Acerca de este libro
Mi biblioteca
Libros en Google Play
To determine the total number of figures which there will be in any
THE PRINCIPLES OF ARITHMETIC.
PAGE
PROP. 1-To explain the common system of Notation
121
PROP. 2.-To shew how to increase a number by an exact number of
units, tens, &c.
122
PROP. 3.-To shew how to diminish a number by an exact number of
units, tens, &c. less than ten.
123
PROP. 4.-To prove and explain the Rule for Addition of numbers
PROP. A.-To explain the Rule for Compound Addition
124
125
126
127
PROP. 5.-To prove and explain the Rule for Subtraction of numbers
PROP. 6. The product of two numbers is the same, whichever be the
multiplier
PROP. 7.-To prove that the product of one number by another is equal
to the sum of the products of each part of the multiplicand and
PROP. 8.-To prove that the product of two numbers is equal to the sum
of the products of the multiplicand by each part of the multiplier
PROP. 9.-To prove the Rule for Multiplication by a composite number
PROP. 10.-To prove the Rule for Multiplication by 10, 100, &c
PROP. 11.-To explain the Rule for Multiplication by any number
PROP. B.-To explain the Rule for Compound Multiplication
PROP. 12.-To prove that the quotient of one number by another is equal
to the sum of the quotients of parts of the dividend divided by the
divisor
PROP. 13.-To shew that the quotient of the product of two or more
numbers may be obtained by dividing one of them, and multiplying
the quotient by the rest
PROP. 14.-To explain the Rule for Division of numbers
PROP. 15. To prove that the quotient obtained by successive division by
several numbers is the same as from the division by their product;
and to prove the Rule for the formation of the total remainder
PROP. 16.-To prove the Rule for Division by any number having ciphers
on the right
PROP. 17.-Every factor of a number is a measure of the same, and every
measure is a factor
PROP. 18.-If one number measure another, the factors of the first are
factors also of the second
PROP. 19.-If one number measure another, the first measures also every
multiple of the second
PROP. 20.-If one number measure each of two others, it measures also
their sum or difference
135
PROP. 21. To prove that the G.C.M. of several numbers is the product
of all the common prime factors
PROP. 22. To prove and explain the Rule for finding the G.C.M. of two
numbers, when their prime factors are not easily obtainable
PROP. 23.-To shew that the G.C.M. of several numbers may be obtained
by finding first the G.C.M. of two, then of this and a third, next
of this last and a fourth, and so on; the last so obtained being the
G.C.M. of the whole
138
PROP. 24.-To prove the Rule for finding the L.C.M. of several numbers 139
PROP. 25.-To shew that the L.C.M. of three or more numbers may be
obtained, by finding first the L.C.M. of two, then of this and a third;
next of this last and a fourth, and so on; the L.C.M. last found being
that required
PROP. 26.-To explain the necessity, method, and meaning of the system
of Fractional Numeration and Notation
PROP. 27.-The numerator and denominator of a fraction may be both
multiplied or divided by any the same number, without altering its
value
PROP. 28.-To prove the Rule for reducing fractions to their L.C.D.
PROP. 29.-To explain the Rule for Addition of fractions
PROP. 30.-To explain the Rule for Subtraction of fractions
PROP. 31.-To prove the Rule for Multiplication of a fraction by an
integer.
PROP. 32. To prove the Rule for division of a fraction by an integer
PROP. 33. To prove the Rule for finding the value of a compound
fraction
140
143
.
144
145
146
PROP. 34.-To explain the meaning of the Multiplication by a fraction,
and to deduce a Rule for finding the product
PROP. 35.-To explain the meaning of Division by a fraction, and to de-
duce a Rule for forming the quotient
PROP. 36.-To explain the meaning of a complex fraction, and to deduce
a Rule for its reduction to a simple fraction
152
PROP. 37. To explain the system of notation of decimal fractions
PROP. 38.-To explain the Rules for Addition and Subtraction of decimals 153
PROP. 39.-To prove the Rule for Multiplication of a Decimal by any
power of 10
PROP. 40.-To prove the Rule for Division of a Decimal by any power
of 10
*
PROP. 41. To prove the Rule for Multiplication of Decimals
PROP. 42. To prove the Rule for Division of Decimals
PROP. 43.-To shew under what circumstances a vulgar fraction is con-
vertible into a finite decimal; and that, in all cases, where the deci-
mal is infinite, the figures recur in a certain order; and to find the
extent of the recurring period
PROP. 44.-To prove the Rule for the conversion of a recurring decimal
into a vulgar fraction
PROP. 45. To shew that the ratio of one number or quantity to another
may be properly represented by the fraction, whose numerator is the
number of units in the former, and denominator the number of the
same kind of units in the latter, quantity
PROP. 46. To shew how to divide a number or quantity into parts, which
shall bear to each other a given ratio
PROP. 47. To shew that if four numbers be proportional in a given
order, the product of the extremes is equal to that of the means, and
conversely
PROP. 48.-To find a fourth proportional to three given numbers
PROP. 49. To find a third proportional to two given numbers
PROP. 50.-To find a mean proportional to two given numbers
PROP. 51.-To shew that, if the corresponding terms of any number of
proportions be multiplied together, they will still form a proportion 161
b
PROP. 52. To shew that, if one quantity vary directly as another, cor-
responding numerical values of the two quantities will form a pro-
portion, the first being to the second value of the one, as the first to
the second value of the other
161
PROP. 53. To shew that, if one quantity vary inversely as another, cor-
portion, the first being to the second value of the one, as the second
to the first value of the other
PROP. 54.-To explain the Rule for Simple Proportion
PROP. 55.-If a quantity be so connected with two sets of other quan-
tities, that its value varies directly as each of the first set, and in-
versely as each of the second, when all the rest remain unaltered;
then, if all are changed, the value will vary directly as the product of
the first set, and inversely as the product of the second
PROP. 56.-To explain the Rule for Compound Proportion
PROP. 57-To explain the Rule for finding the Simple Interest on a given
162
PROP. 58. To explain the Rules for finding any one of the quantities,
Time, Rate, Principal, when the others and the interest are known 165
PROP. 59.-To explain the Rule for finding the true discount on a sum of
money
PROP. 60.-To prove the Rule for finding the amount at Compound
Interest of a given principal; and conversely
169
PROP. 61.-To explain the Rules for the several cases of Stocks
PROP. 62.-To explain the Rules of Commission, Brokerage, and Insur-
PROP. 63.-To explain the several cases in Profit and Loss
PROP. 64.-To explain the Rules of Fellowship
PROP. 65. The product of different powers of the same number is a
power of a degree equal to the sum of the degrees of the several
powers
PROP. 66. A power of a power is another of a degree equal to the pro-
duct of the degrees of the two
PROP. 67.-The power of a product is equal to the product of the powers
of the factors
PROP. 68.-The square of a number is equal to the sum of the squares
of any two parts into which it may be divided, together with twice
the product of these parts
PROP. 69.-The cube of any number is equal to the sum of the cubes of
any two parts into which it may be divided, together with three times
the sum of the products of the square of each into the other
PROP. 70.-A power of a fraction is the fraction formed by raising the
numerator and denominator to the required power
PROP. 71.-To prove the Rule for pointing in extraction of the Square
Root
PROP. 72.-To prove the Rule for pointing in extraction of the Cube
174
PROP. 73.-To prove and explain the Rule for the extraction of the
Square Root
PROP. 74.-If the Square Root of a number contain 2 n + 1 digits, and
n+ 1 of them have been found by the ordinary Rule, the remaining
n may be found by dividing the remainder by the corresponding
trial-divisor
PROP. 75.-To prove and explain the Rule for the extraction of the Cube
Root of a number
177
PROP. 76. If the cube root of a number contain 2 n +2 digits, and n +2
have been found by the ordinary Rule, the remaining n may be
found by dividing the remainder by the corresponding trial-divisor. 179
PROP. 77.-In any Arithmetic Series the sum of any two terms, equi-
distant from the extremes is always the same; and, when the number
of terms is odd, twice the middle term is equal to the sum of the
extremes
PROP. 78.-To find the sum of an Arithmetic Series
PROP. 79.-To find any required term of an Arithmetic Series
180
PROP. 80.-To find any required term in a Geometric Series, from the
first term, and the common ratio
PROP. 81. To find the sum of a Geometric Series
181
PROP 82.-The powers of a number, greater than unity, increase, while
those of a number, less than unity, decrease, continually without limit 182
PROP. 83.-To find the limit of the sum of an infinite decreasing Geo- metrical Series
PROP. 84.-To prove and explain the Rule for finding the time at which
several sums due at different times may be paid together
182
183
184
PROP. 85.-To prove the Rule for finding the amount of an annuity
PROP. 86. To prove the Rule for finding the present value of an annuity 184
PROP. 87.-To prove the Rule for finding the value of a deferred annuity 185
PROP. 88.-To prove the Rule for finding the annuity, which can be pur-
chased for a given sum
185
PROP. 89.-To explain the methods of working questions in Exchanges 185
PROP. 90.-To explain the Rules for questions in Barter
PROP. 91.-To explain the Rule of Alligation
PROP. 92. To explain the method of passing from one scale of Notation
to another
PROP. 93.-The number of superficial units in the area of a rectangle is
the product of the numbers of lineal units of the same kind in the
length and breadth
PROP. 94.-The number of solid units in a rectangular parallelopiped is
length, breadth, and thickness
190
APPENDIX.
1-To explain the Rules of Reduction
192
2-To prove the Rule for conversion of shillings, pence, and farthings,
into decimals of a pound
Note to Prop. 41.-To explain the abbreviated method of Multiplication
of decimals
Note to Prop. 42.-To explain the abbreviated method of Division of
decimals