Cor. The quotient of an odd number by an odd number, is an odd number.

32. If an even number be multiplied by any number whatever, the product will be even.

Let 2A and 2B be any even numbers, and 2c+1 an odd number, 2A X2B-4AB an even number, also 2AX 2c+1 and 2B×2c+1 are even numbers.

Cor. If an even number contain an odd number a certain number of times, the quotient will be an even number. Hence also an even number cannot be contained an exact number of times in an odd number.

Other particular properties of numbers are given at page 6, 10, 15, 66, 70, 105, 106, 200, 202, &c.

ON SQUARE AND CUBE NUMBERS, &c.

33. The sum of any number of terms of the series of odd numbers, 1. 3. 5. 7. 9. 11. &c. is equal to the square of that number.

1.3 • 5. 7.9

11, &c. series of odd numbers.

Then 1+3=22; 1+3+5=32; 1+3+5+7=42; and so on as far as you please.

34. If to the sum of any number of terms of the series of squares 1. 4. 9. 16. 25. 36. 49. &c. you add the square of half the sum of the same number of terms, and increase that sum by an unit, the last sum will always be a square number.

Hence may be found as many square whole numbers as you please, whose sum shall universally be a square number.

+1=1+4+9+16+225+1=256

35. In a series of squares proceeding from an unit, the second differences will be equal to each other; in cubes the third differences; in biquadrates the fourth dif ferences, &c.

Thus, 149 · 16 · 25 · 36, &c. series of squares.

3

And, 1 8

5

17

.

2

911, &c. 1st order of differences. 2. 2. 2, &c. 2d order of differences. 27 · 64 · 125 • 216, &c. series of cubes. 7.1937 61 91, &c. 1st order of diff. 24 30, &c. 2d order of diff.

12 18
6.

6.

6, &c. 3d order of diff. In the same manner the fourth order of differences in the series of biquadrates, 1. 16 .81.256.625 . 1296, &c. will be 24. These orders of differences are obtained, by subtracting the first term from the second, the second from the third, the third from the fourth, &c. in the series; and in each of the orders of differences.

36. If a be the first term of any series, d' the first term of the first order of differences; a" the first term of the second order of differences; d" the first term of the third order of differences; div the first term of the fourth order of differences, &c. The last or nth term will be

-3

n-4

X Laiv &c. And the sum of n terms will be na+nx

page 12th and 13th. Any term of a given series, or the sum of any number of its terms, may be accurately determined, when any of the orders of differences become at last equal to each other.

1. Required the 20th-term of the series 1 48 · 13 • 19 • 26 34, &c.

2. Required the sum of a thousand terms of the series of squares 12. 22

32.42.52 &c.

Here d'=3, d"=2, a=1, and n=100;).

--- Then, na + (n×”—=—-1d'′)+(nx"="x"=2d′′)

X

2

2

3

3. Required the sum of twenty terms of the series of

Lubes 13.

1 8 27

23

19. 37 61

24.

12, 18.

91 &c. first order of diff.
30 &c. second order of diff.

6. 6 6 &c. third order of diff.

37. The sum of any two square numbers whatever ; their difference, and twice the product of their roots; will express the three sides of aright angled triangle in rational numbers.

Let 4 and 9 be the two squares, then 4+9=13 their sum, 0—4—5 their difference, and 4× √/9×2=12, twice the product of their roots; hence the three sides of the triangle will be 13, 5, and 12; for 12+5=132.

38. The cube of any number divided by 6, will leave the same remainder as the number itself, when divided by 6. Or, the difference between any number and its cube will divide even by 6.

Let 47 be proposed, the cube of which is 103823; each of these numbers divided by 6 will leave 5 for a remainder.

Or, 103823—47 will divide by 6 without a remainder.

39. The sum of any number of the cubes of the natural series 1. 2. 3. 4. 5. &c. taken from the beginning always makes a square number; the roots of these squares are 1. 3. 6. 10. 15. 21. &c. whose differences are,

2. 3. 4. 5. 6. &c.

Let 1. 8. 27. 64. 125. 216. 343. &c. be a series of cubes; the sum of the two first is 9, the sum of three 36, of four 100, &c. whose roots are 3. 6. 10. &c.

40. An even square number will divide by 4, and leave no remainder, but an odd square number divided by 4 will leave a remainder of 1.

Since the square of an odd number must be an odd number, let 2R+1 express the root; the square of which is 4RR+4R+1, this divided by 4 leaves 1 for a remainder; and the first term of it, viz. the square of 2R, is divisible by 4.

Other particular properties of numbers may be seen in INVOLUTION, EVOLUTION, PROGRESSION, &c.

END OF THE SECOND PART.

THE

COMPLETE PRACTICAL

ARITHMETICIAN.

PART III.

AN USEFUL COLLECTION OF

BILLS OF PARCELS, &c. &c.

CLASS I.

Exercising the Rules in COMPOUND MULTIPLICATION.

(1.) James Lamb, esq.

Jan, 1, 1822.

Bought of John Simpson,

£, s. d.

74lb. of green tea, at 10s. 4d. per lb.

144 do. finest bloom, at 14s. 8d.
10 dc. fine green, at 16s. 5d.
21 do. hyson, at 10s. 10 d.
19 do. good hyson, at 13s. 94d,
3 do. bohea, at 6s. 9d.

(2.) Sir John Guchim,

Hull, 1822.

To S. Jefferson, Dr.

Jan. 11. For 37 yds. of sheeting, at 1s. 4 d. £. s. d.

per yard

Feb. 3. For 43 yds. of lace, at 4s. Od. per

yard

16. For 753 ells of Irish, at 2s. 3d. per ell

May 12. For 209 do. dowlas, at 9}d.

15. For 730 do. muslin, at 7s. 0žd.

Received the contents,

S. Jefferson.