This little book is the outcome of the effort annually renewed over a long period to make clear to my students the principles on which the Theory of Proportion is based, with a view to its application to the study of the Properties of Similar Figures. Its content formed recently the subject matter of a course of lectures to Teachers, delivered at University College, under an arrangement with the London County Council, and it is now being published in the hope of interesting a wider circle. At the commencement of my career as a teacher I was accus tomed, in accordance with the then established practice, to take for granted the definition of proportion as given by Euclid in the Fifth Definition of the Fifth Book of his Elements and to supply proofs of the other properties of proportion required in the Sixth Book which were valid only when the magnitudes considered were commensurable. Dissatisfied with the results of a method which could have no claim to be considered logical, after trying some other modes of exposition, I turned to the syllabus of the Fifth Book drawn up by the Association for the Improvement of Geometrical Teaching. But again I found this hard to explain, and it was evident that my students could not grasp the method as a whole, even when they were able to understand its steps singly. After prolonged study I found that, in addition to the difficulty arising out of Euclid's notation, which is a matter of form and not of substance, and the difficulty that Euclid does not sufficiently define ratio, two reasons could be assigned for the great difficulty of his argument.