Since all manner of quantities require to be compared together, in mathematical computations, and their various relations searched out and determined and as most of our knowledge in mathematical subjects depends on the proportions of several things to one another: so it is requisite that the nature of proportion, and the methods of reasoning thereby be distinctly laid down and well understood. It is a method of reasoning so very short, subtle, solid, and certain, and likewise so useful in all parts of the mathematics, that it is impossible to make the least progress without it. It is the marrow of the mathematics, and the very soul of geometry and geometrical reasoning. Therefore it is absolutely necessary that every one who expects to succeed in his mathematical studies, should make himself acquainted with the nature of reasoning with proportional quantities, and become ready and quick in the use thereof. Section 1 treats of arithmetical proportion and progression, Section 2 of geometrical proportion, and Section 3 of general proportions. Following those three sections, this volume also contains Books I-VIII of "The Elements of Geometry."