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viii years, so that it is impossible to make definite acknowledgement in every case where assistance has been obtained from other writers. But speaking generally, our acknowledgements are chiefly due to the treatises of Schlömilch, Serret, and Laurent; and among English writers, besides Todhunter's Algebra, we have occasionally consulted the works of De Morgan, Colenso, Gross, and Chrystal.

To the Rev. J. Wolsienholme, D.Sc, Professor of Mathematics at the Royal Indian Engineering College, our thanks are due for his kindness in allowing us to select questions from his unique collection of problems; and the consequent gain to our later chapters we gratefully acknowledge.

It remains for us to express our thanks to our colleagues and friends who have so largely assisted us in reading and correcting the proof sheets; in particular we are indebted to the Rev. H. C Watson of Clifton College for his kindness in revising the whole work, and for many valuable suggestions; in every part of it.

May, 1887.

PREFACE TO THE THIRD EDITION.

this edition the text and examples are substantially the same as in previous editions, but a few articles have been recast, and all the examples have been verified again. We have also added a collection of three hundred Miscellaneous Examples which will be found useful for advanced students. These examples have been selected mainly but not exclusively from Scholarship or Senate House papers; much care has been taken to illustrate every part of the subject, and to fairly represent the principal University and Civil Service Examinations.

March, 1889.