Arithmetical Institutions

then because b=b (In. 20.) :· a=bd (In. 71.) And if a=bd, then because b=b (In. 20.) :• —=d (In. 77.)

THEOREM I.

84. The Product of two Numbers or Quantities, (Ex. gr. a+bxetz is equal to the Product of all the Parts of one multiplied (or drawn) into all the Parts of the other i. e. a+bxe+x=axe+z+bxe+z.

Demonftration.

If equal Quantities are multiplied into equal Quantities, the Products are equal (In. 71.) But every Whole is equal to all its Parts taken together (In 23.) Therefore the Product made by multiplying two Wholes one into. another axe+z is equal to the Product made by multiplying all the Parts of one into all the Parts of the other; ie, a+bXe+z=axe += + bxe+z. or e¥a+b+zXa+b=ae+az+be+bz. QE. D.

COROLLARY VI.

85. Therefore in Multiplication 'tis all one which of the Factors be the Multiplicand, and which the Multiplier.

THEOREM II.

86. If a Dividend and Divifor (Ex. gr.) be both multiplied into (and confequently both divided by) the fame Quantity (Ex. gr. x) the Quotient will continue ftill the fame; i. e. =

ax.

bx Demonftration..

If a Divifor and Dividend be encreased or diminished by like Parts of themselves, the Divifor will be ftill the fame Way contained in the Dividend ;i. e. their Ratio will ftill be the fame (In. 45.) But the Quotient is that which expreffes the Ratio of the Dividend to the Divifor (In. 8o.) Therefore the Quotient is ftill the fame; i. e.

2=22 Q. E. D.
b bz

COROLLARY VII.

87. Therefore wherever the fame Term is found in both the Dividend, and the Divifor, it is to be ftruck out. Or in Numbers, whenever the Dividend and the Divifor can both be divided by the fame Number, it is to be done Which is called a bringing the Expreffion to its lowest Terms. Ex. gr.

abx x Aa

CHA P. V.

Of FRACTIONS.

DEFINITION XXXVIII.

88. F a Divifor be an Aliquant Part of the Dividend, then the Division of the Remainder by the fame Divisor makes a Fraction (In. 35.) As when 14 is to be divided by 3, the Quotient is 4, with the third Part of 2,

i. e. 3 and the 5th Part of 4, or 4 Fifths of 1.

DEFINITION XXXIX.

89. The Numerator of a Fraction is the Number above, expreffing how many Parts are taken in the Fraction. The Denominator is the Number beneath, which fhews the Denomination of the Parts, or the Number of Parts into which the Unit is divided.

PARTITION VIII.

90. Fractions are divided into Proper and Improper.

DEFINITION XL.

91. A Proper Fraction is that whose Numerator is less than the Denominator; as,,, &c. An Improper Fraction is that whofe Numerator is greater, as &c.

8 12

PARTITION IX.

92. Fractions are again divided into Pure and Mixed.

DEFINITION XLI.

93. A Pure Fraction is that which is joined to no Integer, as, &c. } A Mixed Number or Species is that which is made up of an Integer, and a Pure Fraction, as 3 +;; i. e. 3 and a half. 6 + /1/ . b

SCHOLIUM II.

94. In Mixed Numbers the Sign + which connects the Integer and Fraction is ufually omitted. Thus 3+ is writ 3, 15+ is 154, but not so in Species.

PARTITION X.

95. Pure Fractions are divided into Simple, and Compound.

DEFINITION XLII.

96. A Simple Fraction is that which is not divided into more, as the Examples above. A Compound Fraction is the Multiplication of Fractions, or the breaking a Fraction into more; as the Expreffions of of, of of;, of, &c.

PARTITION XI.

97. Simple Fractions are either Homogeneous, or Heterogeneous.

DEFINITION XLIII.

98. Homogeneous Fractions are fuch as have the fame Denominator, or are referred to the fame Unit. Heterogeneous Fractions are fuch as have different Denominators, or are referred to different Units.

COROLLARY VIII.

99. In Proper Fractions the Numerator is to the Denominator in a Ratio of leffer Inequality In Improper Fractions in a Ratio of greater Inequality.

COROLLARY IX.

100. Every Integer may be looked upon as a Fraction whofe Denominator is Unity; thus 56, b = &c.

COROLLARY X.

101. When the Numerator and Denominator of a Fraction are the fame, the Fraction is the fame with Unity (In. 39.)

COROLLARY XI.

102. The greater the Denominator of a Fraction is in refpect of its Numerator, the leffer is the Fraction; and vice verfâ, the leffer the Denominator of a Fraction is in refpect of its Numerator, the greater is the Fraction (In. 41, and 42.)

COROLLARY XII.

103. If both the Numerator and Denominator of a Fraction be multiplied (or divided) by the fame Number or Quantity, the Fraction will still retain the value (In. 86, and 87.)

COROLLARY XIII.

104. If the Numerator of a Fraction be multiplied by any Number or Quantity, it is made fo many times greater, as there are Units in that Number or Quantity; and if divided by it, fo many times lefs (In. 102.)

COROLLARY XIV.

105. If the Denominator of a Fraction be multiplied by any Number or Quantity, it is made fo many times leffer, as there are Units in that Quantity; and if divided by it, fo many times greater (In. 102.)

COROLLARY XV.

106. Whence, to divide the Numerator of a Fraction by any Quantity, is all one as to multiply the Denominator of the fame Fraction by that Quantity And vice verfa, to multiply the Numerator of a Fraction by any Number or Quantity is all one as to divide the Denominator by that Number or or Quantity. Ex. gr.

axb

aa÷÷a

PROBLEM I.

107. To reduce Heterogeneous Fractions into Homogeneous ones retaining the fame Value.

Effection.

Pre. 1. Multiply all the given Denominators together for a new and common Denominator.

2. Multiply each Numerator into all the Denominators, except its own, for new Numerators.

3. Subfcribe the new and common Denominator under each of these new Numerators. Then I fay that each Homogeneous Fraction, thus found, is equal to the respective given Heterogeneous one from whence its Numerator was formed. Q. E. E.

Example.

Let it be required to reduce the given Heterogeneous Fractions

into Homogeneous ones of the fame Value.

By Pre. 1. pXqXr pqr the common Denominator.
By Pre. 2. the new Numerators are bqr, cpr, and dpq.

Therefore by Pre. 3. the Homogeneous Fractions required are