ax3 + bx + cx + a = 0

sum of these remainders is 13, and the remainder 4, which is also the remainder is an adfected equation, containing terms obtained by dividing 31, the sum of the digits which involve different powers of x. See in the sum total, by 9. Hence, we conclude Affected.

that the operation of addition was correctly AD IN-FI-NI'TUM. [L.] To endless experformed. tent, according to the same law. When a None of these methods of proof are strictly series is given, and a sufficient number of perfect, since it is possible that two errors terms are written to indicate the law of the might be committed which would exactly series, the words ad infinitum are added to balance each other; the last one is, however, show that there are an infinite number of sucnearly free from any liability to error. ceeding terms, connected by the same mathematical law, with those already given.

In Algebra, the quantities to be added are represented by symbols arranged according to the rules of algebraic Notation.

ADDITION OF ENTIRE QUANTITIES. Set them down so that similar terms, if there are any, shall fall in the same column. Add the several sets of similar terms, and to the result annex the remaining terms, giving to each its proper sign. To add similar terms, take the numerical sum of the co-efficient of the additive and subtractive terms separately subtract the less from the greater, and give to the remainder the sign of the greater, after which write the common literal part. This will be the sum required.

ADDITION OF FRACTIONS. The rule is the same as that already given for the addition of arithmetical fractions.

ADDITION OF RADICALS. Reduce them, if possible, to equivalent radicals which shall be similar. Add the co-efficients, and to this sum annex the common radical part. This will be the sum required. If the given radicals cannot be reduced to equivalent similar radicals, the addition can only be indicated.

Ad infinitum sometimes means to the limit. For example, if a regular polygon be inscribed in a circle, and the arcs subtended by the sides be severally bisected, and the points of bisection be joined by chords with the adjacent vertices of the polygon, a new regular polygon will be formed, having double the number of sides, and approaching more nearly to an equality with the circle. If the operation be then repeated, we shall have a polygon still nearer in area to the circle, and so on. If the operation be repeated ad infinitum we shall reach the limit, that is, the inscribed polygon will coincide with the circle.

AD-JA'CENT. [From ad, to, and jaceo, to lie]. Contiguous to, or bordering upon.

ADJACENT ANGLES, in a plane, are those which have one side in common. and their other sides in the prolongation of the same straight line.

Two diedral angles are adjacent when they have a common face, and their other faces lying in the same plane produced.

When the quantities are written by means of exponents, reduce them, if possible, to equivalent expressions having the same exponent. Add the co-efficients for a new Two spherical angles are adjacent when co-efficient, after which write the common they have one side in common, and their other part. The result will be the sum required. sides arcs of the same great circle. ADDITION OF RATIOS, is the same as the addition of fractions.

ADD'I-TIVE. A quantity is additive when it is preceded by a positive sign. If it is not preceded by any sign, the sign is always understood.

AD-FECT'ED. Compounded, that is, made up of terms involving different powers of the unknown quantity; thus,

The sum of the two adjacent angles, in each case, is equal to two right angles.

AD-JUST'MENT. [From ad, to, and justus, just.] The operation of bringing all the parts of a mathematical instrument into their proper relative positions. When the parts have these positions, the instrument is said to be in adjustment, and is fit for use. When several independent steps have to be taken,

each step is often called an adjustment: thus, distinction to one which is to be subtracted. in the theodolite we say there are four adjust- The term implies that the quantity is essentially positive, that is, of such a nature that

1 To bring the intersection of the cross when added to another quantity, the latter hairs into the axis of the Y's. will be increased.

AL'GE-BRA. [From the Arabic words al and gabron, reduction of parts to a whole]. That branch of analysis whose object is to investigate the relations and properties of The term affected, is sometimes even ap- numbers by means of symbols. The quantiplied to the numerical co-efficients, in which ties considered are generally representel by case the literal parts are said to be affected letters, and the operations to be performed with positive or negative co-efficients. In on these are indicated by signs. The letthis last case the term is improperly applied. ters and signs are called symbols Algebra AF-FIRM'A-TIVE. [L. from ad, to, and fir- embraces all the operations of Addition, mo, to make firm]. In Algebra, an affirmative Subtraction, Multiplication, Division, raising quantity is one that is to be added, in contra- to powers denoted by constant exponents,

and extraction of roots indicated by constant troduction of a concise system of notation, indices; it also includes the discussion of the the foundation of which was laid by a Gernature and properties of all equations in man named Stifel, or Stifelius, who wrote which the relations between the known and about the middle of the 16th century. unknown quantities can be expressed by the ordinary operations of Algebra. Such equations are called algebraic.

vanced, as far as its general outline is concerned, to nearly its present condition.

In the year 1637, Descartes published his great work on the application of the principles of Algebraic Analysis to the investigation

From this period. improvements, both in the methods of notation, and in the generalization of processes, were rapidly made by Higher or Transcendental Algebra treats such mathematicians as Robert Recorde, of those quantities which cannot be exactly Vieta, Albert Girard, Harriot, and many expressed by a finite number of algebraic others, by whose labors the science was adterms, and which are therefore called transcendental. It also investigates the nature of transcendental equations, that is, all which are not algebraic. Under this branch of Algebra also falls the treatment of logarithms, formation and laws of series, and all that of geometrical truths, and besides opening class of problems which arise in the investi- an entirely new field of mathematical regation of Analytical Trigonometric formulas. search, contributed much to the advancement These two branches form what may be and perfection of pure Algebra. Since his called the Science of Algebra; besides these, time there has been no great revolution in a third might be added, having for its object Algebra, as a science, but it has been vastly the practical application of the principles de- improved in its details, and greatly extended duced, to the solution of all kinds of prob- in its applications. The theory of Series has lems, whether abstract or concrete, which been successfully developed by Euler, Walcome within the range of algebraic analysis. lis, the Bernouillis, Newton, De Moivre, It also includes the formation of rules for many of the higher arithmetical operations, as Interest, Annuities, Alligation, &c.

For an account of the several processes of Algebra, the reader is referred to the several articles, Addition, Subtraction, Multiplication, Division, Equations, &c., under their appropriate headings.

Simpson, and others. The composition of equations has been investigated, and the methods of approximating to their roots systematized and reduced to order.

Amongst the more recent laborers in the field of Algebra, may be mentioned Taylor, M'Laurin, Clairaut, Euler, Legendre, Arbogast, Gauss, Bourdon, and many others.

Perhaps the work containing the most complete exposition of the present state of the science, is the recent edition of L'Algèbre de M. Bourdon.

The most ancient Treatise extant on the subject of Algebra, is that of Diophantus, who wrote about the year 350. His work consists principally of a collection of solutions of problems relating to properties AL-GE-BRA'IC AL-GE-BRA'IC-AL. of numbers, and more particularly to the properties of square and cube numbers, of Appertaining to Algebra: thus, we say algebraic solutions, algebraic symbols, algebraic which some account may be found in the characters, &c. article on Diophantine Analysis. The science was cultivated by the Arabians, and from them a knowledge of its principles was de- relation between the co-ordinates of all its rived by the Italians, about the beginning of points can be expressed by the ordinary the 13th century. Many improvements were operations of Algebra. They are sometimes introduced, and many new processes discov-called geometrical curves, because their dif ered by Ferreas, Cardan, Tartalea, and others ferent points may be constructed by the of the Italian school, amongst the most im- operations of Elementary Geometry. The portant of which may be mentioned the name algebraic is used in contra-distinction method of solving cubic equations. to transcendental.

No great advances, however, were made

ALGEBRAIC CURVE. A curve such that the

ALGEBRAIC EQUATION. One in which the

in systematizing the science till after the in-relation between the known and unknown

quantities is expressed by the ordinary opera- and 15. In like manner any number may be tions of Algebra.

AL-GE-BRA'IST. One learned or skilled in Algebra.

AL'GO-RITHM. The art of computing in any particular way. We speak of the algorithm of numbers, surds, imaginary quantities, &c. The word is of Arabic origin, and properly means the art of numbering readily and correctly.

AL'I-QUANT PART. [L. aliquantum, a little]. In arithmetic, is such a part of a number as will not exactly divide it. Or, it is a part such that being taken any number of times, the result will be either greater or less than the given number: thus, 4 is an aliquant part of 10, because, being taken twice, the result is 8, a number less than 10, and being taken three times, the result is 12, number greater than 10. Again, 6 shillings is an aliquant part of a pound, made up of the two aliquot parts 4 shillings and 2 shillings. The term is used in contra-distinction to Aliquot part.

ever.

resolved into factors, and its aliquot parts
found. The idea of aliquot parts seems to
exclude that of fractions forming any aliquot
part of a whole number; still, in the case of
denominate numbers, there is an apparent
exception; as, for example, we say that 28.
eighth of it; 1s. 4d. is also an aliquot part of
6ð. is an aliquot part of a pound, being one-

a pound, being one-twelfth of it. An aliquot
part should not be confounded with a com-
mensurable part, for although every aliquot
part
of a number is commensurable with it,
every commensurable part is not an aliquot
60, but it is not an aliquot part.
part. Thus 40 is a commensurable part of

AL-LI-GA'TION. [L. From ad, to, and ligo, to bind]. A rule of practical Arithmetic relaadients. The rule is named from the method ting to the compounding or mixing of ingreof connecting or tying together the terms by certain ligature-like signs.

The rule is divided into two parts: Alligation medial, and alligation alternate.

ALLIGATION MEDIAL teaches the method

of finding the price or quality of a mixture of several simple ingredients whose prices or qualities are known.

ALLIGATION ALTERNATE teaches what amount of each of several simple ingredients, whose prices or qualities are known, must be taken to form a mixture of any required price or quality.

AL'I-QUOT PART. Such a part of any number or quantity as will exactly divide that number or quantity. Thus, 2 is an aliquot part of 4. 6, or any even number; and 1 is an aliquot part of any whole number whatTo find all of the aliquot parts of any number: Divide it by the least number except 1, that will exactly divide it; then divide the quotient by its least divisor, except As an example of a problem in alligation 1; and so on, always dividing the last quo-medial, we take the following: tient by its least divisor except 1, till 1 is Having a mixture of 30 bushels of wheat, found as a quotient; the several divisors, to- worth 150 cents per bushel, 72 bushels of gether with 1, are the prime aliquot parts. If rye, worth 90 cents per bushel; and 60 bushwe next form every possible product of these els of barley, worth 60 cents per bushel; divisors, taken in sets of two, in sets of three, required the price of a bushel of the mixture. and so on, in sets of n 1, n being the num30 bush. of wheat, at 150 cts., worth 4500 cts. ber of divisors, the products thus formed, 72" 66 6. rye, 90" 6480" taken with the original divisors, will make up 60"* barley 60" 3600" all the aliquot parts of the number. To 162 bushels of the mixture, worth 14580 find all the aliquot parts of 30: We divide it by 2, which gives a quotient 15; we next Whence, 1 bushel is worth of 14580 divide 15 by 3, which gives a quotient 5, which, on being divided by 5, gives a quotient Again: Suppose a goldsmith to mix gold 1; hence, 1, 2, 3 and 5 are the prime aliquot as follows: 6oz. of 22 carats, with 4oz. of 17 parts; but by multiplying these factors to- carats; required the quality of the mixture.

gether, two and two, we find 6, 10, and 15 for the compound aliquot parts. Hence, all the aliquot parts of 30 are 1, 2, 3, 5, 6, 10,|

cents, or 90 cents.

66

6oz. of 22 carats gives,
4oz. of 17 66
10

132

68

200

If, now, we divide 200 by 10, the whole these restrictions greatly limit the generality number of ounces in the mixture, we shall of the problem.

may be assumed at pleasure, and the value of the third deduced from equation (3).

find 20 carats for the quality of the mixture. | Since there are three simples, there are The principle in the last example is in no three unknown quantities, any two of which wise different from that in the former, the apparent difference lying entirely in the language employed in stating the proposition. We may in this example regard 24 as the value of pure gold per ounce; then 22 and 17 will be the respective values of each specimen mixed, and we shall find, as before, 20 for the value of an ounce of the mixture, that is, an ounce will contain fths of pure gold.

We may then write this rule for solving all questions in alligation medial :

RULE.-Multiply the price or quality of a unit of each simple by the number of such units; take the sum of their products, and divide it by the whole number of units; the quotient will be the price or quality of a unit of the mixture.

ALLIGATION ALTERNATE, as may be seen from the definition, gives rise to the solution of an indeterminate problem in Algebra. According as fewer or more restrictions are imposed, the solutions will be more or less numerous. There will be several cases. We shall first discuss the general one, in which it is required to find the amount of each simple of known value, which must be mixed so that each unit of the mixture shall have a given

value.

If there are n simples, the equation of condition will contain n unknown quantities, and (n − 1) of them may be assumed at pleasure.

In order to deduce a practical rule for solving questions in alligation alternate, let us begin with the case where there are but two simples. Denote the price or quality of a unit of the mixture by a; let a + b and a - c be the respective values of a unit of each simple, and let x and y denote, as before, the number of units of these simples that are taken. We shall have, as before,

(a + b) x + (a−c) y = a(x + y) (1). Whence, by reduction, we find

a relation which shows that b'

other as b is to c, will fulfill the required con-
any two values of y and x which are to each
dition, hence, y = b and x = c, are answers
of b and c.
of the question, as well as any equi-multiples

From a consideration of the notation em

Let there be three simples of the respective values of a, b, and c; let x, y, and z denote ployed, it appears that b is the excess of the the number of units taken from the respectvalue of a unit of the first simple over that ive simples to form m units of the mixture; of a unit of the mixture, and c is the excess and let d denote the price or quality of a unit of value of a unit of the mixture over that of the mixture. Then, from the conditions of of a unit of the second simple. The above the question, we shall have

ax + by + cz = md..
x + y + z = m . . . . . .

(1),
(2);

discussion indicates the following rule, when there are but two simples:

Write down the values of a unit of each simple beginning with the greatest, and link two equations of condition, which can, by them together by a bracket; write on their left the elimination of m be reduced to a single equation: (a − d) x + (b − d) y + (c − d) ≈ = 0... (3). This equation must be satisfied, in all cases, and any set of values of x, y, and z, which will satisfy it, will give a true answer to the question, considered in its most general sense.

Ordinarily, negative solutions are rejected, and the results are required to be integral;

the value of a unit of the mixture; subtract the last value from the first value given, and set the difference opposite the second; subtract the second value from the last, and set the difference opposite the first: these differences, or any equi-multiples of them, will be answers to the question proposed.

1. Required the number of bushels of oats at 50 cents per bushel, and of wheat at 120 cents per bushel, that must be mixed, so that