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MATHE MATICAL AND PHYSICAL 8CIENCE.

rule; and introduced the numeral exponents of the powers—3,-2,1, 0, 1, 2, 3, &c. both positive and negative, as far as integral numbers, but uot fractional ones; calling them by the maine exponens, exponent; and taught the general uses of the exponents in the several operations of powers, as we now use them in the logarithms. He likewise used the general literal notation A, B, C, D, &c. for so many different, unknown, or general quantities. Scheubelius treats pretty largely upon surds, and gives a general rule for extracting the root of any binomial or residual, a + b, where one or both parts are surds, and a the greater quantity; namely, that the square root of it is

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which he illustrates by various examples. A few years after the appear. ance of these treatises in Italy and Germany, Robert Recorde, a celebrated mathematician and physician, born in Wales, published “The Whetstone of Witte, which is the seconde part of Arithmetike: containing the Extraction of Rootes; the Cossike Practice, with the Rule of Equation; and the Works of Surde Nombers.” He introduced the extraction of the roots of compound algebraic quantities; the use of the terms binomial and residual; the use of the sign of equality, or =. eletarius, in 1558, shewed that the root of an equation is one of the divisors of the last, or absolute term. He taught how to reduce trinomials to simple terms, by multiplying them by compound factors. He taught curious precepts and properties concerning square and cube numbers, and the method of constructing a series of each by addition only; namely, by adding successively their several orders of differences. The science received further improvements up to the year 1600, from Ramus, Bombelli, Steven, and others; and a few years after, Schooten published the whole ma*aucal works of Wieta. Wieta

introduced the general use of the letters of the alphabet to denote indefinile given quantities. He uses the vowels A, E, I, O, Y, for the unknown quantities, and the consonants, B, C, D, &c. for known ones. He invented and introduced many expressions or terms, several of which are in use to this day : such as co-efficient, affirmative and negative, pure and adfected, or affected, unciae, homogeneum, &c. and the line, or vinculum, over

compound quantities, thus, A + B. Albert Girard was the first person who understood the general doctrine of the formation of the co-efficients of the powers, from . the sums of their roots, and their products, &c. . He was the first who understood the use of megative roots, in the solution of geometrical problems; and was the first who spoke of the imaginary roots, and understood that every equation might have as many roots, real and imaginary, and no more, as there are units in the index of the highest power. Thomas Harriot flourished about the year 1610. He introduced the uniform use of the small letters a, b, c, d, &c.; viz. the vowels a, e, and o, for unknown quantities, and the consonants, b, c, d.f., &c. for the known ones; which he joins together like the letters of a word, to represent the multiplication or product of any number of these literal quantities, and prefixing the numeral co-efficient as we do at present, except being separated by a point, thus 5.bbc.. For a root he sets the index of the root after the mark V ; as V 3 tor the cube root. He also introduced the characters - and <, for greater and less; and in the reduction of equations he arranged the operations in separate steps, or lines, setting the explanations in the margin on the left-hand for line. He may be considered as the introducer of the modern state of algebra. He also showed the universal generation of all the com

pound or ad fected equations, by the continual multiplication of so many binomial roots.

Oughtred's Clavis appeared in 1631, the same year in which Har.

riot's Algebra was published. He was the first, as far as we can lsarn, who set down the decimals without their denominator; separating them thus, 21(56. In algebraic multiplication, he either joins the letters which represent the factors together like a word, or connects them by the mark X, which is the first introduction of this character. He also seems to be the first who used points to denote proportions, thus 7.9 :: 28.36; and for continued proportion he has this mark + . In his work we likewise neet with the first instance of applying algebra to investigate new geometrical properties. Descartes' Geometry, first published in 1637, was ratiner an application of algebra to geometry, than either algebra or geometry separately considered. Still he made in provements in both. His inventions and discoveries in algebra may be comprehended in the application of algebra to the geometry 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 marima et minima. In 1655, Wallis published his “Arithmetica Infinitorum,” which in a great measure led the way to infinite series, the universal application of the binomial theorem, and the method of fluxions. He first gave an expression for the quadrature of the circle by an infinite series; and it was he who first substituted the fractional exponents in the place of radical signs, by which the operations are in many cases much facilitated and abridged. In 1707, Whiston published the first edition of Sir Isaac Newton's “Arithmetico Universalis.” This was the text-book used by Newton, while he was professor of Mathematics in the university of Cambridge: and although it was never intended for publication, it contains many and great improvements in analytics; particularly in the nature and transmutation of equations; the limits of the roots

of equations; the number of impossible roots; the invention of divisors, both surd and rational ; the resolution of problems, arithmetcal and geometrical ; the linear construction of equations; approximating to the roots of all equations. To the later editions of this work is commonly subjoined, Dr. Halley's method of finding the roots of equations. Commentaries upon this work, by several persons, as P. Gravesande, Castilion, Wilden, and others, and many of Newton’s algebraical discoveries, were farther developed and explained by Halley, Maclaurin, Nicoles, Stirling, Euler, Clairaut, &c. Algebra is sometimes divided into numeral, and specious or literal. Numeral ALG EBRA, is that in which all the given quantities are expressed by numbers. Specious or Literal ALG EBRA, is that in which all quantities, whether known or unknown, are expressed by general characters, as letters, &c. in consequence of which general designation, all the conclusions become universal theorems. o In algebraical inquiries, some quantities are assumed as known or given; and others are unknown and to be found out: the former are commonly represented by the leading letters of the alphabet, a, b, c, d, &c.; the latter by the final letters, w, or, y, z, Though it often tends to relieve the memory, if the initial letter of the subject under consideration be made use of, whether that be known or unknown : thus r may denote a radius, b a base, p a perpendicular, s a side, d density, m mass, &c. The characters used to denote operations and relations are principally as follow : = denotes the equality of the quantities between which it is placed : thus, a = b, denotes that a is equal to b. > denotes the inequality of the quantities between which it is placed, the point being turned toward the less: thus, a P b, means that a is greater than 0. + (plus) denotes the addition of

the quantities between which it is placed : thus, a + b means that a and b are to be added. w means the difference of the quantities between which it is placed, without indicating what the difference is, or which is the greater: thus, a n b 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. × or . denotes the product of the quantities between which it is placed : thus, axb, or a . b, means the product of a and b. When 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, a -b,

or, # means a divided by b,

When two or more quantities having the signs +, or –, are to be considered as oue quantity, or twinculum, , is drawn over them, or they are inclosed in

parenthesis: thus, a + b, or (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 . = 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 V with the exponent when different from 2, or else, with a fractional exponent: thus ao

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5a denotes that the quantity a is to be taken 5 times, and 7,00+c) is 7 times b + c. And these numbers, 5 or 7, she wing how often the quantities are to be taken, or multiplied, are called co-efficients. Like quantities, are those which consist of the same letters and powers. As a and 3a; or 2ab and 4ab ; or 3a*bc and —5a?bc. Unlike quantities are those which consist of different letters or different powers. As a and b; or 2a and ao ; or 3ab? and 3abc. Simple quantities, or monomials, are those which consist of one term only. As 3a, or 5ab, or 6abco. Compound quantities, are those which consist of two or more terms. As a + b, or a + 2b–3C. 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, or + a, 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 —a, or —2ab, or —3ab?. Like signs, are either all positive (+), or all negative (- ). Unlike signs, are when some are positive (+), and others negative

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a. For the several operations in Algebra, see the corresponding articles. ALGEBRAICAL, something relating to Algebra. ALGEBRAICAL Curve, is a curve the nature of which may be expressed by an equation. ALGORITHM, or Algoris M, 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 mumber 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 Medial, and Alligation Alternate. ALLIGATION Medial, is the method of finding the rate or quality of the composition, from i. the rates or qualities of the simple ingredients given. Fule. Multiply each quantity by its rate, and divide the sum of ail these products by the sum of the quantities, and the quotient will be the price or quality required.

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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 lates are less and which greater than the proposed compound. Then connect or link with a crooked 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 opposite any rate, it will be the quantity belonging to that rate: but when there are several differences to any one, their sum will be quantity.

Exam. Let it be required to mix together gold of various degrees of fineness; viz. of 18, 20, and 24 carats fine, so that the mixture may be of 21 carats fine.

results.

Rates. diff. = 3 of 18 car.

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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. ALMANAC, a calendar or table, in which are noted down all the most remarkable phenomena of the heavenly bodies; such as eclipses, occultations, the conjunctions and oppositions of the planets, the risings and settings of the sun and moon, &c. &c. Nautical ALMANAC and Astronomical Ephemeris, is a kind of national Almanac, begun in 1767. This Almanac is generally computed a few years forward, for the convenience of ships going out on long voyages, for which it is highly 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. ALM U CANTARS, or ALMACANTARS, circles parallel to the horizon, passing through every degree of the meridian ; and, having 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. ALTITUDE of a Figure, is the distance of its vertex from its base, or the length of a perpendicular let fall from the vertex to the base. Altitudes arc divided into accessible 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 - roached. .

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To measure an accessible or inaccessible Altitude geometrically.— Under this head are included all those cases in which the calculation depends upon pure geometrical principles, and particularly on the similarity of triangles, of which we propose to give an illustration in the following examples: let A B (Plate I. fig. T.) 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 ED, 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 I D, as also the length EI), we shali have by similar triangles, as

ID : ED = 1A: AB;

that is, by multiplying the second and third terms, and dividing by the first, we shall have the whole altitude of the tower BA: that is,

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then BA=!" × 4 = 50 feet, 8

the altitude of the tower.

If the object AB were inaccessible, two such operations as the above intust be made, in order to ascertain the altitude required: thus,

Let ID = a, and ED = d ; also the unknown distance IA = r, 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

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