y'

To proceed further, we may suppose y = 20 + &c., we shall then obtain, successively, values more and more exact. The several equations transformed into equations in y, y', y", &c. will have only one root greater than unity, unless two or more roots of the proposed equation are comprehended within the same limits a and a + 1; when this is the case, as in the above example, we shall find in some one of the equations in y, y', &c. several values greater than unity. These values will introduce the different series of equations, which show the several roots of the proposed equation, that exist within the limits a and a + 1. The learner may exercise himself upon the following equation; 2x- 5 = 0,

X3

the real root of which is between 2 and 3; we find for the entire values of y, y', &c.

10, 1, 1, 2, 1, 3, 1, 1, 12, &c.

and for the approximate values of X,

223. ARITHMETIC introduces us to a knowledge of the definition and fundamental properties of proportion and equidifference, or of what is termed geometrical and arithmetical proportion. I now proceed to treat of the application of algebra to the principles there developed; this will lead to several results, of which frequent use is made in geometry.

I shall begin by observing, that equidifference and proportion may be expressed by equations. Let A, B, C, D, be the four terms of the former, and a, b, c, d, the four terms of the latter; we have then

b

d

B — A = D — C (Arith. 127), /= (Arith. 111),
.

a

с

equations, which are to be regarded as equivalent to the expres

sions

and which give

A. B: C.D, a: b::c:d,

A+D=B+ C, adbc.

Hence it follows, that, in equidifference, the sum of the extreme terms is equal to that of the means, and, in proportion, the product of the

extremes is equal to the product of the means, as has been shown in Arithmetic (127, 113), by reasonings, of which the above equations are only a translation into algebraic expressions.

The reciprocal of each of the preceding propositions may be easily demonstrated; for from the equations

and, consequently, when four quantities are such, that two among them give the same sum, or the same product, as the other two, the first are the means and the second the extremes (or the converse) of an equidifference or proportion.

When BC, the equidifference is said to be continued; the same is said of proportion, when b = c. We have in this case A+ D = 2B, a d = b2:

that is, in continued equidifference, the sum of the extremes is equal to double the mean; and in proportion, the product of the extremes is equal to the square of the mean. From this we deduce

B =

A+D,

2

b = vad;

the quantity B is the middle or mean arithmetical proportional between A and D, and the quantity b the mean geometrical proportional between a and d.

from which it is evident, that we may change the relative places of the means in the expressions A. B: C. D, a:b:: c: d, and in this way obtain A. C: B. D, a:c::b: d. In general, we may make any transposition of the terms, which is consistent with the equations

A+D=B+C and ad bc (Arith. 114.)

I have now done with equidifference, and shall proceed to consider proportion simply.

224. It is evident, that to the two members of the equation

we may add the same quantity, m, or subtract it from

reducing the terms of each member to the same denominator, we obtain

and may be reduced to the following proportion,

b±ma:d±mc::a: c;

These two proportions may be enunciated thus; The first consequent plus or minus its antecedent taken a given number of times, is to the second consequent plus or minus its antecedent taken the same number of times, as the first term is to the third, or as the second is to the fourth.

Taking the sums separately and comparing them together, and also the differences, we obtain

or rather, by changing the relative places of the means

and if we make m=1, we have simply

baba::d+c:d-c,

which may be enunciated thus;

The sum of the first two terms is to their difference as the sum of

the last two is to their difference.

225. The proportion a: b c d may be written thus ;

we have then

:

a:c::b: d;

c±ma:d±mb:: a b or cd,

from which it follows, that the second antecedent plus or minus the first taken a given number of times, is to the second consequent plus or minus the first taken the same number of times, as any one of the antecedents whatever is to its consequent.

This proposition may also be deduced immediately from that given in the preceding article; for by changing the order of the means in the original proportion

a:b::c:d,

and applying the proposition referred to, we obtain, successively, a:c::b: d,

c±ma: d±mb::a:b or ::c: d,

and denominating the letters, a, b, c, d, in this last proportion, according to the place they occupy in the original proportion, we may adopt the preceding enunciation.

Making m = 1, we obtain the proportions

ca: db::a:b

whence it appears, that the sum or difference of the antecedents is to the sum or difference of the consequents, as one antecedent is to its consequent, and that the sum of the antecedents is to their difference as that of the consequents is to their difference.

then, by adding these equations member to member, we obtain b+d+f+ h = a q + c q + e q + g q, b+d+f+h = q(a + c + e + g).

or

This result is enunciated thus; in a series of equal ratios, ab::cd::e:f::g: h, &c.

the sum of any number whatever of antecedents is to the sum of a like number of consequents, as one antecedent is to its consequent. 226. If we have the two equations

and multiply the first members together and the second together, the result will be

bf dh
= ;
a e eg

an equation equivalent to the proportion

ae: bfcg: dh,

which may be obtained also by multiplying the several terms of the proportion

a:b::c: d,

by the corresponding ones in the proportion

e:f::g: h.

Two proportions multiplied thus term by term are said to be multiplied in order; and the products obtained in this way, are, as will be seen, proportional; the new ratios are the ratios compounded of the original ratios (Arith. 123).

It will be readily perceived also, that if we divide two proportions term by term, or in order, the result will be a proportion.

whence it follows, that the squares, the cubes, and, in general, the similar powers of four proportional quantities are also proportional. The same may be said of fractional powers, for, since Alg.

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