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We take 1, the first term of the continued fraction, for the 1st approximate value. Reducing the first two partial fractions, we have y for the 2d approximate value. Continuing the reduction, we obtain and for the remaining values. Examining the last two reductions, we find that the 3d approximate fraction is obtained by multiplying the terms of the 2d approximate fraction by the denominator of the 3d partial fraction, and adding to these products the corresponding terms of the first approximate fraction. We see also that the 4th approximate value is equal to the product of the terms of the third approximate value by the denominator of the 4th partial fraction, plus the corresponding terms of the 2d approximate fraction. The value obtained by using the last partial fraction is the exact value of the fraction. Hence we derive the following
Rule.-1. For the first approximate value, take the first partial fraction.
II. For the second value, reduce the complex fraction formed by the first two partial fractions.
III. For each succeeding approximate value, multiply both terms of the approximation last obtained by the denominator of the following partial fraction, and add to the products the corresponding terms of the preceding approximation.
IV. The last value thus obtained will be the common fraction required.
3. Reduce the continued fraction }11_ to a common fraction.
Ans. 13507 4. Reduce the continued fraction
to a common fraction.
Ans. 13024. 5. Find the approximate values of 1
tion & FITA
es of tits FTSTE
Ans. 11, 10, 11, m. 6. Find the approximate values of the fraction 2016
Ans. I, I, 29, 121 NOTE.—For a fuller discussion of the subject see Brooks's Philosophy of Arithmetic.
SECTION VI. DENOMINATE NUMBERS. 337. A Denominate Number is a concrete number in which the unit is a measure; as, 3 feet, 4 pounds, etc.
338. A Measure is a unit by which quantity of magnitude or continuous quantity is estimated numerically; as, a yard, a pound, etc.
339. A Compound Number is a number which expresses several different units of the same kind of quantity; as, 4 yd. 8 ft. 6 in.
340. The Terms of a compound number are the numbers of its different units. Thus, in £12 10 s. 8d, the terms are £12, and 16 s., and 8 d.
341. Similar Compound Numbers are compound numbers which express the same kind of quantity.
342. Denominate Numbers may be embraced under four distinct classes: Value, Weight, Extension, and Time.
343. Some of these classes contain several subdivisions of so much importance that the following is regarded as the most convenient classification : 1. Value.
5. Volume. 2. Weight.
6. Capacity. 3. Length.
7. Time. 4. Surface.
8. Angles. ORIGIN.—There are two kinds of quantity; quantity of multitude and quantity of magnitude. Quantity of multitude exists in individual things, and is immediately expressed in numbers. Quantity of magnitude is a mass, and can only be expressed numerically by fixing upon a unit of measure and finding how many times the quantity considered contains the unit of measure. Those who prefer may use the terms discrete and continuous, for multitude and magnitude.
344. A Denominate Number may also be defined to be a numerical expression of quantity of magnitude, or of continuous quantity.
MEASURES OF VALUE. 345. The Value of anything is its worth, or that property which makes it useful or estimable.
346. Value depends principally upon utility and difficulty of attainment. Value is usually estimated in something called Money.
347. Money is the measure or representative of the value of things. It is of two kinds, coin and paper money.
348. Coin, or Specie, is metal prepared and authorized by government to be used as money. The metals generally used are gold, silver, copper, and nickel.
349. Paper Money consists of printed promises to pay the bearer a certain amount, duly authorized to circulate as money.
350. Currency (from curro, I run,) is that wbich circulates as money. It is of two kinds, specie currency and paper currency.
351. Legal Tender is a term applied to money which is required by law to be accepted in payment of debts.
352. An Alloy is a baser metal compounded with either gold or silver for the purpose of rendering it harder and more durable. In coinage the alloy is considered as having no value.
UNITED STATES MONEY. 353. United States Money is the legal currency of the United States.
10 mills (m.) . . equal 1 cent . ct.
« 1 dime. d.
" 1 dollar. $. 10 dollars
“ 1 eagle . E. SCALE.- Ascending and descending uniform by 10. I. NAME.- United States money is so called because it is the money of the United States. It is called Federal Money because it was the money of the Federal Union. It was adopted by Act of Congress, Aug. 8, 1786.
II. TERMS.-The term dollar is from Dale or Daleburg, a town where it was first coined ; or thal, a dale or valley; or from the Anglo-Saxon dael, a portion, it being a portion of a ducat. Dime is from the French disme, meaning a tenth ; cent is from the Latin centum, a hundred ; mill is from the Latin mille, a thousand ; eagle is from the name of the national bird. The cent was proposed by Robert Morris, and named by Thomas Jefferson.
III. SYMBOLS.-There are several theories for the origin of the dollar mark:
1st. That it is a combination of U. S., the initials of United States.
2d. That it is a modification of the figure 8, the dollar being formerly called a piece of eight, and designated by the symbol g.
3d. That it is derived from a representation of the “ Pillars of Hercules," consisting of two pillars connected with a scroll. The old Spanish coins containing this were called " pillar dollars."
4th. That it is a combination of HS. the mark of the Roman money unit.
5th. That it is a combination of P. and S. from the Spanish peso duro, signifying hard dollar. In Spanish accounts peso is contracted by writing the S over the P, and placing it after the sum.
IV. Unit.—The unit is the gold dollar. The currency is founded upon the decimal system, dimes, cents, and mills being written as decimals. This gives great simplicity to the operations.
V, COINS.-.The coins are of gold, silver, nickel, and bronze. The gold coins are the double eagle,
le eagle, eagle. half-eagle. quarter-eagle. three dollars, and one dollar. The silver coins are the trade dollar, half-dollar, quarter-dollar, tinenty-cent piece, and dime. The nickel coins are the three-cent and five-cent pieces. The bronze coin is the cent. The silver half-dime and three-cent piece, the bronze two-cent piece, the nickel cent, and the old copper cent and half cent, although still seen in circulation, are no longer coined. The mill. has never been a coin ; it is merely a convenient name for the tenth part of a cent.
VI. COMPOSITION.-The gold and silver coins consist of 9 parts of pure metal and 1 part alloy. The alloy of the silver coin consists of pure copper ; the alloy of the gold consists of silver and copper, the silver not to exceed to of the alloy. The nickel coins contain 1 nickel and copper. The bronze coins consist of 95 parts copper and 5 parts tin and zinc.
VII. WEIGHT.-The gold dollar weighs 25.8 gr., and the other gold coins proportionally; the silver trade dollar, 420 gr., not intended for circulation in the United States, but for convenience of commerce, especially with China and Japan; the old silver dollar, weighing 4125 gr, is no longer coined ; the half-dollar weighs 192 gr.; the quarter-dollar, 96 gr. ; the twenty-cent piece, 77.16 gr.; the dime, 38% gr.; the nickel 3-cent piece, 30 gr.; the 5-cent piece, 77 gr.; the cent varies for different coinages.
VIII. LEGAL TENDER.--Gold coins are a legal tender for any amount ; silver coins, of the present coinage, for any amount not exceeding $5 in any one payment ; bronze and nickel coins for any amount not exceeding 25 cents in any one payment.
STATE CURRENCIES. 354. Previous to the establishment of the decimal currency, we employed the currency of England, that is, pounds, shillings, and pence. Some of the States still use shillings and pence, though not with the same values..
355. This difference of value was caused by the difference of depreciation of colonial currency in different States when the decimal system was adopted (1786).
NOTE.-In New York currency, used in New York, Michigan, Ohio, and North Carolina, $1=8 s.; hence is.=124. In Pennsylvania currency, used in Pennsylvania, New Jersey, Delaware, and Maryland, $1=7 8. 6 d., and 1 s. =1314. In New England, Virginia, Kentucky, Tennessee, Texas, Mississippi, Indiana, Illinois, $1=6 s., and '1s.=163¢.' In Georgia and South Carolina, $1=4 s. 8 d., and 18.=2134.
ENGLISH, OR STERLING MONEY. 356. English, or Sterling Money, is the legal currency of England.
TABLE. 4 farthings (far. or qr.) = 1 penny . . d.
12 pence . . = 1 shilling . . S. # 20 shillings . = pound or sovereign, £ 21 shillings . . = l guinea . .
1 = SCALE.- Ascending, 4, 12, 20; descending, 20, 12, 4. I. NAME.—The term Sterling is supposed to be derived from Easterling, the name given to early German traders, who came from the east to England. Their money was called Easterling Money, which was contracted into Sterling Money.
II. TERMS.—The term furthing is a modification of " four things,” the old English penny being marked with a cross so deeply impressed that it could be broken into two or four pieces, called respectively half-penny and four things. The pound, as a measure of value, was derived from the pound as a measure of weight, 240 pence formerly weighing a pound. The guinea is so called because it was first made of gold brought from Guinea.
III. SYMBOLS.—The symbols £s. d. gr. are the initials of the Latin words, libra, solidus, denarius, and quadrans, signifying respectively, pound, shilling, penny, and quarter. The old /, the original abbreviation for shillings, was formerly written between shillings and pence. The s has since been changed into /; thus, 78. 6d. are sometimes written 76.
IV. UNIT.-The unit is the pound, represented by the sovereign and £1 bank note. Its value by late act of Congress is fixed at $4.8665.
V. COINS.—The coins are of three classes ; gold, silver, and copper. The goll coins are the sovereign (=£1), and half sovereign (=10s.), guinea (21 s.), and half guinea (10 s. 6 d.). The silver coins are the crown (=5 s.), the half-crown (=2 s. 6 d.), the florin (=2 s.), the shilling, and the sixpenny, four-penny, and three-penny pieces. The copper coins are the penny, half-penny, and farthing
The pound is not a coin ; it is represented by the sovereign and £1 banknote. The guinea (=21 s.) and half-guinea (10 s. 6d.) are old gold coins no longer coined, though some of them are still in circulation. The crown and half-crown, also, although still in circulation, are no longer coined.
VI. COMPOSITION.—The standard for gold coins is 22 carats fine, that is, 11 parts pure gold and 1 part alloy. The standard for silver is 37 parts pure silver and 3 parts alloy ; hence the silver coins are 37 pure, and is copper. Pence and half-pence are made of pure copper.
VII. WEIGHT.-The sovereign weighs 123.274 grains; the shilling weighs 87.27 grains; the penny weighs 240 gr., or } oz. Troy.