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less than the roots of the given equation by i=e, Cor. 4. Prop. 2. When x=ò the

ice result is the absolute term of the given equation. When for x is inserted --I =-e, the result is the absolute term of an equation whose roots exceed the roots of the given equation by 1=e. Hence, if the terms of the series, 1, 0,-1, -2, &c. be inserted successively for x, the results will be the absolute terms of so many equations, of which the roots form an increasing arithmetical series with the difference 1. But, as the commensurate roots of these equations must be among the divisors of their absolute terms, they must also be among the arithmetical progressions found by this rule. The roots of the given equation, therefore, are to be sought for among

the terms of these progressions which are divisors of the result, upon the supposition of x=o, because that result is its absolute


It is plain that the progression must always be increasing, only it is to be observed, that a decreasing series, with the fign t, becomes increasing with the fign Сс



Thus, in the preceding example, ---4, -3, --2, is an increasing series, of which -3 is to be tried, and it succeeds.

If, from the substitution of three terms of the progression, 1,0,-1, &c. there arise

. a number of arithmetical serieses, by substituting more terms of that progression, some of the seriefes will break off, and, of course, fewer trials will be necessary. III. Examples of Questions producing the

bigber Equations.



It is required to divide 16 l. between two

perfons, so that the cube of the one's share

may exceed the cube of the other's by 386. Let the greater share be x pounds, And the less will be 16-*;

By the question, x--16-* = 386. And by Inv. 2x?-48x2+768x---4096=386 Tranf. and

x?—24x2+384x2241=0. divide Suppos. Results. Divisors. If x=+1; - 1880 =

1, 2, 4, 5, 8, 10, 20,

2241 1, 3, 9, 27, 83. x=-; - 2650 - '1, 2, 5, 10, 25, 53. I

5, ,


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are 9

Where 8, 9, 10, differ by i ; therefore +9 is to be tried; and being inserted for x, the equation is = 0. The two shares then and


which succeed. Since x=9; *~9=o is one of the simple equations from which this cubic is produced ; therefore m3—24x2 +384x---2241

=%2— 15x+ 249=0. And the two roots of this quadratic are impossible.

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What two numbers are those, whose product

multiplied by the greater will produce 405, and their difference multiplied by the

less 20 ?

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Then by quen. {xx**

Let the greater number be x, and the less y.

Şxy X *=xży=405

y + 20


g+ +40y2 +400 And



y4+40y'+400 405





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Mult, and trans. y4+40y2-40537400=0.

This 5

This biquadratic resolved by divifors, gives y=5; and therefore x=9. Also, y4+40y?-_405y+400

-=y?+592 +657—80=0. This cubic equation has one positive incommensurate root, viz. 1.114, &c. which

» may be found by the rule in the next fection, and two impoflìble. The incommenfurate root y=1,114, &c. gives x = 19.067,

y= &c. and these two answer the conditions

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very nearly.


The sum of the squares of two numbers 208,

and the sum of their cubes 2240 being given, to find them.

Let the greater be xty, and the less x—%. Then x+31 +3--yl2 = 2x2 + 2y2 = 208.

Hence yż= 104—*2. Also x+y+xm-y = 2x3-6xy? = 2240. Substitute for y2 its value and 2x3 +624x—

6x3 = 2240.

This reduced gives x?-156x+560=0.


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The roots of this equation are +10, +4, _14

If x=10 then y=2, and the numbers fought are 12 and 8, which give the only just solution. If x=4, then yż=88 and y=108. The numbers sought are therefore 4+788 and 4-788. The last is negative, but they answer the conditions. Lastly, if x=-14, then y2 =-92, hence y=v=92, is impossible; but still the two numbers -14 +-92, -14-7-92, being inserted, would answer the conditions. But it has been frequently observed, that such solutions are both useless, and without meaning.


IV. Solution of Equations by Approximation.

By the former rules, the roots of equations when they are commensurate may

be obtained: These, however, more rarely occur; and when they are incommensurate, we can find only an approximate value of them, but to any degree of exactness required. There are various rules for this


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