Mathematicsis the science which treats of all kinds of quantity whatever, that can be numbered, or measured.
Arithmeticis that part which treats of numbering.
Fractionstreat of broken numbers, or parts of numbers.
Algebrais the art of computing by symbols.
In this science, quantities of all kinds are represented by the letters of the alphabet.
Geometryis the science relating to measurement. By the assistance of geometry, engineers, &c., conduct all their works, take the distances of places, and the measure of inaccessible objects, &c.
Characters, marks, or signs,
which are used in arithmetic, and algebra, to denote several of the operations, and propositions:
+signifiesplus, or addition,- minus, or subtraction,× multiplication,÷ division,: :: : proportion,= equality,√ square root,∛ cube root,42denotes that4 is to be squared.43denotes that4 is to be cubed.
REDUCTION.
Reductionis the method of converting numbers from one name, or denomination to another: or the method of finding the value of a quantity in terms of some other higher, or lower quantity.
To reduce from a higher to a lower denomination.
Rule.—Multiply the given number by as many of the lower denomination as make one of the greater;[40]adding to the product as many of the lower denomination as are expressed in the given sum.
Example.—In £6 15s.5d., how many pence?
To convert from a lower to a higher denomination.
Rule.—Divide the given number by as many of the lower denomination as are required to make one of the greater.[41]Should there be any remainder, it will be of the same denomination as the dividend.
Example.—Convert 1625 pence into pounds, shillings, and pence.
THE RULE OF THREE, OR SIMPLE PROPORTION.
It is called theRule of Threebecause three numbers are given to find a fourth. It is also calledSimple Proportion, because the 1st term bears the same proportion to the 2nd, as the 3rd does to the 4th. Of the three given numbers, two of them are always of the same kind, or name, and are to be the 1st, and 2nd terms of the question; the 3rd number is always of the same name, or kind as the 4th, or answer sought; and in stating the question it is always to be made the 3rd term. If the answer will begreaterthan the 3rd term, place the least of the other two given quantities for the 1st term; but if the answer will belessthan the 3rd term, put the greater of the two numbers, or quantities, for the 1st term.
Rule.—State the question according to the above directions, and multiply the 2nd and 3rd terms together, and divide this product by the 1st, for the 4th term, or answer sought.
If the 1st and 2nd terms are not of the same denomination, they must be reduced to it; and if the third term is a compound number, it must be reduced to its lowest denomination before the multiplication, or division of the term takes place.
Note 1.—The operation may frequently be considerably abridged, by dividing the 1st and 2nd, or the 1st and 3rd terms, by any number which will exactly divide them, afterwards using the quotients, instead of the numbers themselves.*
Example.—If 2 tons of iron for ordnance cost £40, how many tons may be bought for £360?
As £40 : £360 :: 2 tons : 18 tons.(Thus 360 × 2) ÷ 40 = 18. The answer.* Or thus, 9 × 2 = 18. The answer.
Note 2.—A concise method of ascertaining the annual amount of a daily sum of money.
Rule.—Bring the daily sum into pence, and then add together as many pounds, half pounds, groats, and pence, as there are pence inthe daily sum, for the amount required. For leap year, add the rate for one day.
Example.—Required the annual amount of 2s.6d.per diem.
Note 3.—To find the amount of any number of days’ pay, the daily rate (under twenty shillings) being given.
The price of any article being given, the value of any number may be ascertained in a similar manner.
Rule 1.When the rate (or price) is an even number, multiply the given number by half of the rate, doubling the first figure to the right hand for the shillings, the remainder of the product will be pounds.
Example.Required the amount of 243 days’ pay, at 4s.per diem.
Rule 2.When the price is an odd number, find for the greatest number as before, to which add one-twentieth of the given number for the odd shilling.
Example.What is the price of 566 pairs of shoes, at 7s.per pair.
A fractionis a quantity which expresses a part, or parts of a unit, or integer. It is denoted by two numbers placed with a line between them.
A Simple fractionconsists of two numbers, called the numerator, and denominator; thus,
The Denominatoris placed below the numerator, and expresses the number of equal parts into which the integer is divided.
The Numeratorexpresses the number of parts of the broken unit, or integer; or shows how many of the parts of the unit are expressed by the fraction.
A Compound fractionis a fraction of a fraction, as ⅔ of ⅘.
A Mixed numberconsists of a whole number with a fraction annexed to it, as 4⅖.
An Improper fractionhas the numerator greater than the denominator, as6/5.
REDUCTION OF FRACTIONS.
is bringing them from one denomination to another.
To reduce a fraction to its lowest terms.
Rule.—Divide the numerator, and the denominator, by any number that exactly divides them, and the quotients by any other number, till they can be no longer divided by any whole number, when the fraction will be in its lowest terms.
Example.—Reduce40326048to its lowest terms.
Thus,440326048=1210081512=684126=71421=23.Answer.
To reduce an improper fraction to a whole, or mixed number.
Rule.—Divide the numerator by the denominator, the quotient will be the whole number; and the remainder (if any) the numerator of the fraction, having the divisor for the denominator.
Example.—Reduce11412to a whole, or mixed number.
To reduce a mixed number to an improper fraction.
Rule.—Multiply the whole number by the denominator, and add the numerator to the product, under which place the given denominator.
Example.—Reduce 17⅝ to an improper fraction.
To reduce a compound fraction to a simple fraction.
Rule.—Multiply all the numerators together for the numerator, and all the denominators for the denominator.
Example.—Reduce ⅜ of ⅙ of ½ of 9 to a simple fraction.
To reduce fractions of different denominators to equivalent fractions, having a common denominator.
Rule.—Multiply each numerator by all the denominators except its own for the new numerators, and multiply all the denominators together for a common denominator.[42]
Example.—Reduce ⅜, ⅔, and ⅘ to fractions having a common denominator.
3 × 3 × 5 = 452 × 8 × 4 = 804 × 8 × 3 = 968 × 3 × 5 = 120Answer,45120,80120,96120.
ADDITION OF FRACTIONS.
Rule.—Bring compound fractions to simple fractions; reduce all the fractions to a common denominator, then add all the numerators together, and place their sum over the common denominator. When mixed numbers are given, find the sum of the fractions, to which add the whole numbers.
Example.—Add together ⅚, ¾, and 6½.
5 × 4 × 2 = 404048+3648+2448+ 6 = 8448.3 × 6 × 2 = 361 × 6 × 4 = 24 or, by cancelling, and dividing,[43]6 × 4 × 2 = 481012+912+612+ 6 = 8112Answer.
SUBTRACTION OF FRACTIONS.
Rule.—Prepare the quantities, as in addition of fractions. Place the less quantity under the greater. Then, if possible, subtract the lower numerator from the upper; under the remainder write the common denominator, and, if there be whole numbers, find their difference as in simple subtraction. But if the lower numerator exceed the upper, subtract it from the common denominator, and to the remainder add the upper numerator; write the common denominator under this sum, and carry 1 to the whole number in the lower line.
Example.—
From 5456or 542530Take 25515or 251030——291530Answer.
MULTIPLICATION OF FRACTIONS.
Rule.—Reduce mixed numbers to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator, which will give the product required.
Example.—Multiply ⅚, ⅜, and 2½ together.
⅚ × ⅜ × (2½ or)52=7596Answer.
DIVISION OF FRACTIONS.
Rule.—Prepare the fractions, as for multiplication; then divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide; but if they will not do so, then invert theterms of the divisor, and multiply the dividend by it, as in multiplication.
Example.—Divide916by 4½.
916÷ (4½ or)92= ⅛Answer.
RULE OF THREE IN FRACTIONS.
Rule.—State the terms, as directed in “Simple proportion;” reduce them (if necessary) to improper, or simple fractions, and thetwo firstto the same denomination. Then multiply together the second and third terms, and the first with its parts inverted, as in division, for the answer.
Example.—If 4⅕ cwt. of sugar cost £19⅞, how much may be bought for £59⅝?
As 19⅞ : 59⅝ :: 4⅕Or,1598:4778::215: 12⅗Answer.8159×4778×215=801366360= 12⅗ cwt.
A decimal fractionis that which has for its denominator an unit (1), with as many ciphers annexed as the numerator has places; and it is usually expressed by setting down the numerator only, with a point before it, on the left hand. Thus,510is ·5;25100is ·25;251000is ·025; ciphers beingprefixed, to make up as many places as are required by the ciphers in the denominator.
A mixed numberis made up of a whole number with some decimal fraction, the one being separated from the other by a point, thus 3·25 is the same as 325100or325100.
Ciphers on the right hand of decimalsmake no alteration in their value; for ·5, ·50, ·500 are decimals having all the same value, each being =510. But when they are placed on the left hand, they decrease the value in a tenfold proportion; thus, ·5 is510; but ·05 is5100.
ADDITION OF DECIMALS.
Rule.—Set the numbers under each other, according to the value of their places, in which state the decimal separating points will all stand exactly under each other. Then beginning at the right hand, add up all the columns of numbers as in integers, and point off as many places for decimals as are in the greatest number of decimal places in any of the lines that are added; or place the point directly below all the other points.
Example.—Required the sum of 29·0146, 3146·5, 14·16, and 165.
SUBTRACTION OF DECIMALS.
Rule.—Place the numbers under each other according to the value of their places. Then, beginning at the right hand, subtract as in whole numbers, and point off the decimals, as in addition.
Example.—Subtract 4·90142 from 214·81.
MULTIPLICATION OF DECIMALS.
Rule.—Place the factors, and multiply them together, the same as if they were whole numbers. Then point off in the product just as many places of decimals as there are decimals in both the factors. But, if there be not so many figures in the product, prefix ciphers to supply the deficiency.[44]
Example.—Multiply 32·108 by 2·5.
DIVISION OF DECIMALS.
Rule.—Divide as in whole numbers, and point off in the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor. When the decimal places of the quotient are not so many as the above rule requires, the deficiency is to be supplied by prefixing ciphers. When there is a remainder after the division, or when the decimal places in the divisor are more than those in the dividend, then ciphers may be annexed to the dividend, and the quotient carried on as far as required.
Example.—Divide 234·7052 by 64·25.
REDUCTION OF DECIMALS.
To reduce a vulgar fraction to its equivalent decimal.
Rule.—Divide the numerator by the denominator, as in Division of Decimals, annexing ciphers to the numerator as far as necessary: and the quotient will be the decimal required.
Example.—Reduce724to a decimal.
To find the value of a decimal, in terms of the inferior denominations.
Rule.—Multiply the decimal by the number of parts in the next lower denomination, and cut off as many places to the right hand for a remainder, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination, again cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer; then the several denominations, separated on the left hand, will make up the answer.
Example.—What is the value of ·775 pounds sterling.
To convert integers, or decimals to equivalent decimals of higher denominations.
Rule.—Divide by the number of parts in the next higher denomination, continuing the operation to as many higher denominations as may be necessary.
When there are several numbers, all to be converted to the decimal of the highest—
Set the given numbers directly under each other for dividends, proceeding from the lowest to the highest; opposite to each dividend, on the left hand, place such a number for a divisor as will bring it to the next higher name. Begin at the uppermost, and perform all the divisions, placing the quotient of each division, as decimal parts, on the right hand of the dividend next below it; so shall the last quotient be the decimal required.
Example.—Convert 15s.9¾d.to the decimal of a pound sterling.
Example.—Convert 1 dwt. to the decimal of a pound, Troy weight.
RULE OF THREE IN DECIMALS.
Rule.—Prepare the terms, by reducing the fractions to decimals; compound numbers to decimals of the higher denominations, or integers of the lower; also the first, and second terms to the same name. Then multiply, and divide, as in the Rule of Three, in whole numbers.
Example.—If ⅜ of a yard of cloth cost £⅖, what will516of a yard cost?
DUODECIMALS.
By Duodecimals, artificers, &c., compute the content of their works.
Rule.—Set down the two dimensions to be multiplied together one under the other, so that feet may stand under feet, inches under inches, &c.
Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner multiply all the multiplicand by the inches, and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand: omitting, however, what is below parts of inches, only carrying to these the proper number of units from the lowest denominations. Or, instead of multiplying by the inches, take such part of the multiplicand as those are of a foot.
Then add the two lines together for the content required.
Example.—Multiply 14 feet 9 inches, by 4 feet 6 inches.
TROY WEIGHT.
AVOIRDUPOIS WEIGHT.
APOTHECARIES’ WEIGHT.
WEIGHTS.
To find the weight, for tonnage.
Rule for ascertaining the weight of Hay.
Measure the length and breadth of the stack; then take its height from the ground to the eaves, and add to this last one-third of the height from the eaves to the top: Multiply the length by the breadth, and the product by the height, all expressed in feet; divide the amount by 27, to find the cubic yards, which multiply by the number of stones supposed to be in a cubic yard (viz., in a stack of new hay, six stones; if the stack has stood a considerable time, eight stones; and if old hay, nine stones), and you have the weight in stones. For example, suppose a stack to be 60 feet in length, 30 in breadth, 12 in height from the ground to the eaves, and 9 (the third of which is three) from the eaves to the top; then 60 × 30 × 15 = 27000; 27000 ÷ 27 = 1000; and 1000 × 9 = 9000 stones of old hay.
LONG MEASURE.
LAND MEASURE (Length).
LAND MEASURE (Surface, or Superficial).
NAUTICAL MEASURE.
SQUARE MEASURE.
CUBIC MEASURE (Measure of solidity).
Note.—A cubic foot is equal to 2200 cylindrical inches, or 3300 spherical inches, or 6600 conical inches.
Timber.
40 feet of round, and 50 feet of hewn timber make 1Ton; 16 cubic feet make 1Footof wood; 8 feet of wood make 1Cord.
Water.
MEASURES OF CAPACITY.
FRENCH MEASURES.
INVOLUTION.
Involutionis the raising of powers from any given number, as a root.
APoweris a quantity produced by multiplying any given number, called theRoot, a certain number of times continually by itself. Thus, 2 × 2 = 4, the 2nd power, or square of 2, expressed thus, 22.
The index, or exponent of a poweris the number denoting the height, or degree of that power. Thus, 2 is the index of the 2nd power.
Powers that are to be raised, are usually denoted by placing the index above the root, or first power.
Thus 22= 4, the 2nd power of 2.
Example.—What is the 2nd power of 45?
45 × 45 = 2025Answer.
EVOLUTION.
Evolutionis the reverse of Involution, being the extracting, or finding the roots of any given powers, or numbers.
The Rootof any number, or power, is such a number as being multiplied into itself a certain number of times, will produce that power.
Thus, 2 is the square root, or 2nd root of 4, because, 22= 2 × 2 = 4; and 3 is the cube root, or third root of 27. But there are many numbers of which a proposed root can never be exactly found; by means of decimals, however, the root may be very nearly ascertained.
Any power of a given number, or root, may be found exactly by multiplying the number continually into itself.
Those roots which only approximate are calledSurd-roots; but those which can be found, quite exactly, are calledRational-roots. Thus, the square root of 3 is a surd root, but the square root of 4 is a rational root, being equal to 2; also the cube root of 8 is rational, being equal to 2, but the cube root of 9 is surd, or irrational. Roots are sometimes denoted by writing the character √ before the power with the index of the root against it. Thus, the 3rd, or cube root of 20 is expressed by ∛20. When the power is expressed by several numbers with the sign + or - between them, a line is drawn from the top of the sign over all the parts of it; thus the cube (or third) root of 45 - 12 is ∛45 - 12or thus ∛(45 - 12).
TO EXTRACT THE SQUARE ROOT.
Rule.—Divide the given number into periods of two figures each, by setting a point over the place of units, and another over the place of hundreds, and so on over every second figure, both to the left hand in integers, and right hand in decimals. Find the greatest square in the first period on the left hand, and set its root on the right hand ofthe given number, after the manner of the quotient figure in division. Subtract the square thus found from the said period, and to the remainder annex the two figures of the next following period for a dividend. Double[45]the root above-mentioned for a divisor, and find how often it is contained in the said dividend, exclusive of its right-hand figure; and set that quotient figure both in the quotient, and divisor. Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to it the next period of the given number, for a new dividend. Repeat the same process over again—viz., find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend find the next figure of the root as before; and so on through all the periods to the last.
To extract the square root of a fraction, or mixed number.
Reduce the fraction to a decimal, and extract its root.
Mixed numbers may be either reduced to improper fractions, and the root extracted; or the fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted.
Example.—To find the square root of 29506624.
TO EXTRACT THE CUBE ROOT.
Rule 1.—By trials, or by the table of roots (vide page280), take the nearest rational cube to the given number, whether it be greater, or less, and call it the assumed cube.
2.—Then (by the Rule of Three),
As the sum of the given number, and double the assumed cube, is to the sum of the assumed cube, and double the given number, so is the root of the assumed cube, to the root required, nearly.
3.—Or as the first sum,is to the difference of the given, and assumed cube,so is the assumed root,to the difference of the roots, nearly.
4.—Again, by using, in like manner, the cube of the root last found as a new assumed cube, another root will be obtained still nearer. Repeat this operation as often as necessary, using always the cube of the last-found root, for the assumed root.
Example.—To find the cube root of 21035·8.
By trials it will be foundfirst, that the root lies between 20, and 30; and,secondly, between 27, and 28. Taking, therefore, 27, its cube is 19683, which will be the assumed cube. Then by No. 2 of the Rule