This book provides an English translation of the work Principles of the Probability Calculus published in 1886 by Johannes von Kries, which discusses the range theory of probability. It offers a novel account of the foundations of probability, an account which was familiar to Keynes, Kneale, Weber, Reichenbach, and von Mises. This account dispenses with the principle of indifference in probability, and it introduces the method of arbitrary functions. Confusions in the history of probability are pinpointed, and a novel theory is developed in which probability is neither entirely subjective nor objective. The book develops what is known as the range theory or Spielraum theory in detail, in a narrative way using few formulas. Von Kries applies range theory to Boltzmann's theory of the statistical behaviour of gases, and to several applications in medical statistics. Many uses of probability are found wanting; very often they are found not to admit any expression of probability in numbers at all. The book will be of first interest to philosophers of science and historians interested in the foundations of probability. It is also of general interest to anyone who applies statistics everyday in such fields as econometrics, psychology, or medicine.