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LONGMAN, HURST, REES, ORME, BROWN, AND GREEN,

PATERNOSTER-ROW,

1824.

LONDON: Printed by A. & R. Spottiswoode,

New-Street-Square.

Page

88

XXIII. Of Pure Quadratic Equations, or Pure Equations

of the Second Degree

101

XXIV. Of Adfected Quadratic Equations

115

XXV. Of Powers in general

133

XXVI. Of the Calculation of Powers

137

XXVII. Of Roots with Relation to Powers in general 138

XXVIII. On the Method of representing Irrational Num-

bers by Fractional Exponents

140

XXIX. On the Reduction of Surds

142

XXX. Of the Calculation of Irrational Quantities

148

XXXI. Of Cubes, and the Extraction of Cube Roots 152

XXXII. Of the Higher Powers of Compound Quantities

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CHAP.

Page

XXXIII. On the Permutation of the Letters upon which

the Demonstration of the preceding Rule is

founded

167

XXXIV. Of the Resolution of Fractions into Infinite

Series

173

XXXV. On the Expansion of Irrational Powers in Infinite

Series

183

XXXVI. Of the Expansion of Negative Powers

187

XXXVII. Of Arithmetical Ratio, or the Difference between

Two Numbers

189

XXXVIII. Of Arithmetical Proportion

191

XXXIX. Of Arithmetical Progression

195

XL. On the Summation of Arithmetical Progressions 198

XLI. Of Geometrical Ratio

204

XLII. Of the greatest Common Measure

209

XLIII. Of Geometrical Proportion

219

XLIV. Of Geometrical Progression

225

XLV. Of the Summation of Fractional Series in Geo-

metrical Progression ad infinitum, and of the

Method of finding the value of Circulating

Decimals

230

XLVI. On the Method of finding any Number of

Arithmetic or Geometric Means between Two

Numbers

234

XLVII. On the Extraction of the Square Root of

Binomial Surds

238

XLVIII. Of Logarithms in general

244

XLIX. On the Application of Logarithms to Arith- :

metical Calculations

254

L. Of the Calculation of Interest

257

LI. Of the Nature of Equations of the Second

Degree

269

LII. Of Pure Equations of the Third Degree

273

LIII. Of the Resolution of Complete Equations of

the Third Degree

276

LIV. Of the Rule of Cardan

283

LV. Of the Resolution of Equations of the Fourth

Degree

289

LVI. Of the Rule of Bombelli, for reducing the

Resolution of Equations of the Fourth Degree

to that of Equations of the Third Degree 296

LVII. Of the Method of Euler for resolving Equations

of the Fourth Degree

301

LVIII. On the different Scales of Notation

306

A A NDIX

311

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ELEMENTS

OF

ALGEBRA.

OF THE DIFFERENT METHODS OF CALCULATING SIMPLE

QUANTITIES.

CHAP. I.

OF MATHEMATICS IN GENERAL.

1.Magnitude or Quantity may be defined to be any thing that is capable of increase and diminution.

A sum of money is therefore a quantity, because we may either add to, or diminish it. It is the same also with weights, and other things of the same nature.

2. It will thus be easily seen, that there must be so many different kinds of magnitude, as to render it difficult to enumerate them, and hence is the origin of the different parts of Mathematics, each of which is directed to a particular kind of quantity. Mathematics, in general, is nothing more than the Science of quantities, or the science which points out the method of measuring quantity.

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