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4. A labourer dug two trenches, one of which was 6 yards longer than the other, for £17 16s., and the digging of each cost as many shillings per yard, as there were yards in its length; what was the length of each ?

Ans. 10 and 16 yards.

5. A. and B. set out from two towns, which were distant from each other 247 miles, and travelled till they met. A. travelled 9 miles a day, and the number of miles walked by B. in a day, increased by 3, was equal to the number of days occupied by the journey. Required the number of days they were travelling, and the number of miles passed over by each?

Ans. 13 days, and A. went 117 miles, and B. 130.

6. A. and B. having bought 41 oxen, for which each of them paid 420 dollars, find A's. worth a dollar a head more than B's.; how must they divide them?

Ans. A. 20, B. 21. 7. Divide the number 14 into two parts, whose product shall be 48. Ans. the numbers are 6 and 8.

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8. Given the sum of two numbers 9, and the sum of the squares 45; what are the numbers? Ans. 6 and 3. 9. What two numbers are those, whose sum, product, and the difference of their squares are equal to each other? Ans. +5, and 1+1√5.

9. 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?

Ans. 2, 5, 8, and 11.

10. There are three numbers in geometrical progression, whose sum is 7, and the sum of their squares 21; what are the numbers? Ans. 1, 2, and 4.

11. Required to find two numbers, such that the less may be to the greater, as the greater is to 12; and the sum of the squares may be 45.

Ans. 3 and 6.

12. What two numbers are those, whose difference is 2, and the difference of their cubes 98?

Ans. 3 and 5.

13. What two numbers are those, whose sum is 6, and the sum of whose cubes is 72? Ans. 2 and 4. 14. Required to find two such numbers, that their product shall be 20, and the difference of their cubes 61? Ans. 4 and 5.

15. Required to divide the number 5 into two such parts, that if each part be divided by the other, the sum of the quotients shall be 21? Ans. 3 and 2.

16. Divide 12 into two parts, so that their product may be equal to 8 times their difference. Ans. 8 and 4. 17. There are two numbers, the sum of whose squares is 89, and their sum multiplied by the greater, is 104; what are the numbers?

Ans. 132, and 2, or 8 and 5.

18. What number is that, which being divided by the product of its two digits, the quotient is 5; but when 9 is subtracted from it, the remainder is expressed by the same digits in an inverted order? Ans. 32.

19. Required to divide 13 into three such parts, that their squares may be equidifferent, and the sum of those squares may be 75. Ans. 1, 5, 7.

20. The sum of three equidifferent numbers is 12, and the sum of their 4th powers 962; what are the numbers? Ans. 3, 4, and 5.

21. There are three equidifferent numbers, such that the square of the least, added to the product of the other two makes 28; but the square of the greatest, added to the product of the other two makes 44; what are the numbers? Ans. 2, 4, 6.

22. Three merchants, A. B. C., on comparing their gains, find they amount collectively, to 1,444 dollars; that B's. gain added to the square root of A's. makes 920 dollars; and that B's. gain added to the square root of C's. makes 912 dollars; how much did they severally gain ? Ans. A. 400, B. 900, C. 144. 23. What two numbers are those, whose sum added

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to the square of the sum, makes 702, and whose difference subtracted from the square of the difference, leaves 56 ? Ans. 17 and 9.

24. The sum of two numbers is 10, and the sum of their 4th powers 1552; what are the numbers ?*

Ans. 6 and 4.

25. There are four numbers in arithmetical progression, the common difference of which is 4, and their continued product 9945; what are the numbers? Ans. 5, 9, 13, 17.

26. The sum of three numbers in harmonical proportion is 191, and the product of the first and last is 4032; what are the numbers? Ans. 56, 63, 72.

27. The sum of two numbers added to their product makes 31; but the sum subtracted from the sum of the squares, leaves 48. Quere the numbers?

Ans. 7 and 3.

28. Required to find three numbers in continued proportion, whose sum shall be 26, and the sum of their squares 364? Ans. 2, 6, and 18.

29. Required to find two numbers, whose product shall be 320, and the difference of their cubes to the cube of their difference, as 61 to 1? Ans. 20 and 16.

30. If 700 dollars be divided among four persons, so that their shares may be in geometrical progression, and the difference of the extremes to the difference of the means as 37 to 12; what will be the several shares ?

Ans. 108, 144, 192, and 256 dollars.

In examples of this nature, we may assume letters to denote the half sum, and half difference of the required numbers; whence, expressions for the numbers themselves, are readily obtained, and these being involved, and the powers added together, the odd powers of the unknown quantity will disappear. Hence, if the unknown quantity does not rise higher than the 5th power, the solution may be effected by quadratic equations.

31. The sum of two numbers is 11, and the sum of their 5th powers 17831; quere the numbers?

Ans. 7 and 4. 32. The difference of two numbers is 8, and the difference of their 4th powers is 14560; required the numbers ?* Ans. 11 and 3.

33. What number is that, from the square of which, if 9 be subtracted, and the remainder be multiplied by the number itself, the product will be 80?

Ans. 5.

34. Required a number, from the square of which, 30 being deducted, and the remainder multiplied by the number itself, the product shall be 56? Ans. 2+3√2.

35. The continued product of five equidifferent numbers is 945, and their sum 25; what are the numbers?

Ans. 1, 3, 5, 7, and 9.

* In this example, assuming for the half sum and half difference, we obtain a cubic equation, whence it appears that questions of this nature are not generally solvible by quadratics. When, however, we have an equation of the form x3+ax=b; if b=mn, and m2+a=n,the equation may be reduced to a quadratic. For multiplying by x, and adding mix to each member, the given equation becomes x++ax2+ m2x2=m2x2 + bx, or x1+ nx2=m2x2+nmx, whence, by completing the squares and ex

tracting the roots,

n

n

+12=
•=mx+· or x-m.

9 2

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half the sum of the re

In the above example, x being made quired numbers, we find x3+16x-455-7×65; where 72+16 =65, whence, x=7. If x3—ax=mn, and m2—a-n; or if x-ax ——mn, and a—m3—n, the equation may be reduced to a quadratic, and found-m, as before. But if x3-ax=mn, and a—m2=n, or x3—ax——mn, and m2-a-n; the equation may be changed to a quadratic, from which a second quadratic x-mx= =±n, m±√m3±4n

will arise, whence x=

36. The sum of three numbers in geometrical progres sion is 35, and the mean is to the difference of the extremes as 2 to 3; what are the numbers ?

Ans. 5, 10, and 20.

37. The sum of three numbers in geometrical progression is 13, and the product of the mean by the sum of the extremes is 30; required the numbers?

Result, 1, 3, 9.

38. There is a number consisting of three digits in geometrical progression; the number itself, is to the sum of its digits, as 124 to 7; and if 594 be added to it, the order of the digits will be inverted; what is the number?

Ans. 248.

Promiscuous examples to exercise the foregoing rules.

1. What number is that, to the double of which, if 44 be added, the sum will be equal to 4 times the number proposed ? Ans. 22.

2. A gentleman meeting 4 poor persons, divided a dollar among them in such manner, that their several shares composed an equidifferent series; and the sum given to the last was 4 times that given to the first. Quere their several shares?

Ans. 10, 20, 30, and 40 cents.. 3. A sum of money being divided among 6 poor persons, the second received 10d., the third 14d., the fourth 25d., the fifth 28d. and the sixth 33d. less than the first. The whole sum divided, was 10d. more than three times what the first received. How much did they severally receive? Ans. 40d., 30d., 26d., 15d., 12d., and 7ď.

4. A mercer having cut 19 yards from each of 3 equal pieces of silk, and 17 yards from another of the same length, finds the four remnants together, measure 142 yards; what was the original length of each piece?

Ans. 54.

5. A grazier having two flocks of sheep, containing the same number; sells 39 from one, and 93 from the

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