This work forms a Key or Companion to the Higher Algebra, and contains full solutions of nearly all the Examples. In many cases more than one solution is given, while throughout the book frequent reference is made to the text and illustrative Examples in the Algebra. The work has been undertaken at the request of many teachers who have introduced the Algebra into their classes, and for such readers it is mainly intended; but it is hoped that, if judiciously used, the solutions may also be found serviceable by that large and increasing class of students who read Mathematics without the assistance of a teacher.


In this edition, the entire manuscript was typeset in a bigger size font [10 pt : `DejaVu Serif'] (honoring readers' suggestions) using the LaTeX document processing system originally developed by Leslie Lamport, based on TeX typesetting system created by Donald Knuth. The typesetting software used the XeLaTeX distribution.

We are grateful for this opportunity to put the materials into a consistent format, and to correct errors in the original publication that have come to our attention. Most of the hard work of preparing this edition was accomplished by Neeru Singh, who expertly keyboarded and edited the text of the original manuscript. She helped us put hundreds of pages of typographically difficult material into a consistent digital format.

The process of compiling this book has given us an incentive to improve the layout, to doublecheck almost all of the mathematical rendering, to correct all known errors, to improve the original illustrations by redrawing them with Till Tantau's marvelous TikZ. Thus the book now appears in a form that we hope will remain useful for at least another generation.


Table of Contents

EXAMPLES I : Ratio
EXAMPLES II : Proportion
EXAMPLES III : Variation
EXAMPLES IV : Arithmetical Progression
EXAMPLES V : Geometrical Progression
EXAMPLES VI : Harmonical Progression
EXAMPLES VII : Scales of Notation
EXAMPLES VIII : Surds and Imaginary Quantities
EXAMPLES IX : The Theory of Quadratic
EXAMPLES X : Miscellaneous Equations
EXAMPLES XI : Permutations and Combinations
EXAMPLES XIII : Binomial Theorem Positive Integral Index
EXAMPLES XIV : Binomial Theorem. Any Index
EXAMPLES XV : Multinomial Theorem
EXAMPLES XVI : Logarithms
EXAMPLES XVII : Exponential and Logarithmic Series
EXAMPLES XVIII : Interest and Annuities
EXAMPLES XIX : Inequalities
EXAMPLES XX : Limiting Values and Vanishing Fractions
EXAMPLES XXI : Convergency and Divergency of Series
EXAMPLES XXII : Undetermined Coefficients
EXAMPLES XXIII : Partial Fractions
EXAMPLES XXIV : Recurring Series
EXAMPLES XXV : Continued Fractions
EXAMPLES XXVI : Indeterminate Equations of the First Degree
EXAMPLES XXVII : Recurring Continued Fractions
EXAMPLES XXVIII : Indeterminate Equations of the Second Degree
EXAMPLES XXIX : Summation of Series
EXAMPLES XXX : Theory of Numbers
EXAMPLES XXXI : The General Theory of Continued Fractions
EXAMPLES XXXII : Probability
EXAMPLES XXXIII : Determinants
EXAMPLES XXXIV : Miscellaneous Theorems and Examples
EXAMPLES XXXV : Theory of Equations
MISCELLANEOUS EXAMPLES