Cor. If an equation exist between terms affected by whole (65) additive powers of a variable quantity, x, capable of indefinite diminution, then the constant coëfficients of the like powers of x, will be respectively equal to each other.

Thus, if A+ A1 • x1 + A1⁄2 • x2 + A3

=

x3+

are constant and inde

a+a1 • x2 + а2 • x2 + αz • x3 +. where A, A1,* A2, A3, ..., α, ɑ1, A2, A3, pendent of x which is variable, then

or

19

... 9

and ... (63) A1 = a1, &c., &c., &c.; which was to be proved.
If, in a problem, ɑ, ɑı, ɑë
will become known.

can by any means be found, A,

A1, A2, ...,

on.

Applications will be given further

EXERCISES.

I. In Simple Equations.

1o. Given x+3=5, to find x. Subtracting 3 from both sides, we have

* Read A sub-one, A sub-two, &c., or simply A first, A second, &c. The advantage of this notation is that it points out the power of x to which the coefficient belongs; thus, An would belong to x".

... g we have

† For, denoting by A, the greatest of the coefficients A1, A2, A3, A1.x + A2.x2 + A3 • 23 +..... < Aq • X + Aq • x2 + Aq • x3 +..

which diminishes without limit as x approaches zero-and the same may be affirmed of

a1 • x+a2. x2 +a3 • x3 + ..... .

6°.

III. In Quadratics.

x2+4x=45, to find x.
x2+4x+4=49,

(x + 2)2 = 49,

x+2=±7,

x=- 2±75, or = — 9. 3x2+6= x2+24, .'. x = 9.

ax2-b

=

· cx2 + d, .. x = = ?

7- x2 - 8x2-6x2+3-21, .. x = ?

x2

Ꮖ

a b

=

= c, .. x = ?

(x − a )2 + 2n (x − a ) = b, .'. x = a − n ±(b+n2)$ 7°.x2+rx2 = a, or y2+2my = a, putting y = x”, 2m = r ;

::. y=—m±(a+m2)*, .. x = y = = [ — m ± ( a + ') * ] ·

8°.

10°.

11°.

12°.

x6 6x3 = 16, .. x = = ?

z2-2az=2ab+b2, ... x = ?

ru2 - 2rmu = 2rnu+p2r - 2rpm - 2nrp, .. x = ? x2 +36 + x3 + 6x 18x2 = 0, .. x = ?

Def. 1. The doctrines of extension constitute the science of Geometry.

Def. 2. Solids have three dimensions, length, breadth, and thick

ness.

Def. 3. The boundaries of a solid are surfaces, the perimeter of a surface are lines, and the extremities of a line are points.

PROPOSITION I. [PRIMARY NOTION].

A Straight Line is such that it does not change its direc- (66) tion at any point in its whole extent.

This truth is not produced as a theorem, for it is incapable of demonstration; not as a problem, for there is nothing to be done; neither as a corollary, for it is the consequence of nothing; nor as an axiom, for it is hardly of the unconditional and absolute character of that enounced in the words, "the whole is equal to the sum of all its parts;" and it is not a definition, for we gain no new idea by the mere terms of the proposition: we only recognize by and in them one of those primary notions which we possess anterior to all instruction, and which, as they are necessary to, lie at the foundation of, every logical deduction. It would doubtless be out of place to enter here into any investigation in regard to the origin of our ideas; but I think it will be apparent, that the notion of continuity is coöriginal with that of personal identity, and therefore, antecedent to argumentation; and continuity measured out on