The introduction to the main facts of geometry has been put into intuitive form and each new subject is introduced in the same manner or by means of its connection with one which has been previously discussed. A large number of illustrations and applications have been given; these are really more illustrations than applications, however. In order to understand and appreciate applications of mathematics one must have already some technical knowledge of the subject in which the application is made and such knowledge has not been assumed here, although this may be done in the higher branches of mathematics. A short Foreword has been included for readers who here approach the subject for the first time. Although the long and severely formal proofs have been omitted the simpler proofs are given fairly complete, though not in formal style, in order to illustrate and cultivate systematic reasoning and logical processes. The numerical parts of the subject, amounting to applications of some of the principles, are an extension and amplification of the work of arithmetic on measurement.