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 ALGARVE- -ALGEBRA ally believed that Bryophyta and vascular plants are descended from an Algal ancestry, it is natural to suppose that, prior to the luxuriant vegetable growths of Occurthe Carboniferous period, there must have existed the rocks. an a§e °1 Algae, it was doubtless this expectation that has led to the description of a number of Silurian and Devonian remains as Algae upon what is now regarded as indequate evidence. The geologic record is, as perhaps is to be expected, exceedingly poor, except as regards the Calcareous Siphonales, which are well represented at various horizons, from the Silurian to the Tertiary ; even the Diatomaceoe, which are found in great quantities in the Tertiary deposits, do not occur at all earlier than the chalk. It is believed, however, that the Devonian fossil, Nematophycus, is a Laminarian Alga, but it is not until the late Secondary and the Tertiary formations that fossil remains of Algse become frequent. (See Paljeobotany). The subjoined list includes the larger standard works on Algse, together with a number of papers to which reference is made in this article. For a detailed catalogue of Algological literature, see the “ Bibliotheca Phycologica ” in De Toni’s Sylloge Algarum, vol. i. (1889), with the addendum thereto in vol. iv. (1897) of the same work. General. Agardh, J. G. Species, genera, et ordines Algarum, vols. i.-iii., Lund, 1848-98 ; Analecta Algologica, Lund, 1892-96 ; Till Algernes Systematik, Lund, 1872-99.—Areschoug, J. E. “ Observationes Phycologicse, ” 2Tova Acta reg. soc. sci. Upsaliensis. Upsala, 1866-75.—Blackman, F. F. “The Primitive Algse and the Flagellata,” Ann. of Botany, vol. xiv. Oxford, 1900. Bornet, E., and Thuret, G. Notes AIgologiqucs. Fasc. i.-ii. Paris, 1876-80.—Dangeard, P. A. “Recherches sur les Algues inferieures,” Ann. des Sci. Naturelles, Bot., vol. vii. Paris, 1888.—Derbes, A., and Solier, A. J. J. Memoire de la Physiologic des Algues. Paris, 1856.—De Toni, J. B. Sylloge Algarum, vol. i., Chlorophyceoe ; vol. ii., Bacillariacece vol. iii., Fucoidece •, vol. iv., Floridece (in progress). Padua, 1889. — Falkenberg, P. “Die Algen im weitesten Sinne,” Schenk’s Handbuch der Botanik, vol. ii., 1882.—Farlow, W. G. Marine Algae of New England. Washington, 1881. — Harvey, W. H. Phycologia Britannica, 4 vols., London, 1846-55; Nereis Boreali-Americana, 3 pts., Washington, 1851-58 ; Phycologia Australica, 5 vols., London, 1858-63.—Hauck, F. “Die Meeresalgen Deutschlands und Oesterreichs,” Rabenhort’s Kryptogamen-Flora. Leipzig, 1885. —Kjellman, F. R. The Algce of the Arctic Sea. Stockholm, 1883. —Kutzing, F. T. Tabulce Phycologicce. 19 vols. Hordhausen, 1845-69.—Kuckcck, P. Beitrdge zur Kenntniss der Meeresalgen. Kiel and Leipzig, 1897-99.—Murray, G. Phycological Memoirs, London, 1892-95 ; An Introduction to the Study of Seaweeds, London, 1895.—Naegeli, C. Die neueren Algensysteme. Zurich, 1847.—Pringsheim, FT. “ Beitriige zur Morphologie der Meeresalgen,” Abhand. K'dnigl. Akad. der Wissensch. Berlin, 1862.— Reinke, J. Atlas Deutscher Meeresalgen. Berlin, 1889-92.— Schutt, F. Das Pflanzenleben der Hochsee. Leipzig, 1893.— Stackhouse, J. Nereis Britannica, ed. i., Bath, 1801 ; ed. ii., Oxford, 1816.—Thuret, G., and Bornet, E. Etudes phy-

Algarve, the southernmost province of Portugal, coinciding with the modern district of Faro, with an area of 1872 square miles and population 228,635—a density of 122 inhabitants to the square mile. The coast is fringed for a distance of 30 miles, from Quarteira to Tavira, with long sandy islands, through which there are six passages, the most important being the New Bar, between Faro and Olhao. In 1896 the fisheries yielded 61,132 tons of herring, valued at £79,100; in 1897, 76,845 tons, valued at £68,200; and in 1898, 65,637 tons, valued at £56,900. The sardines caught in the same years averaged £73,250 in value. The capital is Faro.

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cologiques. Paris, 1878.—Turner, D. Historia Fucorum. 4 vols. London, 1808-19. — Zanardini, G. Iconographia Phycologica Adriatica. Venice, 1860-76. 1. Cyanophyceae. Bornet, E., and Flahault, Ch. “Revision des Nostocacees heterocystees,” Ann. des Sc. Naturelles, Bot., vols. iii.-vii. Paris, 1887 -88.—Gomont, M. “Monographic des Oscillariees.” Ann. des Sc. Naturelles, Bot., vols. xv.-xvi. Paris, 1893. — Hegler. “ Uber Kerntheilungserscheinungen,” lief. Botan. Centralbl., vol. Ixiv. Cassel, 1895.—Kirchner, O. “ Schizophycese, ” in Engler and Prantl’s Pflanzcnfamilicn. Leipzig, 1900. 2. Chlorophyceae. Borzi A. “Studi anamorfici di alcune alghe verdi,” Bull. Soc. Bot. Ital. in N. Giorn. Bot. Ital., vol. xxii. Pisa, 1890.—Chodat, R. “ On the Polymorphism of the Green Algae,” Ann. of Botany, vol. xi. Oxford, 1897.—Cooke, M. C. British Freshwater Algae,, 2 vols., London, 1882-84 ; British Dcsmids, London, 1887.—Klebs, G. Die Bcdingungen der Fortpflanzung bei einigen Algen und Pilzen. Jena, 1896.—SolmsLaubach, H. Graf zu. “Monograph of the Acetabulariaceae,” Trans. Linn. Soc. (Bond.) Bot. London, 1895.—Wille, N. “Chlorophycese,” in Engler and Prantl’s Pflanzenfamilicn. Leipzig, 1897. 3. Phseophycese. Batters, E. A. L. “On Ectocarpus secundus,” Grevillea, vol. xxi. London, 1893.—Berthold, G. “Diegeschlechtliche Fortpflanzung der eigentlichen Phaeosporeen,” Mitth. Zool. Stat. Neapel, vol. ii. Leipzig, 1881.—Brebner, G. “On the Classification of the Tilopteridaceae,” Proc. Bristol Nat. Soc., vol. viii. Bristol, 1896-97. — Church, A. H. “ On the Polymorphy of Cutleria multifida,” Ann. of Botany, vol. xii. Oxford, 1898.—Farmer, J. B., and Williams, J. Ll. “Contributions to our Knowledge of the Life-history and Cytology of Fucacese,” Phil. Trans. Boy. Soc., vol. cxc. London, 1898.— Janczewski, E. “ Observations sur 1’accroissement du thalle des Plueosporees,” Mem. Soc. Nat. de Sc. Cherbourg, 1895.—Kjellman, F. R. “ Phseophycese” in Engler and Prantl’s Pflanzenfamilien. Leipzig, 1897.—Oltmanns, F. “ Beitrage zur Kenntniss der Fucaceen,” Bibliotheca bolanica, xiv. Cassall, 1889.—Sauvageau, C. “ Observations relatives a la sexualite des Phaeosporees,” Journal de Botanique, vol. x. Paris, 1896.—Strasburger, E. “ Kerntheilung und Befruchtung bei Fucus,” Cytologische Studien. Berlin, 1897.—Schutt, F. Die Peridinien der Plankton-Expedition. Keil and Leipzig, 1895.—Valiante, R. Le Cystoseirce del Golfo di Napoli. Leipzig, 1883.—Williams, J. Ll. “On the Antherozoids of Dictyota and Taonia,” Ann. of Botany, vol. xi. Oxford, 1897. 4. Rhodophyceae. Bertholu, G. “ Die Bangiacen des Golfes von Neapel,” Mitth. Zool. Stat. Neapel. Naples, 1882.—Oltmanns, Fr. “Zur Entwickelungsgeschichte der Florideen,” Botanische Zeitung, 1898. — Phillips, R. W. “The Development of the Cystocarp in Rhodymeniales,” i. and ii. Annals of Botany, vols. xi. xii. Oxford, 1897-98.—Schmitz, F. “ Untersuchungen fiber die Befruchtung der Florideen,” Sitzungsber. der kbnigl. Akad. der Wissensch. Berlin, 1883; “ Kleinere Beitrage zur Kenntniss der Florideen,” La Nuova Notarisia, 1892-94. — Schmitz, F., Falkenberg, P., Hauptfleisch, P. “Rhodophyceae,” in Engler and Prantl’s Pflanzenfamilien. 1897.—Schmidle, W. “Die Befruchtung, Keimung und Haarinsertion von Batrachospermum,” Bot. Zeitung. 1899.—Sirodot. Les Batrachospermes. Paris, 1884. Wille, N. “ Ueber die Befruchtung bei Nemalion multifidum,” Ber. d. deutschen bot. Gesellsc. Band xii. Berlin, 1894 ’ (R. W. P.)

the detailed exposition of any special variety of it. It is not easy to give a thoroughly satisfactory definition of algebra; for the purpose of this article “ an algebra ” will be taken to mean a mathematical calculus in which symbolical expressions of certain conventional types are constructed, and their relations investigated according to a fixed consistent system of formal rules of equivalence. This definition, imperfect as it is, will at least serve to indicate the essentially formal and abstract nature of algebraic reasoning. Whether a particular symbolism is to be regarded as an algebra or not depends partly upon its formal development, partly upon the sense attributed to the term “ mathematical.” Thus, for instance, the Algebra., Universal.—1. This term (employed methods of symbolic logic have produced a calculus which by Sylvester in a rather more restricted sense) has come to is generally recognized to be an algebra, because it admits be used to denote the comparative study of algebra in of a formal treatment which is independent of the pargeneral, as distinguished from its applications, and from ticular meaning originally given to its symbols; while, S. 1.-35