Polymorphism.

The difficulty of tracing the relationships of algae is largely due to the inadequacy of our knowledge of the conditions under which they pass through the crucial stages of their life-cycle. Of the thousands of species which have been distinguished, relatively few have been traced from spore to spore, as the flowering plants have been observed from seed to seed. The aquatic habit of most of the species and the minute size of many of them are difficulties which do not exist in the case of most seed-plants. From the analogy of the higher plants observers have justly argued that when they have seen and marked the characters of the reproductive organs they have found the plant at the stage when it exhibits its most noteworthy features, and they have named and classified the species in accordance with these observations. While even in such cases it is obvious that interesting stages in the life of the plant may escape notice altogether, in the cases of those plants the reproduction of which is unknown, and which have been named and placed on the analogy of the vegetative parts alone, there is considerable danger that a plant may be named as a distinct species which is only a stage in the life of another distinct and perhaps already known species. To take an example, Lemanea and Batrachospermum are Florideae which bear densely-whorled branches, but which, on the germination of the carpospore, give rise to a laxly-filamentous, somewhat irregularly-branched plant, from which the ordinary sexual plants arise at a later stage. This filamentous structure has been attributed to the genus Chantransia, which it greatly resembles, especially when, as is said to be the case in Batrachospermum, it bears similar monospores. The true Chantransia, however, bears its own sexual organs as well as monospores. To the specific identity of Haplospora globosa and Scaphospora speciosa, and of Cutleria muitifida and Aglaozonia reptans, reference has already been made. Again, many Green Algae—some unicellular, like Sphaerella and Chlamydomonas; some colonial forms, like Volvox and Hormotila; some even filamentous forms, like Ulothrix and Stigeoclonium— are known to pass into a condition resembling that of a Palmella, and might escape identification on this account.

It is, on the other hand, a danger in the opposite sense to conclude that all Chantransia species are stages in the life-cycle of other plants, and, similarly, that all irregular colonial forms, like Palmella, represent phases in the life of other Green Algae. Long ago Kutzing went so far as to express the belief that the lower algae were all capable of transformations into higher forms, even into moss-protonemata. Later writers have also thought that in all four groups of algae transformations of a most far-reaching character occur. Thus Borzi finds that Protoderma viride passes through a series of changes so varied that at different times it presents the characters of twelve different genera. Chodat does not find so general a polymorphism, but nevertheless holds that Raphidium passes through stages represented by Protococcus, Characium, Dactylococcus and Sciadium. Klebs has, however, recently canvassed the conclusions of both these investigators; and as the result of his own observations declares that algae, so far from being as polymorphic as they have been described, vary only within relatively narrow limits, and present on the whole as great fixity as the higher plants. It certainly supports his view to discover, on subjecting to a careful investigation Botrydium granulatum, a siphonaceous alga whose varied forms had been described by J. Rostafinski and M. Woronin, that these authors had included in the life-cycle stages of a second alga described previously by Kutzing, and now described afresh by Klebs as Protosiphon bolryoides. In Botrydium the chromatophores are small, without pyrenoids, and oil-drops are present; in Protosiphon the chromatophores form a net-work with pyrenoids, and the contents include starch. Klebs insists that the only solution of such problems is the subjection of the algae in question to a rigorous method of pure culture. It is interesting to learn that G. Senn, pursuing the methods described by Klebs, has confirmed Chodat's observation of the passage of Raphidium into a Dactylococcus-stage, although he was unable to observe further metamorphosis. He has also seen Pleurococcus viridis dividing so as to form a filament, but has not succeeded in seeing the formation of zoospores as described by Chodat. While, therefore, there is much evidence of a negative character against the existence of an extensive polymorphism among algae, some amount of metamorphosis is known to occur. But until the conditions under which a particular transformation takes place have been ascertained and described, so that the observation may be repeated by other investigators, scant credence is likely to be given to the more extreme polymorphistic views.

Physiology.

In comparison with the higher plants, algae exhibit so much simplicity of structure, while the conditions under which they grow are so much more readily controlled, that they have frequently been the subject of physiological investigation with a view chiefly to the application of the results to the study of the higher plants. (See PLANTS: Physiology of.) In the literature of vegetable physiology there has thus accumulated a great body of facts relating not only to the phenomena of reproduction, but also to the nutrition of algae. With reference to their chemical physiology, the gelatinization of the cell-wall, which is so marked a feature, is doubtless attributable to the occurrence along with cellulose of pectic compounds. There is, however, considerable variation in the nature of the membrane in different species; thus the cell-wall of Gedogonium, treated with sulphuric acid and iodine, turns a bright blue, while the colour is very faint in the case of Spirogyra, the wall of which is said to consist for the most part of pectose. While starch occurs commonly as a cell-content in the majority of the Green Algae no trace of it occurs in Vaucheria and some of its allies, nor is it known in the whole of the Phaeophyceae and Rhodophyceae. In certain Euphaeophyceae bodies built up of concentric layers, and attached to the chromatophores, were described by Schmitz as phaeophycean-starch; they do not, however, give the ordinary starch reaction. Other granules, easily mistaken for the "starch'' granules, are also found in the cells of Phaeophyceae; these possess a power of movement apart from the protoplasm, and are considered to be vesicles and to contain phloroglucin. The colourless granules of Florideae, which are supposed to constitute the carbohydrate reserve material, have been called floridean-starch. A white efflorescence which appears on certain Brown Algae (Saccorhiza bulbosa, Laminaria saccharina), when they are dried in the air, is found to consist of mannite. Mucin is known in the cell-sap of Acetabularia. Some Siphonales (Codium) give rise to proteid crystalloids, and they are of constant occurrence among Florideae. The presence of tannin has been established in the case of a great number of freshwater algae.

Colouring matters.

By virtue of the possession of chlorophyll all algae are capable of utilizing carbonic acid gas as a source of carbon in the presence of sunlight. The presence of phycocyanin, phycophaein and phycoerythrin considerably modifies the absorption spectra for the plants in which they occur. Thus in the case of phycoerythrin the maximum absorption, apart from the great absorption at the blue end of the spectrum, is not, as in the case where chlorophyll occurs alone, near the Fraunhofer line B, but farther to the right beyond the line D. By an ingenious method devised by Engelmann, it may be shown that the greatest liberation of oxygen, and consequently the greatest assimilation of carbon, occurs in that region of the spectrum represented by the absorption bands. In this connexion Pfeffer points out that the penetrating power of light into a clear sea varies for light of different colours. Thus red light is reduced to such an extent as to be insufficient for growth at a depth of 34 metres, yellow light at a depth of 177 metres and green light at 322 metres. It is thus an obvious advantage to Red Algae, which flourish at considerable depths, to be able to utilize yellow light rather than the red, which is extinguished so much sooner. The experiment of Engelmann referred to deserves to be mentioned here, if only in illustration of the use to which algae have been put in the study of physiological problems. Engelmann observed that certain bacteria were motile only in the presence of oxygen, and that they retained their motility in a microscopic preparation in the neighbourhood of an algal filament when they had come to rest elsewhere on account of the exhaustion of oxygen. After the bacteria had all been brought to rest by being placed in the dark, he threw a spectrum upon the filament, and observed in what region the bacteria first regained their motility, owing to the liberation of oxygen in the process of carbon-assimilation. He found that these places corresponded closely with the region of the absorption band for the algae under experiment.

Although algae generally are able to use carbonic acid gas as a source of carbon, some algae, like certain of the higher plants, are capable of utilizing organic compounds for this purpose. Thus Spirogyra filaments, which have been denuded of starch by being placed in the dark, form starch in one day if they are placed in a 10 to 20% solution of dextrose. According to T. Bokorny, moreover, it appears that such filaments will yield starch from formaldehyde when they are supplied with sodium oxymethyl sulphonate, a salt which readily decomposes into formaldehyde and hydrogen sodium sulphite, an observation which has been taken to mean that formaldehyde is always a stage in the synthesis of starch. With reference to the assimilation of nitrogen, it would seem that algae, like other green plants, can best use it when it is presented to them in the form of a nitrate. Some algae, however, seem to flourish better in the presence of organic compounds. In the case of Scenedesmus acutus it is said that the alga is unable to take up nitrogen in the form of a nitrate or ammoniacal salt, and requires some such substance as an amide or a peptone. On the other hand, it has been held by Bernhard Frank and other observers that atmospheric nitrogen is fixed by the agency of Green Algae in the soil: (For the remarkable symbiotism between algae and fungi see FUNGI and LICHENS.)

Habitat.

Most algae, particularly Phaeophyceae and Rhodophyceae, spend the whole of the life-cycle immersed in water. In the case of the freshwater algae, however, belonging to the Chlorophyceae and Cyanophyceae, although they required to be immersed during the vegetative period, the reproductive cells are often capable of resisting a considerable degree of desiccation, and in this condition are dispersed through great distances by various agencies. Again, as is well known, many species of marine algae growing in the region between the limits of high and low water are so constituted that they are exposed to the air twice a day without injury. The occurrence of characteristic algae at different levels constituting the zones to which reference has already been made, is probably in part an expression of the fact that different species vary in the capacity to resist desiccation from exposure. Thus Laminaria digitata, which characterizes the lowest zone, is only occasionally exposed at all, and then only for short periods of time. On the other hand, Pelvetia canaliculata, which marks the upper belt, is exposed for longer periods, and during neap tides may not be reached by the water for many days. Algae of more delicate texture than either Fucaceae or Laminariaceae also occur in the region exposed by the ebb of the tide, but these secure their exemption from desiccation either by retaining water in their meshes by capillary attraction, as in the case of Pilayella, or by growing among the tangles of the larger Fucaceae, as in the case of Polysiphonia fastigiata, or by growing in dense masses on rocks, as in the case of Laurencia pinnatifida. Such a species as Delesseria sanguinea or Callophyllis laciniata would on the contrary run great risk by exposure for even a short period. A few algae approach the ordinary terrestrial plants in their capacity to live in a sub-aerial habitat subject only to such occasional supphes of water as is afforded by the rainfall. Of this nature are some of the species of Vaucheria. A very few species, like Chroolepus, which grows on rock surfaces, are comparable with the land plants which have been termed xerophilous.

Plankton.

The great majority of the aquatic algae, both freshwater and marine, are attached plants. Some, however, are wanderers, either swimming actively with the aid of cilia, or floating inertly as the result of a specific weight closely approaching that of the medium. To the aggregate of such forms, both animal and vegetable, the term plankton has been applied, and the investigation of the vegetable plankton, both freshwater and marine, has been pursued in recent times with energy and success. The German Plankton Expedition of 1889 added greatly to our knowledge of the floating vegetable life of the North Atlantic Ocean, while many laboratories established on the shores of inland seas and lakes have rendered a similar service in the case of our freshwater phyto-plankton. The quantitative estimate of the amount of this flora has revealed its enormous aggregate amount and therefore its great importance in the economy of oceanic and lacustrine animal life. The organisms constituting this plankton are mostly unicellular, often aggregated together in colonies, and the remarkable structure which they exhibit has added a new chapter to the story of adaptation to environment. The families Diatomaceae, Peridiniaceae and Protococcaceae are best represented in the pelagic plankton, while in addition the Volvocaceae are an important element in freshwater plankton.

Benthos.

The great majority of algae, however, grow like land-plants attached to a substratum, and to these the term benthos is now generally applied. While the root of land-plants serves for the double purpose of attachment and the supply of water, it is attachment only that is usually sought in the case of algae. Immersed as they usually are in a medium containing in solution the inorganic substances which they require for their nutrition, the absorption of these takes place throughout their whole extent. The elaborate provision for the conduct of water from part to part which has played so important a role in the morphological development of land plants is entirely wanting in algae, such conducting tissues as do exist in the larger Phaeophyceae and Rhodophyceae serving rather for the convection of elaborated organic substance, and being thus comparable with the phloem of the higher plants. The attachment organ of algae is thus more properly called a holdfast, and is found to be of very varied structure. It generally takes the form of a single flattened disc as in the Fucaceae, or a group of finger- like processes as in Laminariaceae, or a tuft of filaments as in many instances. When the attachment is in sand or mud, it often simulates the appearance of a true root as in Chara or Caulerpa. It is clear that where the bottom of a lake or sea consists of oozy mud or shifting sand, it is impossible for algae to secure a foothold. Thus a rock emerging from a sandy beach may often be observed to stand covered with vegetation like an oasis in a desert. The rapidity with which walls, piles and pontoons—stone, wood and iron—become covered with marine plants is well known, while the discovery of some effective means of preventing the fouling of the bottoms of ships by the growth of algae would be hailed as a boon by shipowners. While rocks and boulders are the favoured situation for the growth of marine algae, those which readily disintegrate, like the coarser sandstones, are naturally less favoured than the hard and resistant. A large number of algae again live as epiphytes or endophytes. In the case of the freshwater species the host-plants are mostly species of aquatic Graminaceae, Naiadaceae or Nymphaeaceae. In the case of marine algae, the hosts are chiefly the larger Phaeophyceae and Rhodophyceae. A bed of Zostera near the level of low water is, however, on the British coast a favourite collecting ground for the smaller red and brown epiphytes. Of endophytes a distinction must be made between those which occupy the cell-wall only and those which perforate the cells, bringing about their destruction. There can be little doubt that in some cases the epiphytism approaches parasitism. In one case described by Kuckuck the chromaphores of the infesting algae are absent, a circumstance which points to a complete parasitism. Allusion has already been made to the peculiar habit of the shell-boring algae.

Habit.

In many algae certain branches of limited growth bear a remarkable resemblance to leaves. The Characeae among freshwater algae and the Sargassaceae among marine algae might be cited as examples. Surveying the whole range of algae life, Oltmanns distinguishes bush-forms, whip- forms, net-forms, leaf-forms, sack-forms, dorsi-ventral forms, and cushions, plates and crusts. The similarity of outline in many species to that of trees and shrubs will strike any one who examines algae mounted for the herbarium. Cladophora and Bryopsis among monosiphonous forms, Chara, Polysiphonia, Ceramium and Cystoseira among larger algae, are illustrations of this. The whip-forms are represented by Spirogyra, Chaetomorpha, Scytosiphon, Nemalion, Himanthalia and Chorda. Net-forms are found in Hydrodictyon and Microdictyon. The leaf-forms are very varied and owe their existence to the advantage accruing from the exposure of a large surface to the influence of the light. In some cases such as Delesseria, Neurymenia, Fucus, Alaria, the leaf-like structure is provided with a strengthening mid-rib, and when as in Delesseria it is also richly veined the resemblance to the leaf of a flowering plant is striking. Laminaria, Padina, Cutleria, Punctaria, Iridaea, Ulva, Porphyra, are leaf-like with a rigidity varying from a fleshy lamina to the thin and pliable. Agarum, Claudea and Struvea are leaf-forms which are perforated like Aldrovanda among flowering plants. Enteromorpha, Asperococcus and Adenocystis are sack-forms. Dorsi-ventral algae are rare. Leveillea jungermanneoides bears a remarkable resemblance to a leafy liverwort. In the next group of forms the simplest are crusts attached to the substratum throughout their extent, and growing at the margin. Such are Myrionema, Ralfsia, Melobesia and Hildebrandtia. Others are attached throughout their extent, but also grow vertical filaments so as to form a velvety pile. Such are Coleochaete, Ochlochaete, Elachistea, Ascocyclus and Rhododermis. Peysonellia squamaria, Melobesia lichenoides, Leathesia difformis are forms which are not attached throughout but grow in plates like the foliaceous lichens.

Ecology.

When it is sought to consider algae with a view to the correlation of the external form to the conditions of life, a subject the study of which under the name of ecology has been latterly pursued with great success among land plants, it is difficult as yet to arrive at generalizations which are trustworthy. Among land plants, as is well known, similarity of environment has often called forth similar adaptations among plants of widely separated families. The similarity of certain xerophilous Euphorbiaceae to Cactaceae is a ready illustration of this phenomenon. From what has been already said it is evident that among algae also strikingly similar forms exist in widely different groups. Instances might be multiplied. Compare, for example, the blue-green Gloeocapsa with the green Gloeocystis, the red Batrachospermum with the green Draparnaldia, the red Corallina with the green Cymopolia, the green Enteromorpha with the brown Asperococcus, the green Ulva with the red Porphyra, the red Nemalion with the brown Castagnea, and so on. But on the one hand similar forms seem to grow often under different conditions, while on the other hand different forms flourish under the same conditions. The conceivable variations in the conditions which would count in algal life are variations in the chemical character of the water—whether fresh, brackish or salt; or in the rate of movement of the water, whether relatively quiet, or a stream or a surf; or in the degree of illumination with the depth and transparency of the water. But the laws which determine the associations of various algae under one environment are as yet little understood. The occurrence of a plentiful mucilage in many freshwater forms is, however, doubtless a provision against desiccation on exposure. The fine subdivision of filamentous and net-forms is similarly a provision for easy access of water and light to all parts. The calcareous deposits in Characeae, Corallinaceae and Siphonaceae are at once a protection against attack and a means of support. The whip-forms would seem to be designed to resist injury from surf or current. The vesicles of Fucaceae and Laminariaceae prevent the sinking of the bulkier forms. But why certain Fucaceae favour certain zones in the littoral region, why certain epiphytes are confined to certain hosts, why Red and Brown Algae are not better represented in fresh water Or Green Algae in salt,—these are problems to which it is difficult to find a ready answer.

Uses.

Algae cannot be regarded as directly important in the industries. On the coasts of Europe marine algae detached by the autumnal gales are commonly carted on to the land as a convenient manure. Porphyra laciniata and Rhodymenia palmata are locally used as food, the latter being known as dulse. Agar-agar is a gelatinous substance derived from an eastern species of Gracilaria. The ash of seaweeds, known in Scotland as kelp, and in Brittany as varec, was formerly used as a source of iodine to a greater extent than is at present the case.

Occurence in the rocks.

Excepting where the thallus is impregnated with silica, as in Diatomaceae, or carbonate of lime, as in Corallinaceae, Characeae and some Siphonales, it is perhaps not surprising that algae should not have been extensively preserved in the fossil form. Considering, however, that it is generally 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 the Carboniferous period, there must have existed an age of 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 inadequate 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 Diatomaceae, 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 algae become frequent. (See PALAEOBOTANY.)

The subjoined list includes the larger standard works on algae, 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 Tonii's Syllope Algarum, vo1. i. (1889), with the addendum thereto in vol. iv. (1897) of the same work. GENERAL.—J. G. Agardh, Species, genera et ordines Algarum (vols. i-iii., Algernes Systematik (Lund, 1872-1899); J. E. Areschoug, "Observationes Phycologicae,'' Nova Acta reg. soc. sci. Upsaliensis (Upsala, 1866-1875); F. F. Blackman, "The Primitive Algae and the Flagellata,'' Ann. of Botany (vol. xiv., Oxford, 1900); E. Bornet and G. Thuret, Notes agologiques (fasc. i.-ii., Paris, 1876-1880); P. A. Dangeard, "Recherches sur les algues inferieures,'' Ann. des sci. naturelles, Bot. (vol. vii., Paris, 1888); A. Derbes and A. J. J. Solier, Momoire de la physiologie des algues (Paris, 1856); J. B. de Toni, Sylloge Algarum—-vol. i. Chlorophyceae, vol. ii. Bacillariaceae, vol. iii. Fucoideae, vol. iv. Florideae (Padua, 1889-1900); P. Falkenberg, "Die Algen im weitesten Sinne,'' Schenk's Handbuch der Botanik (vol. ii., 1882); W. G. Farlow, Morine Algae of New England (Washington, 1881); W. H. Harvey, Phycologia Britannica (4 vols., London, 1846-1855); Nereis Boreali-Americana (3 pts., Washington, 1851-1858); Phycologia Australica (5 vols., London, 1858-1863); F. Hauck, "Die Meeresalgen Deutschlands und Osterrichs,'' Rabenhort's Kryptogamen-Flora (Leipzig, 1885); F. R. Kjellman, The Algae of the Arctic Sea (Stockholm, 1883); F. T. Kutzing, Tabulae Phycologicae (19 vols., Nordhausen, 1845-1869); P. Kuckuck, Beitrage zur Kenntniss der Meercsalgen (Kiel and Leipzig, 1897-1899); G. Murray, Phycological Memoirs (London, 1892-1895) Naegeli, Die neueren Algensysteme (Zurich, 1847); F. Oltmanns, Morphologie und Biologie der Algen (Jena, Band i. 1904, Band ii. 1905); N. Pringsheim, "Beitrage zur Morphologie der Meeresalgen,'' Abhand. Konigl. Akad. der Wissensch. (Berlin, 1862); J. Reinke, Atlas deutscher Meeresalgen (Berlin, 1889-1892); F. Schutt, Das Pflanzenleben der Hochsee (Leipzig, 1893); J. Stackhouse, Nereis britannica (ed. i., Bath, 1801; ed. ii., Oxford, 1816); G. Thuret and E. Bornet, Etudes phycologiques (Paris, 1878); D. Turner, Historia Fucorum (4 vols., London, 1808-1819); G. Zanardini, Iconographia Phycologia Adriatica (Venice, 1860-1876).

1. CYANOPHYCEAE.—E. Bornet and Ch. Flahault, "Revision des Nostocacees heterocystees,'' Ann. des sc. naturelles, Bot.(vols. iii.-vii., Paris, 1887-1888); M. Gomont, "Monographic des Oscillariees,'' Ann. des sc. naturelles, Bot. (vols. xv.-xvi., Paris, 1893); Hegler, "Uber Kerntheilungserscheinungen,'' Ref. Botan. Centralbl. (vol. lxiv., Cassel, 1895); O. Kirchner, "Schizophyceae'', in Engler and Prantl's Pflanzenfamilien (Leipzig, 1900).

2. CHLOROPHYCEAE.—A. Borzi, "Studi anamorfici di alcune alghe verdi,'' Bull. Soc. Bot. Ital. in N. Giorn. Bot. Ital. (vol. xxii., Pisa, 1890); F. F. Blackman and A. G. Tansley, A Revision of the Classification of the Green Algae, reprinted from the New Phytologist (vol. i., London, 1903); K. Bohlin, "Studier ofver nagra slagten af alggruppen confervales Borzi,'' Bihang till K. Svenska vel. akad. Handlinger (Bd. xxiii. afd. 3, 1897);—Ufkasttill, De grona algernas och arkegomiaternas bylogeni (Upsala, 1901); R. Chodat, "On the Polymorphism of the Green Algae,'' Ann. of Botany (vol. xi., Oxford, 1897); M. C. Cooke, British Freshwater Algae (2 vols., London, 1884), British Desmids (London, 1887); G. Klebs, Die Bedingungen der Fortpflanzung bei einigen Algen und Pilzen (Jena, 1896); A. Luther, "Uber Chlorosaccus, n.g.'' Bihang till K. Svenska vel. akad. Handlinger (Bd. xxiv. afd. 3, 1899); H. Grat zu Solms-Laubach, "Monograph of the Acetabulariaceae,'' Trans. Linn. Soc. (Lond.) Bot. (London, 1895); N. Wille, "Chlorophyceae'', in Engler and Prantl's Pflanzenfamilien (Leipzig, 1897).

3. PHAEOPHYCEAE.—E. A. L. Batters, "On Ectocarpus secundus,'' Grevillea (vol. xxi., London, 1893); G. Berthold, "Die geschlechliche Fortpflanzung der eigentlichen Phaeosooreen,'' Mitth. Zool. Stat. Neapel (vol. ii., Leipzig, 1881); G. Brebner, "On the Classification of the Tilopteridaceae,'' Proc. Bristol Nat. Soc. (vol. viii., Bristol, 1896-1897); A. H. Church, "On the Polymorphy of Cutleria multiflda,'' Ann. of Botany (vol. xii., Oxford, 1898); J. B. Farmer esnd J. Ll. Williams, "Contributions to our Knowledge of the Life- history and Cytology of Fucaceae,'' Phil. Trans. Roy. Soc. (vol. cxc., London, 1898); E. Janczewski, "Observations sur l'accroissement du thalle des Phaeosporees,'' Mem. soc. nat. de sc. (Cherbourg, 1895); F. R. Kjellmann, "Phaeophyceae,'' in Engler and Prantl's Pflanzenfamilian (Leipzig, 1897); F. Oltmanns, "Beitrage zur Kenntniss der Fucaceen,'' Bibliotheca botanica, xiv. (Cassel, 1889); C. Sauvageau, "Observations relatives a la sexualite des Phaeosporees,'' Journal de botanique (vol. x., Paris, 1896); E. Strasburger, "Kerntheilung und Befruchtung bei Fucus,'' Cytologische Studien (Berlin, 1897); F. Schutt, Die Peridinien der Plankton-Expedition (Kiel and Leipzig, 1895); R. Valiante, Le Cystoseirae del Golfo di Napoli (Leipzig, 1883); J. Ll. Williams, "On the Antherozoids of Dictyota and Taonia,'' Ann. of Botany (vol. xi., Oxford, 1897).

4. RHODOPHYCEAE.—G. Berthold, "Die Bangiacen des Golfes von Neapel,'' Mitth. Zool. Stat. Neapel (Naples, 1882); F. Oltmanns, "Zur Entwickelungsgeschichte der Florideen,'' Botanische Zeitung (1898); R. W. Philligs, "The Development of the Cystocarp in Rhodymeniales,'' i. and ii., Annals of Botany (vols. xi. xii., Oxford. 1897-1898); F. Schmitz, "Untersuchungen uber die Befruchtung der Florideen,'' Sitzungsber. der konigl. Akad.der Wissensch. (Berlin, 1883); "Kleinere Beitrage zur Kenntniss der Florideen,'' La Nuova Notarisia, 1892-1894; F. Schmitz, P. Falkenberg, P. Hauptfleisch, "Rhodophyceae,'' in Engler and Prantl's Pflanzenfamilien (1897); W. Schmidle, "Die Befruchtung, Keimung und Haarinsertion von Batrachospermum,'' Bot. Zeitung.. (1899); Sirodot, Les Batrachospermes (Paris, 1884); N. Wille, "Uber die Befruchtung bei Nemalion multifidum,'' Ber. d. deutschen bot. Gesellsc. Band xii. (Berlin, 1894); J. J. Wolfe, "Cytological Studies on Nemalion,'' Annals of Botany (vol. xviii., Oxford, 1904); S. Yamanouchi, "The Life- History of Polysiphonia violacea,'' Botanical Gazette (vol. xli., Chicago, 1906). (R. W. P.)

ALGARDI, ALESSANDRO (1602-1654), Italian sculptor, was born at Bologna in 1602. While he was attending the school of the Caracci his preference for the plastic art became evident, and he placed himself under the instruction of the sculptor Conventi. At the age of twenty he was brought under the notice of Duke Ferdinand of Mantua, who gave him several commissions. He was also much employed about the same period by jewellers and others in modelling in gold, silver and ivory. After a short residence in Venice he went to Rome in 1625 with an introduction from the duke of Mantua to the pope's nephew, Cardinal Ludovisi, who employed him for a time in the restoration of ancient statues. The death of the duke of Mantua left him to his own resources, and for several years he earned a precarious maintenance from these restorations and the commissions of goldsmiths and jewellers. In 1640 he executed for Pietro Buoncompagni his first work in marble, a colossal statue of San Filippo Neri, with kneeling angels. Immediately after, he produced a similar group, representing the execution of St Paul, for the church of the Barnabite Fathers in Bologna. These works, displaying great technical skill, though with considerable exaggeration of expression and attitude, at once established Algardi's reputation, and other commissions followed in rapid succession. The turning point in Algardi's fortune was the accession of Innocent X., of the Bolognese house of Panfili, to the papal throne in 1644. He was employed by Camino Panfili, nephew of the pontiff, to design the Villa Doria Panfili outside the San Pancrazio gate. The most important of Algardi's other works were the monument of Leo XI., a bronze statue of Innocent X. for the capitol, and, above all, La Fuega d'Attila, the largest alto-relievo in the world, the two principal figures being about 10 ft. high. In 1650 Algardi met Velasquez, who obtained some interesting orders for his Italian companion in Spain. Thus there are four chimneys by Algardi in the palace of Aranjuez, where also the figures on the fountain of Neptune were executed by him. The Augustine monastery at Salamanca contains the tomb of the count and countess de Monterey, which was also the work of Algardi. From an artistic point of view, he was most successful in his portrait-statues and groups of children, where he was obliged to follow nature most closely. In his later years he became very avaricious and amassed a great fortune. He died in Rome on the 10th of June 1654.

See Le arti di Bologna disegnate da A. Caracci ed intagliati da S. Giulini, con' assistenza d' Alessandro A. Algardi (1740).

ALGAROTH, POWDER OF, a basic chloride of antimony. It was known to Basil Valentine, and was used medicinally by the Veronese physician Victor Algarotus about the end of the 16th century. Its composition is probably Sb4O5Cl2, and it may be prepared by the addition of much water to a solution of antimony chloride; a bulky amorphous precipitate being formed, which, on standing, gradually becomes crystalline. It is soluble in hydrochloric acid and tartaric acid, but insoluble in alcohol.

On its composition and preparation see E. Peligot, Annalen, 1847, lxiv. 280; L. Schaffer, Annalen, 1869, clii. 314; and R. W. E. MacIvor, Chem. News, 1875, xxxii. 229.

ALGAROTTI, FRANCESCO, COUNT (1712-1764), Italian philosopher and writer on art, was born on the 11th of December 1712 at Venice, and died at Pisa in 1764. He studied at Rome and Bologna, and at the age of twenty went to Paris, where he enjoyed the friendship of Voltaire and produced his great work Neutonianismo per le dame, a work on optics. Voltaire called him his cher cygne de Padoue. Returning from a journey to Russia, he met Frederick the Great who made him a count of Prussia (1740) and court chamberlain (1747). Augustus III. of Poland honoured him with the title of councillor. In 1754, after seven years' residence partly in Berlin and partly in Dresden, he returned to Italy, living at Venice and then at Pisa, where he died on the 3rd of May 1764. Frederick the Great erected to his memory a monument on the Campo Santo at Pisa. He was a man of wide knowledge, a connoisseur in art and music, and the friend of most of the leading authors of his time. His chief work on art is the Saggi sopra le belle arti ("Essays on the Fine Arts''). Among his other works may be mentioned Poems, Travels in Russia, Essay on Painting, Correspondence.

The best complete edition with biography was published by D. Michelessi (1791-1794).

ALGARVE, or ALGARVES, an ancient kingdom and province in the extreme S. of Portugal, corresponding with the modern administrative district of Faro, and bounded on the N. by Alemtejo, E. by the Spanish province of Huelva, and S. and W. by the Atlantic Ocean. Pop. (1900) 255,191; area, 1937 sq. m. The greatest length of the province is about 85 m. from E. to W.; its average breadth is about 22 m. from N. to S. The Serra de Malhao and the Serra de Monchique extend in the form of a crescent across the northern part of the province, and, sweeping to the south-west, terminate in the lofty promontory of Cape St Vincent, the south-west extremity of Europe. This headland is famous as the scene of many sea-fights, notably the defeat inflicted on the Spanish fleet in February 1797 by the British under Admiral Jervis, afterwards Earl St Vincent. Between the mountainous tracts in the north and the southern coast stretches a narrow plain, watered by numerous rivers flowing southward from the hills. The coast is fringed for 30 m. from Quarteira to Tavira, with long sandy islands, through which there are six passages, the most important being the Barra Nova, between Faro and Olinao. The navigable estuary of the Guadiana divides Algarve from Huelva, and its tributaries water the western districts. From the Serra de Malhao flow two streams, the Silves and Odelouca, which unite and enter the Atlantic below the town of Silves. In the hilly districts the roads are bad, the soil unsuited for cultivation, and the inhabitants few. Flocks of goats are reared on the mountain-sides. The level country along the southern coast is more fertile, and produces in abundance grapes, figs, oranges, lemons, olives, almonds, aloes, and even plantains and dates. The land is, however, not well suited for the production of cereals, which are mostly imported from Spain. On the coast the people gain their living in great measure from the fisheries, tunny and sardines being caught in considerable quantities. Salt is also made from sea-water. There is no manufacturing or mining industry of any importance. The harbours are bad, and almost the whole foreign trade is carried on by ships of other nations, although the inhabitants of Algarve are reputed to be the best seamen and fishermen of Portugal. The chief exports are dried fruit, wine, salt, tunny, sardines and anchovies. The only railway is the Lisbon-Faro main line, which passes north-eastward from Faro, between the Monchique and Malhao ranges. Faro (11,789), Lagos (8291), Loule (22,478), Monchique (7345), Olhao (10,009), Silves (9687) and Tavira (12,175), the chief towns, are described in separate articles.

The name of Algarve is derived from the Arabic, and signifies a land lying to the west. The title "king of Algarve,'' held by the kings of Portugal, was first assumed by Alphonso III., who captured Algarve from the Moors in 1253.

ALGAU, or ALLGAU, the name now given to a comparatively small district forming the south-western corner of Bavaria, and belonging to the province of Swabia and Neuburg, but formerly applied to a much larger territory, which extended as far as the Danube on the N., the Inn on the S. and the Lech on the W. The Algau Alps contain several lofty peaks, the highest of which is Madelegabel (8681 ft.). The district is celebrated for its cattle, milk, butter and cheese.

ALGEBRA (from the Arab. af-jebr wa'l-muqabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi), a branch of mathematics which may be defined as the generalization and extension of arithmetic.

The subject-matter of algebra will be treated in the following article under three divisions:—-A. Principles of ordinary algebra; B. Special kinds of algebra; C. History. Special phases of the subject are treated under their own headings, e.g. ALGEBRAIC FORMS; BINOMIAL; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; CONTINUED FRACTION; FUNCTION; GROUPS, THEORY OF; LOGARITHM; NUMBER; PROBABILITY; SERIES.

A. PRINCIPLES OF ORDINARY ALGEBRA

1. The above definition gives only a partial view of the scope of algebra. It may be regarded as based on arithmetic, or as dealing in the first instance with formal results of the laws of arithmetical number; and in this sense Sir Isaac Newton gave the title Universal Arithmetic to a work on algebra. Any definition, however, must have reference to the state of development of the subject at the time when the definition is given.

2. The earliest algebra consists in the solution of equations. The distinction between algebraical and arithmetical reasoning then lies mainly in the fact that the former is in a more condensed form than the latter; an unknown quantity being represented by a special symbol, and other symbols being used as a kind of shorthand for verbal expressions. This form of algebra was extensively studied in ancient Egypt; but, in accordance with the practical tendency of the Egyptian mind, the study consisted largely in the treatment of particular cases, very few general rules being obtained.

3. For many centuries algebra was confined almost entirely to the solution of equations; one of the most important steps being the enunciation by Diophantus of Alexandria of the laws governing the use of the minus sign. The knowledge of these laws, however, does not imply the existence of a conception of negative quantities. The development of symbolic algebra by the use of general symbols to denote numbers is due to Franciscus Vieta (Francois Viete, 1540-1603).This led to the idea of algebra as generalized arithmetic.

4. The principal step in the modern development of algebra was the recognition of the meaning of negative quantities. This appears to have been due in the first instance to Albert Girard (1595-1632), who extended Vieta's results in various branches of mathematics. His work, however, was little known at the time, and later was overshadowed by the greater work of Descartes (1596-1650).

5. The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between arithmetical and geometrical measurement. This involved not only the geometrical interpretation of negative quantities, but also the idea of continuity; this latter, which is the basis of modern analysis, leading to two separate but allied developments, viz. the theory of the function and the theory of limits.

6. The great development of all branches of mathematics in the two centuries following Descartes has led to the term algebra being used to cover a great variety of subjects, many of which are really only ramifications of arithmetic, dealt with by algebraical methods, while others, such as the theory of numbers and the general theory of series, are outgrowths of the application of algebra to arithmetic, which involve such special ideas that they must properly be regarded as distinct subjects. Some writers have attempted unification by treating algebra as concerned with functions, and Comte accordingly defined algebra as the calculus of functions, arithmetic being regarded as the calculus of values.

7. These attempts at the unification of algebra, and its separation from other branches of mathematics, have usually been accompanied by an attempt to base it, as a deductive science, on certain fundamental laws or general rules; and this has tended to increase its difficulty. In reality, the variety of algebra corresponds to the variety of phenomena. Neither mathematics itself, nor any branch or set of branches of mathematics, can be regarded as an isolated science. While, therefore, the logical development of algebraic reasoning must depend on certain fundamental relations, it is important that in the early study of the subject these relations should be introduced gradually, and not until there is some empirical acquaintance with the phenomena with which they are concerned.

8. The extension of the range of subjects to which mathematical methods can be applied, accompanied as it is by an extension of the range of study which is useful to the ordinary worker, has led in the latter part of the 19th century to an important reaction against the specialization mentioned in the preceding paragraph. This reaction has taken the form of a return to the alliance between algebra and geometry (S 5), on which modern analytical geometry is based; the alliance, however, being concerned with the application of graphical methods to particular cases rather than to general expressions. These applications are sometimes treated under arithmetic, sometimes under algebra; but it is more convenient to regard graphics as a separate subject, closely allied to arithmetic, algebra, mensuration and analytical geometry.

9. The association of algebra with arithmetic on the one hand, and with geometry on the other, presents difficulties, in that geometrical measurement is based essentially on the idea of continuity, while arithmetical measurement is based essentially on the idea of discontinuity; both ideas being equally matters of intuition. The difficulty first arises in elementary mensuration, where it is partly met by associating arithmetical and geometrical measurement with the cardinal and the ordinal aspects of number respectively (see ARITHMETIC.) Later, the difficulty recurs in an acute form in reference to the continuous variation of a function. Reference to a geometrical interpretation seems at first sight to throw light on the meaning of a differential coefficient; but closer analysis reveals new difficulties, due to the geometrical interpretation itself. One of the most recent developments of algebra is the algebraic theory at number, which is devised with the view of removing these difficulties. The harmony between arithmetical and geometrical measurement, which was disturbed by the Greek geometers on the discovery of irrational numbers, is restored by an unlimited supply of the causes of disturbance.

10. Two other developments of algebra are of special importance. The theory of sequences and series is sometimes treated as a part of elementary algebra; but it is more convenient to regard the simpler cases as isolated examples, leading up to the general theory. The treatment of equations of the second and higher degrees introduces imaginary and complex numbers, the theory of which is a special subject.

11. One of the most difficult questions for the teacher of algebra is the stage at which, and the extent to which, the ideas of a negative number and of continuity may be introduced. On the one hand, the modern developments of algebra began with these ideas, and particularly with the idea of a negative number. On the other hand, the lateness of occurrence of any particular mathematical idea is usually closely correlated with its intrinsic difficulty. Moreover, the ideas which are usually formed on these points at an early stage are incomplete; and, if the incompleteness of an idea is not realized, operations in which it is implied are apt to be purely formal and mechanical. What are called negative numbers in arithmetic, for instance, are not really negative numbers but negative quantities (S 27 (i.)); and the difficulties incident to the ideas of continuity have already been pointed out.

12. In the present article, therefore, the main portions of elementary algebra are treated in one section, without reference to these ideas, which are considered generally in two separate sections. These three sections may therefore be regarded as to a certain extent concurrent. They are preceded by two sections dealing with the introduction to algebra from the arithmetical and the graphical sides, and are followed by a section dealing briefly with the developments mentioned in S S 9 and 10 above.

[The intermediate portion of this article is typeset in TeX and is available elsewhere.]

C. HISTORY Various derivations of the word "algebra,'' which is of Arabian origin, have been given by different writers. The first mention of the word is to be found in the title of a work by Mahommed ben Musa al-Khwarizmi (Hovarezmi), who flourished about the beginning of the 9th century. The full title is ilm al-jebr wa'l-muqabala, which contains the ideas of restitution and comparison, or opposition and comparison, or resolution and equation, jebr being derived from the verb jabara, to reunite, and muqabala, from gabala, to make equal. (The root jabara is also met with in the word algebrista, which means a "bone-setter,'' and is still in common use in Spain.) The same derivation is given by Lucas Paciolus (Luca Pacioli), who reproduces the phrase in the transliterated form alghebra e almucabala, and ascribes the invention of the art to the Arabians.

Other writers have derived the word from the Arabic particle al (the definite article), and gerber, meaning "man.'' Since, however, Geber happened to be the name of a celebrated Moorish philosopher who flourished in about the 11th or 12th century, it has been supposed that he was the founder of algebra, which has since perpetuated his name. The evidence of Peter Ramus (1515-1572) on this point is interesting, but he gives no authority for his singular statements. In the preface to his Arithmeticae libri duo et totidem Algebrae (1560) he says: "The name Algebra is Syriac, signifying the art or doctrine of an excellent man. For Geber, in Syriac, is a name applied to men, and is sometimes a term of honour, as master or doctor among us. There was a certain learned mathematician who sent his algebra, written in the Syriac language, to Alexander the Great, and he named it almucabala, that is, the book of dark or mysterious things, which others would rather call the doctrine of algebra. To this day the same book is in great estimation among the learned in the oriental nations, and by the Indians, who cultivate this art, it is called aljabra and alboret; though the name of the author himself is not known.,' The uncertain authority of these statements, and the plausibility of the preceding explanation, have caused philologists to accept the derivation from al and jabara. Robert Recorde in his Whetstone of Witte (1557) uses the variant algeber, while John Dee (1527-1608) affirms that algiebar, and not algebra, is the correct form, and appeals to the authority of the Arabian Avicenna.

Although the term "algebra'' is now in universal use, various other appellations were used by the Italian mathematicians during the Renaissance. Thus we find Paciolus calling it l'Arte Magiore; ditta dal vulgo la Regula de la Cosa over Alghebra e Almucabala. The name l'arte magiore, the greater art, is designed to distinguish it from l'arte minore, the lesser art, a term which he applied to the modern arithmetic. His second variant, la regula de la cosa, the rule of the thing or unknown quantity, appears to have been in common use in Italy, and the word cosa was preserved for several centuries in the forms coss or algebra, cossic or algebraic, cossist or algebraist, &c. Other Italian writers termed it the Regula rei et census, the rule of the thing and the product, or the root and the square. The principle underlying this expression is probably to be found in the fact that it measured the limits of their attainments in algebra, for they were unable to solve equations of a higher degree than the quadratic or square.

Franciscus Vieta (Francois Viete) named it Specious Arithmetic, on account of the species of the quantities involved, which he represented symbolically by the various letters of the alphabet. Sir Isaac Newton introduced the term Universal Arithmetic, since it is concerned with the doctrine of operations, not affected on numbers, but on general symbols.

Notwithstanding these and other idiosyncratic appellations, European mathematicians have adhered to the older name, by which the subject is now universally known.

It is difficult to assign the invention of any art or science definitely to any particular age or race. The few fragmentary records, which have come down to us from past civilizations, must not be regarded as representing the totality of their knowledge, and the omission of a science or art does not necessarily imply that the science or art was unknown. It was formerly the custom to assign the invention of algebra to the Greeks, but since the decipherment of the Rhind papyrus by Eisenlohr this view has changed, for in this work there are distinct signs of an algebraic analysis. The particular problem—-a heap (hau) and its seventh makes 19—-is solved as we should now solve a simple equation; but Ahmes varies his methods in other similar problems. This discovery carries the invention of algebra back to about 1700 B.C., if not earlier.

It is probable that the algebra of the Egyptians was of a most rudimentary nature, for otherwise we should expect to find traces of it in the works of the Greek aeometers. of whom Thales of Miletus (640-546 B.C.) was the first. Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an algebraic analysis from their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to algebra. The first extant work which approaches to a treatise on algebra is by Diophantus (q.v.), an Alexandrian mathematician, who flourished about A.D. 350. The original, which consisted of a preface and thirteen books, is now lost, but we have a Latin translation of the first six books and a fragment of another on polygonal numbers by Xylander of Augsburg (1575), and Latin and Greek translations by Gaspar Bachet de Merizac (1621-1670). Other editions have been published, of which we may mention Pierre Fermat's (1670), T. L. Heath's (1885) and P. Tannery's (1893-1895). In the preface to this work, which is dedicated to one Dionysius, Diophantus explains his notation, naming the square, cube and fourth powers, dynamis, cubus, dynamodinimus, and so on, according to the sum in the indices. The unknown he terms arithmos, the number, and in solutions he marks it by the final s; he explains the generation of powers, the rules for multiplication and division of simple quantities, but he does not treat of the addition, subtraction, multiplication and division of compound quantities. He then proceeds to discuss various artifices for the simplification of equations, giving methods which are still in common use. In the body of the work he displays considerable ingenuity in reducing his problems to simple equations, which admit either of direct solution, or fall into the class known as indeterminate equations. This latter class he discussed so assiduously that they are often known as Diophantine problems, and the methods of resolving them as the Diophantine analysis (see EQUATION, Indeterminate.) It is difficult to believe that this work of Diophantus arose spontaneously in a period of general stagnation. It is more than likely that he was indebted to earlier writers, whom he omits to mention, and whose works are now lost; nevertheless, but for this work, we should be led to assume that algebra was almost, if not entirely, unknown to the Greeks.

The Romans, who succeeded the Greeks as the chief civilized power in Europe, failed to set store on their literary and scientific treasures; mathematics was all but neglected; and beyond a few improvements in arithmetical computations, there are no material advances to be recorded.

In the chronological development of our subject we have now to turn to the Orient. Investigation of the writings of Indian mathematicians has exhibited a fundamental distinction between the Greek and Indian mind, the former being pre-eminently geometrical and speculative, the latter arithmetical and mainly practical. We find that geometry was neglected except in so far as it was of service to astronomy; trigonometry was advanced, and algebra improved far beyond the attainments of Diophantus.

The earliest Indian mathematician of whom we have certain knowledge is Aryabhatta, who flourished about the beginning of the 6th century of our era. The fame of this astronomer and mathematician rests on his work, the Aryabhattiyam, the third chapter of which is devoted to mathematics. Ganessa, an eminent astronomer, mathematician and scholiast of Bhaskara, quotes this work and makes separate mention of the cuttaca ("pulveriser''), a device for effecting the solution of indeterminate equations. Henry Thomas Colebrooke, one of the earliest modern investigators of Hindu science, presumes that the treatise of Aryabhatta extended to determinate quadratic equations, indeterminate equations of the first degree, and probably of the second. An astronomical work, called the Surya-siddhanta ("knowledge of the Sun''), of uncertain authorship and probably belonging to the 4th or 5th century, was considered of great merit by the Hindus, who ranked it only second to the work of Brahmagupta, who flourished about a century later. It is of great interest to the historical student, for it exhibits the influence of Greek science upon Indian mathematics at a period prior to Aryabhatta. After an interval of about a century, during which mathematics attained its highest level, there flourished Brahmagupta (b. A.D. 598), whose work entitled Brahma-sphuta-siddhanta ("The revised system of Brahma'') contains several chapters devoted to mathematics. Of other Indian writers mention may be made of Cridhara, the author of a Ganita-sara ("Quintessence of Calculation''), and Padmanabha, the author of an algebra.

A period of mathematical stagnation then appears to have possessed the Indian mind for an interval of several centuries, for the works of the next author of any moment stand but little in advance of Brahmagupta. We refer to Bhaskara Acarya, whose work the Siddhanta-ciromani ("Diadem of anastronomical System''), written in 1150, contains two important chapters, the Lilavati ("the beautiful [science or art]'') and Viga-ganita ("root-extraction''), which are given up to arithmetic and algebra.

English translations of the mathematical chapters of the Brahma-siddhanta and Siddhanta-ciromani by H. T. Colebrooke (1817), and of the Surya-siddhanta by E. Burgess, with annotations by W. D. Whitney (1860), may be consulted for details.

The question as to whether the Greeks borrowed their algebra from the Hindus or vice versa has been the subject of much discussion. There is no doubt that there was a constant traffic between Greece and India, and it is more than probable that an exchange of produce would be accompanied by a transference of ideas. Moritz Cantor suspects the influence of Diophantine methods, more particularly in the Hindu solutions of indeterminate equations, where certain technical terms are, in all probability, of Greek origin. However this may be, it is certain that the Hindu algebraists were far in advance of Diophantus. The deficiencies of the Greek symbolism were partially remedied; subtraction was denoted by placing a dot over the subtrahend; multiplication, by placing bha (an abbreviation of bhavita, the "product'') after the factom; division, by placing the divisor under the dividend; and square root, by inserting ka (an abbreviation of karana, irrational) before the quantity. The unknown was called yavattavat, and if there were several, the first took this appellation, and the others were designated by the names of colours; for instance, x was denoted by ya and y by ka (from kalaka, black).

A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the existence of two roots of a quadratic equation, but the negative roots were considered to be inadequate, since no interpretation could be found for them. It is also supposed that they anticipated discoveries of the solutions of higher equations. Great advances were made in the study of indeterminate equations, a branch of analysis in which Diophantus excelled. But whereas Diophantus aimed at obtaining a single solution, the Hindus strove for a general method by which any indeterminate problem could be resolved. In this they were completely successful, for they obtained general solutions for the equations ax(+ or -)by=c, xy=ax+by+c (since rediscovered by Leonhard Euler) and cy2=ax2+b. A particular case of the last equation, namely, y2=ax2+1, sorely taxed the resources of modern algebraists. It was proposed by Pierre de Fermat to Bernhard Frenicle de Bessy, and in 1657 to all mathematicians. John Wallis and Lord Brounker jointly obtained a tedious solution which was published in 1658, and afterwards in 1668 by John Pell in his Algebra. A solution was also given by Fermat in his Relation. Although Pell had nothing to do with the solution, posterity has termed the equation Pell's Equation, or Problem, when more rightly it should be the Hindu Problem, in recognition of the mathematical attainments of the Brahmans.

Hermann Hankel has pointed out the readiness with which the Hindus passed from number to magnitude and vice versa. Although this transition from the discontinuous to continuous is not truly scientific, yet it materially augmented the development of algebra, and Hankel affirms that if we define algebra as the application of arithmetical operations to both rational and irrational numbers or magnitudes, then the Brahmans are the real inventors of algebra.

The integration of the scattered tribes of Arabia in the 7th century by the stirring religious propaganda of Mahomet was accompanied by a meteoric rise in the intellectual powers of a hitherto obscure race. The Arabs became the custodians of Indian and Greek science, whilst Europe was rent by internal dissensions. Under the rule of the Abbasids, Bagdad became the centre of scientific thought; physicians and astronomers from India and Syria flocked to their court; Greek and Indian manuscripts were translated (a work commenced by the Caliph Mamun (813-833) and ably continued by his successors); and in about a century the Arabs were placed in possession of the vast stores of Greek and Indian learning. Euclid's Elements were first translated in the reign of Harun-al-Rashid (786-809), and revised by the order of Mamun. But these translations Were regarded as imperfect, and it remained for Tobit ben Korra (836-901) to produce a satisfactory edition. Ptolemy's Almagest, the works of Apollonius, Archimedes, Diophantus and portions of the Brahmasiddhanta, were also translated. The first notable Arabian mathematician was Mahommed ben Musa al-Khwarizmi, who flourished in the reign of Mamun. His treatise on algebra and arithmetic (the latter part of which is only extant in the form of a Latin translation, discovered in 1857) contains nothing that was unknown to the Greeks and Hindus; it exhibits methods allied to those of both races, with the Greek element predominating. The part devoted to algebra has the title al-jeur wa'lmuqabala, and the arithmetic begins with "Spoken has Algoritmi,'' the name Khwarizmi or Hovarezmi having passed into the word Algoritmi, which has been further transformed into the more modern words algorism and algorithm, signifying a method of computing.

Tobit ben Korra (836-901), born at Harran in Mesopotamia, an accomplished linguist, mathematician and astronomer, rendered conspicuous Service by his translations of various Greek authors. His investigation of the properties of amicable numbers (q.v.) and of the problem of trisecting an angle, are of importance. The Arabians more closely resembled the Hindus than the Greeks in the choice of studies; their philosophers blended speculative dissertations with the more progressive study of medicine; their mathematicians neglected the subtleties of the conic sections and Diophantine analysis, and applied themselves more particularly to perfect the system of numerals (see NUMERAL), arithmetic and astronomy (q.v..) It thus came about that while some progress was made in algebra, the talents of the race were bestowed on astronomy and trigonometry (q.v..) Fahri des al Karbi, who flourished about the beginning of the 11th century, is the author of the most important Arabian work on algebra. He follows the methods of Diophantus; his work on indeterminate equations has no resemblance to the Indian methods, and contains nothing that cannot be gathered from Diophantus. He solved quadratic equations both geometrically and algebraically, and also equations of the form x2n+axn+b=0; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes.

Cubic equations were solved geometrically by determining the intersections of conic sections. Archimedes' problem of dividing a sphere by a plane into two segments having a prescribed ratio, was first expressed as a cubic equation by Al Mahani, and the first solution was given by Abu Gafar al Hazin. The determination of the side of a regular heptagon which can be inscribed or circumscribed to a given circle was reduced to a more complicated equation which was first successfully resolved by Abul Gud. The method of solving equations geometrically was considerably developed by Omar Khayyam of Khorassan, who flourished in the 11th century. This author questioned the possibility of solving cubics by pure algebra, and biquadratics by geometry. His first contention was not disproved until the 15th century, but his second was disposed of by Abul Weta (940-908), who succeeded in solving the forms x4=a and x4+ax3=b.

Although the foundations of the geometrical resolution of cubic equations are to be ascribed to the Greeks (for Eutocius assigns to Menaechmus two methods of solving the equation x3=a and x3=2a3), yet the subsequent development by the Arabs must be regarded as one of their most important achievements. The Greeks had succeeded in solving an isolated example; the Arabs accomplished the general solution of numerical equations.

Considerable attention has been directed to the different styles in which the Arabian authors have treated their subject. Moritz Cantor has suggested that at one time there existed two schools, one in sympathy With the Greeks, the other with the Hindus; and that, although the writings of the latter were first studied, they were rapidly discarded for the more perspicuous Grecian methods, so that, among the later Arabian writers, the Indian methods were practically forgotten and their mathematics became essentially Greek in character.

Turning to the Arabs in the West we find the same enlightened spirit; Cordova, the capital of the Moorish empire in Spain, was as much a centre of learning as Bagdad. The earliest known Spanish mathematician is Al Madshritti (d. 1007), whose fame rests on a dissertation on amicable numbers, and on the schools which were founded by his pupils at Cordoya, Dama and Granada. Gabir ben Allah of Sevilla, commonly called Geber, was a celebrated astronomer and apparently skilled in algebra, for it has been supposed that the word "algebra', is compounded from his name.

When the Moorish empire began to wane the brilliant intellectual gifts which they had so abundantly nourished during three or four centuries became enfeebled, and after that period they failed to produce an author comparable with those of the 7th to the 11th centuries.

In Europe the decline of Rome was succeeded by a period, lasting several centuries, during which the sciences and arts were all but neglected. Political and ecclesiastical dissensions occupied the greatest intellects, and the only progress to be mcorded is in the art of computing or arithmetic, and the translation of Arabic manuscripts. The first successful attempt to revive the study of algebra in Christendom was due to Leonardo of Pisa. an Italian merchant trading in the Mediterranean. His travels and mercantile experience had led him to conclude that the Hindu methods of computing, were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for nulnbers, and solves many problems, both arithmetical and algebraical. But it contains little that is original, and although the work created a great sensation when it was first published, the effect soon passed away, and the book was practically forgotten. Mathematics was more or less ousted from the academic curricula by the philosophical inquiries of the schoolmen, and it was only after an interval of nearly three centuries that a worthy successor to Leonardo appeared. This was Lucas Paciolus (Lucas de Burgo), a Minorite friar, who, having previously written works on algebra, arithmetic and geometry, published, in 1494, his principal work, entitled Summa de Arithmetica, Giometria, Proportioni et Proportionalita. In it he mentions many earlier writers from whom he had learnt the science, and although it contains very little that cannot be found in Leonardo's work, yet it is especially noteworthy for the systematic employment of symbols, and the manner in which it reflects the state of mathematics in Europe during this period. These works are the earliest printed books on mathematics. The renaissance of mathematics was thus effected in Italy, and it is to that country that the leading developments of the following century were due. The first difficulty to be overcome was the algebraical solution of cubic equations, the pons asinorum of the earlier mathematicians. The first step in this direction was made by Scipio Ferro (d. 1526), who solved the equation x3+ax=b. Of his discovery we know nothing except that he declared it to his pupil Antonio Marie Floridas. An imperfect solution of the equation x3+px2=q was discovered by Nicholas Tartalea (Tartaglia) in 1530, and his pride in this achievement led him into conflict with Floridas, who proclaimed his own knowledge of the form resolved by Ferro. Mutual recriminations led to a public discussion in 1535, when Tartalea completely vindicated the general applicability of his methods and exhibited the inefficiencies of that of Floridas. This contest over, Tartalea redoubled his attempts to generalize his methods, and by 1541 he possessed the means for solving any form of cubic equation. His discoveries had made him famous all over Italy, and he was earnestly solicited to publish his methods; but he abstained from doing so, saying that he intended to embody them in a treatise on algebra which he was preparing. At last he succumbed to the repeated requests of Girolamo or Geronimo Cardano, who swore that he would regard them as an inviolable secret. Cardan or Cardano, who was at that time writing his great work, the Ars Magna, could not restrain the temptation of crowning his treatise with such important discoveries, and in 1545 he broke his oath and gave to the world Tartalea's rules for solving cubic equations. Tartalea, thus robbed of his most cherished possession, was in despair. Recriminations ensued until his death in 1557, and although he sustained his claim for priority, posterity has not conceded to him the honour of his discovery, for his solution is now known as Cardan's Rule.

Cubic equations having been solved, biquadratics soon followed suit. As early as 1539 Cardan had solved certain particular cases, but it remained for his pupil, Lewis (Ludovici) Ferrari, to devise a general method. His solution, which is sometimes erroneously ascribed to Rafael Bombelh, was published in the Ars Magna. In this work, which is one of the most valuable contributions to the literature of algebra, Cardan shows that he was familiar with both real positive and negative roots of equations whelher rational or irrational, but of imaginary roots he was quite ignorant, and he admits his inability to resolve the so-called "irreducible case'' (see EQUATION.) Fundamental theorems in the theory of equations are to be found in the same work. Clearer ideas of imaginary quantities and the "irreducible case'' were subsequently published by Bombelli, in a work of which the dedication is dated 1572, though the book was not published until 1579.

Contemporaneously with the remarkable discoveries of the Italian mathematicians, algebra was increasing in popularity in Germany, France and England. Michael Stifel and Johann Scheubelius (Scheybl) (1494-1570) flourished in Germany, and although unacquainted with the work of Cardan and Tartalea, their writings are noteworthy for their perspicuity and the introduction of a more complete symbolism for quantities and operations. Stifel introduced the sign (+) for addition or a positive quantity, which was previously denoted by plus, piu, or the letter p. Subtraction, previously written as minus, mone or the letter m, was symbolized by the sign (-) which is still in use. The square root he denoted by (sqrt. ), whereas Paciolus, Cardan and others used the letter R.

The first treatise on algebra written in English was by Robert Recorde, who published his arithmetic in 1552, and his algebra entitled The Whetstone of Witte, which is the second part of Arithmetik, in 1557. This work, which is written in the form of a dialogue, closely resembles the works of Stifel and Scheubelius, the latter of whom he often quotes. It includes the properties of numbers; extraction of roots of arithmetical and algebraical quantities, solutions of simple and quadratic equations, and a fairly complete account of surds. He introduced the sign (=) for equality, and the terms binomial and residual. Of other writers who published works about the end of the 16th century, we may mention Jacques Peletier, or Jacobus Peletarius (De occulta parto Numerorum, quare Algebram vocant, 1558); Petrus Ramus (Arithmeticae Libri duo et totidem Algebrae, 1560), and Christoph Clavius, who wrote on algebra in 1580, though it was not published until 1608. At this time also flourished Simon Stevinus (Stevin) of Bruges, who published an arithmetic in 1585 and an algebra shortly afterwards. These works possess considerable originality, and contain many new improvements in algebraic notation; the unknown (res) is denoted by a small circle, in which he places an integer corresponding to the power. He introduced the terms multinomial, trinomial, quadrinomial, &c., and considerably simplified the notation for decimals.

About the beginning of the 17th century various mathematical works by Franciscus Vieta were published, which were afterwards collected by Franz van Schooten and republished in 1646 at Leiden. These works exhibit great originality and mark an important epoch in the history of algebra. Vieta, who does not avail himself of the discoveries of his predecessors—the negative roots of Cardan, the revised notation of Stifel and Stevin, &c.—introduced or popularized many new terms and symbols, some of which are still in use. He denotes quantities by the letters of the alphabet, retaining the vowels for the unknown and the consonants for the knowns; he introduced the vinculum and among others the terms coefficient, affirmative, negative, pure and adjected equations. He improved the methods for solving equations, and devised geometrical constructions with the aid of the conic sections. His method for determining approximate values of the roots of equations is far in advance of the Hindu method as applied by Cardan, and is identical in principle with the methods of Sir Isaac Newton and W. G. Horner.

We have next to consider the works of Albert Girard, a Flemish mathematician. This writer, after having published an edition of Stevin's works in 1625, published in 1629 at Amsterdam a small tract on algebra which shows a considerable advance on the work of Vieta. Girard is inconsistent in his notation, sometimes following Vieta, sometimes Stevin; he introduced the new symbols ff. for greater than and sec. for less than; he follows Vieta in using the plus (+) for addition, he denotes subtraction by Recorde's symbol for equality (=), and he had no sign for equality but wrote the word out. He possessed clear ideas of indices and the generation of powers, of the negative roots of equations and their geometrical interpretation, and was the first to use the term imaginary roots. He also discovered how to sum the powers of the roots of an equation.

Passing over the invention of logarithms (q.v.) by John Napier, and their development by Henry Briggs and others, the next author of moment was an Englishman, Thomas Harriot, whose algebra (Artis analyticae praxis) was published posthumously by Walter Warner in 1631. Its great merit consists in the complete notation and symbolism, which avoided the cumbersome expressions of the earlier algebraists, and reduced the art to a form closely resembling that of to-day. He follows Vieta in assigning the vowels to the unknown quantities and the consonants to the knowns, but instead of using capitals, as with Vieta, he employed the small letters; equality he denoted by Recorde's symbol, and he introduced the signs > and < for greater than and less than. His principal discovery is concerned with equations, which he showed to be derived from the continued multiplication of as many simple factors as the highest power of the unknown, and he was thus enabled to deduce relations between the coefficients and various functions of the roots. Mention may also be made of his chapter on inequalities, in which he proves that the arithmetic mean is always greater than the geometric mean.

William Oughtred, a contemporary of Harriot, published an algebra, Clavis mathematicae, simultaneously with Harriot's treatise. His notation is based on that of Vieta, but he introduced the sign X for multiplication, @ for continued proportion, :: for proportion, and denoted ratio by one dot. This last character has since been entirely restricted to multiplication, and ratio is now denoted by two dots (:). His symbols for greater than and less than (@ and @) have been completely superseded by Harriot's signs'

So far the development of algebra and geometry had been mutually independent, except for a few isolated applications of geometrical constructions to the solution of algebraical problems. Certain minds had long suspected the advairages which would accrue from the unrestricted application of algebra to geometry, but it was not until the advent of the philosopher Rene Descartes that the co-ordination was effected. In his famous Geometria (1637), which is really a treatise on the algebraic representation of geometric theorems, he founded the modern theory of analytical geometry (see GEOMETRY), and at the same time he rendered signal service to algebra, more especially in the theory of equations. His notation is based primarily on that of Harriot; but he differs from that writer in retaining the first letters of the alphabet for the known quantities and the final letters for the unknowns.

The 17th century is a famous epoch in the progress of science, and the mathematics in no way lagged behind. The discoveries of Johann Kepler and Bonaventura Cavalieri were the foundation upon which Sir Isaac Newton and Gottfried Wilhelm Leibnitz erected that wonderful edifice, the Infinitesimal Calculus (q.v..) Many new fields were opened up, but there was still continual progress in pure algebra. Continued fractions, one of the earliest examples of which is Lord Brouncker's expression for the ratio of the circumference to the diameter of a circle (see CIRCLE), were elaborately discussed by John Wallis and Leonhard Euler; the convergency of series treated by Newton, Euler and the Bernoullis; the binomial theorem, due originally to Newton and subsequently expanded by Euler and others, was used by Joseph Louis Lagrange as the basis of his Calcul des Fonctions. Diophantine problems were revived by Gaspar Bachet, Pierre Fermat and Euler; the modern theory of numbers was founded by Fermat and developed by Euler, Lagrange and others; and the theory of probability was attacked by Blaise Pascal and Fermat, their work being subsequently expanded by James Bernoulli, Abraham de Moivre, Pierre Simon Laplace and others. The germs of the theory of determinants are to be found in the works of Leibnitz; Etienne Bezout utilized them in 1764 for expressing the result obtained by the process of elimination known by his name, and since restated by Arthur Cayley.

In recent times many mathematicians have formulated other kinds of algebras, in which the operators do not obey the laws of ordinary algebra. This study was inaugurated by George Peacock, who was one of the earliest mathematicians to recognize the symbolic character of the fundamental principles of algebra. About the same time, D. F. Gregory published a paper "on the real nature of symbolical algebra.'' In Germany the work of Martin Ohm (System der Mathematik, 1822) marks a step forward. Notable service was also rendered by Augustus de Morgan, who applied logical analysis to the laws of mathematics.

The geometrical interpretation of imaginary quantities had a far-reaching influence on the development of symbolic algebras. The attempts to elucidate this question by H. Kuhn (1750-1751) and Jean Robert Argand (1806) were completed by Karl Friedrich Gauss, and the formulation of various systems of vector analysis by Sir William Rowan Hamilton, Hermann Grassmann and others, followed. These algebras were essentially geometrical, and it remained, more or less, for the American mathematician Benjamin Peirce to devise systems of pure symbolic algebras; in this work he was ably seconded by his son Charles S. Peirce. In England, multiple algebra was developed by James Joseph Sylvester, who, in company with Arthur Cayley, expanded the theory of matrices, the germs of which are to be found in the writings of Hamilton (see above, under (B); and QUATERNIONS.)

The preceding summary shows the specialized nature which algebra has assumed since the 17th century. To attempt a history of the development of the various topics in this article is inappropriate, and we refer the reader to the separate articles.

REFERENCES.—-The history of algebra is treated in all historical works on mathematics in general (see MATHEMATICS: References.) Greek algebra can be specially studied in T. L. Heath's Diophantus. See also John Wallis, Opera Mathematica (1693-1699), and Charles Sutton, Mathematical and Philosophical Dictionary (1815), article "Algebra.'' (C. E.*)

[The article on Algebraic Forms is typeset in TeX and is available elsewhere.]

ALGECIRAS, or ALGEZIRAS, a seaport of southern Spain in the province of Cadiz, 6 m. W. of Gibraltar, on the opposite side of the Bay of Algeciras. Pop. (1900) 13,302. Algeciras stands at the head of a railway from Granada, but its only means of access to Gibraltar is by water. Its name, which signifies in Arabic the island, is derived from a small islet on one side of the harbour. It is supplied with water by means of a beautiful aqueduct. The fine winter climate of Algeciras attracts many invalid visitors, on whom the town largely depends for its prosperity. The harbour is bad, but at the beginning of the 20th century it became important as a fishing-station. Whiting, soles, bream, bass and other fish are caught in great quantities by the Algeciras steam-trawlers, which visit the Moroccan coast, as well as Spanish and neutral waters. There is also some trade in farm produce and building materials which supplies a fleet of small coasters with cargo.

Algeciras was perhaps the Portus Albus of the Romans, but it was probably refounded in 713 by the Moors, who retained possession of it until 1344. It was then taken by Alphonso XI. of Castile after a celebrated siege of twenty months, which attracted Crusaders from all parts of Europe; among them being the English earl of Derby, grandson of Edward III. It is said that during this siege gunpowder was first used by the Moors in the wars of Europe. The Moorish city was destroyed by Alphonso; it was first reoccupied by Spanish colonists from Gibraltar in 1704; and the modern town was erected in 1760 by King Charles III. During the siege of Gibraltar in 1780- 1782, Algeciras was the station of the Spanish fleet and floating batteries. On the 6th of July 1801 the English admiral Sir James Saumarez attacked a Franco-Spanish fleet off Algeciras, and sustained a reverse; but on the 12th he again attacked the enemy, whose fleet was double his own strength, and inflicted on them a complete defeat. The important international conference on Moroccan affairs, which resulted in an agreement between France and Germany, was held at Algeciras from the 16th of January to the 7th of April 1906. (See MOROCCO.)

ALGER OF LIEGE (d c. 1131), known also as ALGER OF CLUNY and ALGERUS MAGISTER, a learned French priest who lived in the first half of the 12th century. He was first a deacon of the church of St Bartholomew at Liege, his native town, and was then appointed (c. 1100) to the cathedral church of St Lambert. He declined many offers from German bishops and finally retired to the monastery of Cluny, where he died about 1131 at a great age and leaving a good reputation for piety and intelligence. His History of the Church of Liege, and many of his other works, are lost. The most important of those still extant are: 1. De Misericordia et Justitia, a collection of biblical and patristic extracts with a commentary (an important work for the history of church law and discipline), which is to be found in the Anecdota of Martene, vol. v. 2. De Sacramentis Corporis et Sanguinis Domini; a treatise, in three books, against the Berengarian heresy, highly commended by Peter of Cluny and Erasmus. 3. De Gratia et Libero Arbitrio; given in B. Pez's Anecdota, vol. iv. 4. De Sacrificio Missae; given in the Collectio Scriptor. Vet. of Angelo Mai, vol. ix. p. 371.

See Migne, Patrol Ser. Lat. vol. clxxx. pp. 739-.972; Herzog- Hauck, Realencyk.fur prot. Theol., art. by S. M. Deutsch.