7. A certain sum of money is given to four persons: to A , B., and C., and D. draws the rest, which is D28; re quired the sum. (See question 2, Single Position.)

In the above case the suppositions are varied to illustrate the rule, but in both statements the errors are alike, that is, both too small, or minus. In the following example the errors are unlike, that is, one is minus and the other plus.

REMARKS.

In the above examples, I find the errors to be alike—that is, both too small; because the sum in either case is less than D500 the given number; therefore, after proceeding according to rule, till I find the difference of the errors, and then make the statement X for obtaining the answers, you will observe that the statement or terms are multiplied crosswise; then take the di ference of the products and divide it by the difference of th errors, which is 40, and you get the first or A.'s answer; from this you can easily get the others. Had the errors been unlike, that is, one too great and the other too small, you would divide the sum of the products by the sum of the errors, and the quotient would be the answer. It is immaterial what numbers are used for the suppositions, provided you observe the principles of the rule and the examples above, for the result will always be the same. As has been before observed, this rule has its origin in proportion; the reason for changing the errors or suppositions to obtain a different result or answer, is accounted for on the same principles that the 1st and 2d terms in Proportion are varied, or changed, to gain the required result; in that rule we change the first and second terms; if the answer is to be less, then the greater of the two terms is the divisor; if the answer is to be more, then the less of the two terms is the divisor, &c.

Algebraical demonstration.-Let A and B be two numbers, produced from a and b by similar operations; it is required to find the number from which N is produced by a like operation. Put a number required, and let N-A-r, and N-B=s. Then, according to the supposition on which the rule is founded, r: s :: x—a : x-b; whence, by multiplying means and extremes, rx―rb=sx-sa; and by transposition, rx —: ―sx=rb― sa; and by division, rb—as

=the number sought; and if r and

s be both negative, the theorem is the same; and if r or s be

negative, a will be equal to

rb+sa

rt s.

REVIEW.

which is the rule.

What is Position? How many kinds are there? What is Single Position? What is the rule? What is Double Position? What is first to be done when you commence an operation in

Single Position? After having ascertained the result of the operation, how will you proceed? How will you first begin an operation in Double Position? After having obtained the first error, how will you proceed? When you have obtained the second error, what is to be done? What will you do after you have multiplied the second supposition by the first error, and the first supposition by the second error? When you have ascertained whether the errors are both of the same kind, how do you proceed? If they are not of the same kind, how will you proceed? What is the rule for Double Position?

8. A laborer was hired for 60 days upon this condition: that for every day he wrought, he should receive 75 cents, and for every day he was idle, he should forfeit 37 cents; at the expiration of the time he received D18; how many days did he work, and how many was he idle?

Ans. worked 36 days, was idle 24 days.

9 Two persons, A. and B., have the same income; A. saves of his yearly; but B., by spending D150 per annum more than A., at the end of 8 years finds himself D400 in debt; what is their income, and what does each spend per annum?

Ans. income D400; A spends D300; B. D450.

INVOLUTION, OR THE RAISING OF POWERS, Teaches the method of finding the powers of numbers.

A POWER is the product arising from multiplying any given number into itself continually a certain number of times; thus, 2×2=4, is the second power or square of 2; 2×2×2=8, the third power or cube of 2; 2×2×2×2=16, the fourth power. of 2, &c. The number which denotes a power is called its index. If two or more powers are multiplied together, their product is that power whose index is the sum of the exponents of the factor; thus 2x2=4, square of 2; 4×4=16, fourth power of 2; and 16x16=256, eighth power of 2, &c. The power often denoted by a figure placed at the right and a little above he number, which figure is called the index or exponent of that power (thus, 22, 38), and is always one more than the number of multiplications to produce the power, or is equal to the num

ber of times the given number is used as a factor in producing the power. In producing the square of 2, there is only one multiplication, or two factors; in producing the cube, there are two multiplications or three factors, 33×3×3=27, &c. This subject will be more fully illustrated in progression.

RULE.

Multiply the given number or first power continually by itself till the number of multiplications be 1 less than the index of the power to be found, and the last product will be the power required. Fractions are multiplied by taking the products of their numerators, and of their denominators; they will be involved by raising each of their terms to the power required, and if a mixed number be proposed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule.

EXAMPLES.

1. Required the third power or cube of 35.

Here we see that in raising a fraction to a higher power, we de

crease its value.

6. What is the third power of .263 ?

7. What is the eighth power of ? 8. What is the square of 60?

9. What is the square of

?

10. What is the square of .01?

11. What is the cube of 24?

12. What is the ninth power of 747?

13. What is the seventh power of 298.75 ?

REVIEW.

Ans. .018191447.

Ans. 636

Ans. 3600.

Ans. Ans. .0001.

Ans. 112.

What is a power? How do you raise a number to any pow.

er? What is the rule!

EVOLUTION, OR THE EXTRACTION OF ROOTS.

EVOLUTION, or the extraction of roots, properly belongs to mathematics, and without a knowledge of that science, it will require strict attention and close application to arrive at any degree of perfection in the use and principles of those rules. The most correct and convenient method of extracting the roots of the several powers, particularly those of the higher order, is by logarithmic tables, as far preferable to any rules that can be given in common arithmetic.

The root of a number, or power, is such a number as, being multiplied into itself a certain number of times, will produce that number or power, and is denominated the square, cube, biquadrate, &c., or 2d, 3d, and 4th root, accordingly as it is, when raised to the 2d, 3d, and 4th power, equal to that power. Thus 4 is the square root of 16, because 4×4 16 and 4 is the cube root of 64, because 4×4×4=64 : and 4 is the fourth root or biquadrate of 256, because 4×4×4x4=250, &c. The roots are proportional, but their proportion is different from simple or compound proportion; the raising of powers increase in a uniform ratio, but this will not always-indeed but seldomoccur in the extraction of roots. Although there is no number of which we can not find any power exactly, yet there are many numbers of which precise or exact roots can never be determined; but by the use of decimals we can approximate toward the root to any assigned degree of accuracy; those roots are called surds; and those which are perfectly accurate, rational roots; surd roots sometimes have their origin in circulating decimals, or vulgar fractions. As few numbers are complete powers, surds must very often occur in arithmetical operations, but the result can be obtained nearly by continuing the extraction of the root.

SQUARE ROOT.

RULE 1.

1 SEPARATE the given number into periods of two figures each, beginning at the right hand or place of units.