To find two mean proportionals between two given numbers.

RULE. Divide the greater extreme by the lesser; and the cube root of the quotient, multiplied by the lesser extreme, gives the lesser mean. Multiply the said cube root by the lesser mean, and the product will be the greater mean proportional.

34. What are the two mean proportionals between 6 and 162? Ans. 18 and 54. 35. What are the two mean proportionals between 4 and 108 ? Ans. 12 and 36.

NOTE. If a number cannot be divided by any number less than the square root thereof, that number is a prime. All prime numbers, except 2 and 5, have 1, 3, 7 or 9, in the place of units. If the two right-hand figures of any number be divisible by 4, the whole is divisible by 4. If the three right-hand figures of any number be divisible by 8, the whole is divisible by 8. If the sum of the digits, constituting any number, be divisible by 3 or 9, the whole is divisible by 3 or 9. The product of any two numbers can have at most but as many places of figures as are in both their factors, and at least but one less. A square number cannot have more places of figures than double the places of the root, and at least but one less. A cube number cannot have more places of figures than triple the places of the root, and at least but one less.

DUODECIMALS,

OR MULTIPLICATION OF FEET, INCHES AND PARTS, Is a Rule made use of by workmen and artificers, in casting up the contents of their work.

RULE 1. Under the multiplicand place the corresponding denominations of the multiplier. Multiply each term into the multiplicand, beginning at the lowest in the multiplicand, by the highest in the multiplier, and write the result of each under its respective term, observing to carry an unit for every 12, from each lower denomination to its next superior, on the left hand. In the same manner multiply the multiplicand by the inches or seconds in the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand. Do the same with the seconds in the multiplier, setting the result of each term two places to the right hand of those in the multiplicand.

RULE 2. By Practice.-Multiply the multiplicand by the highest denomination of the multiplier, and take parts for the remaining inferior denominations of the multiplier.

RULE S. By whole numbers.-Reduce the given numbers to the lowest denomination in the question, and multiply them together, and that product divide by the several denominations, according to the table, as far as inches: then, when you have the whole amount in inches, divide by 144, and you have the answer in square feet.

RULE 4. By Vulgar Fractions.-Bring the inferior denominations to the fraction of a foot, and annex it to the feet in the given question, and multiply as in multiplication of vulgar fractions, and you will have the answer in feet, and the fractional part of a foot.

RULE 5. By Decimals.-Reduce the inferior denominations to the decimal of a foot, and annex it to the feet in the given question, and multiply as in multiplication of decimals, and the product will be the answer in feet.

TABLE 2nd.

1. Feet multiplied by feet, give feet.

2. Feet multiplied by inches, give inches.
3. Feet multiplied by seconds, give seconds, &c.
4. Inches multiplied by inches, give seconds
5. Inches multiplied by seconds, give thirds, &c.
6. Seconds multiplied by seconds, give fourths.
7. Seconds multiplied by thirds, give fifths, &c.

NOTE. In adding feet, inches, &c. place the several denominations, whicit are alike, under each other, and add as in compound addition, carrying 1 for every 12 to the next denomination, towards the left hand. In subtraction of feet, inches, &c. place the several denominations, which are alike, under each other, and subtract as in compound subtraction, borrowing 12 when necessary.

EXAMPLES.

1. Add 7 ft. 9 inc. 14 ft. 4 inc. 12 ft. 8 inc. and 6 ft. 6 inç. together. Ans. 41 ft. 3 inc. 2. Subtract 21 feet 9 inches from 100 feet 2 inches.

Ans. 78 feet 5 inches.

5. Let it be required to multiply 2 feet 6 inches by 2 feet

5 inches, and find the content?

OF

MENSURATION,

ACCORDING TO

EUCLID, EVERÅRD, WARD, PARDIE, F. R. S. VAN CULEN, HARRIS, AND WILLIAM HAWNEY.

MENSURATION OF SUPERFICIES.

SUPERFICIAL figures are all such as have only length and breadth, and not having any commensurable thickness.

1st. OF A SQUARE.

A square is a geometrical figure, having four equal sides, and as many right (or square) angles. To find the superficial content thereof, this is the

RULE. Multiply the side into itself, and the product is the content.

Let the side of a geometrical square be 14 yards, poles, or feet; required the superficial content?

14×14=196 yards, poles, or feet, answer.

2nd. OF A PARALLELOGRAM.

A parallelogram is a figure having four sides, and as many right angles, the opposite sides thereof being equal and parallel. To find the superficial content thereof,

RULE. Multiply the length by the breadth, and the product is the superficial content.

Let a parallelogram be a long square, whose length is 18 feet, and breadth 9 feet; required the superficial content? Ans. 162 feet.