The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

x³ − 3x ³ f ² + g² − 2f = 0,

the general formula of the irreducible case, for

¹ (2f )² + ^{1 .}−3^√3 f ² + g^2Σ3 = −g².
4

27

If g = 0 we shall have x = 2^√3 f . The sole desideratum, therefore,

is to demonstrate that if g have any value whatever, x has a corresponding real value. Now the second last equation gives

and cubing we get

− −


√_{3 f 2}

+ g²

₌ x³ − 2f

3x

,

f ² + g² =

x⁹ 6x⁶f + 12x³f ² 8f ³

27x^{3 ,}

whence

g² =

x⁹ − 6x⁶f − 15x³f ² − 8f ³

27x

₃ ,

an equation which may be written as follows

or, better, thus:

g² =

(x³ 8f )(x³ + f )²

27x^{3 ,}

−

_{g2 =} 1

27

.1 − ^8f Σ (x³ + f )².

It is plain from the last expression that g is zero when x³ = 8f ; further, that g constantly and uninterruptedly increases as x increases; for the factor (x³ + f )² augments constantly, and the

_x3

8f

other factor 1 − _x3 also keeps increasing, seeing that as the de-

nominator x³ increases the negative part ^8f , which is originally

_x3

equal to 1, keeps constantly growing less than 1. Therefore,

if the value of x³ be increased by insensible degrees from 8f to infinity, the value of g² will also augment by insensible and corresponding degrees from zero to infinity. And therefore, re- ciprocally, to every value of g² from zero to infinity there must correspond some value of x³ lying between the limits of 8f and infinity, and since this is so whatever be the value of f we may legitimately conclude that, be the values of f and g what they may, the corresponding value of x³ and consequently also of x is always real.

But how is this value of x to be assigned? It would seem that it can be represented only by an imaginary expression or by a series which is the development of an imaginary expression. Are

we to regard this class of imaginary expressions, which corre- spond to real values, as constituting a new species of algebraical expressions which although they are not, like other expressions, susceptible of being numerically evaluated in the form in which they exist, yet possess the indisputable advantage—and this is the chief requisite—that they can be employed in the opera- tions of algebra exactly as if they did not contain imaginary expressions. They further enjoy the advantage of having a wide range of usefulness in geometrical constructions, as we shall see in the theory of angular sections, so that they can always be ex- actly represented by lines; while as to their numerical value, we can always find it approximately and to any degree of exactness that we desire, by the approximate resolution of the equation on which they depend, or by the use of the common trigonometrical tables.

It is demonstrated in geometry that if in a circle having the radius r an arc be taken of which the chord is c, and that if the chord of the third part of that arc be called x, we shall have for the determination of x the following equation of the third degree