Again, x2+y2 xx+y=s2-2pxs by multiplication,

Or x2+xyxx+y+y3=s3-2sp,

Or x2+sp+y=s3-2sp by substituting sp for its equal

-

xyxx+y;

And therefore x3 + y2 =s3 -3sp= sum of the cubes.

In like manner, x3 +y3 xx+y=s3-35px sby multiplication,

Or x4+xy xx2+y2+y=s4-352p,

Or x2+pxs22p+y=s4-3s2p by substituting px s2-2p

for its equal xyxx2+y2;

And consequently, x+y-s4-352p-pxs2 -2p=84-4s2 p+2p2 sum of the biquadrates, or fourth powers; and so on, for any power whatever.

10. The sum (a) and the sum of the squares (b) of four numbers in geometrical progression being given; to find those numbers.

Then will

1

Also, let the sum of the two means =s, and their

product = p.

And then will the sum of the two extremes =a-s by

the question,

And their product = p by the nature of proportion.

Or x3+y=xy Xa-spxa-s.
But x2+y23-3sp by the last problem,
And therefore pxa-s=s3-3sp by equality,

And from this value of s all the rest of the quantities p, x and y may be readily determined.

QUESTIONS FOR PRACTICE.

1. What two numbers are those, whose sum is 20, and their product 36? Ans. 2 and 18.

2. To divide the number 60 into two such parts, that their product may be to the sum of their squares in the ratio of 2 to 5. Ans. 20 and 40.

3. The difference of two numbers is 3, and the difference of their cubes is 117; what are those numbers ?

Ans. 2 and 5.

4. A company at a tavern had 31. 15s. to pay for their reckoning; but, before the bill was settled, two of them sneaked off, and then those, who remained, had ros. a piece more to pay than before; how many were there in the company ?

Ans. 7. 5. A

5. A grazier bought as many sheep as cost him 60l. and after reserving 15 out of the number, he sold the remainder for 541. and gained 2s. a head by them; how many sheep did he buy? Ans. 75.

6. There are two numbers, whose difference is 15, and half their product is equal to the cube of the less number; what are those numbers ? Ans. 3 and 18.

7. A person bought cloth for 331. 15s. which he sold again at 21. 8s. per piece, and gained by the bargain as much as one piece cost him; required the number of pieces. Ans. 15.

8. What number is that, which being divided by the product of its two digits, the quotient is 33 and if 18 be added to it, the digits will be inverted ? Ans. 24.

9. What two numbers are those, whose sum multiplied by the greater is equal to 77; and whose difference multiplied by the less is equal to 12 ? Ans. 4 and 7.

10. When will the hour, minute and second hands of a clock be all together next after 12 o'clock ?

Ans. Only at 12 o'clock.

II. The sum of two numbers is 8, and the sum of their cubes is 152; what are the numbers ?

Ans. 3 and 5. 12. The sum of two numbers is 7, and the sum of their fourth powers is 641; what are the numbers ?

Ans. 2 and 5.

13. The sum of two numbers is 6, and the sum of their fifth powers is 1056; what are the numbers ?

Ans. 2 and 4.

14. The sum of four numbers in arithmetical progression is 56, and the sum of their squares is 864; what are the numbers ?

Ans. 8, 12, 16 and 20.

Yr

15. Το

15. To find four numbers in geometrical progression, whose sum is 15, and the sum of their squares 85.

A CUBIC EQUATION, or equation of the third 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 fourth power of the unknown quantity : as xax2+bx-cx=d.

An equation of the fifth power, or degree, is one, that contains the fifth power of the unknown quantity: as as -ax+bx3-cx2+dx=e.

6

An equation of the sixth power, or degree, is one, that contains the sixth power of the unknown quantity: as a -ax+bx+cx3 +dx2-ex-f.

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 various particular rules for the resolution of cubic and higher equations; but they may be all easily resolved by the following rule of Double Position.

RULE.

RULE.*

1. Find by trial two numbers, as near the true root as possible, and substitute them separately in the given equation, instead of the unknown quantity ; marking the errors, which arise from each of them.

2. Multiply the difference of the two numbers, found by trial, by the least error, 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 the least error to the correction of its supposed number.

3. Add the quotient last found to the number belonging to the least error, when that 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. Each new operation commonly doubles the number of true figures in the root.

NOTE I. 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.

EXAMPLES.

* This rule may be used for solving the questions of Double Position, as well as that given in the Arithmetic, and is preferable for the present purpose. Its truth is easily deduced from the same supposition.

For, by the supposition, r : s :: x-a: x-b, therefore, by division, r-s: s::b-a: x-b; which is the rule.