Chapter XVII: Part 17

HARI-RUD, a river of Afghanistan. It rises in the northern slopes of the Koh-i-Baba to the west of Kabul, and finally loses itself in the Tejend oasis north of the Trans-Caspian railway and west of Merv. It runs a remarkably straight course westward through a narrow trough from Daolatyar to Obeh, amidst the bleak wind-swept uplands of the highest central elevations in Afghanistan. From Obeh to Kuhsan 50 m. west of Herat, it forms a valley of great fertility, densely populated and highly cultivated; practically all its waters being drawn off for purposes of irrigation. It is the contrast between the cultivated aspect of the valley of Herat and the surrounding desert that has given Herat its great reputation for fertility. Three miles to the south of Herat the Kandahar road crosses the river by a masonry bridge of 26 arches now in ruins. A few miles below Herat the river begins to turn north-west, and after passing through a rich country to Kuhsan, it turns due north and breaks through the Paropamisan hills. Below Kuhsan it receives fresh tributaries from the west. Between Kuhsan and Zulfikar it forms the boundary between Afghanistan and Persia, and from Zulfikar to Sarakhs between Russia and Persia. North of Sarakhs it diminishes rapidly in volume till it is lost in the sands of the Turkman desert. The Hari-Rud marks the only important break existing in the continuity of the great central water-parting of Asia. It is the ancient Arius. (T. H. H.*)

HARISCHANDRA, in Hindu mythology, the 28th king of the Solar race. He was renowned for his piety and justice. He is the central figure of legends in the Aitareyabrahmana, Mahabharata and the Markandeyapurana. In the first he is represented as so desirous of a son that he vows to Varuna that if his prayer is granted the boy shall be eventually sacrificed to the latter. The child is born, but Harischandra, after many delays, arranges to purchase another's son and make a vicarious sacrifice. According to the Mahabharata he is at last promoted to Paradise as the reward for his munificent charity.

HARITH IBN HILLIZA UL-YASHKURI, pre-Islamic Arabian poet of the tribe of Bakr, famous as the author of one of the poems generally received among the Mo'allakat (q.v.). Nothing is known of the details of his life.

HARIZI, JUDAH BEN SOLOMON (13th cent.), called also al-Harizi, a Spanish Hebrew poet and traveller. He translated from the Arabic to Hebrew some of the works of Maimonides (q.v.) and also of the Arab poet Hariri. His own most considerable work was the _Tahkemoni_, composed between 1218 and 1220. This is written in Hebrew in unmetrical rhymes, in what is commonly termed "rhymed prose." It is a series of humorous episodes, witty verses, and quaint applications of Scriptural texts. The episodes are bound together by the presence of the hero and of the narrator, who is also the author. Harizi not only brought to perfection the art of applying Hebrew to secular satire, but he was also a brilliant literary critic and his _makame_ on the Andalusian Hebrew poets is a fruitful source of information.

See, on the _Tahkemoni_, Kaempf, _Nicht-andalusische Poesie
andalusischer Dichter_ (Prague, 1858). In that work a considerable
section of the _Tahkemoni_ is translated into German. (I. A.)

HARKNESS, ALBERT (1822-1907), American classical scholar, was born at Mendon, Massachusetts, on the 6th of October 1822. He graduated at Brown University in 1842, taught in the Providence high school in 1843-1853, studied in Berlin, Bonn (where in 1854 he was the first American to receive the degree of Ph.D.) and Gottingen, and was professor of Greek language and literature in Brown University from 1855 to 1892, when he became professor emeritus. He was one of the founders in 1869 of the American Philological Association, of which he was president in 1875-1876, and to whose _Transactions_ he made various contributions; was a member of the Archaeological Institute's committee on founding the American School of Classical Studies at Athens, and served as the second director of that school in 1883-1884. He studied English and German university methods during trips to Europe in 1870 and 1883, and introduced a new scholarly spirit into American teaching of Latin in secondary schools with a series of Latin text-books, which began in 1851 with a _First Latin Book_ and continued for more than fifty years. His _Latin Grammar_ (1864, 1881) and _Complete Latin Grammar_ (1898) are his best-known books. He was a member of the board of fellows of Brown University from 1904 until his death, and in 1904-1905 was president of the Rhode Island Historical Society. He died in Providence, Rhode Island, on the 27th of May 1907.

His son, ALBERT GRANGER HARKNESS (1857- ), also a classical scholar, was born in Providence, Rhode Island, on the 19th of November 1857. He graduated at Brown University in 1879, studied in Germany in 1879-1883, and was professor of German and Latin at Madison (now Colgate) University from 1883 to 1889, and associate professor of Latin at Brown from 1889 to 1893, when he was appointed to the chair of Roman literature and history there. He was director of the American School of Classical Studies in Rome in 1902-1903.

HARKNESS, ROBERT (1816-1878), English geologist, was born at Ormskirk, Lancashire, on the 28th of July 1816. He was educated at the high school, Dumfries, and afterwards (1833-1834) at the university of Edinburgh where he acquired an interest in geology from the teachings of Robert Jameson and J. D. Forbes. Returning to Ormskirk he worked zealously at the local geology, especially on the Coal-measures and New Red Sandstone, his first paper (read before the Manchester Geol. Soc. in 1843) being on _The Climate of the Coal Epoch_. In 1848 his family went to reside in Dumfries and there he commenced to work on the Silurian rocks of the S.W. of Scotland, and in 1849 he carried his investigations into Cumberland. In these regions during the next few years he added much to our knowledge of the strata and their fossils, especially graptolites, in papers read before the Geological Society of London. He wrote also on the New Red rocks of the north of England and Scotland. In 1853 he was appointed professor of geology in Queen's College, Cork, and in 1856 he was elected F.R.S. During this period he wrote some articles on the geology of parts of Ireland, and exercised much influence as a teacher, but he returned to England during his vacations and devoted himself assiduously to the geology of the Lake district. He was also a constant attendant at the meetings of the British Association. In 1876 the syllabus for the Queen's Colleges in Ireland was altered, and Professor Harkness was required to lecture not only on geology, palaeontology, mineralogy and physical geography, but also on zoology and botany. The strain of the extra work proved too much, he decided to relinquish his post, and had retired but a short time when he died, on the 4th of October 1878.

"Memoir," by J. G. Goodchild, in _Trans. Cumberland Assoc._ No. viii.
(with portrait). In memory of Professor Harkness his sister
established two Harkness scholarships. One scholarship (of the value
of about L35 a year, tenable for three years) for women, tenable at
either Girton or Newnham College, Cambridge, is awarded triennially to
the best candidate in an examination in geology and palaeontology,
provided that proficiency be shown; the other, for men, is vested in
the hands of the university of Cambridge, and is awarded annually, any
member of the university being eligible who has graduated as a B.A.,
"provided that not more than three years have elapsed since the 19th
day of December next following his final examination for the degree of
bachelor of arts."

HARLAN, JAMES (1820-1899), American politician, was born in Clark county, Illinois, on the 26th of August 1820. He graduated from Indiana Asbury (now De Pauw) University in 1845, was president (1846-1847) of the newly founded and short-lived Iowa City College, studied law, was first superintendent of public instruction in Iowa in 1847-1848, and was president of Iowa Wesleyan University in 1853-1855. He took a prominent part in organizing the Republican party in Iowa, and was a member of the United States Senate from 1855 to 1865, when he became secretary of the interior. He had been a delegate to the peace convention in 1861, and from 1861 to 1865 was chairman of the Senate committee on public lands. He disapproved of President Johnson's conservative reconstruction policy, retired from the cabinet in August 1866, and from 1867 to 1873 was again a member of the United States Senate. In 1866 he was a delegate to the loyalists' convention at Philadelphia. One of his principal speeches in the Senate was that which he made in March 1871 in reply to Sumner's and Schurz's attack on President Grant's Santo Domingan policy. He was presiding judge of the court of commissioners of Alabama claims (1882-1885). He died in Mount Pleasant, Iowa, on the 5th of October 1899.

HARLAN, JOHN MARSHALL (1833- ), American jurist, was born in Boyle county, Kentucky, on the 1st of June 1833. He graduated at Centre College, Danville, Ky., in 1850, and at the law department of Transylvania University, Lexington, in 1853. He was county judge of Franklin county in 1858-1859, was an unsuccessful candidate for Congress on the Whig ticket in 1859, and was elector on the Constitutional Union ticket in 1860. On the outbreak of the Civil War he recruited and organized the Tenth Kentucky United States Volunteer Infantry, and in 1861-1863 served as colonel. Retiring from the army in 1863, he was elected by the Union party attorney-general of the state, and was re-elected in 1865, serving from 1863 to 1867, when he removed to Louisville to practise law. He was the Republican candidate for governor in 1871 and in 1875, and was a member of the commission which was appointed by President Hayes early in 1877 to accomplish the recognition of one or other of the existing state governments of Louisiana (q.v.); and he was a member of the Bering Sea tribunal which met in Paris in 1893. On the 29th of November 1877 he became an associate justice of the United States Supreme Court. In this position he showed himself a liberal constructionist. In opinions on the Civil Rights cases and in the interpretation of the 13th, 14th and 15th Amendments to the Constitution, he dissented from the majority of the court and advocated increasing the power of the Federal government. He supported the constitutionality of the income tax clause in the Wilson Tariff Bill of 1894, and he drafted the decision of the court in the Northern Securities Company Case, which applied to railways the provisions of the Sherman Anti-Trust Law. In 1889 he became a professor in the Law School of the Columbian University (afterwards George Washington University) in Washington, D.C.

HARLAND, HENRY (1861-1905), American novelist, was born in St Petersburg, Russia, in March 1861, and was educated in New York and at Harvard. He went to Europe as a journalist, and, after publishing several novels, mainly of American-Jewish life (under the name of Sidney Luska), first made his literary reputation in London as editor of the _Yellow Book_ in 1894. His association with this clever publication, and his own contributions to it, brought his name into prominence, but it was not till he published _The Cardinal's Snuff-box_ (1900), followed by _The Lady Paramount_ (1902), that his lightly humorous touch and picturesque style as a novelist brought him any real success. His health was always delicate, and he died at San Remo on the 20th of December 1905.

HARLAY DE CHAMPVALLON, FRANCOIS DE (1625-1695), 5th archbishop of Paris, was born in that city on the 14th of August 1625. Nephew of Francois de Harlay, archbishop of Rouen, he was presented to the abbey of Jumieges immediately on leaving the College de Navarre, and he was only twenty-six when he succeeded his uncle in the archiepiscopal see. He was transferred to the see of Paris in 1671, he was nominated by the king for the cardinalate in 1690, and the domain of St Cloud was erected into a duchy in his favour. He was commander of the order of the Saint Esprit and a member of the French Academy. During the early part of his political career he was a firm adherent of Mazarin, and is said to have helped to procure his return from exile. His private life gave rise to much scandal, but he had a great capacity for business, considerable learning, and was an eloquent and persuasive speaker. He definitely secured the favour of Louis XIV. by his support of the claims of the Gallican Church formulated by the declaration made by the clergy in assembly on the 19th of March 1682, when Bossuet accused him of truckling to the court like a valet. One of the three witnesses of the king's marriage with Madame de Maintenon, he was hated by her for using his influence with the king to keep the matter secret. He had a weekly audience of Louis XIV. in company with Pere la Chaise on the affairs of the Church in Paris, but his influence gradually declined, and Saint-Simon, who bore him no good will for his harsh attitude to the Jansenists, says that his friends deserted him as the royal favour waned, until at last most of his time was spent at Conflans in company with the duchess of Lesdiguieres, who alone was faithful to him. He urged the revocation of the edict of Nantes, and showed great severity to the Huguenots at Dieppe, of which he was temporal and spiritual lord. He died suddenly, without having received the sacraments, on the 6th of August 1695. His funeral discourse was delivered by the Pere Gaillard, and Mme de Sevigne made on the occasion the severe comment that there were only two trifles to make this a difficult matter--his life and his death.

See Abbe Legendre, _Vita Francisci de Harlay_ (Paris, 1720) and _Eloge
de Harlay_ (1695); Saint-Simon, _Memoires_ (vol. ii., ed. A. de
Boislisle, 1879), and numerous references in the _Lettres_ of Mme de
Sevigne.

HARLECH (perhaps for _Hardd lech_, fair slate, or _Harleigh_, an Anglicized variant), a town of Merionethshire, Wales, 38 m. from Aberystwyth, and 29 from Carnarvon on the Cambrian railway. Pop. 900. Ruins of a fortress crown the rock of Harlech, about half a mile from the sea. Discovery of Roman coins makes it probable that it was once occupied by the Romans. In the 3rd century Bronwen (white bosom), daughter of Bran Fendigaid (the blessed), is said to have stayed here, perhaps by force; and there was here a tower, called Twr Bronwen, and replaced about A.D. 550 by the building of Maelgwyn Gwynedd, prince of North Wales. In the early 10th century, Harlech castle was, apparently, repaired by Colwyn, lord of Ardudwy, founder of one of the fifteen North Wales tribes, and thence called Caer Colwyn. The present structure dates, like many others in the principality, from Edward I., perhaps even from the plans of the architect of Carnarvon and Conway castles, but with the retention of old portions. It is thought to have been square, each side measuring some 210 ft., with towers and turrets. Glendower held it for four years. Here, in 1460, Margaret, wife of Henry VI., defeated at Northampton, took refuge. Dafydd ap Ieuan ap Einion held it for the Lancastrians, until famine, rather than Edward IV., made him surrender. From this time is said to date the air "March of the men of Harlech" (_Rhyfelgerdd gwyr Harlech_). The castle was alternately Roundhead and Cavalier in the civil war. Edward I. made Harlech a free borough, and it was formerly the county town. It is in the parish of Llandanwg (pop. in 1901, 931). Though interesting from an antiquarian point of view, the district around, especially Dyffryn Ardudwy (the valley), is dreary and desolate, e.g. Drws (the door of) Ardudwy, Rhinog fawr and Rhinog fach (cliffs); an exception is the verdant Cwm bychan (little combe or hollow). The Meini gwyr Ardudwy (stones of the men of Ardudwy) possibly mark the site of a fight.

HARLEQUIN, in modern pantomime, the posturing and acrobatic character who gives his name to the "harlequinade," attired in mask and parti-coloured and spangled tights, and provided with a sword like a bat, by which, himself invisible, he works wonders. It has generally been assumed that Harlequin was transferred to France from the "Arlecchino" of Italian medieval and Renaissance popular comedy; but Dr Driesen in his _Ursprung des Harlekins_ (Berlin, 1904) shows that this is incorrect. An old French "Harlekin" (Herlekin, Hellequin and other variants) is found in folk-literature as early as 1100; he had already become proverbial as a ragamuffin of a demoniacal appearance and character; in 1262 a number of harlekins appear in a play by Adam de la Halle as the intermediaries of King Hellekin, prince of Fairyland, in courting Morgan le Fay; and it was not till much later that the French Harlekin was transformed into the Italian Arlecchino. In his typical French form down to the time of Gottsched, he was a spirit of the air, deriving thence his invisibility and his characteristically light and aery whirlings. Subsequently he returned from the Italian to the French stage, being imported by Marivaux into light comedy; and his various attributes gradually became amalgamated into the latter form taken in pantomime.

HARLESS (originally HARLES), GOTTLIEB CHRISTOPH (1738-1815), German classical scholar and bibliographer, was born at Culmbach in Bavaria on the 21st of June 1738. He studied at Halle, Erlangen and Jena. In 1765 he was appointed professor of oriental languages and eloquence at the Gymnasium Casimirianum in Coburg, in 1770 professor of poetry and eloquence at Erlangen, and in 1776 librarian of the university. He held his professorship for forty-five years till his death on the 2nd of November 1815. Harless was an extremely prolific writer. His numerous editions of classical authors, deficient in originality and critical judgment, although valuable at the time as giving the student the results of the labours of earlier scholars, are now entirely superseded. But he will always be remembered for his meritorious work in connexion with the great _Bibliotheca Graeca_ of J. A. Fabricius, of which he published a new and revised edition (12 vols., 1790-1809, not quite completed),--a task for which he was peculiarly qualified. He also wrote much on the history and bibliography of Greek and Latin literature.

His life was written by his son, Johann Christian Friedrich Harless
(1818).

HARLESS, GOTTLIEB CHRISTOPH ADOLF VON (1806-1879), German divine, was born at Nuremberg on the 21st of November 1806, and was educated at the universities of Erlangen and Halle. He was appointed professor of theology at Erlangen in 1836 and at Leipzig in 1845. He was a strong Lutheran and exercised a powerful influence in that direction as court preacher in Dresden and as president of the Protestant consistory at Munich. His chief works were _Theologische Encyklopadie und Methodologie_ (1837) and _Die christliche Ethik_ (1842, Eng. trans. 1868). He died on the 5th of September 1879, having, a few years earlier, written an autobiography under the title _Bruchstucke aus dem Leben eines suddeutschen Theologen_.

HARLINGEN, a seaport in the province of Friesland, Holland, on the Zuider Zee, and the terminus of the railway and canal from Leeuwarden (15-1/2 m. E.). It is connected by steam tramway by way of Bolswaard with Sneek. Pop. (1900) 10,448. Harlingen has become the most considerable seaport of Friesland since the construction of the large outer harbour in 1870-1877, and in addition to railway and steamship connexion with Bremen, Amsterdam, and the southern provinces there are regular sailings to Hull and London. Powerful sluices protect the inner harbour from the high tides. The only noteworthy buildings are the town hall (1730-1733), the West church, which consists of a part of the former castle of Harlingen, the Roman Catholic church, the Jewish synagogue and the schools of navigation and of design. The chief trade of Harlingen is the exportation of Frisian produce, namely, butter and cheese, cattle, sheep, fish, potatoes, flax, &c. There is also a considerable import trade in timber, coal, raw cotton, hemp and jute for the Twente factories. The local industries are unimportant, consisting of saw-mills, rope-yards, salt refineries, and sail-cloth and margarine factories.

HARMATTAN, the name of a hot dry parching wind that blows during December, January and February on the coast of Upper Guinea, bringing a high dense haze of red dust which darkens the air. The natives smear their bodies with oil or fat while this parching wind is blowing.

HARMODIUS, a handsome Athenian youth, and the intimate friend of Aristogeiton. Hipparchus, the younger brother of the tyrant Hippias, endeavoured to supplant Aristogeiton in the good graces of Harmodius, but, failing in the attempt, revenged himself by putting a public affront on Harmodius's sister at a solemn festival. Thereupon the two friends conspired with a few others to murder both the tyrants during the armed procession at the Panathenaic festival (514 B.C.), when the people were allowed to carry arms (this licence is denied by Aristotle in _Ath. Pol._). Seeing one of their accomplices speaking to Hippias, and imagining that they were being betrayed, they prematurely attacked and slew Hipparchus alone. Harmodius was cut down on the spot by the guards, and Aristogeiton was soon captured and tortured to death. When Hippias was expelled (510), Harmodius and Aristogeiton became the most popular of Athenian heroes; their descendants were exempted from public burdens, and had the right of public entertainment in the Prytaneum, and their names were celebrated in popular songs and scolia (after-dinner songs) as the deliverers of Athens. One of these songs, attributed to a certain Callistratus, is preserved in Athenaeus (p. 695). Their statues by Antenor in the agora were carried off by Xerxes and replaced by new ones by Critius and Nesiotes. Alexander the Great afterwards sent back the originals to Athens. It is not agreed which of these was the original of the marble tyrannicide group in the museum at Naples, for which see article GREEK ART, Pl. I. fig. 50.

See Kopp in _Neue Jahrb. f. klass. Altert._ (1902), p. 609.

HARMONIA, in Greek mythology, according to one account the daughter of Ares and Aphrodite, and wife of Cadmus. When the government of Thebes was bestowed upon Cadmus by Athena, Zeus gave him Harmonia to wife. All the gods honoured the wedding with their presence. Cadmus (or one of the gods) presented the bride with a robe and necklace, the work of Hephaestus. This necklace brought misfortune to all who possessed it. With it Polyneices bribed Eriphyle to persuade her husband Amphiaraus to undertake the expedition against Thebes. It led to the death of Eriphyle, of Alcmaeon, of Phegeus and his sons. Even after it had been deposited in the temple of Athena Pronoia at Delphi, its baleful influence continued. Phayllus, one of the Phocian leaders in the Sacred War (352 B.C.) carried it off and gave it to his mistress. After she had worn it for a time, her son was seized with madness and set fire to the house, and she perished in the flames. According to another account, Harmonia belonged to Samothrace and was the daughter of Zeus and Electra, her brother Iasion being the founder of the mystic rites celebrated on the island (Diod. Sic. v. 48). Finally, Harmonia is rationalized as closely allied to Aphrodite Pandemos, the love that unites all people, the personification of order and civic unity, corresponding to the Roman Concordia.

Apollodorus iii. 4-7; Diod. Sic. iv. 65, 66; Parthenius, _Erotica_,
25; L. Preller, _Griech. Mythol._; Crusius in Roscher's _Lexikon_.

HARMONIC. In acoustics, a harmonic is a secondary tone which accompanies the fundamental or primary tone of a vibrating string, reed, &c.; the more important are the 3rd, 5th, 7th, and octave (see SOUND; HARMONY). A harmonic proportion in arithmetic and algebra is such that the reciprocals of the proportionals are in arithmetical proportion; thus, if a, b, c be in harmonic proportion then 1/a, 1/b, 1/c are in arithmetical proportion; this leads to the relation 2/b = ac/(a + c). A harmonic progression or series consists of terms whose reciprocals form an arithmetical progression; the simplest example is: 1 + 1/2 + 1/3 + 1/4 + ... (see ALGEBRA and ARITHMETIC). The occurrence of a similar proportion between segments of lines is the foundation of such phrases as harmonic section, harmonic ratio, harmonic conjugates, &c. (see GEOMETRY: II. _Projective_). The connexion between acoustical and mathematical harmonicals is most probably to be found in the Pythagorean discovery that a vibrating string when stopped at 1/2 and 2/3 of its length yielded the octave and 5th of the original tone, the numbers, 1-2/3, 1/2 being said to be, probably first by Archytas, in harmonic proportion. The mathematical investigation of the form of a vibrating string led to such phrases as harmonic curve, harmonic motion, harmonic function, harmonic analysis, &c. (see MECHANICS and SPHERICAL HARMONICS).

HARMONICA, a generic term applied to musical instruments in which sound is produced by friction upon glass bells. The word is also used to designate instruments of percussion of the Glockenspiel type, made of steel and struck by hammers (Ger. _Stahlharmonika_).

The origin of the glass-harmonica tribe is to be found in the fashionable 18th century instrument known as musical glasses (Fr. _verrillon_), the principle of which was known already in the 17th century.[1] The invention of musical glasses is generally ascribed to an Irishman, Richard Pockrich, who first played the instrument in public in Dublin in 1743 and the next year in England, but Eisel[2] described the _verrillon_ and gave an illustration of it in 1738. The _verrillon_ or _Glassspiel_ consisted of 18 beer glasses arranged on a board covered with cloth, water being poured in when necessary to alter the pitch. The glasses were struck on both sides gently with two long wooden sticks in the shape of a spoon, the bowl being covered with silk or cloth. Eisel states that the instrument was used for church and other solemn music. Gluck gave a concert at the "little theatre in the Haymarket" (London) in April 1746, at which he performed on musical glasses a concerto of his composition with full orchestral accompaniment. E. H. Delaval is also credited with the invention. When Benjamin Franklin visited London in 1757, he was so much struck by the beauty of tone elicited by Delaval and Pockrich, and with the possibilities of the glasses as musical instruments, that he set to work on a mechanical application of the principle involved, the eminently successful result being the glass harmonica finished in 1762. In this the glass bowls were mounted on a rotating spindle, the largest to the left, and their under-edges passed during each revolution through a water-trough. By applying the fingers to the moistened edges, sound was produced varying in intensity with the pressure, so that a certain amount of expression was at the command of a good player. It is said that the timbre was extremely enervating, and, together with the vibration caused by the friction on the finger-tips, exercised a highly deleterious effect on the nervous system. The instrument was for many years in great vogue, not only in England but on the Continent of Europe, and more especially in Saxony, where it was accorded a place in the court orchestra. Mozart, Beethoven, Naumann and Hasse composed music for it. Marianne Davies and Marianna Kirchgessner were celebrated virtuosi on it. The curious vogue of the instrument, as sudden as it was ephemeral, produced emulation in a generation unsurpassed for zeal in the invention of musical instruments. The most notable of its offspring were Carl Leopold Rollig's improved harmonica with a keyboard in 1786, Chladni's euphon in 1791 and clavicylinder in 1799, Ruffelsen's melodicon in 1800 and 1803, Franz Leppich's panmelodicon in 1810, Buschmann's uranion in the same year, &c. Of most of these nothing now remains but the name and a description in the _Allgemeine musikalische Zeitung_, but there are numerous specimens of the Franklin type in the museums for musical instruments of Europe. One specimen by Emanuel Pohl, a Bohemian maker, is preserved in the Victoria and Albert Museum, London.

For the steel harmonica see GLOCKENSPIEL. (K. S.)

FOOTNOTES:

[1] See G. P. Harsdorfer, _Math. und philos. Erquickstunden_
(Nuremberg, 1677), ii. 147.

[2] _Musicus_ [Greek: autodidaktos] (Erfurt, 1738), p. 70.

HARMONIC ANALYSIS, in mathematics, the name given by Sir William Thomson (Lord Kelvin) and P. G. Tait in their treatise on _Natural Philosophy_ to a general method of investigating physical questions, the earliest applications of which seem to have been suggested by the study of the vibrations of strings and the analysis of these vibrations into their fundamental tone and its harmonics or overtones.

The motion of a uniform stretched string fixed at both ends is a periodic motion; that is to say, after a certain interval of time, called the fundamental period of the motion, the form of the string and the velocity of every part of it are the same as before, provided that the energy of the motion has not been sensibly dissipated during the period.

There are two distinct methods of investigating the motion of a
uniform stretched string. One of these may be called the wave method,
and the other the harmonic method. The wave method is founded on the
theorem that in a stretched string of infinite length a wave of any
form may be propagated in either direction with a certain velocity, V,
which we may define as the "velocity of propagation." If a wave of any
form travelling in the positive direction meets another travelling in
the opposite direction, the form of which is such that the lines
joining corresponding points of the two waves are all bisected in a
fixed point in the line of the string, then the point of the string
corresponding to this point will remain fixed, while the two waves
pass it in opposite directions. If we now suppose that the form of the
waves travelling in the positive direction is periodic, that is to
say, that after the wave has travelled forward a distance l, the
position of every particle of the string is the same as it was at
first, then l is called the wave-length, and the time of travelling a
wave-length is called the periodic time, which we shall denote by T,
so that l = VT.

If we now suppose a set of waves similar to these, but reversed in
position, to be travelling in the opposite direction, there will be a
series of points, distant 1/2l from each other, at which there will be
no motion of the string; it will therefore make no difference to the
motion of the string if we suppose the string fastened to fixed
supports at any two of these points, and we may then suppose the parts
of the string beyond these points to be removed, as it cannot affect
the motion of the part which is between them. We have thus arrived at
the case of a uniform string stretched between two fixed supports, and
we conclude that the motion of the string may be completely
represented as the resultant of two sets of periodic waves travelling
in opposite directions, their wave-lengths being either twice the
distance between the fixed points or a submultiple of this
wave-length, and the form of these waves, subject to this condition,
being perfectly arbitrary.

To make the problem a definite one, we may suppose the initial
displacement and velocity of every particle of the string given in
terms of its distance from one end of the string, and from these data
it is easy to calculate the form which is common to all the travelling
waves. The form of the string at any subsequent time may then be
deduced by calculating the positions of the two sets of waves at that
time, and compounding their displacements.

Thus in the wave method the actual motion of the string is considered
as the resultant of two wave motions, neither of which is of itself,
and without the other, consistent with the condition that the ends of
the string are fixed. Each of the wave motions is periodic with a
wave-length equal to twice the distance between the fixed points, and
the one set of waves is the reverse of the other in respect of
displacement and velocity and direction of propagation; but, subject
to these conditions, the form of the wave is perfectly arbitrary. The
motion of a particle of the string, being determined by the two waves
which pass over it in opposite directions, is of an equally arbitrary
type.

In the harmonic method, on the other hand, the motion of the string is
regarded as compounded of a series of vibratory motions (_normal
modes_ of vibration), which may be infinite in number, but each of
which is perfectly definite in type, and is in fact a particular
solution of the problem of the motion of a string with its ends fixed.

A simple harmonic motion is thus defined by Thomson and Tait (S
53):--When a point Q moves uniformly in a circle, the perpendicular
QP, drawn from its position at any instant to a fixed diameter AA' of
the circle, intersects the diameter in a point P whose position
changes by a _simple harmonic motion_.

The amplitude of a simple harmonic motion is the range on one side or
the other of the middle point of the course.

The period of a simple harmonic motion is the time which elapses from
any instant until the moving-point again moves in the same direction
through the same position.

The phase of a simple harmonic motion at any instant is the fraction
of the whole period which has elapsed since the moving-point last
passed through its middle position in the positive direction.

In the case of the stretched string, it is only in certain particular
cases that the motion of a particle of the string is a simple harmonic
motion. In these particular cases the form of the string at any
instant is that of a curve of sines having the line joining the fixed
points for its axis, and passing through these two points, and
therefore having for its wave-length either twice the length of the
string or some submultiple of this wave-length. The amplitude of the
curve of sines is a simple harmonic function of the time, the period
being either the fundamental period or some submultiple of the
fundamental period. Every one of these modes of vibration is
dynamically possible by itself, and any number of them may coexist
independently of each other.

By a proper adjustment of the initial amplitude and phase of each of
these modes of vibration, so that their resultant shall represent the
initial state of the string, we obtain a new representation of the
whole motion of the string, in which it is seen to be the resultant of
a series of simple harmonic vibrations whose periods are the
fundamental period and its submultiples. The determination of the
amplitudes and phases of the several simple harmonic vibrations so as
to satisfy the initial conditions is an example of harmonic analysis.

We have thus two methods of solving the partial differential equation
of the motion of a string. The first, which we have called the wave
method, exhibits the solution in the form containing an arbitrary
function, the nature of which must be determined from the initial
conditions. The second, or harmonic method, leads to a series of terms
involving sines and cosines, the coefficients of which have to be
determined. The harmonic method may be defined in a more general
manner as a method by which the solution of any actual problem may be
obtained as the sum or resultant of a number of terms, each of which
is a solution of a particular case of the problem. The nature of these
particular cases is defined by the condition that any one of them must
be conjugate to any other.

The mathematical test of conjugacy is that the energy of the system
arising from two of the harmonics existing together is equal to the
sum of the energy arising from the two harmonics taken separately. In
other words, no part of the energy depends on the product of the
amplitudes of two different harmonics. When two modes of motion of the
same system are conjugate to each other, the existence of one of them
does not affect the other.

The simplest case of harmonic analysis, that of which the treatment of
the vibrating string is an example, is completely investigated in what
is known as Fourier's theorem.

Fourier's theorem asserts that any periodic function of a single
variable period p, which does not become infinite at any phase, can be
expanded in the form of a series consisting of a constant term,
together with a double series of terms, one set involving cosines and
the other sines of multiples of the phase.

Thus if [phi]([xi]) is a periodic function of the variable [xi] having
a period p, then it may be expanded as follows:

__[oo] 2i[pi][xi] __[oo] 2i[pi][xi]
[phi]([xi]) = A0 + \ ^i A_i cos ---------- + \ ^i B_i sin ----------. (1)
/__1 p /__1 p

The part of the theorem which is most frequently required, and which
also is the easiest to investigate, is the determination of the values
of the coefficients A0, A_i, B_i. These are
_ _
1 /p 2 /p 2i[pi][xi]
A0 = -- | [phi]([xi])d[xi]; A_i = -- | [phi]([xi]) cos ---------- d[xi];
p _/0 p _/0 p
_
2 /p 2i[pi][xi]
B_i = -- | [phi]([xi]) sin ---------- d[xi].
p _/0 p

This part of the theorem may be verified at once by multiplying both
sides of (1) by d[xi], by cos (2i[pi][xi]/p)/d[xi] or by sin
(2i[pi][xi]/p)/d[xi], and in each case integrating from 0 to p.

The series is evidently single-valued for any given value of [xi]. It
cannot therefore represent a function of [xi] which has more than one
value, or which becomes imaginary for any value of [xi]. It is
convergent, approaching to the true value of [phi]([xi]) for all
values of [xi] such that if [xi] varies infinitesimally the function
also varies infinitesimally.

Lord Kelvin, availing himself of the disk, globe and cylinder
integrating machine invented by his brother, Professor James Thomson,
constructed a machine by which eight of the integrals required for the
expression of Fourier's series can be obtained simultaneously from the
recorded trace of any periodically variable quantity, such as the
height of the tide, the temperature or pressure of the atmosphere, or
the intensity of the different components of terrestrial magnetism. If
it were not on account of the waste of time, instead of having a curve
drawn by the action of the tide, and the curve afterwards acted on by
the machine, the time axis of the machine itself might be driven by a
clock, and the tide itself might work the second variable of the
machine, but this would involve the constant presence of an expensive
machine at every tidal station. (J. C. M.)

For a discussion of the restrictions under which the expansion of a
periodic function of [xi] in the form (1) is valid, see FOURIER'S
SERIES. An account of the contrivances for mechanical calculation of
the coefficients A_i, B_i ... is given under CALCULATING MACHINES.

A more general form of the problem of harmonic analysis presents
itself in astronomy, in the theory of the tides, and in various
magnetic and meteorological investigations. It may happen, for
instance, that a variable quantity [f](t) is known theoretically to be
of the form

[f](t) = A0 + A1 cos n1t + B1 sin n1t + A2 cos n2t + B2 sin n2t + ... (2)

where the periods 2[pi]/n1, 2[pi]/n2, ... of the various
simple-harmonic constituents are already known with sufficient
accuracy, although they may have no very simple relations to one
another. The problem of determining the most probable values of the
constants A0, A1, B1, A2, B2, ... by means of a series of recorded
values of the function [f](t) is then in principle a fairly simple
one, although the actual numerical work may be laborious (see TIDE). A
much more difficult and delicate question arises when, as in various
questions of meteorology and terrestrial magnetism, the periods
2[pi]/n1, 2[pi]/n2, ... are themselves unknown to begin with, or are
at most conjectural. Thus, it may be desired to ascertain whether the
magnetic declination contains a periodic element synchronous with the
sun's rotation on its axis, whether any periodicities can be detected
in the records of the prevalence of sun-spots, and so on. From a
strictly mathematical standpoint the problem is, indeed,
indeterminate, for when all the symbols are at our disposal, the
representation of the observed values of a function, over a finite
range of time, by means of a series of the type (2), can be effected
in an infinite variety of ways. Plausible inferences can, however, be
drawn, provided the proper precautions are observed. This question has
been treated most systematically by Professor A. Schuster, who has
devised a remarkable mathematical method, in which the action of a
diffraction-grating in sorting out the various periodic constituents
of a heterogeneous beam of light is closely imitated. He has further
applied the method to the study of the variations of the magnetic
declination, and of sun-spot records.

The question so far chiefly considered has been that of the
representation of an arbitrary function of the _time_ in terms of
functions of a special type, viz. the circular functions cos nt, sin
nt. This is important on dynamical grounds; but when we proceed to
consider the problem of expressing an arbitrary function of
_space-co-ordinates_ in terms of functions of specified types, it
appears that the preceding is only one out of an infinite variety of
modes of representation which are equally entitled to consideration.
Every problem of mathematical physics which leads to a linear
differential equation supplies an instance. For purposes of
illustration we will here take the simplest of all, viz. that of the
transversal vibrations of a tense string. The equation of motion is of
the form

[dP]^2y [dP]^2y
[rho] ------- = T -------, (3)
[dP]t^2 [dP]x^2

where T is the tension, and [rho] the line-density. In a "normal mode"
of vibration y will vary as e^(int), so that

[dP]^2y
------- + k^2y = 0, (4)
[dP]x^2

where

k^2 = n^2[rho]/T. (5)

If [rho], and therefore k, is constant, the solution of (4) subject to
the condition that y = 0 for x = 0 and x = l is

y = B sin kx (6)

provided

kl = s[pi], [s = 1, 2, 3, ...]. (7)

This determines the various _normal modes_ of free vibration, the
corresponding periods (2[pi]/n) being given by (5) and (7). By analogy
with the theory of the free vibrations of a system of _finite_ freedom
it is inferred that the most general free motions of the string can be
obtained by superposition of the various normal modes, with suitable
amplitudes and phases; and in particular that any arbitrary initial
form of the string, say y = [f](x), can be reproduced by a series of
the type

[pi]x 2[pi]x 3[pi]x
[f](x) = B1 sin ----- + B2 sin ------ + B3 sin ------ + ... (8)
l l l

So far, this is merely a restatement, in mathematical language, of an
argument given in the first part of this article. The series (8) may,
moreover, be arrived at otherwise, as a particular case of Fourier's
theorem. But if we no longer assume the density [rho] of the string to
be uniform, we obtain an endless variety of new expansions,
corresponding to the various laws of density which may be prescribed.
The normal modes are in any case of the type

y = Cu(x) e^(int) (9)

where u is a solution of the equation

d^2u n^2[rho]
---- + -------- u = 0. (10)
dx^2 T

The condition that u(x) is to vanish for x = 0 and x = l leads to a
transcendental equation in n (corresponding to sin kl = 0 in the
previous case). If the forms of u(x) which correspond to the various
roots of this be distinguished by suffixes, we infer, on physical
grounds alone, the possibility of the expansion of an arbitrary
initial form of the string in a series

[f](x) = C1u1(x) + C2u2(x) + C3u3(x)+ ... (11)

It may be shown further that if r and s are different we have the
_conjugate_ or _orthogonal_ relation
_
/l
| [rho] u_r(x) u_s(x) dx = 0. (12)
_/0

This enables us to determine the coefficients, thus
_ _
/l / /l
C_r = | [rho][f](x)u_r(x) dx / | [rho] {u_r(x)}^2 dx. (13)
_/0 / _/0

The extension to spaces of two or three dimensions, or to cases where
there is more than one dependent variable, must be passed over. The
mathematical theories of acoustics, heat-conduction, elasticity,
induction of electric currents, and so on, furnish an indefinite
supply of examples, and have suggested in some cases methods which
have a very wide application. Thus the transverse vibrations of a
circular membrane lead to the theory of Bessel's Functions; the
oscillations of a spherical sheet of air suggest the theory of
expansions in spherical harmonics, and so forth. The physical, or
intuitional, theory of such methods has naturally always been in
advance of the mathematical. From the latter point of view only a few
isolated questions of the kind had, until quite recently, been treated
in a rigorous and satisfactory manner. A more general and
comprehensive method, which seems to derive some of its inspiration
from physical considerations, has, however, at length been
inaugurated, and has been vigorously cultivated in recent years by D.
Hilbert, H. Poincare, I. Fredholm, E. Picard and others.

REFERENCES.--Schuster's method for detecting hidden periodicities is
explained in _Terrestrial Magnetism_ (Chicago, 1898), 3, p. 13; _Camb.
Trans._ (1900), 18, p. 107; _Proc. Roy. Soc._ (1906), 77, p. 136. The
general question of expanding an arbitrary function in a series of
functions of special types is treated most fully from the physical
point of view in Lord Rayleigh's _Theory of Sound_ (2nd ed., London,
1894-1896). An excellent detailed historical account of the matter
from the mathematical side is given by H. Burkhardt, _Entwicklungen
nach oscillierenden Funktionen_ (Leipzig, 1901). A sketch of the more
recent mathematical developments is given by H. Bateman, _Proc. Lond.
Math. Soc._ (2), 4, p. 90, with copious references. (H. Lb.)

HARMONICHORD, an ingenious kind of upright piano, in which the strings were set in vibration not by the blow of the hammer but by indirectly transmitted friction. The harmonichord, one of the many attempts to fuse piano and violin, was invented by Johann Gottfried and Johann Friedrich Kaufmann (father and son) in Saxony at the beginning of the 19th century, when the craze for new and ingenious musical instruments was at its height. The case was of the variety known as _giraffe_. The space under the keyboard was enclosed, a knee-hold being left in which were two pedals used to set in rotation a large wooden cylinder fixed just behind the keyboard over the levers, and covered with a roll-top similar to those of modern office desks. The cylinder (in some specimens covered with chamois leather) tapered towards the treble-end. When a key was depressed, a little tongue of wood, one end of which stopped the string, was pressed against the revolving cylinder, and the vibrations produced by friction were transmitted to the string and reinforced as in piano and violin by the soundboard. The adjustment of the parts and the velocity of the cylinder required delicacy and great nicety, for if the little wooden tongues rested too lightly upon the cylinder or the strings, harmonics were produced, and the note jumped to the octave or twelfth. Sometimes when chords were played the touch became so heavy that two performers were required, as in the early medieval organistrum, the prototype of the harmonichord. Carl Maria von Weber must have had some opinion of the possibilities of the harmonichord, which in tone resembled the glass harmonica, since he composed for it a concerto with orchestral accompaniment. (K. S.)

HARMONIUM (Fr. _harmonium_, _orgue expressif_; Ger. _Physharmonika_, _Harmonium_), a wind keyboard instrument, a small organ without pipes, furnished with free reeds. Both the harmonium and its later development, the American organ, are known as free-reed instruments, the musical tones being produced by tongues of brass, technically termed "vibrators" (Fr. _anche libre_; Ger. _durchschlagende Zunge_; Ital. _ancia_ or _lingua libera_). The vibrator is fixed over an oblong, rectangular frame, through which it swings freely backwards and forwards like a pendulum while vibrating, whereas the beating reeds (similar to those of the clarinet family), used in church organs, cover the entire orifice, beating against the sides at each vibration. A reed or vibrator, set in periodic motion by impact of a current of air, produces a corresponding succession of air puffs, the rapidity of which determines the pitch of the musical note. There is an essential difference between the harmonium and the American organ in the direction of this current; in the former the wind apparatus forces the current upwards, and in the latter sucks it downwards, whence it becomes desirable to separate in description these varieties of free-reed instruments.

FIG. 1.--Free Reed Vibrator, Alexandre Harmonium.]