The present is the first volume of a brief attempt to place the reader in touch with the main ideas dominant in contemporary geometry, excluding the consideration of higher algebraic irrationalities. For this purpose an account of many of the preliminary facts of geometry is indispensable, but in later stages a good deal of variety was possible in the selection, and no complete recital of geometrical theorems is aimed at. In several respects the views taken are not those usual in current textbooks; but it is believed that the system here suggested is logically complete, and does not require that long preliminary study of elementary geometry to which at present so much time is devoted. In particular it is desired to enter a protest against the custom of regarding so-called projective geometry as based upon metrical geometry; in the present account distance, as a primary conception, does not enter at all. And an attempt is made to include as soon as possible the indispensable ideas of geometry of more than three dimensions, and of geometry of so-called imaginary points. While the view is taken that all geometrical deduction should finally be synthetic, it is also held that to exclude algebraic symbolism would be analogous to preventing a physicist from testing his theories by experiment; and it becomes part of the task to justify the use of this symbolism.