a province in the kingdom of Portugal, 67 miles in length and 20 in breadth; bounded on the W. and S. by the sea, on the E. by the river Guadiana, and on the N. by Alentejo. It is very fertile in figs, almonds, dates, olives, and excellent wines; besides, the fishery brings in large sums. The capital town is Pharo. It contains four cities, 12 towns, 67 parishes, and 61,000 inhabitants.

---

**ALGEBRA**

A General method of computation, wherein signs and symbols, commonly the letters of the alphabet, are made use of to represent numbers, or any other quantities.

This science, properly speaking, is no other than a kind of short-hand, or ready way of writing down a chain of mathematical reasoning on any subject whatever; so that it is applicable to arithmetic, geometry, astronomy, mensuration of all kinds of solids, &c. and the great advantages derived from it appear manifestly to arise from the conciseness and perspicuity with which every proposition on mathematical subjects can be written down in algebraic characters, greatly superior to the tedious circumlocutions which would be necessary were the reasoning to be written in words at length.

With regard to the etymology of the word algebra, it is much contested by the critics. Menage derives it from the Arabic *al-jabr*, which signifies the restitution of anything broken; supposing that the principal part of algebra is the consideration of broken numbers. Others rather borrow it from the Spanish, *algebra*, a person who replaces dislocated bones; adding, ding, that algebra has nothing to do with fraction.

Some, with M. d'Herbelot, are of opinion, that algebra takes its name from Gebar, a celebrated philosopher, chemist, and mathematician, whom the Arabs call Giaber, and who is supposed to have been the inventor. Others from gaf, a kind of parchment made of the skin of a camel, whereon Ali and Giafer Sadek wrote, in mystic characters, the fate of Mahometanism, and the grand events that were to happen till the end of the world. But others, with more probability, derive it from geber; a word whence, by prefixing the article al, we have formed algebra; which is pure Arabic, and properly signifies the reduction of fractions to a whole number. However, the Arabs, it is to be observed, never use the word algebra alone, to express what we mean by it; but always add to it the word macabelah, which signifies opposition and comparison; thus algebra-atlmacabelah, is what we properly call algebra.

Some authors define algebra, The art of solving mathematical problems; but this is rather the idea of analysis, or the analytic art. The Arabs call it, The art of restitution and comparison; or, The art of resolution and equation. Lucas de Burgo, the first European who wrote of algebra, calls it, Regula rei et censur; that is, the rule of the root and its square; the root with them being called rer, and the square censur. Others call it Specious Arithmetic; and some, Universal Arithmetic.

It is highly probable that the Indians or Arabians first invented this noble art: for it may be reasonably supposed, that the ancient Greeks were ignorant of it; because Pappus, in his mathematical collections, where he enumerates their analysis, makes no mention of any thing like it; and, besides, speaks of a local problem, begun by Euclid, and continued by Apollonius, which none of them could fully resolve; which doubtless they might easily have done, had they known any thing of algebra.

Diaphantus was the first Greek writer of algebra; who published 13 books about the year 800, though only five of them were translated into Latin, by Xylander, in 1575; and afterwards, viz., anno 1621, in Greek and Latin, by M. Bachet and Fermat, with additions of their own. This algebra of Diaphantus's only extends to the solution of arithmetical indeterminate problems.

Before this translation of Diaphantus came out, Lucas Paciolus, or Lucas de Burgo, a Minorite friar, published at Venice, in the year 1494, an Italian treatise of algebra. This author makes mention of Leonardus Pisanus, and some others, of whom he had learned the art; but we have none of their writings. He adds, that algebra came originally from the Arabs, and never mentions Diaphantus; which makes it probable, that that author was not then known in Europe. His algebra goes no farther than simple and quadratic equations.

After Paciolus appeared Stifelius, a good author; but neither did he advance any farther.

After him came Scipio Ferreus, Cardan, Tartaglia, and some others, who reached as far as the solution of some cubic equations. Bombelli followed these, and went a little farther. At last came Nunius, Ramus, Schoner, Salignac, Clavius, &c. who all of them took different courses, but none of them went beyond quadratics.

In 1590, Vieta introduced what he called his Specious Arithmetic, which consists in denoting the quantities, both known and unknown, by symbols or letters. He also introduced an ingenious method of extracting the roots of equations, by approximations; since greatly improved and facilitated by Raphson, Halley, Maclaurin, Simpson, and others.

Vieta was followed by Oughtred, who, in his Clavis Mathematica, printed in 1631, improved Vieta's method, and invented several compendious characters, to show the sums, differences, rectangles, squares, cubes, &c.

Harriot, another Englishman, contemporary with Oughtred, left several treatises at his death; and among the rest, an Analysis, or Algebra, which was printed in 1631, where Vieta's method is brought into a still more commodious form, and is much esteemed to this day.

In 1657, Des Cartes published his geometry, wherein he made use of the literal calculus and the algebraic rules of Harriot; and as Oughtred in his Clavis, and Marin. Ghetaldus in his books of mathematical composition and resolution published in 1630, applied Vieta's arithmetic to elementary geometry, and gave the construction of simple and quadratic equations; so Des Cartes applied Harriot's method to the higher geometry, explaining the nature of curves by equations, and adding the constructions of cubic, biquadratic, and other higher equations.

Des Cartes's rule for constructing cubic and biquadratic equations, was farther improved by Thomas Baker, in his Clavis Geometrica Catholica, published in 1684; and the foundation of such constructions, with the application of algebra to the quadratures of curves, questions de maximis et minimis, the centrobaryc method of Guldinus, &c. was given by R. Slusius, in 1668; as also by Fermat in his Opera Mathematica, Roberval in the Mem. de Mathem. et de Physique, and Barrow in his Lect. Geomet. In 1708, algebra was applied to the laws of chance and gaming, by R. de Montmort; and since by de Moivre and James Bernouilli.

The elements of the art were compiled and published by Kersey, in 1671; wherein the specious arithmetic, and the nature of equations, are largely explained, and illustrated by a variety of examples: the whole substance of Diaphantus is here delivered, and many things added concerning mathematical composition and resolution from Ghetaldus. The like has been since done by Prefet in 1694, and by Ozanam in 1703: but these authors omit the application of algebra to geometry; which defect is supplied by Guifnee in a French treatise expressly on the subject published in 1704, and l'Hospital in his analytical treatise of the conic sections in 1707. The rules of algebra are also compendiously delivered by Sir Isaac Newton, in his Arithmetica Universalis, first published in 1707, which abounds in select examples, and contains several rules and methods invented by the author.

Algebra has also been applied to the consideration and calculus of infinites; from whence a new and extensive branch of knowledge has arisen, called the Doctrine of Fluxions, or Analysis of Infinites, or the Calculus Differentials. INTRODUCTION.

A Quantity which can be measured, and is the object of mathematics, is of two kinds, Number and Extension. The former is treated of in Arithmetic; the latter in Geometry.

Numbers are ranged in a scale, by the continued repetition of some one number, which is called the Root; and, in consequence of this order, they are conveniently expressed in words, and denoted by characters. The operations of arithmetic are easily derived from the established method of notation, and the most simple reasonings concerning the relations of magnitude.

Investigations by the common arithmetic are greatly limited, from the want of characters to express the quantities that are unknown, and their different relations to one another, and to such as are known. Hence letters and other convenient symbols have been introduced to supply this defect; and thus gradually has arisen the science of Algebra, properly called Universal Arithmetic.

In the common arithmetic too, the given numbers disappear in the course of the operation, so that general rules can seldom be derived from it; but, in algebra, the known quantities, as well as the unknown, may be expressed by letters, which, through the whole operation, retain their original form; and hence may be deduced, not only general canons for like cases, but the dependence of the several quantities concerned, and likewise the determination of a problem, without exhibiting which, it is not completely resolved. This general manner of expressing quantities also, and the general reasonings concerning their connections, which may be founded on it, have rendered this science not less useful in the demonstration of theorems than in the resolution of problems.

If geometrical quantities be supposed to be divided into equal parts, their relations, in respect of magnitude, or their proportions, may be expressed by numbers; one of these equal parts being denoted by the unit. Arithmetic, however, is used in expressing only the conclusions of geometrical propositions; and it is by algebra that the bounds and application of geometry have been of late so far extended.

The proper objects of mathematical science are number and extension; but mathematical inquiries may be instituted also concerning any physical quantities that are capable of being measured or expressed by numbers and extended magnitudes: And, as the application of algebra may be equally universal, it has been called The science of quantity in general.

DEFINITIONS.

1. Quantities which are known are generally represented by the first letters of the alphabet, as \(a\), \(b\), \(c\), &c., and such as are unknown by the last letters, as \(x\), \(y\), \(z\), &c.

2. The sign \(+\) (plus) denotes, that the quantity before which it is placed is to be added. Thus \(a + b\) denotes the sum of \(a\) and \(b\); \(3 + 5\) denotes the sum of 3 and 5, or 8. When no sign is expressed, \(+\) is understood.

3. The sign \(-\) (minus) denotes, that the quantity before which it is placed is to be subtracted. Thus \(a - b\) denotes the excess of \(a\) above \(b\); \(6 - 2\) is the excess of 6 above 2, or 4. Note, These characters \(+\) and \(-\) from their extensive use in algebra, are called the signs; and the one is said to be opposite or contrary to the other.

4. Quantities which have the sign \(+\) prefixed to them are called positive or affirmative; and such as have the sign \(-\) prefixed to them are called negative.

5. Quantities which have the same sign, either \(+\) or \(-\), are also said to have like signs, and those which have different signs are said to have unlike signs. Thus \(+a + b\), have like signs, and \(+a - c\), are said to have unlike signs.

6. The juxtaposition of letters as in the same word, expresses the product of the quantities denoted by these letters. Thus \(ab\) expresses the product of \(a\) and \(b\); \(bcd\) expresses the continued product of \(b\), \(c\), and \(d\). The sign \(\times\) also expresses the product of any two quantities between which it is placed.

7. A number prefixed to a letter is called a numeral coefficient, and expresses the product of the quantity by that number, or how often the quantity denoted by the letter is to be taken. When no number is prefixed, unit is understood.

8. The quotient of two quantities is denoted by placing the dividend above a small line and the divisor below it. Thus \(\frac{18}{3}\) is the quotient of 18 divided by 3, or 6; \(\frac{a}{b}\) is the quotient of \(a\) divided by \(b\). This expression of a quotient is also called a fraction.

9. A quantity is said to be simple, which consists of one part or Term, as \(+a\), \(-abc\); and a quantity is said to be compound, when it consists of more than one term, connected by the signs \(+\) or \(-\). Thus \(a + b\), \(a - b + c\), are compound quantities. If there are two terms, it is called a binomial; if three, a trinomial, &c.

10. Simple quantities, or the terms of compound quantities, are said to be like, which consist of the same letter or letters, equally repeated. Thus \(+ab\), \(-5ab\), are like quantities; but \(+ab\), and \(+a + ab\), are unlike.

11. The equality of two quantities is expressed, by placing the sign \(=\) between them. Thus \(x = a = b = c\), means that the sum of \(x\) and \(a\) is equal to the excess of \(b\) above \(c\).

When quantities are considered abstractly, then \(+\) and \(-\) denote addition and subtraction only, according to Def. 2. and 3., and the terms positive and negative express the same ideas. In that case, a negative quantity by itself is unintelligible. The sign \(+\) also is unnecessary before simple quantities, or before the leading term of a compound quantity which is not negative; though, when such a quantity or term is to be added to another, \(+\) must be placed before it, to express that addition; and hence in Def. 2. it is said, that \(+\) is understood when no sign is expressed.

In geometry, however, and in certain applications of terms in the quantities to be added may be united, so as to render the expression of the sum more simple.

**Prob. II. To subtract Quantities.**

**General Rule.** Change the signs of the quantity to be subtracted into the contrary signs, and then add it, so changed, to the quantity from which it was to be subtracted (by Prob. I.); the sum arising by this addition is the remainder.

**Examp.** From \(+5a\) Subtract \(+3a\) Rem. \(+2a\)

From \(5a - 7b + 9c + 8\) Subt. \(2a - 4b + 9c - d\) Rem. \(3a - 3b + 8 + d\)

When a positive quantity is to be subtracted, the rule is obvious from Def. 3.: In order to show it, when the negative part of a quantity is to be subtracted, let \(c - d\) be subtracted from \(a\), the remainder, according to the rule, is \(a - c + d\). For if \(c\) is subtracted from \(a\), the remainder is \(a - c\) (by Def. 3.); but this is too small, because \(c\) is subtracted instead of \(c - d\), which is less than it by \(d\); the remainder therefore is too small by \(d\); and \(d\) being added, it is \(a - c + d\), according to the rule.

**Otherwise** If the quantity \(d\) be added to these two quantities \(a\) and \(c - d\), the difference will continue the same; that is, the excess of \(a\) above \(c - d\) is equal to the excess of \(a + d\) above \(c - d + d\); that is, to the excess of \(a + d\) above \(c\), which plainly is \(a + d - c\); and is therefore the remainder required.

**Prob. III. To multiply Quantities.**

**General Rule for the Signs.** When the signs of the two terms to be multiplied are like, the sign of the product is \(+\); but, when the signs are unlike, the sign of the product is \(-\).

**Case 1.** To multiply two terms.

**Rule.** Find the sign of the product by the general rule; after it place the product of the numeral coefficients, and then set down all the letters one after another, as in one word.

| Mult. | \(+a\) | \(+5b\) | \(-5ab\) | |-------|--------|---------|----------| | By | \(-b\) | \(-3c\) | \(-7ab\) | | | \(+ab\) | \(-15bc\) | \(+35aabx\) |

The reason of this rule is derived from Def. 6., and from the nature of multiplication, which is a repeated addition of one of the quantities to be multiplied as often as there are units in the other. Hence also the letters in two terms multiplied together may be placed in any order, and therefore the order of the alphabet is generally preferred.

**Case 2.** To multiply compound quantities.

**Rule.** Multiply every term of the multiplicand by all the terms of the multiplier, one after another, according to the preceding rule, and then collect all the products into one sum; that sum is the product required.

---

**CHAP. I.**

**Sect. I. Fundamental Operations.**

The fundamental operations in algebra are the same as in common arithmetic, Addition, Subtraction, Multiplication, and Division; and from the various combinations of these four, all the others are derived.

**Prob. I. To add Quantities.**

Simple quantities, or the terms of compound quantities, to be added together, may be like with like signs, like with unlike signs, or they may be unlike.

**Case 1.** To add terms that are like and have like signs.

**Rule.** Add together the coefficients, to their sum prefix the common sign, and subjoin the common letter or letters.

**Examp.** To \(5ab\) Add \(4ab\) Sum \(9ab\)

\(3aa - ab\) \(7aa - 2ab\) \(4aa - sab\) \(14aa - 8ab\)

**Case 2.** To add terms that are like, but have unlike signs.

**Rule.** Subtract the less coefficient from the greater; prefix the sign of the greater to the remainder, and subjoin the common letter or letters.

**Examp.** \(-4a\) \(+7bc\) \(-5ab\) \(+7a\) \(-3bc\) \(+2ab\) \(+3a\) \(+bc\) \(+3ab\) \(+5bc\) \(0\)

**Case 3.** To add terms that are unlike.

**Rule.** Set them all down, one after another, with their signs and coefficients prefixed.

**Examp.** \(2a + 3b\) \(-5c + 8\) \(2a + 3b - 5c + 8\)

Compound quantities are added together, by uniting the several terms of which they consist by the preceding rules.

**Examp.** The sum of \(5ab - 3xy - 12cd\) \(7xy - ab + 15\) \(9cd - xy - mn\) is \(4ab - 3cd + 15 - mn + 3xy\)

The rule for case 3. may be considered as the general rule for adding all algebraical quantities whatsoever; and, by the rules in the two preceding cases, the like Examp. Mult. \(2a + 3b\)

By \(3ax - 4by\)

\(6aax + 9abx - 8aby - 12bby\)

Prod. \(6aax + 9abx - 8aby - 12bby\)

Mult. \(a - b\)

By \(c - d\)

\(ac - cb - ad + db\)

Prod. \(ac - cb - ad + db\)

Of the general Rule for the Signs.

The reason of that rule will appear by proving it, as applied to the last mentioned example of \(a - b\) multiplied by \(c - d\), in which every case of it occurs.

Since multiplication is a repeated addition of the multiplicand as often as there are units in the multiplier, hence, if \(a - b\) is to be multiplied by \(c\), \(a - b\) must be added to itself as often as there are units in \(c\), and the product therefore must be \(ca - cb\) (Prob. I.).

But this product is too great; for \(a - b\) is to be multiplied, not by \(c\), but by \(c - d\) only, which is the excess of \(c\) above \(d\); \(d\) times \(a - b\) therefore, or \(da - db\), has been taken too much; hence this quantity must be subtracted from the former part of the product, and the remainder, which (by Prob. II.) is \(ca - cb - da + db\), will be the true product required.

Def. 12. When several quantities are multiplied together, any of them is called a factor of the product.

13. The products arising from the continual multiplication of the same quantity are called the powers of that quantity, which is the root. Thus, \(aa, aaa, aaaa, \&c.\) are powers of the root \(a\).

14. These powers are expressed, by placing above the root, to the right hand, a figure, denoting how often the root is repeated. This figure is called an index, or exponent, and from it the power is denominated. Thus,

\[ \begin{align*} aa & \text{ is called the } 1\text{st power of the root } a^1 \text{ or } a \\ aaa & \text{ is the } 2\text{nd power, and is otherwise } a^2 \\ aaaa & \text{ is the } 3\text{rd power, written } a^3 \\ aaaaa & \text{ is the } 4\text{th power, by } a^4, \&c. \end{align*} \]

The 2nd and 3rd powers are generally called the square and cube; and the 4th, 5th, and 6th, are also sometimes respectively called the biquadrates, sursolid, and cubocube.

Cor. Powers of the same root are multiplied by adding their exponents. Thus, \(a^3 \times a^2 = a^5\), or \(aaa \times aa = aaaaa, b^5 \times b = b^6\).

Scholium.

Sometimes it is convenient to express the multiplication of quantities, by setting them down with the sign (\(\times\)) between them, without performing the operation according to the preceding rules; thus \(a^3 \times b\) is written instead of \(a^3b\); and \(a - b \times c - d\) expresses the product of \(a - b\), multiplied by \(c - d\).

Def. 15. A vinculum is a line drawn over any number of terms of a compound quantity, to denote those which are understood to be affected by the particular sign connected with it.

Thus, in the last example, it shows that the terms \(+a\) and \(-b\), and also \(c\) and \(-d\) are all affected by the sign (\(\times\)). Without the vinculum, the expression \(a - b \times c - d\) would mean the excess of \(a\) above \(bc\) and \(d\); and \(a - b \times c - d\) would mean the excess of the product of \(a - b\) by \(c\), above \(d\). Thus also, \((a + b)^2\) expresses the second power of \(a + b\), or the product of that quantity multiplied by itself; whereas \(a + b^2\) would express only the sum of \(a\) and \(b^2\); and so of others.

By some writers a parenthesis (\()\) is used as a vinculum, and \((a + b)^2\) is the same thing as \(a + b\).

Prob. IV. To divide Quantities.

General Rule for the Signs. If the signs of the divisor and dividend are like, the sign of the quotient is \(+\); if they are unlike, the sign of the quotient is \(-\).

This rule is easily deduced from that given in Prob. III.; for, from the nature of division, the quotient must be such a quantity as, multiplied by the divisor, shall produce the dividend with its proper sign.

From Def. 8. the quotient of any two quantities may be expressed, by placing the dividend above a line and the divisor below it. But a quotient may often be expressed in a more simple and convenient form, as will appear from the following distinction of the cases.

Case I. When the divisor is simple, and is a factor of all the terms of the dividend. This is easily discovered by inspection; for then the coefficient of the divisor measures that of all the terms of the dividend, and all the letters of the divisor are found in every term of the dividend.

Rule. The letter or letters in the divisor are to be expunged out of each term in the dividend, and the coefficients of each term to be divided by the coefficient of the divisor; the quantity resulting is the quotient.

Ex. \(a) ab(b, 2aab)\)

\[ \frac{6a^3bc - 4a^2bdm}{(3ac - 2dm)} \]

The reason of this is evident from the nature of division, and from Def. 6. Note. It is obvious from corollary to Prob. III. that powers of the same root are divided by subtracting their exponents.

Thus \(a^3/a^2 = (a^3/a^2)(a^4)\). Also \(a^3b^6/(ab^5)\).

Case II. When the divisor is simple, but not a factor of the dividend.

Rule. The quotient is expressed by a fraction, according to Def. 8. viz. by placing the dividend above a line and the divisor below it.

Thus the quotient of \(3ab^2\) divided by \(2mbc\) is the fraction \(\frac{3ab^2}{2mbc}\).

Such expressions of quotients may often be reduced to a more simple form, as shall be explained in the second part of this chapter.

Case III. When the divisor is compound.

Rule. Rule 1. The terms of the dividend are to be ranged according to the powers of some one of its letters; and those of the divisor, according to the powers of the same letter.

Thus, if $a^2 + 2ab + b^2$ is the dividend, and $a + b$ the divisor, they are ranged according to the powers of $a$.

2. The first term of the dividend is to be divided by the first term of the divisor (observing the general rule for the signs); and this quotient being set down as a part of the quotient wanted, is to be multiplied by the whole divisor, and the product subtracted from the dividend. If nothing remain, the division is finished: the remainder, when there is any, is a new dividend.

Thus, in the preceding example, $a^2$ divided by $a$, gives $a$, which is the first part of the quotient wanted: and the product of this part by the whole divisor $a + b$, viz. $a^2 + ab$ being subtracted from the given dividend, there remains in this example $ab + b^2$.

3. Divide the first term of this new dividend by the first term of the divisor as before, and join the quotient to the part already found, with its proper sign: then multiply the whole divisor by this part of the quotient, and subtract the product from the new dividend; and thus the operation is to be continued till no remainder is left, or till it appear that there will always be a remainder.

Thus, in the preceding example, $+ab$, the first term of the new dividend divided by $a$, gives $b$; the product of which, multiplied by $a + b$, being subtracted from $ab + b^2$, nothing remains, and $a + b$ is the true quotient. The entire operation is as follows.

\[ \begin{array}{c} a + b) \quad a^2 + 2ab + b^2 \\ \hline \end{array} \]

\[ \begin{array}{c} a^2 + ab \\ \hline ab + b^2 \\ \hline ab + b^2 \\ \hline \end{array} \]

\[ \begin{array}{c} 3a - b) \quad 3a^3 - 12a^2 + a^2b + 10ab - 2b^2 \\ \hline \end{array} \]

\[ \begin{array}{c} 3a^3 - a^2b \\ \hline - 12a^2 + 10ab \\ \hline - 12a^2 + 4ab \\ \hline + 6ab - 2b^2 \\ \hline + 6ab - 2b^2 \\ \hline \end{array} \]

\[ \begin{array}{c} 1 - a) \quad 1 + a + a^2 + a^3, \text{ &c.} \\ \hline \end{array} \]

\[ \begin{array}{c} 1 - a \\ \hline +a \\ +a - a^2 \\ +a^2 \\ +a^2 - a^3 \\ +a^3, \text{ &c.} \\ \hline \end{array} \]

It often happens, as in the last example, that there is still a remainder from which the operation may be continued without end. This expression of a quotient is called an infinite series; the nature of which shall be considered afterwards. By comparing a few of the first terms, the law of the series may be discovered, by which, without any more division, it may be continued to any number of terms wanted.

Of the General Rule.

The reason of the different parts of this rule is evident; for, in the course of the operation, all the terms of the quotient obtained by it are multiplied by all the terms of the divisor, and the products are successively subtracted from the dividend till nothing remains: that, therefore, from the nature of division, must be the true quotient.

Notes. The sign $\div$ is sometimes used to express the quotient of two quantities between which it is placed: Thus, $a^2 + x^2 \div a + x$, expresses the quotient of $a^2 + x^2$ divided by $a + x$.

§ 2. Of Fractions.

Definitions.

1. When a quotient is expressed by a fraction, the dividend above the line is called the numerator; and the divisor below it is called the denominator.

2. If the numerator is less than the denominator, it is called a proper fraction.

3. If the numerator is not less than the denominator, it is called an improper fraction.

4. If one part of a quantity is an integer, and the other a fraction, it is called a mixed quantity.

5. The reciprocal of a fraction, is a fraction whose numerator is the denominator of the other; and whose denominator is the numerator of the other. The reciprocal of an integer is the quotient of 1 divided by that integer. Thus, $\frac{b}{a}$ is the reciprocal of $\frac{a}{b}$; and $\frac{1}{m}$ is the reciprocal of $m$.

The distinctions in Def. 2, 3, 4, properly belong to common arithmetic, from which they are borrowed, and are scarcely used in algebra.

The operations concerning fractions are founded on the following proposition:

If the divisor and dividend be either both multiplied or both divided by the same quantity, the quotient is the same; or, if both the numerator and denominator of the fraction be either multiplied or divided by the same quantity, the value of that fraction is the same.

Thus, let $\frac{a}{b} = c$, then $\frac{ma}{mb} = c$. For, from the nature of division, if the quotient $\frac{a}{b} (= c)$ be multiplied by the divisor $b$, the product must be the dividend $a$. Hence $\left(\frac{a}{b} \times b\right) \times c = a$, and likewise $ma = mbc$, and dividing both by $mb$, $\frac{ma}{mb} = c$. Conversely, if $\frac{ma}{mb} = c$, then also $\frac{a}{b} = c$. Cor. 1. Hence a fraction may be reduced to another of the same value, but of a more simple form, by dividing both numerator and denominator by any common measure.

Thus, \(\frac{30ax - 54ay}{12ab} = \frac{5x - 9y}{2b}\).

\(\frac{8ab + 6ac}{4a^2} = \frac{4b + 3c}{2a}\).

Cor. 2. A fraction is multiplied by any integer, by multiplying the numerator, or dividing the denominator by that integer; and conversely, a fraction is divided by any integer, by dividing the numerator, or multiplying the denominator by that integer.

**Prob. I. To find the greatest common Measure of two Quantities.**

1. Of pure numbers.

**Rule.** Divide the greater by the less; and, if there is no remainder, the less is the greatest common measure required. If there is a remainder, divide the last divisor by it; and thus proceed, continually dividing the last divisor by its remainder, till no remainder is left, and the last divisor is the greatest common measure required.

The greatest common measure of 45 and 63 is 9; the greatest common measure of 187 and 391 is 17.

Thus,

\[ \begin{array}{c} 45)63(1 \\ 45 \\ \hline 18 \\ \end{array} \]

\[ \begin{array}{c} 187)391(2 \\ 187 \\ \hline 187 \\ \end{array} \]

From the nature of this operation, it is plain that it may always be continued till there be no remainder. The rule depends on the two following principles:

1. A quantity which measures both divisor and remainder must measure the dividend. 2. A quantity which measures both divisor and dividend must also measure the remainder.

For a quantity which measures two other quantities, must also measure both their sum and difference; and, from the nature of division, the dividend consists of the divisor repeated a certain number of times, together with the remainder. By the first it appears, that the number found by this rule is a common measure; and, by the second, it is plain there can be no greater common measure; for, if there were, it must necessarily measure the quantity already found less than itself, which is absurd.

When the greatest common measure of algebraical quantities is required, if either of them be simple, any common simple divisor is easily found by inspection. If they are both compound, any common simple divisor may also be found by inspection. But, when the greatest compound divisor is wanted, the preceding rule is to be applied; only,

2. The simple divisors of each of the quantities are to be taken out, the remainders in the several operations are also to be divided by their simple divisors, and the quantities are always to be ranged according to the powers of the same letter.

The simple divisors in the given quantities, or in the remainders, do not affect a compound divisor which is wanted; and hence also, to make the division succeed, any of the dividends may be multiplied by a simple quantity. Besides the simple divisors in the remainders not being found in the divisors from which they arise, can make no part of the common measure sought; and for the same reason, if in such a remainder there be any compound divisor which does not measure the divisor from which it proceeds, it may be taken out.

**Examples.**

\[ \begin{array}{c} a^2 - b^2 \\ \hline a^2 - b^2 \\ \end{array} \]

\[ \begin{array}{c} -2ab + 2b^2 \\ \text{Remainder, which divided by } -2b \text{ is } a - b \\ \end{array} \]

\[ \begin{array}{c} a^2 - b^2 \\ \hline a^2 - b^2 \\ \end{array} \]

If the quantities given are \(8a^2b^2 - 10ab^3 + 2b^4\) and \(9a^2b^2 - 9a^3b^2 + 3a^2b^3 - 3ab^4\). The simple divisors being taken out, viz. \(2b^2\) out of the first, it becomes \(4a^2 - 5ab + b^2\); and \(3ab\) out of the second, it is \(3a^2 - 3a^2b + ab^2 - b^3\). As the latter is to be divided by the former, it must be multiplied by 4, to make the operation succeed, and then it is as follows:

\[ \begin{array}{c} 4a^2 - 5ab + b^2 \\ \hline 12a^3 - 12a^2b + 4ab^2 - 4b^3 \\ \end{array} \]

This remainder is to be divided by \(b\), and the new dividend multiplied by 3, to make the division proceed. Thus,

\[ \begin{array}{c} 3a^2 + ab - 4b^2 \\ \hline 12a^3 - 15ab + 3b^2 \\ \end{array} \]

and this remainder, divided by \(-19b\), gives \(a - b\), which being made a divisor, divides \(3a^2 + ab - 4b^2\) without a remainder, and therefore \(a - b\) is the greatest compound divisor; but there is a simple divisor \(b\), and therefore \(a - b \times b\) is the greatest common measure required.

**Prob. II. To reduce a Fraction to its lowest Terms.**

**Rule.** Divide both numerator and denominator by their greatest common measure, which may be found by prob. I.

Thus, \(\frac{75abc}{125bcx} = \frac{3a}{5x}\), \(25bc\) being the greatest common measure, \(a^4 + b^4 = a^4 + b^4\) also,

\[ \begin{array}{c} 9a^4b - 9a^3b^2 + 3a^2b^3 - 3ab^4 \\ \hline 8a^2b^2 - 10ab^3 + 2b^4 \\ \end{array} \]

the greatest common measure being \(a - b \times b\), by Prob. I. Part I.