17. Three merchants, A, B, C, on comparing their gains find, that among them all they have gained 1444/.; and that B's gain added to the square root of A's made 9201.; but if added to the square root of c's it made 912. What were their several gains? • Ans. A 400, в 900, c 144.

18. To find three numbers in arithmetical progression, so that the sum of their squares shall be 93; also if the first be multiplied by 3, the second by 4, and the third by 5, the sum of the products may be 66. Ans. 2, 5, 8.

19. To find four numbers such, that the first may be to the second as the third to the fourth; and that the first may be to the fourth as 1 to 5; also the second to the third as 5 to 9; and the sum of the second and fourth may be 20.

Ans. 3, 5, 9, 15. 20. To find two numbers such, that their product added to their sum may make 47, and their sum taken from the sum of their squares may leave 62. Ans. 5. and 7.

RESOLUTION OF CUBIC AND HIGHER
EQUATIONS.

A CUBIC Equation, or Equation of the 3d degree or power, is one that contains the third power of the unknown» quantity. As x3 — ax2 + bx = c.

A Biquadratic, or Double Quadratic, is an equation that contains the 4th Power of the unknown quantity:

As x-ax3 + bx2

cx= d.

An Equation of the 5th Power or Degree, is one that centains the 5th power of the unknown quantity :

As x5 ax + bx3 - cx2 + dx = e.

And so on, for all other higher powers. Where it is to be noted, however, that all the powers, or terms, in the equation, are supposed to be freed from surds or fractional exponents.

There are many particular and prolix rules usually given for the solution of some of the above-mentioned powers or equations. But they may be all readily solved by the following easy rule of Double Position, sometimes called Trial-and-error.

RULE.

1. FIND, by trial, two numbers, as near the true root as you can, and substitute them separately in the given equation, instead of the unknown quantity; and find how much the

VOL. I.

terms

terms collected together, according to their signs + or, differ from the absolute known term of the equation, marking whether these errors are in excess or defect.

2. Multiply the difference of the two numbers, found or taken by trial, by either of the errors, and divide the product by the difference of the errors, when they are alike, but by their sum when they are unlike. Or say, As the difference or sum of the errors, is to the difference of the two numbers, so is either error to the correction of its supposed number.

3. Add the quotient, last found, to the number belonging to that error, when its supposed number is too little, but subtract it when too great, and the result will give the true root nearly.

4. Take this root and the nearest of the two former, or any other that may be found nearer; and, by proceeding in like manner as above, a root will be had still nearer than before. And so on to any degree of exactness required.

Note 1. It is best to employ always two assumed numbers that shall differ from each other only by unity in the last figure on the right hand; because then the difference, or multiplier, is only 1. It is also best to use always the least error in the above operation.

Note 2. It will be convenient also to begin with a single figure at first, trying several single figures till there be found the two nearest the truth, the one two little, and the other too great; and in working with them, find only one more figure. Then substitute this corrected result in the equation, for the unknown letter, and if the result prove too little, substitute also the number next greater for the second supposition; but contrarywise, if the former prove too great, then take the next less number for the second supposition: and in working with the second pair of errors, continue the quotient only so far as to have the corrected number to four places of figures. Then repeat the same process again with this last corrected number, and the next greater or less, as the case may require, carrying the third corrected number to eight figures; because each new operation commonly doubles the number of true figures. And thus proceed to any extent that may be wanted.

EXAMPLES.

Ex. 1. To find the root of the cubic equation x3+x2 + 100, or the value of x in it.

X=

Here

Here it is soon found that

Again, suppose 4.2 and 4.3,

lies between 4 and 5. As- and repeat the work as folsume therefore these two num-lows:

Again, suppose 4.264, and 4.265, and work as follows:

the sum of which is 064087.

Then as 064087: 001 :: 027552: 0·0004299

The work of the example above might have been much shortened, by the use of the Table of Powers in the Arithmetic, which would have given two or three figures by inspection. But the example has been worked out so particularly as it is, the better to show the method.

Ex. 2. To find the root of the equation x3 50, or the value of x in it.

15 x2 + 63x

Here it soon appears that r is very little above 1.

Suppose

LET the given equations be so multiplied, or divided, &c, and by such numbers or quantities, as will make the terms which contain one of the unknown quantities the same in both equations; if they are not the same when first proposed.

Then by adding or subtracting the equations, according as the signs may require, there will remain a new equation, with only one unknown quantity, as before. That is, add the two equations, when the signs are unlike, but subtract them when the signs are alike, to cancel that common term.

Note. To make two unequal terms become equal, as above, multiply each term by the co-efficient of the other.

Here we may either make the two first terms, containing , equal, or the two 2d terms, containing y, equal. To make the two first terms equal, we must multiply the 1st equation by 2, and the 2d by 5; but to make the two 2d terms equal, we must multiply the 1st equation by 5, and the 2d by 3; as follows.

Subtr. the upper from the under, gives 31y = 62;
And dividing by 31, gives

y = 2.

6x + 15y 31x = 93;

x= 3;

= 2.

5.x 9

To Exterminate Three or More Unknown Quantities; Or, te Reduce the Simple Equations, containing them, to a Single

one.

RULE.

THIS may be done by any of the three methods in the last problem: viz.

1. AFTER the manner of the first rule in the last problem, find the value of one of the unknown letters in each of the given equations: next put two of these values equal to each other, and then one of these and a third value equal, and so on for all the values of it; which gives a new set of equations, VOL. I.

I i

with