This book contains translations of two essays by Henri Lebesgue. The first, which takes up most of the book, is entitled "Measure of Magnitudes" and was published seri- ally in l'Enseignement Mathematique during the years 1931-1935. With the French school system primarily in mind, Lebesgue discusses possible ways to present the subject on the elementary, secondary and collegiate levels "in as simple and specific a manner as possi- ble but without sacrificing logical rigor." The first four chapters deal with the notions of whole numbers, length and real numbers, area and volume. Much of what Lebesgue sug- gests could be used profitably in a high school geometry course. Chapter V deals with curve length and surface area and contains some interesting remarks relating to Lebes- gue's own work on the problem of defining the area of a surface. In the final two chapters the notion of a set function is introduced and used as the context for an elegant treatment of differentiation and integration along the lines found in Buck's Advanced Calculits. The essay concludes with a lucid discussion of oriented integrals and volumes.