2. To divide the number 14 into two such parts, that their product may be 48.

Let x and y denote the two numbers.

Then the 1st condition gives x+y=14,

And the 2d gives xy = 48.

Then transp. y in the 1st gives x = 14-y;

This value subst. for x in the 2d, is 14y-y2 = 48;
Changing all the signs, to make the square positive,
gives y 14y=-48;
y2-14y

Then compl. the square gives y−14y + 49 = 1;
And extrac. the root gives y-7 = ±1;

Then transpos. 7, gives y = & or 6, the two parts.

3. Given the sum of two numbers 9, and the sum of their squares 45; to find those numbers.

Let x and y denote the two numbers.

Then by the 1st condition x + y = 9.
And by the 2d x2 + y2

45.

Then transpos. y in the 1st gives x = 9-y;

This value subst. in the 2d gives 81-18y+2y=45;
Then transpos. 81, gives 2y-18y =-36;

And dividing by 2 gives y2-9y=-18;'

Then compl. the sq. gives y2 − 9y + 87 = 4;
And extrac. the root gives y-= ± ÷ ;

Then transpos. gives y 6 or 3, the two numbers.

=

4. What two numbers are those, whose sum, product, and difference of their squares, are all equal to each other?

Let x and y denote the two numbers.

Then the 1st and 2d expression give x+y=xy,
And the 1st and 3d give x + y = x2 —y2.

Then the last equa. div. by xy, gives 1x-y;
And transpos. y, gives y + 1 = x;

H

This val. substit. in the 1st gives 2y + 1 y2+y;
And transpos. 2y, gives 12-y ;

Then complet. the sq. gives y2 −y + 4 ;
&
And extracting the root gives 5 = y;
And transposing gives√5 + 1 = y;

And therefore = y + 1 = √5 + 2.

And if these expressions be turned into numbers, by extracting the root of 5, &c, they give a 2.6180+,

and y

= 1.6180 +.

5. There are four numbers in arithmetical progression, of

which the product of the two extremes is 22, and that of the means 40; what are the numbers?

Let the less extreme,

and y the common difference;

Then x, x+y, x+2y, x+3y, will be the four numbers.
Hence by the 1st condition a2 + 3xy #22,

And by the 2d x-3xy+2y2 = 40.

Then subtracting the first from the 2d gives 2y2 = 18;
And dividing by 2 gives y2 = 9;

And extracting the root gives y = 3.

Then substit. 3 for y in the 1st, gives x2+9x= 22;
And completing the square gives x2 + 9.x + 1 = 169;
Then extracting the root gives x+2=3;
And transposing gives r 2 the least number.
Hence the four numbers are 2, 5, 8, 11.

6. To find 3 numbers in geometrical progression, whose sum shall be 7, and the sum of their squares 21.

Let x, y, and z denote the three numbers sought.
Then by the 1st condition z = y2,

And by the 2d x + y + z =7,

And by the 3d x2 + y2+z2 = 21.

Transposing y in the 2d gives x + z = 7−y;

Sq. this equa. gives x2 + 2xz +22 = 49-14y+ y2;
Substi. 2y for 2xz, gives x2+2y+49-14y+ y2;
Subtr. y from each side, leaves.2+ y2+z2=49-14y;
Putting the two values of x2 + y2 + z2
21-49-14y;

equal to each other, gives S

Then transposing 21 and 14y, gives 14y = 28;
And dividing by 14, gives y = 2.

Then substit. 2 for y in the 1st equa. gives xz = 4,

And in the 4th, it gives x + z = 5;

Transposing in the last, gives x = 5−z;

This substit. in the next above, gives 52-z2 = 4;
Changing all the signs, gives z2—5% = -4;

Then completing the square, gives 2-5% + 2 = 4;

And extracting the root gives 2-==;

Then transposing 2, gives x and x = 4 and 1, the two other numbers;

So that the three numbers are 1, 2, 4.

2. To find two numbers such, that the less may be to the greater as the greater is to 12, and that the sum of their squares may be 45.

3. What two numbers are those, whose and the difference of their cubes 98 ?

Ans. S and 6.

difference is 2,

Ans. 3 and 5.

4. What two numbers are those whose sum is 6, and the sum of their cubes 72 ?

5. What two numbers are those, whose and the difference of their cubes 61 ?

Ans. 2 and 4.

product is 20,

Ans. 4 and 5.

6. To divide the number 11 into two such parts, that the product of their squares may be 784. Ans. 4 and 7. 7. To divide the number 5 into two such parts, that the sum of their alternate quotients may be 43, that is of the two quotients of each part divided by the other.

Ans. 1 and 4.

8. To divide 12 into two such parts, that their product may be equal to 8 times their difference.

Ans. 4 and 8.

9. To divide the number 10 into two such parts, that the square of 4 times the less part, may be 112 square of 2 times the greater.

more than the

Ans. 4 and 6.

10. To find two numbers such, that the sum of their squares may be 89, and their sum multiplied by the greater may produce 104. Ans. 5 and 8.

11. What number is that, which being divided by the product of its two digits, the quotient is 54; but when 9 is subtracted from it, there remains a number having the same digits inverted? Ans. 32.

12. To divide 20 into three parts such, that the continual product of all three may be 270, and that the difference of the first and second may be 2 less than the difference of the second and third. Ans. 5, 6, 9.

13. To find three numbers in arithmetical progression, such that the sum of their squares may be 56, and the sum arising by adding together 3 times the first and 2 times the second and 3 times the third, may amount to 28. Ans. 2, 4, 6. 14. To divide the number 13 into three such parts, that their squares may have equal differences, and that the sum of those squares may be 75. Ans. 1, 5, 7.

15. To find three numbers having equal differences, so that their sum may be 12, and the sum of their fourth powers

962.

Ans. 3, 4, 5.

16. To find three numbers having equal differences, and such that the square of the least added to the product of the two greater may make 28, but the square of the greatest added to the product of the two less may make 44.

Ans. 2, 4, 6.

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 9207.; but if added to the square root of c's it made 912. What were

their several gains?

Ans. A 400, B 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 = ċ.

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

Âs x1 — ax3 + bx2—cx = d.

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

As x-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

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 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 too 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