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Fundamen- Rule I. The terms of the dividend are to be ranged is still a remainder from which the operation may be Fundamen

according to the powers of some one of its letters ; continued without end. This expression of a quotient tal operations.

and those of the divisor, according to the powers is called an infinite feries; the nature of which shall be
of the same letter.

considered afterwards. By comparing a few of the first
Thus, if a*+206+61 is the dividend, and a th the which, without any more division, it may be continued

terms, the law of the series may be discovered, by divisor, they are ranged according to the powers of a.

to any number of terms wanted.
2. The first term of the dividend is to be divided by

of the General Rule.
the first term of the divisor (observing the gencral
rule for the figns); and this quotient being set down The reason of the different parts of this rule is evi-
as a part of the quotient wanted, is to be multiplied dent; for, in the course of the operation, all the terms
by the whole divisor, and the product fubtracted of the quotient obtained by it are multiplied by all
from the dividend. If nothing remain, the division the terms of the divisor, and the products are succes-
is finished : the remainder, when there is any, is a fively subtracted from the dividend till nothing re-
new dividend.

main: that, therefore, from the nature of division, must

be the true quotient.
Thus, in the preceding example, a* divided by a,

Note. The fign • is sometimes used to express the
gives a, which is the first part of the quotient wanted:
and the product of this part by the whole divifor ati, quotient of two

quantities between which it is placed:
viz. a* tab being subtracted from the given dividend, Thus, q*+*+a+*, expresses the quotient of a'tx'
there remains in this example ab+b.

divided by att.
3. Divide the first term of this new dividend by the

first term of the divisor as before, and join the quo-
tient to the part already found, with its proper sign:

then multiply the whole divisor by this part of the 1. WHEN a quotient is expressed by a fraction, the di-
quotient, and subtract the product from the new visor above the line is called the numerator; and
dividend ; and thus the operation is to be continued the divisor below it is called the denominator.
till no remainder is left, or till it appear that there 2. If the numerator is less than the denominator, it is
will always be a remainder.

called a proper fraction.
Thus, in the preceding example, tab, the first 3. If the numerator is not less than the denominator,
term of the new dividend divided by a, gives b; the

it is called an improper fraction.
product of which, multiplied by a tó, being subtracted 4. If one part of a quantity is an integer, and the o.

ther a fraction, it is called a mixt quantity.
from ab +b, nothing remains, and atb is the true quo-
tient. The entire operation is as follows.

5. The reciprocal of a fraction, is a fraction whose nu.

merator is the denominator of the other; and whose a+b)2 + 2ab + b(a+b

denominator is the numerator of the other. The a't ab

reciprocal of an integer is the quotient of divided

by that integer. Thus, ab + b*

b ab to bi

is the reciprocal of g; and

is the reciprocal





of th.

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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 fame quantity, the value of that fraction is the faine.

+6ab--26 +6ab-25*

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Fundamen- Cor. 1. Hence a fraction may be reduced to another 2. The fimple divisors of each of the quantities are Fundamen-
tal opera- of the same value, but of a more fimple form, by di- to be taken out, the remainders in the several opera- tal opera-

viding both numerator and denominater by any com- tions are also to be divided by their fimple divisors, and
mon measure.

the quantities are always to be ranged according to the

powers of the same letter.
Thus, 300x-54ay_54-9y.

The fimple divisors in the given quantities, or in the
12 ab

remainders, do not affect a compound divisor which is
8ab + bac_4b+ 3c.

wanted ; and hence also, to make the division succeed,

any of the dividends may be multiplied by a simple

Cor, 2. A fraction is multiplied by any integer, by quantity, Besides the fimple divisors in the remainders
multiplying the numerator, or dividing the denomina. not being found in the divisors from which they arise,
tor by that integer; and conversely, a fra&ion is di- for the same reafon, if in such a remainder there be.

can make no part of the common measure sought; and
vided by any integer, by dividing the numerator, or

any compound divisor which does not measure the die multiplying the denominator by that integer.

vifor from which it proceeds, it may be taken out.
Prob. I. ia find the greatest commor Measure of two


ai-b)a'- 2ab +6°(1:
1. Of pure numbers.

Rule. Divide the greater by the less; and, if there is
no remainder, the less is the greatest common mea-

- 2ab +26* Remainder, which
sure required. If there is a remainder, divide the

divided by - 2b is a-ba-b'(a+b
laft divifor by it, and thus proceed, continually di.

viding the last divisor by its remainder, till no re-
mainder is left, and the last divisor is the greatest
common measure required.

If the quantities given are 8a2b-10ab3 +- 264, and
The greatest common measure of 45 and 63 is 9; ga*baga%b2+3ab3—3ab4. The fimple divisors. be-
the greatest common measure of 187 and 391 is 17. ing taken out, viz. 262 out of the first, it becomes

4a2-5ab +67, and 3ab out of the second, it is,

3a1-3ab + ab-13. As the latter is to be divided by 45)63(1

187)391(2 the former, it must be multiplied by 4, to make the 45


operation succeed, and then it is as follows: 18)45(2


402-5ab2) 12a1-12a*b+4ab-4b3(39

1.29-15a*b+ 3ab2

3ab+ab-463 18.

This remainder is to be divided by b, and the new dividend multiplied by 3, to make the division pro

ceed. Thus,
From the nature of this operation, it is plain that

za'+ab-4b4)12a-15ab +362( 4
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 re-

- 1gab + 1962 mainder must measure the dividend,

2. A quantity which measures both divisor and divi. and this remainder, divided by -19b, givesa-b, which
dend must also measure the remainder.

being made a divisor, divides 3a' to ab-462 without a
For a quantity which measures two other quantities, remainder, and therefore amb is the greatest compound.
must alla meafure both their sum and difference ; and, divisor : but there is a simple divisor b, and therefore:
from the nature of division, the dividend consists of the 2–6xb is the greatest common measure required.
divifor repeated a certain number of times, together
with the remainder. By the first it

appears, that the

Prob. II. To reduce a Fraction to its lowest Terms.
number found by this rule is a common measure ; and, Rule. Divide both numerator and denominator by
by the second, it is plain there can be no greater com-
mon measure ; for, if there were, it muft necessarily

their greateft common measure, which may be found
measure the quantity already found less than itself,

by Prob. 1, which is absurd.

75abc_3a When the greatest common measure of algebraical


125bex Š x 2gbc being the greateft common quantities is required, if either of them be limple, any


a4-64 a' +1
cominon fimple divisor is easily found by inspection. measure,

If they are both compound, any common simple divi-

for may also be found by inspection. But, when the

9a4b-9a3b2 + 3a*b3--3ab4 _9a3+3ab2 greateś compound divisor is wanted, the preceding

8a?be-loab3 +26*

the greatest

Sab---262 rule is to be applied ; only,

common measure being a-bxb, by Prob. 1.


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Fundamen. PROB. III. To reduce an Integer to the Form of a Cor. 2. A fraction, whose numerator is a compound Fundamen-
tal opera-

quantity, may be distinguished into parts, by dividing al operations.

tions. Rule. Multiply the given integer by any quantity for the numerator into several parts, and setting each over

a numerator, and set that quantity under the pro. the original denominator, and uniting the new frac-
duct for a denominator.

tions (reduced if necessary) by the figns of their nu

Thus, a=-

a-2ab + b? a

2ab f?

Cor. Hence, in the following operations concerning
fractions, an integer may be introduced; for, by this

PROB. VI. To multiply Fractions,
problem, it may be reduced to the form of a fraction.
The denominator of an integer is generally made i.


Rule. Multiply their numerators into one another, to
PROB. IV. To reduce Fractions with different Denomi. obtain the numerator of the product; and the deno-

nators to Fractions of equal Value, that all have minators, multipled into one another, shall give the
fame Denominator.

denominator of the product.
Rule. Multiply each numerator, separately taken, into

atb amb a_a
all the denominators but its own, and the products

Ex. xarici

a od
the denominators into one another

, and the product For, if is to be multiplied by c, the product is on shall give the common denominator.

but if it is to be multiplid only by the former pro.
Example. Let the fractions be ő a f they are

duct must be divided by d, and it becomes (Cor. 2.
adf, bcf, bdr.
respectively equal to
bdf bdf bar

to the preceding problem.)
The reason of the operation appears from the prece- Or, let g zm, and ĝ=n.

Then a=bm, and cdn,,
ding proposition ; for the numerator and denominator
of each fraction are multiplied by the fame quantities; and ac=bdmn, and (mn=)x = d.

and the value of fractions therefore is the same.

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Prob. V. To add and subtract Fraclions.

Prob. VII. To divide Fractions.
Rule. Reduce them to a common denominator, then

add or subtract the numerators ; and the sum or dif- Rule. Multiply the numerator of the dividend by the
ference set over the common denominator is the sum denominator of the divisor; their product shall give
or remainder required.

the numerator of the quotient. Then multiply the

denominator of the dividend by the numerator of
the fum is

the divisor, and their product shall give the denomi.


ad bc From fubt.the difference is

Or, Multiply the dividend by the reciprocal of the dibd

visor; the product will be the quotient wanted. From the nature of division it is evidenty that, when

bc b
several quantities are to be divided by the same divi-

for, the sum of the quotients is the same with the quo-
tient of the sum of the quantities divided by that com. For, if á is to divided by a, the quotient is adi
mon divisor.
In like manner, the difference of two fractions ha-

but ais
is to be divided, not by a, but by ; therefore

, s;
ving the same denominator, is equal to the difference of
the numerators divided by that common denominator.

the former quotient must be multiplied by b, and it

Cor. 1. By Cor. Prob. 3. integers may be reduced is

to the form of fractions, and hence integers and frac-
tions may be added and subtracted by this rule. Hence
also what is called a mixed quantity may be reduced

Or, let 7 =m, and a =n; then a=bm, and c=dn;


as the fractional part, and adding or subtracting the
numerators according as the two parts are connected

by the signs + or -:


By these problems, the four fundamental operations

may be performed, when any terms of the original. 2a*--a?+62 _a* + b3

quantities, or of those which arise in the course of the

are fractional



into the form of a fraction, with the fame denominator also ad=bdm and be=bdn; therefore (odym =)

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A L G E B R 406 L

Part I.

A. Of Propose


Cor. 1. Of four arithmetical proportionale, ary three of Propor, tion. Example. Mult.

tion. 26

being given, the fourth may be found.

Thus, let a, b, c, be the ift, 2d, and 4th terms, and
By 33

let x be the 3d which ia fought.

Then by def. atc=b+x, and x=a+c-b. a33 a?

бах? Prod.

- 20'+7 6x*

Cor. 2. If three quantities be arithmetical propor2**

tionals, the sum of the extremes is double of the midB**)a' *x(aast

dle term; and hence, of three such proportionals, any 4 + ax

two being given, the third may be found.

2. Of Geometrical Proportion. -axtx

Definition. If of four quantities, the quotient of the firft and second is equal to the quotient of the third

and fourth, these quantities are said to be in geometri-
3x + 2x3

cal proportion. They are also called proportionals.
Thus, if a, b, c, d, are the four quantities, then
= and their ratio is thus denoted a:b::c

Cor. Three quantities may be geometrical propor.
tionals, viz. by supposing the iwo middle terms of the
four to be equal. If the quantities are a, b, c, then


ő and the proportion is expressed thus, a : 6:c. +

Prop. I. The product of the extremes of four quan.

tities geometrically proportional is equal to the product
This quotient becomes a serious, of which the law of the means; and conversely.
of continuation is obvious, without any farther opera- .

Let a:b::c: d.
In such cases, when we arrive at a remainder of one

Then by Def.7=
term, it is commonly set down with the divisor below
it, after the other terms of the quotient, which then be- and multiplying both by bd, ad=bc.
comes a mixt quantity. Thus the last quotient is

If ad=bc, then dividing by bd, 6 = áo that is,

also expressed by @-tatx


Cor, 1. The product of the extremes of three quan-

tities, geometrically proportional, is equal to the square

of the middle term.
Of Proportion.

Cor. 2. Of four quantities geometrically proportional,
By the preceding operations quantities of the fame any three being given, the fourth may be found.
kind may be compared together.

Ex. Let a, b, c, be the three firit ; to find the 4th.
The relation arising from this comparison is called Let it be x, then a : 6::C:x, and by this proposition,
ratio or proportion, and is of two kinds. If we confi-

ax=bc der the difference of the two quantities, it is called

bc arithmetical proportion; and if we consider their

quo- and divining both by a, x= tient, it is called geometrical proportion. This laft be

This coincides with the Rule of Three in arithmetic, ing moft generally useful, is commonly called fimply and may be considered as a demonftration of it. In proportionals.

applying the rule to any particular case, it is only to
1. Of Arithmetical Proportion.

be observed, that the quantities must be so connected
Definition. When of four quantities the difference of and so arranged, that they be proportional, according
the first and second is equal to the difference of the to the preceding definition.
third and fourth, the quantities are called arithmetical Cor.


Of three geometrical proportionals, any two

being given, the third may be found.
Cor. Three quantities may be arithmetically propor.
tional, by supposing the two middle terms of the four

Prop. II. If four quantities be geometrically pro-
to be equal.

portional, then if any equimultiples whatever be taken Prop. In four quantities arithmetically proportional, of the firk and third, and also any equimultiples whatthe sum of the extremes is equal to the sum of the first be greater than that of the second, the multiple of

of the fourth

the third will be greater than that of the fourth; and
Let the four be a, b, c, d. Therefore from Def. if equal, equal ; and if less, less.
4-6=-d; to these add 6+d and a+d=b+c. For, let a, b, c, d, be the four proportionals. Of




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Of Equa- the first and third, ma and mc may represent any equi- and is expressed by placing the fign = between Of Equae
multiples whatever, and also nb, nd, may represent them.

tions. any equimultiples of the second and fourth. Since 2. When a quantity stands alone upon one side of an

a:b::c:d, ad=bc; and hence multiplying by mn, equation, the quantities on the other side are said to
mnad=mnbe, and therefore (Conv. Prop. 1.) be a value of it. Thus in the equation x=bty-d,
ma : nb :: me: nd; and from the definition of pro- x stands alone on one side, and btyed is a value
portionals, it is plain, that if ma is greater than nb, of it.
me must be greater than nd; and if equal, equal; and 3. When an unknown quantity is made to stand alone
if lefs, less.

on one side of an equation, and there are only known

quantities on the other, that equation is said to be
Prop. 111. If four quantities are proportionals, they resolved; and the value of the unknown quantity is
will also be proportionals when taken alternately or in- called a root of the equation.
versely, or by composition, or by division, or by con- 4. Equations containing only one unknown quantity
version. See Def. 13. 14. 15. 16. 17. of Book V. of and its powers, are divided into orders, according to
Euclid, Simson's edition.

the highelt power of the unknown quantity to be
By Prop. II. they will also be proportionals accord- found in any of its terms.
ing to Def. 5. Book V. of Euclid ; and therefore this if the highest power of) ift,
proposition is demonstrated by propofitions 16, B, 18,

The E. S Simple,

the unknown quanti- ad, quation Quadrat, 17, E, of the fame book.

ty in any term be the ) 3d, &c. ) is called Gubic, &c.
Otherwise algebraically.

But the exponents of the unknown quantities are sup-
Let a : b::c:d, and therefore ad=bc.

posed to be integers, and the equation is supposed to
Altern. 2:6::6:d

be cleared of fractions, in which the unknown quanti-
Invert. 6:a::d:c

ty, or any of its powers, enter the denominators.


is a simple equation; 3x Comp.

Thus, x tau 4+b:b::c+d:d

; Convert. 2:4- -6::6-

when cleared of the fraction by multiplying both

fides by 2x, becomes 6x-5=24% a quadratic. For fince adzbc, it is obvious, that in each of these x3_-27+=x®-20 is an equation of the sixth order, &c. cases the product of the extremes is equal to the pro

As the general relations of quantity which may be duct of the means ; the quantities are therefore pro- treated of in algebra, are almost universally either that portionals. (Prop. 1.)

of equality, or such as may be reduced to that of equa-
Prop. IV. If four numbers be proportionals, accord- lity, the doctrine of equations becomes one of the chief

branches of the science.
ing to Def. 5. V. B. of Euclid, they will be geome.
trically proportional, according to the preceding defi- is in the investigation of quantities that are unknown,

The most common and useful application of algebra

from certain given relations to each other, and to such
1/4, Let the four numbers be integers, and let them

as are known; and hence it has been called the analy.
be a, b, c, d. Then if b cimes a and b times c be ta.
ken, and also a times b and a times d, since ba the relations must therefore contain one or more unknown

tical art. The equations employed for expressing these
multiple of the first is equal to ab the multiple of the quantities; and the principal buliness of this art will
second, bc, the multiple of the third, must be equal to
ad the multiple of the fouth. And fince be=ad, by known quantity, and resolving them.

be, the deducing equations containing only one un-
Prop. 1. a, b, c, and d, must be geometrical propor-

The solution of the different orders of equations will tionals.

be successively explained. The preliminary rules in the
four being multiplied by the denominator of the frac- Lufficient for the solution of fimple equations.

four being multiplied by the denominator of the frase following fection are useful in all orders, and are alone
tions, they continue proportionals, aecording to Def.5.
B. V. Euclid (by Prop. 4. of that book); and the
four integer quantities produced being such propor-

Ś 1. Of fimple Equations, and their Resolution.
zionals, they will be geometrical proportionals, by the Simple equations are resolved by the four fundamen.
first part of this prop.; and therefore being reduced tal operations already explained ; and the application
by divifion to their original form, they manifeftly will of them to this purpose is contained in the following
remain proportionals, according to the algebraical de rules.

Rule 1. Any quantity may be transposed from one side

of an equation to the other, by changing its figa. CHAP III.

Thus, if 3x-10=2x+5

Then, 3x-2x=10+5 or x=15
SECT. I. Of Equations in general, and of the Solution

Thus also, 5x+b=a to 2x
of fimple Equations.

By trapfp. 3x=1-b.

This rule is obvious from prob. 1. and 2.; for it is

equivalent to adding equal quantities to both sides of 1. An Equation may in general be defined to be a the equation, or to subiracting equal quantities from propofition afferting the equality of two quantities; both sides.


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