These alterations do not imply any appreciable change of view on my part as to the foundations and province of Probability. Some of them are of course due to the necessary changes involved in the attempt to write up to date upon a subject which has not been stationary during the last eleven years. For instance the greatly increased interest now taken in what may be called the Theory of Statistics has rendered it desirable to go much more fully into the Nature and treatment of Laws of Error. The omissions are mainly due to a wish to avoid increasing the bulk of this volume more than is actually necessary, and to a feeling that the portions treating specially of Inductive Logic (which occupied some space in the last edition) would be more suitable to a regular work on that subject. I am at present engaged on such a work. The publications which I have had occasion to notice have mostly appeared in various scientific journals. The principal authors of these have been Mr F. Galton and Mr F. Y. Edgeworth: to the latter of whom I am also personally much obliged for many discussions, oral and written, and for his kindness in looking through the proof-sheets. His published articles are too numerous for separate mention here, but I may say generally, in addition to the obligations specially noticed, that I have been considerably indebted to them in writing the last two chapters. Two authors of works of a somewhat more substantial character, viz. Prof. Lexis and Von Kries, only came under my notice unfortunately after this work was already in the printer's hands. With the latter of these authors I find myself in closer agreement than with most others, in respect of his general conception and treatment of Probability. _December_ 1887. 1. I have indicated the new chapters and sections by printing them in italics in the Table of Contents. TABLE OF CONTENTS.[1] 1. Chapters and sections which are nearly or entirely new are printed in _italics_. PART I. PHYSICAL FOUNDATIONS OF THE SCIENCE OF PROBABILITY. Chh. I-V. CHAPTER I. THE SERIES OF PROBABILITY. §§ 1, 2. Distinction between the proportional propositions of Probability, and the propositions of Logic. 3, 4. The former are best regarded as presenting a series of individuals, 5. Which may occur in any order of time, 6, 7. And which present themselves in groups. 8. Comparison of the above with the ordinary phraseology. 9, 10. These series ultimately fluctuate, 11. Especially in the case of moral and social phenomena, 12. Though in the case of games of chance the fluctuation is practically inappreciable. 13, 14. In this latter case only can rigorous inferences be drawn. 15, 16. The Petersburg Problem. CHAPTER II. ARRANGEMENT AND FORMATION OF THE SERIES. LAWS OF ERROR.