riot's Algebra was published. He of equations; the number of im was the first, as far as we can possible roots; the invention of dilearn, who set down the decimals visors, both surd and rational; the without their denominator; sepa-resolution of problems, arithmetirating them thus, 21(56. In alge-cal and geometrical; the linear braic multiplication, he either construction of equations; approxjoins the letters which represent |imating to the roots of all equa the factors together like a word, tions. To the later editions of or connects them by the mark X, this work is commonly subjoined, which is the first introduction of Dr. Halley's method of finding the this character. He also seems to roots of equations. Commentaries be the first who used points to de- upon this work, by several pernote proportions, thus 7.9: 28.36; sons, as P. Gravesande, Castilion, and for continued proportion he Wilden, and others, and many of has this mark. In his work Newton's algebraical discoveries, we likewise meet with the first were farther developed and exinstance of applying algebra to in-plained by Halley, Maclaurin, Nivestigate new geometrical proper- coles, Stirling, Euler, Clairaut, &c. ties. Algebra is sometimes divided Descartes' Geometry, first pub-into numeral, and specious or lilished in 1637, was rather an appli- teral. cation of algebra to geometry, than either algebra or geometry separately considered. Still he made iniprovements in both. His Specious or Literal ALGEBRA, is inventions and discoveries in alge- that in which all quantities, whebra may be comprehended in the ther known or unknown, are exapplication of algebra to the geo-pressed by general characters, as metry of curved lines; the construction of equations of the higher orders; a rule for resolving biquadratic equations by means of a cubic and two quadratics and his method of maxima et minima. Numeral ALGEBRA, is that m which all the given quantities arë expressed by numbers. letters, &c. in consequence of which general designation, all the conclusions become universal the orems. In algebraical inquiries, some quantities are assumed as known In 1655, Wallis published his or given; and others are unknown "Arithmetica Infinitorum," which and to be found out: the former in a great measure led the way to are commonly represented by the infinite series, the universal appli-leading letters of the alphabet, a, cation of the binomial theorem, b, c, d, &c. ; the latter by the final and the method of fluxions. He letters, w, x, y, z. Though it often first gave an expression for the tends to relieve the memory, if the quadrature of the circle by an in- initial letter of the subject under finite series; and it was he who consideration be made use of, whe. first substituted the fractional ex-ther that be known or unknown: ponents in the place of radical signs, by which the operations are in many cases much facilitated and abridged. thus r may denote a radius, ba base, p a perpendicular, s a side, a density, m mass, &c. The characters used to denote operations and relations are principally as follow: In 1707, Whiston published the first edition of Sir Isaac Newton's "Arithmetico Universalis." This denotes the equality of the was the text-book used by Newton, quantities between which it is while he was professor of Mathe- placed: thus, ab, denotes that matics in the university of Cam-a is equal to b. bridge: and although it was never > denotes the inequality of the intended for publication, it con- quantities between which it is tains many and great improve-placed, the point being turned toments in analytics; particularly in ward the less: thus, ab, means the nature and transmutation of that a is greater than o. equations; the limits of the roots +(plus) denotes the addition of the quantities between which it is placed thus, a+b means that a and b are to be added. means the difference of the quantities between which it is placed, without indicating what the difference is, or which is the greater: thus, ab means a subtracted from b, or b subtracted from a, according as the one or the other is the greater. (minus) means that the quantity after it is to be subtracted: thus, a+b-c means that c is to be subtracted from the sum of a and b. A quantity having this sign is called a negative quantity, in opposition to one which has no sign, or the sign+, which is called an affirmative quantity. X or denotes the product of the quantities between which it is placed: thus, axb, or a . b, means When the product of a and b. each quantity is expressed by a single letter, the product is also denoted by writing the letters without any sign. denotes that the quantity before it is to be divided by the quantity after; and the same thing is indicated by writing the divisor before the dividend: thus, ab, means the square of a; and Ja, or a, means the square root of a. at, 5a denotes that the quantity a is to be taken 5 times, and 7,(b+c) is 7 times b+c. And these numbers, 5 or 7, shewing how often the quantities are to be taken, or mul tiplied, are called co-efficients. Like quantities, are those which consist of the same letters and powers. As a and 3a; or Lab and 4ab; or 3a2bc and -5a2bc. Unlike quantities are those which consist of different letters or different powers. As a and b; or 2a and a2; or 3ab2 and 3abc. Simple quantities, or monomials, are those which consist of one term only. As 3a, or 5ab, or 6abc2. Compound quantities, are those which consist of two or terms. As a+b, or a +2b-3c. more And when the componnd quantity consists of two terms, it is called a binomial; when of three terms, it is a trinomial; when of four terms, a quadrinomial; more than four terms a multinomial or polynomial. Positive or affirmative quantities, are those which are to be added, or have the sign +. As a, ora, or ab; for when a quantity is found without a sign, it is understood to be positive, or to have the sign + prefixed. Negative quantities, are those which are to be subtracted. As in-a, or -2ab, or -3ab2. Like signs, are either all posiortive (+), or all negative (—). When two or more quantities having the signs +, or, are to be considered as one quantity, or vinculum, is drawn over them, or they are inclosed parenthesis: thus, a + b, (a+b), means that the two quantities are to be taken as one whole. means the proportion or ratio of one quantity to another; and when two quantities have the same ratio as other two, the sign of equality is introduced between the pairs thus, a b c d means that a has the same ratio to b, that c has to d. A power of any quantity is denoted by writing the exponent over it; and a root by the radical sign with the exponent when different from 2, or else with a fractional exponent: thus a2 Unlike signs, are when some are positive (+), and others negative (-). A residual quantity, is a binomial having one of the terms negative. As a-2b. The power of a quantity (a), is its square (a), or cube (a), or biquadrate (a4), &c.; called also the 2d power, or 3d power, or 4th power, &c. The index or exponent, is the number which denotes the power or root of a quantity. So 2 is the exponent of the square or 2d power a2; and 3 is the index of the cube or 3d power; and is the in dex of the square root or a; and is the index of the cube root as or va. A rational quantity, is that which has no radical sign or index annexed to it. As a, or 3ab. An irrational quantity, or surd, is that which has not an exact root, or is expressed by means of the radical sign As√2, or √a, or va2, or ab. The reciprocal of any quantity, is that quantity inverted, or unity divided by it. So, the reciprocal and the reciprocal a of a, or is ALGEBRAICAL, something relating to Algebra. ALGEBRAICAL Curve, is a curve the nature of which may be expressed by an equation. ALGORITHM, or ALGORISM, an Arabic term expressive of numeri cal computation; or the common rules of computing in any branch of analysis; as the algorithm of numbers, of surds, of imaginary quantities, &c. ALIQUOT Part, is such a part of a number as is contained in it a certain number of times exactly. ALLIGATION, one of the rules in Arithmetic, relating generally to the compounding or mixing together of various ingredients, of which the prices or qualities are given, so that the compound may be of a certain value or quality. This rule is usually divided into two distinct cases, viz. Alligation Mediul, and Alligation Alternate. ALLIGATION Medial, is the method of finding the rate or quality of the composition, from having the rates or qualities of the simple ingredients given. Rule. Multiply each quantity by its rate, and divide the sum of all these products by the sum of the quantities, and the quotient will be the price or quality required. ALLIGATION Alternate, is the method of finding the quantities of ingredients necessary to form a compound of a given rate. Rule. Place the given rates of the ingredients in a line under each other; noting which rates are less and which greater than connect or link with a crooked the proposed compound. Then line, each rate which is less than the proposed compound with one or any number of those that are greater than the same; and every one which is greater with one or any number of those that are less. Take the difference between the given compound rate, and that of each simple rate; and set this difference opposite every rate with which that one is linked. Then if only one difference stand oppo site any rate, it will be the quantity belonging to that rate: but when there are several differences to any one, their sum will be Note. If the compound is to consist of a certain quantity, then take the sum of the above results, and say, as that quantity or sum is to the proposed quantity, so is each of the above results to the quantity of each required. Questions of this kind are better solved by Algebra, in which they form a species of indeterminate problems. There are several methods of measuring the height or altitude of bodies, viz. by Geometry, Trigonometry, by Optical Reflection, by means of the Barometer, &c. The instruments commonly used in measuring altitudes, are, the Geometrical Square, Quadrant, and Theodolite; a description of each of which will be found under the respective articles. of ALMANAC, a calendar or table, To measure an accessible or inin which are noted down all the accessible Altitude geometrically.most remarkable phenomena of Under this head are included all the heavenly bodies; such as those cases in which the calculaeclipses, occultations, the con- tion depends upon pure geometri junctions and oppositions of the cal principles, and particularly on planets, the risings and settings of the similarity of triangles, the sun and moon, &c. &c. which we propose to give an illusNautical ALMANAC and Astro-tration in the following examples: nomical Ephemeris, is a kind of national Almanac, begun in 1767. This Almanac is generally computed few years forward, for the convenience of ships going out on long voyages, for which it is high ly useful. Besides most things essential to general use, and which are found in other Almanacs, it contains many important matters, particularly the distance of the moon from the sun and fixed stars, computed to the meridian of Greenwich, for every three hours of time, which is of great use in computing the longitude at sea. ID: ED IA: AB; ALMUCANTARS, OR ALMA- that is, by multiplying the second CANTARS, circles parallel to the and third terms, and dividing by horizon, passing through every the first, we shall have the whole degree of the meridian; and, hav-altitude of the tower BA: that is, ing the zenith for their common centre. They are the same as parallels of altitude. ALTITUDE, in Geometry, the third dimension of a body, considered with regard to its elevation above the plane of its base. let A B (Plate I. fig. 1.) represent an object of which the altitude is required. Being provided with two rods or staves of different lengths, plant the longest of them, as FC, at a certain measured distance from the base of the object. Then, at a farther distance, plant the second or shorter staff E D, in such a manner that the tops of the two, E and F, may be in a line with the top of the tower B. Then having measured the distance ID, as also the length ED, we shalĺ have by similar triangles, as ВА - IA X ED = 8 feet. For example, suppose IA 100 feet, ID = 8 feet, and ED 4 feet, being the height of the staff; 100 X 4 then BA= = 50 feet, 8 ALTITUDE of a Figure, is the distance of its vertex from its base, or the length of a perpendicular let the altitude of the tower. fall from the vertex to the base. If the object AB were inaccesAltitudes are divided into accessible, two such operations as the sible and inaccessible. Accessible Altitude, of an object, is that to whose base we can have access, so as to measure the distance between it and the station from which the measure is to be taken. Inaccessible Altitude, is when the base of the object cannot be proached. above must be made, in order to ascertain the altitude required: thus, Let ID a, and ED=d; also the unknown distance IA = x, and the required altitude of the object =y; then, in a second operation, in which both the staves must be replanted, make the new distance ID=al, and the second unknown |