Circular or Angular Measuresare used in surveying, navigation, astronomy, geography, reckoning latitude and longitude, and computing differences in time.
ACircleis a plane figure bounded by a curved line, every point of which is equally distant from a point within, called the center.
TheCircumferenceis the bounding line of a circle.
TheRadiusof a circle is a straight line drawn from the circumference to the center.
TheDiameteris a straight line drawn through the center, with the ends terminating in the circumference.
AnArcof a circle is any portion of the circumference.
AnAngleis the difference in direction between two straight lines which meet.
If two diameters divide a circle into four equal parts, these diameters makeright angleswith each other.
An angle less than a right angle is anacute angle.
The circumference of a circle may be divided into 360 equal parts, calleddegrees. If the circle is large, the degree is large, and if the circle is small, the degree is small, but the degree is always1⁄360part of the circumference, whatever the size of the circle.
An angle at the center of a circle is measured by the arc which bounds it.
If the angle is a right angle, it is measured by1⁄4of 360 degrees, or 90 degrees; hence, any angle of 90 degrees is a right angle.
An acute angle is always less than 90 degrees.
An obtuse angle is always more than 90 degrees.
Table of Circular Measure
Aquadrantis1⁄4of a circumference, or 90°; asextantis1⁄6of a circumference, or 60°.
The length of a degree of longitude on the earth’s surface at the Equator is 69.16 miles.
In astronomical calculation 30° are called asign, and there are therefore 12 signs in a circle.
The earth’s circumference (which has the form of a circle) at the equator is (3.1416 × 7926), 24900 miles; which divided by 360, gives 69.17 miles for 1 degree of longitude at the equator. Leaving the equator, degrees of longitude gradually diminish, since all meridians converge at the poles. Thus, 1 degree of longitude, at 10 degrees of latitude, is 68.1 miles; at 20 degrees 65 miles; at 30 degrees 59.9 miles; at 40 degrees 53 miles; at 50 degrees 44.5 miles; at 60 degrees 34.6 miles; at 70 degrees 23.7 miles; at 80 degrees 12 miles; at 90 degrees 0.
Imaginary lines running north and south, through these degrees, from pole to pole, are calledmeridians. Those east and west,parallels.
One meridian which runs through Greenwich, near London, England, is called theprime meridian, and all the other meridians are reckoned as east or west of it.
Longitudeis distance east or west of the prime meridian. When we say that the longitude of[865]Paris is 2° 20′ East, we mean that the meridian running through Paris is 2° 20′ east of the prime meridian that runs through Greenwich, England. The longitude of Washington, D. C., is 77° 7′ West. That means that the meridian which passes through Washington is 77° 7′ west of the prime meridian.
The longitude of a place tells in degrees, minutes, and seconds, the distance it is east or west of the prime meridian.
Rule.—To find the difference of time between two places, when the difference of longitude is known, or vice versa, multiply the given longitude, expressed in degrees, by 4. This gives the equivalent time in minutes. Dividing the given time, expressed in minutes, by 4, gives the equivalent longitude in degrees.
Example: The difference of longitude between Boston and San Francisco is nearly 511⁄4°, what is the difference of time?
511⁄4× 4 = 205. 205 minutes equal 3 hours and 25 minutes.Ans.3 hours and 25 minutes.
The difference of time between London and New York is about 4 hours and 551⁄2minutes, what is the difference of longitude?
4 hours and 551⁄2minutes equals 2951⁄2minutes. 2951⁄2÷ 4 = 737⁄8.Ans.737⁄8°.
The unit of time measurement is the same among all nations. Practically it is 1-86400 of the mean solar day, but really it is a perfectly arbitrary unit, as the length of the mean solar day is not constant for any two periods of time. There is no constant natural unit of time.
Time measures are used in telling the time of day, in problems in longitude and time, in figuring interest on notes and bills, and in numerous other ways.
TABLE OF THE DIVISIONS OF TIME
Centennial years exactly divisible by 400, and other years exactly divisible by 4, areLeap Years.
The average time it takes the earth to revolve once around the sun (oneyear) is 365 days, 5 hours, 48 minutes, 47.8 seconds, or about 3651⁄4days.
The change in the length of the mean sidereal day,i.e., of the time of the earth’s rotation upon its axis, amounts to 0.01252 seconds in 2400 mean solar years.
Instead of reckoning this part of a day each year, it is disregarded, and an addition is made when this amounts to one day, which is very nearly every fourth year. This addition of one day is made to the month of February. Since the part of a day that is disregarded when 365 days are considered as a year, is a little less than one quarter of a day, the addition of one day every fourth year is a little too much, and, to correct this excess, addition is made to only every fourth centennial year.
By this is meant time which differs from Greenwich mean time by whole hours.
The earth revolves on its axis from west to east, nearly 17.3 miles in 1 minute at the equator; at the latitude of New Orleans, nearly 15 miles in 1 minute; at Memphis, 14 miles; at Chicago, 13 miles; at London, 10.8 miles; at St. Petersburg, 8.6 miles. That is, a watch would gain one minute going west, or lose one minute going east that distance, in the latitudes of the respective cities.
The globe is divided into zones of 15 degrees or one hour breadth, the Greenwich meridian being in the center of the zero zone. Thus Belgium and Holland (since 1892) keep Greenwich time; Denmark, Sweden, Switzerland (1894), Austrian railroads, Germany, and Italy (1893) keep the time of longitude 15 degrees East—i.e. one hour earlier than Greenwich. In North America again five zones are distinguished. The corresponding times are distinguished as Eastern (671⁄2to 821⁄2degrees), Central (821⁄2to 971⁄2degrees), Mountain (971⁄2to 1121⁄2degrees), and Pacific (1121⁄2to 1271⁄2degrees) times.
New York people are in the Eastern Time Belt. If they rise at six o’clock in the morning,[866]they will be up a whole hour before Chicago people, who get up at the same hour.
The clock at Greenwich, near London, England, from which the standard time of the world is reckoned.
The clock at Greenwich, near London, England, from which the standard time of the world is reckoned.
Thus, each day begins an hour sooner in New York than in Chicago, two hours before Denver, and three hours before San Francisco.
Standard time in Japan is nine hours earlier than Greenwich time.
In the western parts of Canada the twenty-four hour system has been adopted, under which four P. M. becomes sixteen o’clock and so on. Steps are being taken to introduce it generally in India, Belgium, and the United States. It is of special convenience in the construction of railroad time tables; and it has long been used by the Italians and by astronomers.
SIMULTANEOUS TIME IN LONDON, NEW YORK, AND BOMBAYThis diagram illustrates the curious fact that a telegram despatched from London may be delivered in New York apparently before the time it was sent off, and why a telegram apparently takes so long to reach Bombay.
SIMULTANEOUS TIME IN LONDON, NEW YORK, AND BOMBAY
This diagram illustrates the curious fact that a telegram despatched from London may be delivered in New York apparently before the time it was sent off, and why a telegram apparently takes so long to reach Bombay.
COMPARATIVE TIME ALL OVER THE WORLD WHEN NOON AT GREENWICH
COMPARATIVE TIME ALL OVER THE WORLD WHEN NOON AT GREENWICH
The reckoning of time among the ancients was very inaccurate. This was owning to their ignorance of astronomy, and also to changes that were made from time to time for political reasons. The calendar was reformed by Julius Cæsar, 46 B. C., who made the year consist of 3651⁄4days, adding one day every fourth year. In 1582, the error in the calendar established by him had increased to 10 days; that is, too much time had been reckoned as a year, until the civil year was 10 days behind the solar year. To correct this error, Pope Gregory XIII. decreed that 10 days should be stricken from the calendar, that the day following the 3d day of October, 1582, should be made the 14th, and that henceforth only those centennial years should be leap years which are divisible by 400.
Most Catholic countries adopted the Gregorian Calendar soon after it was established. Great Britain did not adopt it until 1752, when the error amounted to 11 days. By Act of Parliament, the 3d of September was called the[867]14th. The civil year by the same act was made to commence on the 1st of January, instead of the 25th of March, as was previously the case.
THE COMPARATIVE TIME ZONES OF THE WORLD
THE COMPARATIVE TIME ZONES OF THE WORLD
Dates reckoned by the Julian calendar are called Old Style (O.S.), and those reckoned by the Gregorian calendar are called New Style (N.S.). The difference now amounts to 12 days.
To find the day of the week for any given date.
1. Take the last two figures of the year, add one-fourth of them (neglecting remainder). Thus: 1949 = 49 + 12 = 61.
2. Add for the month, if for January or October, 1; May, 2; August, 3; February, March, or November, 4; June, 5; September or December, 6; April or July, 0; if leap year (that is, if it be divisible by 4 without remainder) January 0; February 3.
3. Add day of month.
Divide the sum of these three by 7, and remainder gives the number of the day of the week.
Thus:—What day of the week is 15th July, 1908?
4th day of the week = Wednesday.
What day of the week was December 25th, 1905?
2nd day of the week = Monday.
The above only applies to 20th Century. For 19th Century, add 2; for 21st Century, add 6; 18th Century, 4; but before 1752 the “old style” was used.
The day begins earlier as you go east until you meet the 180th meridian. This is where the day begins. Starting here, it travels westward, giving the whole world a new day. The 180th meridian is called theInternational Date Line(I. D. L.) but in reality, the date line is a crooked line which zigzags across the 180th meridian.
From the time the day starts at the International Date Line, until the sun again reaches that line, the same day is in progress the world over.
As marked now, the International Date Line passes southward through Behring Sea, then westerly, then returns to the 180th meridian at about 40 degrees north. It then follows the 180th meridian to 10 degrees south, where it swerves east but returns again to the 180th meridian at about 50 degrees south. It then follows that meridian.
International Data Line
Time on Shipboard.—The twenty-four hours are divided on board ship into seven parts, and the crew is divided into two parts or watches, designated port and starboard watches. Each watch is on duty four hours, except from four to eight p. m., which time is divided into two watches of two hours each, called dog watches, by means of which the watches are changed every day, and each watch gets a term of eight hours’ rest at night. First watch, eight p. m. to midnight; middle watch, midnight to four a. m.; morning watch, four to eight a. m.; forenoon watch, eight a. m. to noon; afternoon watch, noon to four p. m.; first dog watch, four to six p. m.; second dog watch, six to eight p. m. The bell is struck every half-hour to indicate the time, as follows:
January, from Janus, was the sacred month of the year to the Romans. To them, Janus was the god of the year. During the 18th century, the Europeans started to recognize it as the first month, but previous to this, March was considered the first.
February comes from februa, the name of a Roman festival celebrated on the 15th of the second month.
March is from Mars, the god of war. March was the first month of the year to the Romans.
April, from the Latinaperire, “to open,” was probably so called because during this month buds begin to open.
May is from Maia, the mother of Mercury. The Romans offered sacrifices to this goddess on the first day of May.
The sixth month in our calendar, June, got its name from Juno, the wife of Jupiter.
July was so named in honor of Julius Cæsar, who was born in this month.
Emperor Augustus Cæsar commanded that the eighth month be named August after him.
September is from the Latin septem, meaning seven. At the time when March was the first month of the year, September was the seventh.
October, November, and December were originally the eighth, ninth and tenth months.Octo,novem, anddecemare Latin numerals for eighth, ninth, and tenth.
Sunday (that is, day of the sun, like Monday day of the moon), the first day of the week, the Lord’s day, was sacred to Sol or the Sun.
Monday (that is, moon-day; Anglo-Saxon,Monandæg, German,Montag), the second day of our week, was formerly sacred to the moon.
Tuesday, the third day of the week, is so called fromTiwesdæg, the day of Tiw or Tiu, the old Saxon name for the god of war. The day bears a corresponding name in the other Germanic dialects.
Wednesday, the fourth day of the week, theDies Mercuriiof the Romans, theMittwochof the modern Germans. The name Wednesday is derived from the Northern mythology, and signifies Woden’s or Odin’s day. The Anglo-Saxon form wasWôdanesday, the Old GermanWoutanestac. The Swedish and Danish isOnsdag.
Thursday, (SwedishThorsdag, GermanDonnerstag), the fifth day of the week, is so called from Donar, or Thor (seeDictionary of Myths), who, as god of the air, had much in common with the Roman Jupiter, to whom the same day was dedicated. (LatinJovis dies, FrenchJeudi).
Friday, the sixth day of the week, from the Anglo-SaxonFrige-dæg, is the day sacred toFriggaor toFreya, the Saxon Venus.
Saturday (Anglo-SaxonSæterdæg,Sæterndæg—Sæter,Sætern, for Saturn, anddæg, a day—the day presided over by the planet Saturn), is the seventh or last day of the week; the day of the Jewish Sabbath.
The common measure of value isMoney.
It is also called Currency, and is of two kinds, viz.: coin and paper money.
Stamped pieces of metal having a value fixed by law areCoinorSpecie.
Notes and bills issued by the government and banks, and authorized to be used as money, arePaper Money.
All moneys which, if offered, legally satisfy a debt are aLegal Tender.
The unit of United States or Federal money is the Dollar.
The dollar mark is probably a combination of U. S., the initials of the words “United States.”
The standard of United States money is the gold dollar. Gold is used because in itself it has great worth and little bulk, and because it varies very little in value.
Names of United States Coins
It may be interesting to know that the worddollaris supposed to have come fromDale, the name of a small town where dollars were first coined.
Dimeis from the French worddisme, which means tenth.
Centcomes from the Latin wordcentum, meaning hundred.
Millis also from the Latin, coming frommille, a thousand.
Eagleswere named after our national bird.
Weights of the United States CoinsAnd the Amounts for Which They are Legal Tender
Besides the coins there is paper money, founded on credit. It represents value, but in itself has no value.
This paper money is made up of paper promises to pay the amounts named, in gold or silver, on demand.
It includes bank bills, United States treasury notes, government bonds, etc. They represent the values $1, $2, $5, $10, $20, $50, $100, $500, $1,000 and $10,000.
Notation of United States Money
Dollars and cents are written together. Thus, two dollars and sixteen cents is written, $2.16.
The dollars are separated from the cents by a period. If the number of cents is less than ten, the tens’ place is filled by a 0. Thus, we write twenty dollars and two cents, $20.02.
Mills, or tenths of a cent, are written to the right of the cents. Five dollars, six cents, four mills is written, $5.064.
Note.—The rules and processes of decimals apply to the addition, subtraction, multiplication, and division of United States money.
Sterling Moneyis currency of Great Britain and Ireland.
Table of Sterling Money
The standard unit of Sterling Money is 1 pound or sovereign, whose value in our money is $4.8665.
The coins of Great Britain in general use are:—
Gold: Sovereign, half-sovereign, and guinea, which is equal to 21 shillings.
Silver: The crown (equal to 5 shillings), half-crown, florin (equal to 2 shillings), shilling, six-penny and three-penny pieces.
Copper: Penny and half-penny.
Example: I have £5 sterling. What is the value in United States money?
Solution:The value is 5 × $4.8665, or $24.33
In France the currency is decimal. The unit is theFranc.
TABLE
The value of the franc, as determined by the Secretary of the Treasury, is $.193 in United States money.
The coins of France are of gold, silver, bronze, and copper. The gold coins are thehundred,forty,twenty,ten, andfivefranc pieces; the silver coins are thefive,two, andonefranc pieces; also thefiftyandtwenty-fivecentime pieces. The bronze coins are theten,five,two, andonecentime pieces. There are also copper coins intenandfivecentime pieces.
Example: When in France, I bought goods as follows:—
3 books at 2 francs,1⁄2dozen pipes at 1 franc,2 pictures at 4 francs.
What was the cost in United States money?
Work:
German money is legal currency of the German Empire.
TABLE
1. The unit is themark. Its value is $.2385 in United States money.
2. The coins of the German Empire are of gold, silver, nickel, and copper. The gold coins are the 20-mark piece, the 10-mark piece, and the 5-mark piece. The silver coins are thetwoandonemark pieces; the nickel coins are thetenandfivepfennig pieces; and the copper coins are thetwoand onepfennigpieces.
Although a ream contains 480 sheets, 500 sheets are usually sold as a ream.
The expression “per cent,” which is an abbreviation of the Latin words “per centum,” means “for each hundred.”
The symbol % is often used to denote “per cent.” Thus, 7 per cent, or 7%, means 7 parts out of every 100 parts,i.e.,7⁄100of the whole.
Since per cent means hundredths, we may write any fraction whose denominator is 100 as so many per cent. In some cases the corresponding common fractions are so simple that it is advisable to remember them. For example:
and so on.
Thenumberper cent is called therateper cent.
Table of Additional Values
Here are a few others that should be learned:—
Write the decimal as hundredths, and the number expressing the number of hundredths is the per cent.
Examples:
If the decimal has more than two decimal places, the figures after the second one are written as a fraction of a per cent, as,—
To change a common fraction to per cent:
1.Change the fraction to a decimal.
2.Express the decimal as hundredths.
3.The result is the per cent desired.
Examples:
Or, they may be written this way:
InPercentage, there are five terms or quantities considered; namely, theBase,Rate per cent,Percentage,AmountandProceedsorDifference; any two being given, a third one may be found.
The base and rate given, to find the percentage.
Rule.—Multiply the base by the rate per cent expressed decimally.
Example: How many dollars is 6% of $50?
When the rate per cent is an aliquot part of 100, the percentage is readily found by taking such a part of the base as the rate per cent is part of 100. Thus, at 10%, take1⁄10of base; at 121⁄2%,1⁄8; at 162⁄3%,1⁄6, etc.
The base and percentage given, to find the rate.
Rule.—Divide the percentage by 1% of the base
Example: Bought a watch for $15 and sold it for $18; what per cent did I make?