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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

124

PROP. A.-To explain the Rule for Compound Addition

125

PROP. 5.—To prove and explain the Rule for Subtraction of numbers 125

PROP. 6.-The product of two numbers is the same, whichever be the

multiplier

126

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

multiplier

127

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 127

PROP. 9.-To prove the Rule for Multiplication by a composite number 127

PROP. 10.—To prove the Rule for Multiplication by 10, 100, &c

128

PROP. 11.–To explain the Rule for Multiplication by any number

128

PROP. B.—To explain the Rule for Compound Multiplication

129

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

129

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

130

PROP. 14.-To explain the Rule for Division of numbers

130

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 132

Prop. 16.–To prove the Rule for Division by any number having ciphers

on the right

134

PROP. 17.-Every factor of a number is a measure of the same, and every

measure is a factor

134

PROP. 18.-If one number measure another, the factors of the first are

factors also of the second

135

PROP. 19.-If one number measure another, the first measures also every

multiple of the second

135

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

136

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

137

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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

140

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

143

PROP. 28.—To prove the Rule for reducing fractions to their L.C.D. 144

PROP. 29.-To explain the Rule for Addition of fractions

144

PROP. 30.-To explain the Rule for Subtraction of fractions

145

PROP. 31.—To prove the Rule for Multiplication of a fraction by an

integer .

145

PROP. 32.—To prove the Rule for division of a fraction by an integer 146

PROP. 33.—To prove the Rule for finding the value of a compound

fraction

146

PROP. 34.-To explain the meaning of the Multiplication by a fraction,

and to deduce a Rule for finding the product

147

PROP. 35.—To explain the meaning of Division by a fraction, and to de-

duce a Rule for forming the quotient

149

PROP. 36.-To explain the meaning of a complex fraction, and to deduce

a Rule for its reduction to a simple fraction

150

PROP. 37.-To explain the system of notation of decimal fractions 152

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

154

PROP. 40.-To prove the Rule for Division of a Decimal by any power

of 10

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154

PROP. 41.- To prove the Rule for Multiplication of Decimals

154

PROP. 42.–To prove the Rule for Division of Decimals

155

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

155

PROP. 44.-To prove the Rule for the conversion of a recurring decimal

into a vulgar fraction

157

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

158

PROP. 46.-To shew how to divide a number or quantity into parts, which

shall bear to each other a given ratio

159

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

159

PROP. 48.—To find a fourth proportional to three given numbers

160

PROP. 49.-To find a third proportional to two given numbers

160

Prop. 50.—To find a mean proportional to two given numbers

161

PROP. 51.-To shew that, if the corresponding terms of any number of

proportions be multiplied together, they will still form á proportion 161

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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-

responding numerical values of the two quantities will form a pro-

portion, the first being to the second value of the one, as the second

to the first value of the other

162

Prop. 54.—To explain the Rule for Simple Proportion

162

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

162

Prop. 56.—To explain the Rule for Compound Proportion

163

PROP. 57—To explain the Rule for finding the Simple Interest on a given

164

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

166

PROP. 60.- To prove the Rule for finding the amount at Compound

Interest of a given principal ; and conversely

166

PROP. 61.–To explain the Rules for the several cases of Stocks

167

PROP. 62.- To explain the Rules of Commission, Brokerage, and Insur-

ance

169

PROP. 63.—To explain the several cases in Profit and Loss

PROP. 64.–To explain the Rules of Fellowship

171

PROP. 65.—The product of different powers of the same number is &

power of a degree equal to the sum of the degrees of the several

powers

17

Prop. 66.-A power of a power is another of a degree equal to the pro-

duct of the degrees of the two

171

PROP. 67.-The power of a product is equal to the product of the powers

of the factors

172

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

172

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

173

PROP. 70.-A power of a fraction is the fraction formed by raising the

numerator and denominator to the required power

173

Prop. 71.-To prove the Rule for pointing in extraction of the Square

Root

174

PROP. 72.-To prove the Rule for pointing in extraction of the Cube

Root

174

Prop. 73.-To prove and explain the Rule for the extraction of the

Square Root

175

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

177

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

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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

180

PROP. 78.--To find the sum of an Arithmetic Series

180

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

181

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

182

PROP. 84.- To prove and explain the Rule for finding the time at which

several sums due at different times may be paid together

183

PROP: 85.—To prove the Rule for finding the amount of an annuity 184

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

187

Prop. 91.—To explain the Rule of Alligation

187

PROP. 92.-To explain the method of passing from one scale of Notation

to another

187

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

189

PROP. 94.-The number of solid units in a rectangular parallelopiped is

the product of the numbers of lineal units of the same kind in the

length, breadth, and thickness

190

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