And the sum of all the roots is found to be 15, being equal to the coefficient of the second term of the equation, which the sum of the roots always ought to be, when they are right.

NOTE 3. It is also a particular advantage of the foregoing rule, that it is not necessary to prepare the equation, as for other rules, by reducing it to the usual final form and state of equations. Because the rule may be applied at once to an unreduced equation, though it be ever so much embarrassed by surd and compound quantities. As in the following example.

3. Let it be required to find the root x of the equation √144x3 ++ +√ 196x3—x2+24 114, or the

2

value of x in it.

=

By a few trials it is soon found, that the value of x is but little above 7. Suppose therefore first, that x=7, and then that x=8.

Suppose again r=7'2, and, because it turns out too

great, suppose also x=71,

Therefore r=7 124 nearly, the root required,

NOTE 4. The same rule also, among other more difficult forms of equations, succeeds very well in what are called exponential equations, or those, which have an unknown quantity for the exponent of the power; as in the following example.

4. To find the value of x in the exponential equation

**= 100.

For the more easy resolution of this kind of equations, it is convenient to take the logarithms of them, and then compute the terms by means of a table of logarithms. Thus, the logarithms of the two sides of the present equation are, xx log. of x=2, the log. of 100. Then by a few trials it is soon perceived, that the value of x is somewhere between the two numbers 3 and 4, and indeed nearly in the middle between them, but rather nearer the latter than the former. By taking therefore first r=3'5, and then x= 3'6, and working with the logarithms, the operation will be as follows.

'098451 sum of the errors. Then, As 098451 : 1 :: 002689 : 0 ̊00273

Which correction, taken from

3'60000

On trial, this is found to be very little too small.

Take therefore again x=3.59727, and next x=3′59728, and repeat the operation as follows.

5. To find the value of x in the equation x2+10x+ 5r=2600. Ans. x 11'00673."

6. To find the value of x in the equation x3—2x=5. Ans. 2094551.

7. To find the value of x in the equation x3+2x3— 23x=70. Ans. x=51349.

8. To find the value of x in the equation x3-17xa + 54x=350. Ans. x=14 95407.

9. To find the value of x in the equation x-3x3— 75x=10000. Ans. x=102615.

10. To find the value of x in the equation 2x-16x3 +40x3-30x=-1. Ans. x=1'284724.

11. To find the value of x in the equation x+2x+ 3x3+4x2+5x=54321. Ans. x 8'414455.

12. To find the value of in the equation a*= 123456789. Ans. x 8 6400268.

GEOMETRY.*

DEFINITIONS.

1. A POINT is that, which has position,

but not magnitude.

2. A line is length, without breadth or thickness.

3. A surface, or superficies, is an extension, or a figure, of two dimensions, length and breadth, but without thickness.

4. A body, or solid, is a figure of three dimensions, namely, length, breadth, and thick

ness.

Hence surfaces are the extremities of solids; lines the extremities of surfaces; and points the extremities of lines.

5. Lines are either right, or curved, or mixed of these two.

* A TUTOR teaches, in Harvard College, PLAYFAIR'S "Elements of Geometry; containing the first six Books of EUCLID, with two Books on the Geometry of Solids." Of this work Mr. F. Nichols of Philadelphia has given a good American Edition: Vol. I. Tt