stem are one.

ALE. A pleasant common liquor, brewed tatum, published first in 1494, he mentions by pouring hot water upon malt; this is several writers, and particularly Leonardus strained off, and again boiled with hops; Pisanus, otherwise called Bonacci, an Italian which liquor is then fermented with yest, and merchant, who, in the thirteenth century, used stowed in vats or casks. Porter is made from to trade to the seaports, and thence introduced high-dried malt. the science of algebra into Italy. After Lucas A-LEE. A sea term, signifying to the lee- de Burgo, many other Italian writers took up side. the subject, and treated it more at large, as ALEMBIC. A vessel formerly used for dis- Scipio Ferreus, who found out a rule for retilling; in the place of which retorts are now solving one case of a compound cubic equamostly in use. tion; but more especially Hieronymus CarALEXANDRINE. A verse in modern poe-dan, who, in ten books published in 1539-45, try consisting of ten, twelve, or thirteen syl- has given the whole doctrine of cubic equalables. tions; for part of which, however, he was inALGE. A natural order of plants in the debted to Nicholas Tartalea, or Tartaglea, of Linnæan system, containing flags, seaweeds, Brescia, a contemporary of Cardan's, who and other marine plants, whose root, leaf, and published a book on cubic equations, entitled, Quesite Invenzioni diverse, which appeared ALGEBRA. The science of computing ab- in 1596. Cardan often used the literal notastract quantities by means of symbols or signs. tion of a, b, c, d, &c., but Tartalea made no It is called Specious Arithmetic by Vieta, and alteration in the forms of expression used by Universal Arithmetic by Newton. The first Lucas de Burgo, calling the first power of the letters of the alphabet, a, b, c, d, &c. are made unknown quantity in his language cosa, the to represent known quantities; and the last second censa, the third cubo, &c. writing the letters, x, y, z, to represent those that are un-names of all the operations in words at length, known. The operations with these letters are without using any contractions, except the iniperformed by means of the characters (+) for tial B, for root, or radicality. About this time addition, (-) for subtraction, (X) for multi- the science of algebra also attracted the attenplication, (÷) for division, (=) for equality. tion of the Germans, among whom we find ALGEBRA, HISTORY OF. The term alge- the writers Stifelius and Scheubelius. Stifebra is of Arabic original, and is derived by lius, in his Arithmetica Integra, published at some from algabar almocabalen, signifying Nuremberg in 1544, introduced the characters, restitution and comparison, or resolution,+, — for plus, minus, and radix, or which properly expresses the nature of the as he called it; also the initials 4, 3, V, for thing; others have derived it from Geber, a the power 1, 2, 3, &c. and the numeral expocelebrated mathematician. This science is not nents 0, 1, 2, 3, &c. which he called by the of very ancient date, although it is not possible name of exponens exponent. He likewise to fix the exact period of its commencement. uses the literal notation, A, B, C, D, &c. for The earliest treatise on this subject now extant the unknown or general quantities. John is that of Diophantus, a Greek author of Alex-Scheubelius, who wrote about the same time andria, who flourished about the year 350, and as Cardan and Stifelius, treats largely on wrote thirteen books of Arithmeticorum, of surds, and gives a general rule for extracting which six only are preserved. These books the root of any binomial or residual, a+b, do not contain the elementary parts of algebra, where one or both parts are surds. These only some difficult problems respecting square writers were succeeded by Robert Recorde, a and cube numbers, and the properties of num- mathematician and physician of Wales, who bers in general, to which the writings of the in his works, in 1552 and 1557, on Arithmemore ancient authors, as Euclid, Archimedes, tic, showed that the science of algebra had not and Apollonius, might naturally be supposed to been overlooked in England. He first gave have given birth. Whether the Arabians took rules for the extracting of the roots of comtheir hints from this and similar works among pound algebraic quantities, and made use of the Greeks, and drew out the science of Alge- the terms binomial and residual, and introbra for themselves, or whether they more im- duced the sign of equality, or =. Peletarius, mediately derived it, as they did their notation, a French algebraist, in his work, which apfrom the Hindoos, is a matter of doubt. It is peared at Paris in 1558, made many improvecertain, however, that the science was first ments on those parts of algebra which had transmitted by the Arabians or Saracens to already been treated of. He was followed by Europe, about the year 1100; and that after Peter Ramus, who published his Arithmetic its introduction the Italians took the lead in its and Algebra in 1560; Raphael Bombelli, cultivation. Lucas Paciolus, or Lucas de whose Algebra appeared at Bologna in 1579; Burgo, was one of the first who wrote on the and Simon Steven, of Bruges, who published subject, and has left several treatises, publish- his Arithmetic in 1585, and his Algebra a lited between the years 1470 and 1509. In his tle after. This latter invented a new characprincipal work, entitled Summa Arithmetica ter for the unknown quantity, namely, a small et Geometria Proportionumque Proportionali-circle (O), within which he placed the nume

ral exponent of the power; and also denoted operations. Huygens, Barrow, and other roots, as well as powers, by numeral expo- mathematicians, employed the algebraical calnents. The algebraical works of Vieta, the culus in resolving many problems which had next most distinguished algebraist, appeared hitherto baffled mathematicians. Sir Isaac about the year 1600, and contain many im- Newton, in his Arithmetica Universalis, made provements in the methods of working alge- many improvements in analytics, which subbraical questions. He uses the vowels, A, E, ject, as well as the theory of infinite series, I, O, Y, for the unknown quantities, and the was further developed by Halley, Bernoulli, consonants, B, C, D, &c. for the known quan- Taylor, Maclaurin, Nicole, Stirling, De Moivre, tities; and introduced many terms which are Clairaut, Lambert, Waring, Euler, &c. in present use, as coefficient, affirmative and ALGOL. A fixed star of the second magnegative, pure and adfected, &c.: also the line, nitude in the constellation of Pereius or Meor vinculum, over compound quantities (A+B). dusa's Head. Albert Girard, an ingenious Flemish mathematician, was the first person who, in his In- quently used to denote the practical rules of vention Nouvelle en l'Algebre, &c. printed in 1629, explained the general doctrine of the ALIAS (in Law.) A word signifying, liteformation of the coefficients of the powers from rally, otherwise; and employed in describing the sums of their roots, and their products. the defendant, who has assumed other names He also first understood the use of negative besides his real one.

ALGORITHM. An Arabic word, fre

algebra.

roots, in the solution of geometrical problems, ALIBI (in Law.) A term signifying, liteand first spoke of imaginary roots, &c. The rally, elsewhere; and used by the defendant celebrated Thomas Harriot, whose work on in a criminal prosecution, when he wishes to this subject appeared in 1631, introduced the prove his innocence, by showing that he was uniform use of the letters a, b, c, &c.; that is, in another place, or elsewhere, when the act the vowels a, e, and o, for the unknown quan- was committed. tities, and the consonants, b, c, d, &c. for the ALICONDA. An Ethiopian tree, from known quantities; these he joins together the bark of which flax is spun. like the letters of a word, to represent the mul- ALICONDA-TREE. A native of Congo, tiplication or product of any number, of these on the coast of Africa, and supposed to be the literal quantities, and prefixing the numeral largest tree that grows. It bears a melon-like coefficient, as is usual at present, except being fruit, which affords pulpy nutritious food, and separated by a point, thus 5.bbc. For a root the bark yields a coarse thread, with which he sets the index of the root after the mark V, the Africans weave a kind of cloth. as V3 for the cube root, and introduces the ALIEN (in Law.) One born in a foreign characters and, for greater and less; country, out of the allegiance of the governand in the reduction of equations he arranged ment under which he is residing. An alien the operations in separate steps or lines, setting is incapable of inheriting lands until he is natuthe explanations in the margin, on the left ralized by a legislative act. He has likewise hand, for each line. In this manner he no right to vote at elections, or to enjoy any brought algebra nearly to the form which it office, nor to be returned on any jury, unless now bears, and added also much information where an alien is to be tried. on the subject of equations. Oughtred, in his Clavis, which was first published in 1631, set down the decimals without their denominator, separating them thus 21(56. In algebraic ALIQUANT PARTS. Such numbers in multiplications he either joins the letters which arithmetic as will not divide or measure a represent the factors, or connects them with whole number exactly, as 7, which is the alithe sign of multiplication X, which is the quant part of 16.

ALIMONY (in the Civil Law.) The allowance made to a married woman upon her separation from her husband.

first introduction of this character. He also ALIQUOT PARTS. Such part of a numseems to have first used points to denote pro- ber as will divide or measure a whole numportion, as 7. 9:: 28. 36; and for continued ber exactly, as 2 the aliquot part of 4, 3 of 9, proportion has the mark. In his work we and 4 of 16.

likewise meet with the first instance of apply- ALKALI, or Kali, sometimes called natron, ing algebra to geometry, so as to investigate or nitre, a very important salt in soap and new geometrical properties: which latter sub-glass-making. Potash and soda are called

ject is treated at large by Descartes, in his fixed alkalis, and ammonia, volatile alkali. work on Geometry, published in 1637, and ALLAH. The Arabian name of God. also by several other subsequent writers. Wal- ALLEGIANCE. The duty of subjection lis, in his Arithmetica Infinitorum, first led to law, under which subjects lay themselves the way to infinite series, particularly to the in establishing their own protection under the expression of the quadrature of the circle by law.

an infinite series. He also substituted the ALLEGORY. A series or chain of metafractional exponents in the place of radical phors continued through a whole discourse; signs, which in many instances facilitate the thus the prophets represent the Jews under the

ALMONER. An officer appointed to dis-Jews were the altar of burnt offerings and the tribute alms to the poor.

in a language, which vary in number in diffe- to idolatrous purposes those rites which were rent languages. The Hebrew contains 22 let- intended to typify the one atoning sacrifice of ters, as also the Chaldee, Samaritan, Syriac, the gospel. This was the true origin of altars Persian, Ethiopic, Saracen, &c.; but the Irish, erected to lords many and gods many. which is the same as the Pelasgian, or Scythian, still retains only 17; the Greek alpha- which the changes in any number of things ALTERNATION. A rule in arithmetic by bet, which was brought by Cadmus into may be determined. It consists of multiplying Greece from Phoenicia, and was also Pelasgian the numbers one into another, and the product in its original, consisted of 16 or 17, to which is the number of possible changes. were afterwards added 7 or 8 more, to make ALTIMETRY. The art of taking heights up 24. The ancient Arabic alphabet consisted by means of a quadrant, and founded on the of 24, to which 4 more letters have since been principle, that the sides of triangles having added; the Coptic alphabet consists of 32, the equal angles, are in exact proportion to one Turkish of 33, the Georgian of 36, the Rus- another. When the object is accessible, its sian of 39, the Spanish of 27, the Italian of 20, height is considered as one of the sides of the the Latin of 22, the French of 23, and the Eng-triangle; but when it is inaccessible, then two lish of 26. See more on this subject under

the head of WRITING. The Chinese have no proper alphabet, unless we reckon as such their keys to classes of words, distinguished by the number of strokes combined in each, of which they have 214 in number. As to the written characters of these alphabets, see WRI