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in iaio. The city has a Board of Trade and various commerical and social organizations.

Government.— The city is governed by a mayor, elected annually, and a bicameral council. The municipal expenses aggregate about $1,500,000 annually, including $375,000 for schools; $195,000 for streets; $115,000 for fire department; $86,000 for police; $74,000 for lighting, and $54,000 for the poor. The waterworks are owned and operated by The system cost over $2,coo,ooo. In 1910 the city had a net funded debt of $4,240,161. The assessed valuation of real and personal property in the same year was $119,081,778.

Education.— The public school system embraces high, grammar, primary, manual training, trade, and kindergarten schools, occupying 34 buildings. The evening trade school is said to be the first of its kind in the United States. There are 374 teachers and 11,000 pupils. The parochial schools have 1,500 pupils. Springfield is the seat of the French-American College, opened in 1885, and the International Y. M. C. A. Training School. There is a public library here containing 130.000 volumes, and the Hampden County Law Library dating from 1813. The facilities and collections of the art and science museums,— departments of the City Library,— are perhaps unrivaled in any city of similar size in the country; the library itself is of unusual efficiency. The system of public education is considered one of the most advanced in the Unit'd States. The city is the home of two famous publishing enterprises— 'Webster's Dictionary* and the Springfield Republican.

Religion.— There are upward of 50 churches in Springfield, of which 12 are Congregational, 7 Methodist, 7 Baptist, 6 Roman Catholic, 3 Universalist, 2 Episcopalian, 2 Lutheran, and also Unitarian, Hebrew, and other denominations. The charitable institutions include the City Almshouse, Springfield Hospital, Mercy Hospital, and the Hampden Homoeopathic Hospital.

History.— Springfield was first settled in 1636 by a party of emigrants from Roxbury under the leadership of William Pynchon. Until 1640, when it received the name of Springfield (from Springfield, England, the home of Pynchon), the town was known as Agawam. West Springfield, Chicopee, and several of the neighboring towns were then included in its boundaries. On 5 Oct. 1675, during King Philip's War, the town was attacked by Indians and burned. In September 1786, during Shays' Rebellion (q.v.) it was the scene of a riot headed by Daniel Shays. In January 1787 occurred a skirmish between the State militia and 1,000 insurgents led by Shays, in which the latter were defeated. The city was incorporated in 1852.

Population.— Springfield had a population of 2.312 in 1800; (1850) 11,766; (1880) 33.340; (1890) 44,179; (1900) 62,059; (1910) 88,926.

References.— Burt, 'The First Century of the History of Springfield' (1888); Green, •Springfield, i686-i886> (1888).

Springfield, Mo., city, county-seat of Green County; on the Saint Louis & S. F., and the Kansas City, F. S. & M. R.R.'s; about 130 miles south of Jefferson City, the capital of the State. In the early part of the Civil War, there oc

curred in the city and vicinity several battlesOne, which is known as the battle of Wilson's Creek, took place 10 Aug. 1861; the Union forces were defeated and the Union general, Nathaniel Lyon, was killed. The city is on the ridge of the Ozark Mountains in a region known for its valuable deposits of zinc and lead. The climate is healthful and warm temperate about all the year. Its industries are chiefly connected with the mining and marketing of lead and zinc. The chief industrial establishments are two large railroad-shops, iron works, machine shops, wagon and carriage factories, furniture factories, flour mills, and novelty works. The government census of 1900 gives as the total number of manufacturing establishments 245; the amount of capital invested $2,111,048; number of wage-earners, 2,127; average annual wages, $1,017,345; cost: 01 material used, $2,274,705; value of products, $4,126,871. It has an extensive trade in manufactures, lead and zinc and as a jobbing centre for a large part of the southern counties. The principal public buildings are the government buildings, county court-house, municipal buildings, Saint John's Hospital, Springfield Frisco Hospital, the bank buildings, and several fine building blocks. The educational institutions are a State Normal School, Loretto Academy (R.C.), Drury College (Cong.), public and parish elementary schools, and a public library. There are the National and Confederate cemeteries, parks, and in the vicinity many places where the scenery is most beautiful. Pop. (1890) 21,850; (1900) 23,267; (1910) 35,201.

Springfield (Mo.), Military Operations

at. As the central point of southwestern Missouri, Springfield was an important place in military operations during the Civil War. After the engagement at Carthage (q.v.), 5 July 1861, Col. Sigel retreated to Springfield, where he was joined by Gen. Nathaniel Lyon (q.v.) on the 13th, Lyon assuming command of all the Union forces, and calling for more to make head against the combined forces of Gens. Sterling Price and Ben. McCulloch, a call that brought no response. On I August, hearing of the Confederate advance, Lyon marched from Springfield with 6,000 men and 18 guns against it, and next day at Dug Springs, 19 miles from the city, his advance encountered and drove back the Confederates under Gen. Rains, with slight loss, and returned to Springfield on the 5th, which he thought he should be compelled to abandon, falling back either to Saint Louis or Kansas, as he was largely outnumbered. He reported the condition of affairs to Gen. Fremont (q.v.) in command at Saint Louis, held his position, and advancing on 10 August, fought the battle of Wilson's Creek (q.v.), nine miles southwest of Springfield, and was killed. His army, under Col. Sigel, retreated through Springfield to Rolla, the Confederates following only to Springfield, which they held until the 25th, when they moved against Lexington. On 13 August Fremont, hearing of the Wilson's Creek disaster, Lyon's death, and the unobstructed retreat of Sigel on Rolla, sent reinforcements to Rolla, and at the end of September he left Saint Louis with an army of 38,000 men and 86 guns to take the field in southwestern Missouri, against Sterling Price (q.v.). His five columns were directed on Springfield. On 23 October, when about 50 miles from Springfield, Fremont sent Maj. Zagonyi, with about 250 men of his bodyguard and of the t(Irish Brigade* (37th Illinois) to advance on the place. Zagonyi reached Springfield on the 25th. It was then held by about 1,000 recruits, some mounted, and some on foot, and after several charges Zagonyi routed them, his loss being 15 killed, 27 wounded, and 10 missing. Fremont occupied the place on the 27th with 21,000 effective men, and was about to advance and give battle to the Confederates, when (2 November) he was relieved in command by Gen. Hunter, who withdrew the army to Rolla. Price reoccupied Springfield and began to recruit an army to recover Saint Louis, and had gathered some 8,000 men by 12 Feb. 1862, when he was driven from the place by Gen. Curtis, Price retreating into Arkansas. Springfield now became a depot of supplies, and was well fortified. Early in January 1863, Gen. Marmaduke (q.v.) with 4,000 mounted men and 3 guns, started from northern Arkansas on a raid into Missouri, and with a part of his command appeared before Springfield on the morning of the 8th. Gen. E. B. Brown, in command of the place, had but short notice of his approach, but gathered some militia from the adjoining country, which, with his regular command, convalescents from the hospitals, and some citizens, gave him about 2,100 men. The fight began about 10 A.m. of the 8th and continued until dark. Marmaduke's men drove in parts of the line, captured one gun, carried some of the works, and seized part of the town, but Brown held the strongest fort against all efforts to take it. On the morning of the 9th Marmaduke withdrew in the direction of Rolla, after a loss of 20 killed, 105 wounded, and 26 missing. The Union loss was 14 killed, 146 wounded, and 5 missing. E. A. Carman.


Springfield, N. J., township in Union County; on the Rahway River, and on the Delaware & Lackawanna Railroad; eight miles s -uthwest of Newark. The chief manufactories are boot and shoe factories, paper mills, and hat factory. Springfield is celebrated as the scene of a battle between the American and British forces, 23 June 1780. The British, under General Knyphausen, advanced from Elizabethtown about 5 o'clock in the morning. They were opposed by General Greene, but owing to the superior number of the enemy he was compelled to evacuate Springfield, which was then burned by the British. During the action the Rev. James Caldwell, chaplain in the New Jersey brigade, is said to have distributed the hymn books from the neighboring Presbyterian church among the soldiers for wadding, saying at the same time, "Now put Watts into them, boys." This battle prevented further advance on the part of the British. The American loss was about 72 and that of the British about 150. Pop. (1910) 1,246.

Springfield, Ohio, city, county-seat of Clark County, on the Lagonda Creek and the Mad River, and on the Pittsburg, Cincinnati, Chicago & Saint Louis, the Cleveland, Cincinnati, Chicago & Saint Louis, the Erie, and the Detroit Southern R.R's; about 45 miles west of Columbus. Five lines of electric railways provide direct

connections with the cities of Columbus, Dayton, Xenia, Urbana, Bellefontaine, Troy, and Piqua.

Manufacturing.— The surroundings of the city were naturally rich in the possibilities of agriculture. An abundance of water power, utilized in woolen, cotton, flour and powder mills, enabled the early settlers to supply many of the needs of a pioneer community. In later years native genius turned the drift of the city's principal energies into the line of manufactures with the result that the fame of Springfield, as a manufacturing centre, is literally world-wide, and, especially, in respect to the fabrication of implements designed for the planting, cultivation and harvesting of the great staple crops. The developments of more recent years have given these manufactures a varied character embracing gas and steam engines, piano plates, machinery in vast variety, tools, mechanical appliances and factory supplies, in iron, steel and brass. Periodical publications of high class and national renown, and floral industries exceeded in aggregate extent by those of few cities in the Union, contribute to Springfield's commercial prestige and distinction. Flour, medicine, food, wearing apparel, chemical, coffin, rubber and paper manufactures add variety to the list of Springfield's notable products. The government census of 1900 gives the total number of manufacturing establishments as 305; the amount of capital invested, $14,091,175, the number of employees, 6,638; the average annual wages, $3,160,119; and the average value of the annual products $12,777A73- There were seven large manufactories of agricultural implements, with an invested capital of $8,194,543. The 30 foundry and machine-shops had a combined capital of $2,772,036. They had 2,197 employees, to whom was paid annually, on an average, wages amounting to $1,084,681. The value of the annual products was $3,097,910. Since 1900 there has been a notable increase in the. amount of manufacturing. In 1010 there were estimated 10,000 employees in the various industries and the value of the annual output is about $18,000,000.

Transportation and Commerce.— Ample shipping facilities afforded by the steam lines mentioned, ready access to the coal fields of various States and an excellent supply of natural gas fuel are prominent factors in the city's advancement. At an early stage of the development of electric traction lines Springfield took rank as one of the chief centres of Ohio.

Local mercantile interests in lines both of wholesale and retail trade reflect the conditions of a thrifty, growing community.

The Commercial Club, embracing a membership of 300 representative citizens, is the agency for organized effort in the promotion of the growth of the city, and the conservation of all its material interests. Four live and progressive daily newspapers give voice and direction to public-spirited sentiment and action.

Buildings and Municipal Improvements.— The United States post-office building was erected and equipped in 1890, at a cost of $150,000. The II rural free delivery routes add over 6,000 the number served by the Springfield office. The Clark County court-house, a handsome structure is supplemented by a County office building, or striking architectural design and modern construction, the two buildings representing an expenditure of more than $200,000. The City Building cost $250,000, and is one of the finest edifices of its kind in the State, providing a city market 300 feet in length, and a spacious auditorium in addition to quarters for municipal officers. Fronting the City Building are the handsome and graceful Kelly fountain and esplanade which cost $8,000, and were presented to the city by Oliver S. Kelly. The city hospital, founded by the liberal donations of Ross Mitchell and J. H. Thomas and endowed in the sum of $100,000 in government bonds by John Snyder, was erected and furnished at a cost to the city of $100,000 It is equipped with every appliance and convenience essential to its use. The city prison and patrol station, the cost of which was $20,000, is of modern design and construction. Three State fraternal homes have been erected on the hills which border the city. These homes are maintained by the brotherhood of the Masonic craft, the Independent Order of Odd Fellows, and the Knights of Pythias. The city donated the sites on which the institutions are located, the funds being provided by voluntary contributions amounting to $100,000. Asa Smith Bushnell gave to the Masonic Home the sum of $10,000. Thousands of representatives of the fraternal orders annually visit these homes. The Lagonda Club has a handsome building which cost $25,000. The Young Men's Christian Association building, erected by the gifts of the people of Springfield, cost $90,000. Asa Smith Bushnell and Edwin S. Kelly each contributed $5,000.


Snyder Park, a tract of 217 acres, was the gift of John and David L. Snyder. Other donations by the same public benefactors were those of $25,000 for a memorial bridge, and $200,000 in government bonds in endowment of the park. A memorial arch, erected to the memory of the Messrs. Snyder, by the people of Springfield, at a cost of $6,000, is to span the entrance to Snyder Park, and was dedicated 4 July 1904. Ferncliff, Springfield's principal burial ground, occupies a site of surpassing beauty. Springfield has an abundant supply of pure water. The waterworks system cost $800,000. Ample fire pressure is maintained, but the city fire department equipment includes three steam engines of large power.

Churches, Schools, Libraries.—The city has 50 churches, a large number of which are of modern construction and striking artistic beauty. There are 19 public school buildings, four Roman Catholic parish schools, two private business colleges and several private schools. The.Springfield Seminary is one of the large city schools. The Wittenberg College, founded in 1845, by the Lutheran Church, has in attendance about 400 students. The Warder public library, which cost $100,000, is a gift from Benjamin H. Warder. It contains 20,000 volumes and is maintained by an annual appropriation. More than a score of literary clubs contribute to the culture and refinement which are marked characteristics of the people of the city.

Banks and Finances.— There are six banks, five national and one savings bank. The national banks have a combined capital of $1,000,000; the six banks have deposits amounting to $5,214,520. There are four building and loan associations with asset? in excess of $1,500,000. The bank clearings increased from $14,362,664 in

1899, to $27,197,800 in 1910. The gross receipts of the post-office were, in 1899, $117,695 and in 1903, $163,041. The aggregate financial transactions of the post-office, in 1910, were $1,000,000. The aggregate value of the real and personal property, as assessed for taxation, for the year 1910, was $22,581,030.

History.— The plat of the town was made in 1801 by James Demint. The city was chartered in 1850. The life of the city is told by the growth of its industries, charitable institutions, churches, and schools. It has been the home of many noted men, among whom may be mentioned Joseph Warren Keifer (q.v.) and Asa Smith Bushnell (q.v.). Pop. (1900) 38,253; C1910) 46,921. jAMES H. Rabbitts.

Springfield Rifle. See Ordnance; Small Arms.

Spring'tails, a family of wingless insects of the order Collembola (q.v.), distinguished by the possession of an elastic forked caudal appendage, folded under the body when at rest, by the sudden extension of which these insects are enabled to make considerable leaps. Their scales are favorite test-objects for microscopes. Compare Bristletails.

Sprit, a small boom or pole which crosses the sail of a boat diagonally from the mast to the upper aftmost corner, which it is used to extend and elevate. These sails are accordingly called sprit-sails. A sail attached to a yard hanging under the bow-sprit was also formerly so called.

Spruce, or Spruce-fir, a coniferous tree of the subfamily Abietina? and especially of the genus Abies. One of the best known and typical is the Norway spruce (A. excelsa), which is indigenous in the north of Europe where it is the loftiest (often 125 feet) of forest trees, but has been transplanted as an ornament to all cool climates. The American spruces are described under Fir.

Spruce-Beer. Essence of spruce is simply a decoction of the young green tops of the Black Spruce, boiled and evaporated to the consistence of a thick syrup. Spruce beer of good quality may be prepared as follows:— Essence of spruce one half pint; pimento and ginger (bruised) of each five ounces; hops, one half pound; water, three gallons; boil the whole for ten minutes, then add of moist sugar 12 pounds (or good treacle 14 pounds) ; warm water, 11 gallons; mix well, and when lukewarm add a pint of yeast; after the liquid has fermented for about 24 hours it is ready for bottling. This beer is regarded as a diuretic and antiscorbutic, and is relished by many as an agreeable summer beverage.

Spruce Partridge. See Grouse.

Spur, a metal instrument composed of a shank, neck, and prick or rowel, fastened to the heel of a horseman to goad his horse to greater speed. Its use cannot with certainty be traced further back than Roman times. Rowels first appeared early in the 14th century. The spurs of mediaeval knights were gilt and those of esquires silvered. "To win his spurs* meant to gain knighthood. The Mexicans and Spanish-Americans generally wear large spurs, often of silver, except the rowel, and sometimes beautifully ornamented. The tendency in North America and western Europe is to use spurs much less than formerly.

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Spurge. See Euphorbiace.e.

Spurge-laurel. See Daphne.

Spurgeon, sper'jon, Charles Haddon, English Baptist preacher: b. Kelvedon, Essex, 19 June 1834; d. Mentone, France, 21 Jan. 1892. He was the son of a Congregationalist minister and was educated at Colchester and Maidstone. In 1849 he was appointed usher in a school at Newmarket, and soon after engaged in religious work at Cambridge and the neighborhood, being known locally as the "boy preacher." Having joined the Baptist body he accepted the pastorate of a small Baptist congregation at Waterbeach at 18. Becoming known for his eloquence he was called, in 1854, to the pastorate of the Baptist chapel in New Park Street, Southwark, and this, becoming too small for his audience, his congregation successively removed to Exeter Hall and the Surrey Music Hall, and ultimately built, in 1861, the great Metropolitan Tabernacle, which could accommodate 6,000 persons. Here he preached and labored for the rest of his life, his discourses attracting hearers from all parts of the world. Besides his ordinary ministrations, and the publication from 1855 of a weekly sermon, he founded the Pastors' College, at which the ministers of 36 London chapels were trained by him, the Stockwell Orphanage, almshouses, schools, etc. Earnestness, simplicity, directness, liveliness, and not infrequently a genuine touch of humor were the chief characteristics of his sermons. Sagacity, common sense, straightforwardness, hatred for sham and falsity, were prominent traits of his character as a man. He was the author of numerous volumes, among which the best-known are: 'The Saint and his Saviour' (1867); 'John Ploughman's Talk' (1868); 'Feathers for Arrows' (1870); 'The Treasury of David,' a commentary on the Psalms — extending to seven volumes (1865-80) ; 'Types and Emblems' (1875); 'The Metropolitan Tabernacle and its Work' (1876); 'Farm Sermons' (1882); 'The Present Truth' (1883); 'Storm Signals' (1886); 'Salt Cellars' (1889); and he edited the monthly magazine 'Sword and Trowel.' Consult 'Life' by Shindler (1892).

Spurgeon, Thomas, English Baptist clergyman, son of C. H. Spurgeon (q.v.) : b. London, England, 20 Sept. 1856. He was educated at the Pastors' College, studied art, and visited Australia and Tasmania in 1877 and in 1879. In 1880-9 he was in charge of the Baptist Church at Auckland, New Zealand, and was evangelist of the New Zealand Baptist Union in 1880-93. Since 1894 he has been pastor of the Baptist Church, Metropolitan Tabernacle, London. He is editor of 'Sword and Trowel' and has published: 'The Gospel of the Grace of God' sermons (1884); 'Scarlet Threads and Bits of Blue,' poems (1892); 'My Gospel,' sermons (1902); etc.

Spuny, a plant of the genus Spergula, order Caryophyllacece; annual herbs, with the narrow or awl-shaped stipulate leaves fascicled at the swollen nodes of the succulent stems, and thus appear as if in whorls. The flowers are small and white in terminal, loose cymes, and have their parts in fives. S. arvensis is the common corn-spurry or yarr, a weakly spreading slender

plant, generally glabrous, and with numerous flowers. It has been naturalized from Europe, where it is found in sandy fields, and is sometimes cultivated for green manure, and for a rapidly growing fodder, or soiling crop, well liked by cattle. It is also eaten by hens, and a lamp-oil has occasionally been extracted from its seeds. The sand-spurries are delicate plants of the closely allied genus Tissa, haunting saltmarshes and sea-beaches, with small pink flowers.

Spurzheim, spoorts'hlm, Johann Friedrich Kaspar, German phrenologist: b. Longwich, near Treves, Rhenish Prussia, 31 Dec. 1776; d. Boston, Mass., 10 Nov. 1832. He was educated at Vienna, where he became acquainted with Franz Joseph Gall (q.v.), the founder of the system of phrenology. To this study Spurzheim became exceedingly partial; and he soon joined Gall in making inquiries into the anatomy of the brain. They quitted Vienna in 1805 to travel, and went in 1807 to Paris. From that period Spurzheim traveled and lectured in England, Scotland, and Ireland. In 1832 he visited the United States and began his lectures in Boston, but death soon interrupted nis labors. He published 'The Physiognomical System of Drs. Gall and Spurzheim' (1815) ; 'Sur la Folie' (1818); 'Essai philosophique sur la Nature morale et intellectuelle de l'Homme' (1820); 'A View of the Elementary Principles of Education' (1821) etc. Consult: Carmichael, 'Memoir of the Life and Philosophy of Spurzheim' (1833). See also Phrenology.

Spuyten Duyvil (spi'tn di'vil) Creek, a tidal channel connecting the Hudson River with the Harlem, and forming the northern boundary of Manhattan Island, New York. Its name is derived from the Dutch aSpyt den duivel" (in spite of the devil), and is supposed to have been derived from the following circumstance:— When the English fleet appeared in New Amsterdam (New York) harbor, the governor's trumpeter was sent to warn the farmers up the Hudson and summon them to the defense of the city; on reaching this creek he found no ferryman willing to take him across on account of the high wind, and swore to cross the stream ttspyt den duiveP; but was drowned in the attempt to swim across. The creek is crossed by a railway drawbridge near the Hudson. _ Many fish are found in its waters, large quantities of shad being caught here every year.

Spy, a secret emissary sent into the enemy's territory or encampment to inspect their works, ascertain their strength or movements, and report thereon to the proper officer. As the service is most dangerous, for it is the custom when a spy is caught to put him to an ignominious death, a general has no right to compel any person, whether a subject of his own or the enemy's country, to undertake it. The proper business of a spy is to obtain intelligence, and he must not be employed to take the lives of any of the enemy. An officer or soldier found within the enemy's lines should not be treated as a spy if he is clothed in his own uniform, but is dealt with either as a deserter or prisoner of war; but if wearing the enemy's uniform or civil dress, he is liable to be hanged. The American Civil War offered many instances of great daring on the part of spies, Union and Confederate, and in numerous cases their exploits resulted in a tragic death. While spies are undoubtedly induced to perform their perilous work, in numerous cases, by hope of reward, yet there have also been many examples among them of pure and devoted patriotism. In the Russo-Japanese war several Japanese officers of rank and family position, who sought, in the disguise of coolies, to blow up a railway oridge in Manchuria, were premptly hanged as spies.


Spy, The, a famous romance by James Fenimore Cooper, published in 1821. It was immediately successful and has been vary widely read, not only in English but in various European languages. The time of the tale is about 1780 and the scene southeastern New York.

Squad, in the army, a small body of troops assembled for drill, inspection, or other purposes. A troop of cavalry or a company of infantry is usually divided into as many squads as there are sergeants or drill-instructors to train them. The awkward squad is composed of those recruits who have not received sufficient training to take part in regimental drill.

Squadron, (1) the principal division of a cavalry regiment. This body is subdivided into two troops. Three or four squadrons form a regiment. When in line one yard in the length of the front is allotted for each man and horse; the space in line between every two squadrons is equal to one quarter of the extent occupied by each squadron. (2) In the navy a squadron is a number of vessels employed on any particular service or station, under the command of a commodore or junior flag-officer.

Squarcione, Francesco, fran-ches'ko skwarche-6'ne, Italian painter: b. Padua 1394; d. there 1474. In early life he traveled in Greece and Italy, where he became acquainted with the masterpieces of ancient sculpture. He formed a collection of busts, torsos, and bas-reliefs (probably casts) and founded the Padovan school of painting, which produced Mantegna and Zoppo. One of his latest works is (A Madonna and Child,' now in the possession of the Laz/ard family at Padua.

Square, (1) in geometry, a quadrilateral figure, both equilateral and equiangular, of, in other words, a figure with four equal sides and equal angles. In measuring superficial areas it is only necessary to multiply one side by itself to have the area of the square, because each of the sides may be considered as the basis or as the perpendicular height. Thus a square the sides of which measure 4 feet is equal to 16 square feet, that is, 16 squares each 1 foot high and 1 foot long. To square a figure (for example, a polygon) is to reduce the surface to a square of equivalent area by mathematical means. It has often been attempted to square the circle, but this cannot be done. (2) In arithmetic and algebra the square of a number is the number or quantity produced by multiplying a number or quantity by itself. Thus 64 is the square of 8, for 8X8 = 64. (3) In military tactics, a body of infantry formed into a rectangular figure with several ranks or rows of men facing on each side, with officers, horses, colors, etc., in the centre. The front rank kneels, the second and third stoop, and the remaining ranks (generally two) stand. This formation is usually employed to resist a cavalry charge. Hollow squares are frequently formed with the faces

fronting inward when orders and instructions, etc., are to be read, and the like.

Squaring the Circle. See Quadrature.

Square Root. Arithmetical.—The square root of a number is one of its two equal factors. It is indicated by the fractional exponent OA) placed at the right, and above the number, thus

, or by the radical sign (-\/~l6). The two equal factors of 16 are 4 and 4, either one of which may be taken as its square root. The square roots of many numbers are approximate only, and are represented by a whole number and a decimal, the latter carried out as many places as the approximation is desired, as example, ^19 = 4.358899 +. The square root of fractions may be found by extracting the root of the numerator and denominator, but a more practical method is to extract the root of the resulting decimal. Illustration of the method employed in finding the square root of 576: 576(20 20 = 400 4

2 X 20 = 40)176 24 (40 + 4) X 4=176

Since the number 576 has three figures its square root will be composed of tens, and units. The number of tens in the root will be 2, and the square of 2 tens, or 20, will be 400. (See

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F10. 1. Fig. 2.

Fig. I.) But inasmuch as there is still a remainder of 176, such additions must be made to the square as will take up this remainder, and still keep the figure a perfect square. The necessary additions are the two rectangles B and C. and the small square D (see Fig. 2). The remainder 176 divided by the length of the rectangles, 2 X 20, will give the width of the additions, which is 4, and this width is also the side of the small square D; therefore the total length of the additions will be 40 + 4, and the area of the additions 4 times this length, or 176, which completes the square whose area is 576, and whose square root is 24.

Algebraic.— The preceding rules, with the exception of those relating to decimals, are applicable to algebraic quantities. The square root of an algebraic quantity, however, may be positive or negative.

The square root of a negative quantity is imaginary, and is usually factored into two quantities, one of which is real, and the other expressed by V — 1. Thus the square root

V — ab = Vab V —-1.

The square root of algebraic quantities affected by other roots is indicated by multiplying its exponent, or index by 2, wherever possible

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