43. The number of cyphers annexed to the first part of the divifor is lefs by unity than the exponent of the power propofed; for 2=a, the first part of the root, (in regard another figure is to follow) occupies the place of tens, it is therefore equal 20. Hence it is plain, that 3a2 or 202 × 3=1200, the first part of the divifor; and fince 5b, that 3ab, or 3X20X5 =300, the second part of the divifor; alfo that b2=52=25, the last part; likewise that the first part of the root is confiderably greater than any other figure.

44. The reafon of the rule will appear, if we take the cafe where the root confifts of 2 places a+b. Let the given number be represented by a3+3a2b+3ab2+b3; and if we place a, the cube root of a3, in the root, and fubtract a3 from the given number, the remainder or refolvend will be 3a2b+3ab2+b3; and fince it has been fhewn, that a in the root is confiderably greater than 6, it will follow that 3a2b, the first part of the refolvend, will be the greatest part of it. If, therefore, 3426 be divided by 32, it will quot b, the other part of the root fought, by the help of which the divifor may be completed; but fince all the parts of the divifor are multiplied by the last figure of the root, the divifor will be 3a2+3ab+b2, which is obtained by dividing each term by b.

N. B. If the root confifts of more than two places, a reprefents all the places found; and, by repeating the operation for a new divifor, the other part b may be found as before; and fo on.

EXAMPLE

b×3a2+3ab+b2=638949×3 =1916847

In this example the given number is a decimal, and decimals are pointed by beginning at the decimal point, and paffing over as many places towards the right hand as there are units in the exponent of the root required.

45. General Rule for extracting any rost.

Having divided the given number into proper periods, involve a like power of a+b with the number propofed. Put n

3 G

equal

equal to the exponent of the root.

Place the value of a in the

root, and fubtract the value of a" from the first period; then expunge b out of every term of the refolvend. This will give a general divifor that will answer a power of any dimension.

Note. a is always the trial divifor.

The divifor 5a4+10a3b+10a2b2+5ab3+b*

Required the root of 887503681, being the fixth power.

The divifor 6a+15a4b+2ca3 b2+15a2b3+6aba+bs

If the foregoing examples be well understood, the learner will be able to investigate theorems for extracting higher roots. We will now fubjoin a few mixt examples for practice.

Anf. 32.

9. biquadratic root of 6612111737853987761

10

II.

12.

13.

furfolid root of 33554432

the eighth root of 28179280429056

the fquare root of 2 nearly

the cube root of 2 nearly

14. An army of 7744 men was drawn up in quired the number of men in cach rank. 3 G 2

Anf. 48

Anf. 1.41424

Auf. 1.148699+

a fquare. Re

Anf. 88 men.

15.

15. A maltster had a round malt-kiln of 16 feet diameter, but is to build a fquare one that will contain 3 times as The fide of the new kiln is required.

much.

Anf 24.5557 feet. 16. The folidity of a sphere is 47016 cubic inches. Requi red the fide of a cube whofe content is equal to it.

Anf. 36 inches.

46. PROPORTION.

When two quantities of the fame kind are compared, their relation or ratio is obtained by enquiring how often the first contains the fecond. Thus, the ratio of 12 to 4 is 3; of 4 to 3 is 13; and of 3 to 8 is, or .375.

47. When four quantities, a, b, c, d, are proportional, it is ufually.expreffed by faying, the firft is to the fecond as the third is to the fourth; or, a:b::c: d, and the quantities are faid to be in geometrical proportion.

48. The quantity whofe ratio is enquired into is called the antecedent, and the quantity, with which it is compared, the confequent.

49. The first and third terms, a and c, are called antecedents.

The fecond and fourth, b and d, are called the confequents. The first and fourth terms, a and d, are called the extremes. The fecond and third terms, b and c, are called the means.

:

50. If a b c d, the product of the means, is equal to the product of the extremes, Euclid vi. 16. thus ad=bc.

51. If the product of two quantities, ad, be equal to the product of two others, be, the quantities are proportional, and a:b::c:d; that is, a factor of the first is to a factor of the

fecond