29 1950 1954 1959 1963 1968 1972
1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 0 11 2
31 2042 2046 2051 2056 2061
2065 2070 2075 2080 2084 0
2818 2825 2831 2838 2844 2851 2858 2864 2871 2877 1 1
2884 2891 2897 2904 2911 2917 2924 2931 2938 2944
2992 2999 3006 3013
3062
3069 3076 3083
3112 3119 3126 3133 3141 3148 3155 1 1
Page 86, column 1, line 38, for (x+a)3 read (x+a)®
Page 87, column 2, line 27, for a read "
Page 88, column 2, line 23 of note, for a read «
line 24 ditto, for (1+ z)m+a read (1 + 2)m+a
line 27 ditto, for (1+2)m+a read (1+ 2)m+a
Page 94, column 1, line 2, dele comma at the end of the line.
line 5, for (a+n)m read (a+x)m
Page 96, column 2, line 22, for [292] read [297].
INDEX OF THE SUBJECTS TREATED OF UNDER
EACH ARTICLE.
Notation and Definitions, p. 1.
ART. 1. Use of numbers and their refer-
ence to a certain unit. 2. Quantity; dif-
ference between the subjects of arithmetic
and algebra; notation. 3. Examples of the
manner of expressing algebraically unknown
and indefinite quantities. 4. Addition; the
positive sign; a positive quantity. 5.
Subtraction; the negative sign—; a nega-
tive quantity. 6. When no sign is pre-
fixed, the quantity is positive. 7. Multi-
plication-product, factors, multiplier, mul-
tiplicand. 8. Exponential notation-am,
or the mth power of a. 9. Numerical coeffi-
cient. 10. Division-dividend, divisor,
quotient. 11. Square root, cube root, mth
root. 12, 13. Terms of an algebraical ex-
pression-vinculum. 14. Sign of equality.
Of Addition and Subtraction, p. 4.
Art. 15. Like and unlike quantities-
addition and subtraction of each, when
consisting of one term only. 16. A nega
tive result, how explained. 17. Addition
and subtraction of quantities consisting of
more than one term. 18-21. Rules for
addition and subtraction of numbers; and
explanation of them. 22. Arithmetical
complement: example.
Of Multiplication, p. 6.
Art. 23. The product of any number of
factors is the same, however the factors be
arranged. 24. The product of any two
powers of the same quantity, is that quan-
tity raised to the power expressed by the
sum of the exponents in the factors. 25.
Multiplication of quantities consisting of
one term. 26. Multiplication of quantities
consisting of several terms. 27. Reason of
the rule which determines the signs of the
terms of the product. 28. Examples of Al-
gebraical multiplication. 29. Continued
product: example. 30. Sign of the product
of any number of negative factors. 31-33.
Multiplication table, first used by Pytha
goras-rule for multiplication of numbers--
reason of rule: examples.
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36. When the divisor and dividend have a
has more than one term, and the divisor
common factor. 37. When the dividend
only one term. 38. Sign of the quotient;
how determined.
braical division.
39. Examples of alge
of numbers-short division
40, 41. Rule for division
42. Rule for the division of algebraical
remainder.
quantities, when the divisor and dividend
have more than one term. 43. Examples.
44, 45. an- 1 always divisible by a -
always divisible by a + 1; a2n + 1 not di-
a" + 1 not divisible by a -1; a2n+1 + 1
visible by a + 1. 46. Division of 1 by
when numbers are substituted for the let-
47. Algebraical results are true
ters care must be taken not to neglect the
remainder.
Art. 48-50. Whole numbers or integers
-measure-multiple- sub-multiple. 51.
If one quantity measure another, it mea-
sures any multiple of that quantity. If
measured by all the factors of that other.
one number be measured by another, it is
52, 53. Prime number. How to find whe-
ther a given number be a prime number.
54. What is meant by numbers prime to
each other. Common measure. 55. The
common measure of two numbers measures
also their sum and difference. 56. If a
number measure two others, it will also
measure the remainder arising from the
division of one of these by the other.
57, 58. Rule for finding the greatest com-
mon measure of two numbers; reason of
the rule: example. 59. Application of the
rule to algebraical expressions: example.
60. If a and b be any two numbers, and
c any prime number, that neither measures
a nor b, then c does not measure their pro-
duct a b. 61, 62. Every number can be re-
duced into only one set of prime factors.
Example of this reduction. 63. All the
numbers that measure any number are the
products of some of its prime factors. 64.
Rule for finding the greatest common mea-
sure of three or more numbers. 65. The
least common multiple of two or more
numbers: method of finding it. 66. pro-
the real value of the several digits. The
perties of numbers deduced from considering
remainder, after dividing a number by 2
or 5, is the same as that after dividing
the units digit by 2 or 5. 67. When any
number is divided by 9, the remainder is
the same as when the sum of its digits is
divided by 9. 68. Rule for verifying the
multiplication of two numbers, by casting
out the nines. Proof of the rule. 69-72.
Other properties of numbers. 73. Decimal
notation base of the scale of notation.
74, 75. Rule for transforming a number
from the decimal scale of notation into any
other, and the converse: examples. 76.
If A., A1 A2, &c. &c. be the numbers ex-
pressed respectively by the separate periods
of the first n digits, second n digits, third
n digits, &c., beginning at the right, of any
number expressed in a system of notation
whose base is r; then that number is mea-
sured first by any factor of r", which mea-
sures A.; secondly, by any factor of - 1
which measures A. + A1 + A2 + &c.;
Thirdly, by any factor of + 1 that mea-
sures (A. + A2 + A + &c.) — (A1 +
Aз + As + &c.)}.
Art. 77. Fractions ; derivation of the word.
78. Method of representing a fraction: de-
nominator: numerator. 79. Reduction of
fractions to their lowest terms: irreducible
fraction. 80. Reduction of several frac
tions to the same denominator. 81. The
signs of all the terms in the numerator and
denominator of a fraction may be changed
without altering its value. 82. Addition
and subtraction of fractions: examples.
83. Proper fraction; mixed number. 84.
Reduction of a mixed number to an equiva
lent fraction, and the converse. 85. Rule
for multiplying a fraction by a whole num
ber: examples. 86. Rule for multiplying
by a fraction: examples. 87. Rule for di-
viding a fraction by any quantity: example.
88. Rule for multiplying one fraction by
another examples. 89. The product is
the same, however the fractional factors be
arranged. 90. Rule for raising a fraction
to any power. 91. Rule for dividing an in-
teger by a fraction: examples. 92. Rule
for dividing one fraction by another: ex-
amples. 93. Fractions whose numerators
and denominators are themselves fractions.
94. Mixed numbers must be reduced to im-
proper fractions previous to multiplication
and division. 95. Multiplying any quantity
by a proper fraction diminishes its value;
by an improper fraction increases it. Di-
viding any quantity by a proper fraction
increases its value; by an improper one,
diminishes it. 96. Reciprocal of a fraction
or integer. 97. The sum of two irreducible
fractions, whose denominators are prime to
each other, cannot be a whole number.
Of Compound Numbers, p. 24.
Art. 98. Units of different denominations
-for what purpose used-compound num-
bers. 99-102. Rules for reduction from
one denomination into another: examples.
103. Rule for addition and subtraction of
compound numbers. 104. Rule for multi-
plying a compound number by a simple
number. 105. Rule of practice. Aliquot
parts: examples. 106, 107. Observations
on the multiplication and division of com-
pound numbers by each other. Duodeci-
mal multiplication.
Of Simple Equations, p. 27.
Art. 108. Equation-members of an equa
tion-simple equations. 109. A quantity may
be transposed from one side of an equation
to the other, provided its sign be changed.
110. How to clear an equation of fractions.
111. General rule for the solution of a sim-
ple equation with only one unknown quan-
tity. Such an equation admits of only one
solution. 112. Problem producing a simple
equation-a symmetrical expression. 113.
Explanation of the meaning of a negative
answer. 114, 115. Problems producing
simple equations. 116. Further explana-
tion of a negative answer-of an answer
with a denominator equal to zero.
117. An
infinitely great quantity-infinity, its alge-
braical symbol. 118. No general rule for
the reduction of a problem into an algebrai
equations involving two unknown quanti-
cal equation. 119. Example of two simple
ties. There is only one pair of values
General rule for the solution of these
which will satisfy both equations. 120.
equations, involving two unknown quanti-
121. Problem producing two
equations.
ties-its solution. 122. Problem producing
quantities-its solution.
three equations, involving three unknown
123. In order
that several equations, involving several un-
known quantities, may be all satisfied by
the same values of the unknown quantities,
and by only one system of such values, the
number of unknown quantities must be
equal to the number of equations-indepen-
dent equations.
Art. 124. When numbers are said to be in
direct proportion. 125. Examples of direct
proportion-caution necessary in deciding
that quantities are proportional. 126. What
is meant by the proportion of two quanti-
ties-ratio. 127. Manner of representing
proportion-extremes-means. 128. Vari-
ous relations between the several terms of a
proportion. 129. Mean proportional-third
proportional-duplicate ratio continued
proportion-triplicate ratio quadruplicate
ratio. 130, 131. Inverse or reciprocal pro-
portion examples. 132. Properties of
numbers in inverse proportion. 133. Direct
and inverse rule of three. 134. Compound
proportion: examples. 135. General rule
of compound proportion: examples. 136,
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