in Cayley’s Collected papers, Vol. VIII.

⁵⁵Klein, Evanston Lectures, Lect. I.

⁵⁶Dyck, W., Katalog mathematischer und mathematisch-physikalischer Modelle, München, 1892; Deutsche Universitätsausstellung, Mathematical Papers of Chicago Congress, p. 49.

⁵⁷Wiener, Chr., Lehrbuch der darstellenden Geometrie, Leipzig, 1884-87; Geschichte der darstellenden Geometrie, 1884.

⁵⁸On recent development of graphic methods and the influence of Monge upon this branch of mathematics, see Eddy, H. T., Modern Graphical Developments, Mathematical Papers of Chicago Congress (New York, 1896), p 58.

⁵⁹Klein, F., Erlangen Programme of 1872, Haskell’s translation, Bulletin of New York Mathematical Society, Vol. II, p. 215.

⁶⁰Todhunter, I., History of certain formulas of spherical trigonometry, Philosophical Magazine, 1873.

⁶¹Gunther, S., Die Lehre von den gewöhnlichen und verallgemeinerten Hyperbelfunktionen, Halle, 1881; Chrystal, G., Algebra, Vol. II, p. 288.

⁶²Smith, D. E., Biography of Lemoine, American Mathematical Monthly, Vol. III, p. 29; Mackay, J. S., various articles on modern geometry in Proceedings Edinburgh Mathematical Society, various years; Vigarié, É., Géométrie du triangle. Articles in recent numbers of Journal de Mathématiques spéciales, Mathesis, and Proceedings of the Association française pour l’avancement des sciences.

⁶³Klein, F., Vorträge über ausgewählten Fragen; Rudio, F., Das Problem von der Quadratur des Zirkels. Naturforschende Gesellschaft Vierteljahrschrift, 1890; Archimedes, Huygens, Lambert, Legendre (Leipzig, 1892).

⁶⁴Stäckel and Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss, Leipzig, 1895; Halsted, G. B., various contributions: Bibliography of Hyperspace and Non-Euclidean Geometry, American Journal of Mathematics, Vols. I, II; The American Mathematical Monthly, Vol. I; translations of Lobachevsky’s Geometry, Vasiliev’s address on Lobachevsky, Saccheri’s Geometry, Bolyai’s work and his life; Non-Euclidean and Hyperspaces, Mathematical Papers of Chicago Congress, p. 92. Loria, G., Die hauptsächlichsten Theorien der Geometrie, p. 106; Karagiannides, A., Die Nichteuklidische Geometrie vom Alterthum bis zur Gegenwart, Berlin, 1893; McClintock, E., On the early history of Non-Euclidean Geometry, Bulletin of New York Mathematical Society, Vol. II, p. 144; Poincaré, Non-Euclidean Geom., Nature, 45:404; Articles on Parallels and Measurement in Encyclopædia Britannica, 9th edition; Vasiliev’s address (German by Engel) also appears in the Abhandlungen zur Geschichte der Mathematik, 1895.

⁶⁵Fink, E., Kant als Mathematiker, Leipzig, 1889.

⁶⁶For an excellent summary of the results of the hypothesis, see an article by McClintock, The Non-Euclidian Geometry, Bulletin of New York Mathematical Society, Vol. II, p. 1.

⁶⁷Klein. Evanston Lectures. Lect. IX.

⁶⁸Bulletin of the American Mathematical Society (N. S.), Vol. XI, p. 115.

⁶⁹Compte rendu du deuxième congrès international des mathématiciens tenu à Paris, 1900. Paris, 1902, p. 115.

⁷⁰Göttinger Nachrichten, 1900, p. 253; Archiv der Mathematik und Physik, 1901, pp. 44, 213; Bulletin of the American Mathematical Society, 1902, p. 437.

⁷¹Bulletin of the American Mathematical Society (N. S.), Vol. X, p. 443.

⁷²It was founded as the New York Mathematical Society six years earlier, in 1888.

⁷³It is now, in 1905, approximately 500.

⁷⁴For students wishing to investigate the work of mathematicians who died in the last two decades of the nineteenth century, Eneström’s ”Bio-bibliographie der 1881-1900 verstorbenen Mathematiker,” in the Bibliotheca Mathematica Vol. II (3), p. 326 (1901), will be found valuable.

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