Page:The New International Encyclopædia 1st ed. v. 14.djvu/135

MTJLREABY. 105 MULTIPLICATION. more than sixty years. An early marriage con- tracted soon after his arrival in London was proihative of much unhappincss. He was, throuj^lunit liis life, a successful and painstaking teacher. Mulready died in London. .July 7, 18G3. Consult Stcpiiens,U(«ionu/s of Mulrcady (Lon- don, 18(17). MTJLTAN, or MOOLTAN, moTrl'tiin'. The capital (if a district of the Punjab, British India, 190 miles southwest of Lahore, and 4 miles from the left hank of the Chenah. the inundations of which sometimes reach the city (Map: India, B 2). It possesses railway comniuuication with all the principal cities of India, and has. in the Indus Valley Railwa,v. a commercial outlet from Central Asia, the Punjab, and the L'nitcd Prov- inces, to the Arabian Sea at Karachi. Steamers ply to Hyderabad, a distance of .570 miles. The city, situated in a district remarkable for its fertility, is built on a mound consisting of the ruins of ancient cities, and is surrounded by a dilapidated wall from 40 to .50 feet high. The vicinity abounds in mosques, tombs, and slirines, attesting the city's antiquity and former mag- nificence. The most important of these, situated in the old fort, is the tomb of Rukn-i-Alam, 'pil- lar of the world.' dating from 1340, an octagonal redbrick structure covered with nnilticolored glazed tiles and raised mosaics, and forming a conspicuous object in the surrounding landscape, being 100 feet high and built on an elevation. The tomb of Bhawal Hakk. dating from 12(i4. and the tomb of Shams-i-Tabriz, are also noteworthy. The bazaars are numerous, extensive, and well sfoeked. and the stores are adequately supplied with European and Asiatic connnodities. There are manufactures of silks, cottons, shawls, scarfs, brocades, tissues, etc., and extensive banking interests. The local merchants are proverbially rich. JIultan is a military station, with an important cantonment one and one-half miles to the east, ilultan was taken by the British in the second Sikh War. in .Tanuarv. 1840. Population, in 1801. 74..5(;2; in 1901, 87,394. MULTIPLE (ML. mulfipli/s. manifold, from Lat. nudltifi. man,v + -plus. fold). An integral number divisible without remainder by another integral number. Thus. 3.5 is a multiple of 7, and also a multiple of .5. EquimuUipJcs are multiples containing different numbers an equal number of times; e.g. 21 and 33 are equinniltiples of 7 and 11. Tlie least common multiiilr of sev- eral numbers is the least number containing each of them without a remainder. The least com- mon multiple contains all prime factors not common to all of the numbers, and all common prime factors with their highest exponents. Thus, the least conmion nuiltiple of 5-7 (i.e. 35), 3^-2 (i.e. 18), and 7- -2" .5 (i.e. 980) is 5-7^-2^-3- (i-e. 8820). Srathematieal symbols which satisfy given con- ditions for different numerical values are said to be 'nniltiple valued' : e.g. sin~'x ( i.e. the angle whose sine is x) is two-valued betw'cen 0" and 3C0"; thus, if the sine equals ^■.. the angle has two values, 60°. 120". (See TiticoNOMETHY.) Contacts above the first order between curves or surfaces are called 'multiple eontaets.' A 'multiple point' of a curve is a singularitv com- posed of several coincident points; e.g. if the curve crosses itself twice at the same place, the intersection is called a triple point. A double point admits of two tangents, a trijjle point ad- mits of three tangents, and so on, and these are called 'nuiltiple lan^eMls.' MULTIPLE POINDING. A form of action in Scotland, by which competing claims to one and the same fund are set at rest. Its purjiose is to avoid double [winding or double distress; and it corresponds to what is known in England as interpleader ( q.v. ) . MULTIPLICATION (Lat. multiplicatio, from iniilliplicarc, to nuiltiply, from multiplex, manifold, from )nullu.s, niaii.y -f pliaire, to fold, Gk. TrKfii', /(/r/,:ei», to twine). A fundamental process in arithmetic and algebra. In arithme- tic, the symbols for niultiidication are ., X. In algebra, they are ., X, and simple juxtaposition. E.g. ab, a X b, ab, are synd)ols for <i times 6. It ma.v be defined as the process by which a num- ber called the product is formed from a number called the multiiilicr. in the same wa.v that this multiplier is formed from unity. E." the num- ber — 3 may be formed from unity by the pro- cess symbolized as — (1 -(- 1 -f- 1), and simi- larly, the product — 3-4 may be formed by the process - (4-|-4-t-4) = — 12. Elementary multiplication is subject to the associative and distributive laws (qcpv. ); but there are branches of higher mathematics in which excep- tions occur. (See SmsTiTi'TioN ; Qihternions.) For a method of checking multiplication, see Checking. In a series of operations, multiplication takes precedence over addition and subtraction. E.g. 2 + 3 • 6 — - 4 equals 2 + 18 — 4, not 5 ■ 2. The operation of multiplication can be abbreviated by the use of logarithms (q.v.), the slide rule (q.v.). or tables of products and factors or of quarter squares. The plan of nuiltiplication by means of Napier's rods (lidhdoloiiitv sire numrra- tionis per rircjulus lihri duo, Edinburgh, 1017) has been revived through the manufacture of sets of Ileglettes multiplicdlriccs planned by Genaille and Lucas (Paris, 1883). Crowing out of the demand for a system by which [irime numbers could be detected, there appeared, in the seven- teenth century, numerous tables, of service in the theory of" nunibers. In 1728 Poetius pub- lished a table of factors for numbers up to 100,- 000. In 1770 Lambert arranged such a table in modern form for numbers up t« 102.000. Burk- hardt's table (1814-17) includes factors of num- bers to 36.000. and Crelle. Pase, and Glaisher have carried these to 9,000,000. Tile oldest of the large tables is that of Crelle (7th ed. with an in- troduction by Bremikcr. Berlin, 189.5). This gives the products to 1000 • 1000. Zimmermann's Keehentafel (Berlin. 1889) and Midler's iluUi- plicationstabellen (Karlsruhe, 1897) give the products to 100 • 1000. ;ind are well arranged. For the products to 100-100. .Tordan's Mnthc- mnlische und (jeodiHischc Hilfstafeln (9th ed., Hanover. 189.5) is one of the best. Products have also been tabulated by means of quarter squares, a relation known to the Arabs and doubt- less of Hindu origin. The construction of these tables depends upon the identity ab = — t (a — by thus the product of any two num- bers is given by subtracting the quarter square of their difTerenee from the quarter square of their sum. Among the various tables of this