How

on taking a lease of 40 acres more, he engaged 2 additional servants; after which, he had a servant for every 50 acres. many acres and servants had he at first ?

After the errors are found, the answer may be obtained more easily, (on some occasions,) by the following proportion: As the sum of the errors, when they are of different kinds, or the difference of the errors, when they are of the same kind, is to the difference of the suppositions, so is the least error to a fourth number, which is to be added to the supposition by which that error is produced, if the errors be of the same kind, and this supposition, greater than the other; or, if the errors be of different kinds, and this supposition less than the other, in every other case it is subtractive*.

Thus 30 and 40 were the errors in the last Example, which are of the same kind.

* Such questions as form Simple Equations, in Algebra, can only be performed by Position: but it is, nevertheless, very useful, and often saves much trouble, in approximating to the roots of the higher equations, &c. &c.

Here the 4th number is added to 180, the supposition, from which 30, the least error, is produced; because the errors are of the same kind, and this supposition greater than the other.

EXERCISES.

2. What number is that, which being multiplied by 3, the product increased by 4, and that sum divided by 8, quotes 32?

3. A person had 2 horses and a saddle, worth £50; when the saddle was placed on the first horse, it made his value double the second; but, when placed on the second horse, made his value triple the first. Required the value of each horse?

4. A gentleman hired a labourer for 40 days, and agreed to give him 8d. for every day he worked, but he was to return 4d. for every day he was idle: at the end of the 40 days the labourer received 10s. 5d. How many days did he work?

5. A vessel, which could just contain 63 gallons, was filled with wine of two sorts; the one at 8s. per gallon, and the other at 10s. The mixture sold at £28 16s. per hhd. How much was there of each sort?

6. A merchant allows £100 per annum for the expenses of his family, and augments, yearly, that part of his stock which remains, by a third part of itself; at the end of 3 years his original stock was doubled. With what sum did he begin trade?

7. Required a number, consisting of 2 digits, which is equal to 4 times the sum of its digits; and, if to that number 27 were added, the digits would be inverted?

INVOLUTION.

INVOLUTION is the operation of raising powers, and is formed by successive multiplication. Any number, multiplied into itself, produces the square or second power of that number, and that product, multiplied into the original number, produces the cube or third power of that number, &c. See page 2, Art. 13 and 14. The number given to be involved is called the root or first power. The number expressing the power is called the index or exponent of that power. See page 3, Art. 25 and the note at page 102.

TO INVOLVE ANY NUMBER TO ANY POWER.

RULE.

Multiply the number as many times into itself as is denoted by the exponent of the power.

EXAMPLE.

Required to involve 4 to the 4th power*. 4x4x4x4−4 −256

EXERCISES.

2. Involve 9 to the 3d power.

3. Involve 234 to the square.

The fourth power is called the biquadratic; the fifth power, sursolid power, &c., but these names are obsolete.

the

4. Involve 54 to the 3d power.

5. Involve 35 to the 4th power.

6. Involve 55 to the 3d power.

TO INVOLVE A SIMPLE FRACTION TO ANY POWER.

RULE.

Involve the numerator and denominator separately to the proposed power, and the results will be the respective terms of the fraction raised to the power required*.

* Decimals are involved like whole numbers, and mixed numbers may be reduced to improper fractions, and then involved as directed in the rule.

EVOLUTION.

EVOLUTION is the reverse of Involution, and is that operation by which any proposed root of any given number is discovered.

Although every number may be involved or raised to any power, yet there are many numbers of which the first power, or exact root, cannot be found.

The roots which are exactly found, are termed rational roots; and those which cannot be accurately found, are termed irrational, or surd roots

Roots are often denoted by placing the character ✔ before the power, with the index or exponent of the root over it; but now, more frequently, by placing the index of the required root above the number whose root is required, in the form of a vulgar fraction: the numerator denotes the power the given number is to be raised to, and the denominator the root required. Thus ✓12, 12, or 12 denotes the square root of 12;

m

√2+6, or 2+6) the cube root of 2+6; and 24 or 24 that 24 is raised to the mth power, and then the nth root extracted.

TO EXTRACT THE SQUARE ROOT.

RULE.

Point off the given number into periods of two figures each, beginning at the place of units, and pointing off to the left hand in integers, and to the right in decimals.

Then find a number whose square is either equal to, or the next less than, the figure or figures in the left hand period: place this figure in the quotient, as in Division, and place its square under, and subtract it from, the above-mentioned period, and to the remainder annex the next period, for a dividend.