688. How long is the edge of the greatest cube that can be cut from a 6-inch globe?

689. The weights of 3 stacks of hay of similar form are respectively 3, 5, and 7 tons, and the slant height of the least is 12 feet. Find the slant height of the other two.

690. What are the dimensions of a box similar to, and that will hold 3 times as much as, a box 8 by 6 by 4 inches?

691. If an iron ball 4 inches in diameter weighs 8 pounds, what is the weight of a ball 50 inches in diameter ?

692. If a stack of hay 26 feet high weighs 29 tons, what is the height of a similar stack which weighs 9 tons?

693. How many bullets of an inch in diameter will be required to make a ball 4 inches in diameter?

694. If a loaf of sugar 10 inches high weighs 8 lb., what is the height of a similar loaf weighing 1 lb.?

695. Suppose the diameter of the earth is 7912 miles, and that it takes 1404928 bodies like the earth to make one as large as the sun, what is the diameter of the sun?

696. A bin is 15 feet long, 5 feet wide, and 3 feet deep; what is the edge of a cubical box that will hold the same quantity of grain ?

697. If a ball 8 inches in diameter weighs 50 pounds, what is the diameter of a ball that weighs 164 pounds?

698. If a bell 6 inches high, 4 inches in diameter, and of an inch thick weighs 4 lb., what are the dimensions of a similar bell that weighs 740 lb. ?

699. If a globe of gold 1 inch in diameter is worth $100, what is the diameter of a globe worth $ 3600 ?

700. A, B, and C own equal shares of a conical stack of hay 20 feet in height. If A takes his from the top, B next, and C last, what is the altitude of the part that each takes?

APPENDIX.

NOTATION AND NUMERATION.

505. In addition to the table on page 4 the periods above billions are trillions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, duodecillions, tredecillions, etc.

506. And also, in addition to the table on page 102, the names of the decimal periods below billions are formed by changing the s of the names of the integral periods to ths.

507. The method of numeration, given on pages 4 and 102, is called the French method, and is the one generally used in this country.

508. In the English Method a period is six figures.

509. In the French and English methods the written expressions are the same as far as, and including, hundred millions. But an English billion is a thousand French billions, an English trillion a million French trillions, etc.

510. The Roman Notation, or that used by the Romans, employs seven capital letters to express numbers, namely:

L, C, D, M. 10, 50, 100, 500, 1000.

I,

X,

5,

1,

All other numbers are expressed by combining and repeating these letters.

511. The Roman Notation is based on the following principles:

1. When two or more numbers of equal value are united, or when a letter of less value follows one of greater, the sum of their values is indicated; thus, XXX stands for 30, LXV for 65, CC for 200, MDCLXVII for 1667.

2. When a letter of less value is placed before one of greater, the difference of their values is indicated; as, IX stands for 9, XL for 40, XC for 90.

3. When a letter of less value stands between two of greater value, the less is to be taken from the sum of the other two; as, XIV stands for 14, XIX for 19, CXL for 140.

4. A letter with a line over it represents a number one thousand times as great as the same letter without the line; thus, X stands for ten, but stands for one thousand times ten, that is, ten thousand; M stands for one thousand, but M for one thousand times one thousand.

512. The Roman Numerals are seldom used, except for numbering the pages of a preface, the divisions of a discourse, and the sections, chapters, and other divisions of a book.

CONTRACTIONS IN MULTIPLICATION.

513. To multiply by 9, 99, or any number whose figures are 9's,

Annex as many O's to the multiplicand as there are 9's in the multiplier, and from the number so formed subtract the multiplicand; the remainder will be the product sought.

1. Multiply 786 by 999.

786 X 1000 = 786000

786 X

1: = 786

786 X 999 = 785214, Ans.

2. Multiply 437 by 99.

3. Multiply 8764 by 999.

4. Multiply 18432 by 9999.

514. To multiply by aliquot parts of 100,

Move the decimal point two places to the right, and take such fraction of the result as the multiplier is of 100.

Thus, 160 x 12 = 16000 × = 2000.

515. To multiply by aliquot parts of 1000,

Move the decimal point three places to the right, and take such fraction of the result as the multiplier is of 1000.

Thus, 273 x 333 = 273000 X

= 91000.

NOTE. The same principle may be used when the multiplier is an aliquot part of 1 with any number of ciphers annexed. (See Art. 172.)

CONTRACTIONS IN DIVISION.

516. To divide by a composite number,

Divide the dividend by one factor of the divisor, and the quotient so obtained by another factor, and so on till all the factors of the divisor have been used. The last quotient will be the required quotient.

NOTE. If the divisor, greater than 12, is composed of factors not greater than 12, the work of division is lessened.

517. To divide by 9, 99, or any number whose figures are 9's,

Cut off from the right of the dividend as many places as there are 9's in the divisor; the number at the left of the point will be the quotient; and the remainder will be the same as the quotient plus the figures at the right of the point. If the remainder exceeds the divisor, divide this remainder again as before; and add the second quotient to the first for the true quotient.

Thus, 9 in 10, once and one remainder; 9 in 50, 5 times and 5 remainder; 9 in 71, 7 times and 7 + 1 remainder; 9 in 67, 6 times and 6+7: = 13 remainder, and 9 in 13, once and 4 remainder, therefore, 9 in 67, (6 + 1) times and 4 remainder.

99 in 100, once and one remainder; 99 in 700, 7 times and 7 remainder; 99 in 817, 8 times and 8 + 1725 remainder.

999 in 1000, once and one remainder; 999 in 314526, 314 times and 314526 = 840 remainder; 999 in 7869843, 7869 times and 8712 remainder; and 999 in 8712, 8 times and 8 + 712 720 remainder; therefore, 999 in 7869843, (7869 +8=) 7877 times, and 720 remainder.

7869+843=

=