Miscellaneous Questions on the Measures of Capacity. (1) Taking the weight of a cubic foot of water as 1000 oz., and the contents of a gallon as 2774 cubic inches, find the weight of 3456 gallons.

(2) Find how many gallons of water will fill a cistern 75 ft. long, 38 ft. broad, and 23 ft. 1 in. deep.

(3) How many tuns of wine will occupy the same space

as 1512 tuns of beer?

(4) How many tuns of ale will occupy the same space as 1512 tuns of wine?

(5) How many hogsheads of ale will occupy the same space as 1890 hogsheads of wine?

(6) The weight of a gallon of water is 10 lbs., what is the weight in tons of 277274 cubic feet of water?

(7) By how much does the number of gallons in 3150000 pints, exceed the number of hogsheads in 3150000 gallons of wine ?

(8) What quantity of water must I add to a pipe of wine which cost £45, that its price may be reduced to 5s. a gallon?

Time.

115. Of all the magnitudes we have to consider, time is the most easily measured. Different nations select different standards of weight, length, surface, volume, and value; but nature has provided all men alike with three natural units for the measurement of time.

(a) The period of the earth's rotation about its axis, (b) The period of the moon's revolution round the earth, and

(c) The period of the earth's revolution round the sun, termed respectively,

(a) The Day, (b) the Lunar Month, and (c) the Year. These are invariable units, as far as we know.

In all the countries of the world the day is the principal unit of time, and this common standard makes it an easy matter to compare foreign methods of computing time with our own. The chief difficulty encountered in forming a table of the units of time arises from the fact that the three natural units are incommensurable-no number of times one being an exact number of times another. We can neither express a year nor a lunar month exactly by days and parts of a day. The solar year is nearly 365

days 5 hours 48 minutes 49 seconds; and the time between the two new moons nearly 29 days.†

116. Previous to 1752 the civil year in England was that introduced by Julius Cæsar. The years that were divisible by 4 were made to consist of 366 days-other years of 365 days only; so that the average length of the civil year was 365 days 6 hours. This differed from the actual duration of the year by an excess of 11 minutes. These excesses had throughout the lapse of eighteen centuries accumulated to eleven days. In the above-named year this method was adopted in England, which Pope Gregory XIII., with a view to correct the defects of the Julian year, had established on the continent in the sixteenth century. According to this plan the leap year is omitted every hundredth year, except the four hundredth. The years 1700, 1800, 1900, &c., not being multiple of 400, have 365 days; but 2000, 2400, 2800, &c., being multiples of 400, have 366 days. It is calculated that this arrangement reduces the error to less than one day in 6000 years. The rule for finding leap year, therefore, is as follows:-if the year be divisible by 4, it is leap year, except when it is an exact number of hundreds, and then only when the number of hundreds is divisible by 4.

117. The year is divided into 12 months, called calendar months; the number of days in a calendar month varies with the month. The following doggerel will make them easily remembered :

Thirty days are in September.

April, June, and dull November:
All the rest have one and thirty
Save the month of February;
Twenty-eight are all its store,
But in Leap Year one day more.

Write out all the leap years from 1867 to 2010.

*More nearly 365-2422414 days.

† More nearly 29.5305887.

H

(1) To minutes, 276 hrs.; 3154 dys.; 7253 secs. ;

wks.

5697

(2) To seconds, 297 mins.; 385 hrs. 3 mins. 27 secs.; 516 wks. 3 dys. 11 hrs.

(3) To seconds, 7163 dys.; 415 wks. 3 dys. 21 hrs. 31 secs.; 173 wks. 19 hrs. 12 mins.

(4) To seconds, 715 years of 365 days each; a century. How many days are there :

(5) From Jan. 1, 1860, to Jan. 1, 1870, and from June 8, 1801 to Dec. 25, 1867 ?

(6) From Jan. 1, 1868, to July 12, 2010, and from Feb. 1, 1753, to Feb. 1, 1870?

How many hours :

(7) From 12 P.M. June 8, 1820, to 6 P.M. April, 8, 1880? (8) From 5 P.M. March 7th, 1760, to 7 P.M. May 7, 1860? (9) From 3 A.M. April 29, 1825, to 3 P.M. July 19, 1865? How many minutes :

(10) From 8 A.M. Dec. 11, 1767, to 11 P.M. June 24, 1890? (11) From 1 P.M. Jan. 31, 1748, to 9 a.м, Aug. 31, 1920? (12) From 12 A.M. Feb. 9, 1799, to 5 P.M. Sept. 24. 1834 ? How many seconds:

(13) From 7 P.M. Mar. 3, 1824, to 7 A.M. Dec. 30, 1880?
(14) From 9 P.M. Nov. 9, 1785, to 2 P.M. Nov. 1, 1879?
(15) From 11 A.M, Oct. 15, 1752, to 1 P.M. Mar. 9, 1900?

(1) How many more times will a clock tick in March than in April, if it tick twice in a second?

(2) How many seconds were there in the year 1867 ?

(3) How does the solar year differ from the civil year ? What is the arrangement adopted to correct the error arising from this difference?

(4) How many days will elapse from the 25th of December, 1867, to the 25th of December, 1967 ?

(5) How many hours will there be in the 5 centuries commencing January 1st, 1868 ?

(6) How many times will a pendulum vibrate in a week, which vibrates 4 times a second?

(7) Two men start at 6 A.M. to run round a circular field 3 miles in circumference, in opposite directions, at the rate respectively of 5 and 7 miles an hour; in the race they met 5 times. At what times did they meet, and how far did they run?

(8) How many times will a clock which chimes the quarters strike in the year 1868 ?

(9) How is leap year determined? Write out a table of all the leap years from 1899 to 2001.

(10) How many lunar months are there in 27 years ?

(11) In how long a time would a million of millions of money be counted, if £100 were to be counted every minute without intermission, and the year to consist of 365 dys. 5 hrs. 49 min. ?

(12) Light travels at the rate of 192,000 miles a second. How long would it be travelling from the Sun to Neptune and back, the distance being 2,800,000,000 miles?

(13) What is the average length of the civil year? If the solar year were 365 dys. 5 hrs. 48 min. 49 secs., what would the error caused by the present arrangement amount to in 1000 years?

H 2

Common Measures.

118. Any number which will divide another without remainder is called a measure of that number.

119. Any number which will divide each of two others without remainder is a common measure of these numbers, and the greatest of the common measures of two or more numbers is termed their greatest common measure.

120. If two numbers be used as divisor and dividend, every common measure of the numbers is a measure of the remainder,

121. On this fact is based the rule for finding the G. C. M. of two numbers.

EXAMPLE.-Find the G. C. M. of 504 and 4194.

4032

162)504(3
486

Divide 4194 by 504. The remainder is 162. 504)4194(8 By the statements above, the G. C. M. of 504 and 4194 also measures 162. Divide 504 by 162 and we have remainder 18. Applying the same reasoning as before, the G. C. M. of 162 and 504, that is of 504 and 4194, is also the G. C. M. of 18 and 162. But 18 measures 162, and it is the greatest measure of itself, therefore 18 is the G. C. M. of 504 and 4194.

18)162(9 162

If there be

122. RULE.-Divide the greater by the less. no remainder the less of the two numbers is the G. C. M. If there be a remainder divide the first divisor by it. Afterwards divide the second divisor by the second remainder, and so on until there is no remainder. The last divisor is the greatest common measure of the two original numbers.

If the last divisor be 1, the numbers have no common

measure.

The

123. Often, however, the G. C. M. of two or more numbers can be found by resolving them into factors. G. C. M. is the product of the common factors.

EXAMPLE. Find the G. C. M. of 78 and 42.

78=13×3×2 and 42=7×3×2, therefore the G. C. M =3×2.