"The author's object in this tract is to defend Euclid from the charges of inconsistency which have been brought against him by Sir John Leslie and others, in consequence of the introduction of the doctrine of ratio and proportion as part of his system of geometry. Most of the best writers of geometry (as Legendre) omit this part in their elementary systems, and most teachers in this country pass over the 5th book, and adopting the doctrine of proportionals from algebra, proceed to apply it to the theorems of the 6th book. Professor Powell treats the subject in detail, stating the objections which have been urged against Euclid, and presenting answers to these objections. He begins with a general statement of the question; he then proceeds to the consideration of Euclid's method, or the doctrine of commensurables and incommensurables. He shews that Euclid, in his earlier books, does not even imply the idea of incommensurability. Neither is this introduced in the 5th and 6th books, and it is not till we arrive at the 10th that this edition in geometrical magnitudes, expressed by numerical measures, is broached. In the 11th and 12th books all reference to this distinction is dropped, recurrence being made to the principles of the 5th book. It is again, however, resumed in the 13th book, and is applied to various properties. The author observes, "that much of the confusion of ideas which has arisen on these subjects, has been occasioned by not observing that when we say two lines are incommensurable, the phrase is, in fact, elliptical, and we ought always to consider as understood, if not expressed, that two lines if referred to numbers are incommensurable. The deficiency of exact comparison in such cases is not in the geometrical relation of the quantities, but in the powers and capabilities of our numerical system to express them. Mr. Powell then proceeds to discuss the views of the earlier geometers and of later mathematicians. He points out the misapprehension under which they all labour, from the common mistake of considering that definitions describe the thing defined instead of fixing the meaning of terms. He shews that the mistake must be corrected before reasoning can be admitted on the subject. The nature of abstract quantity is next ably treated of, and the paper concluded in the same philosophic spirit which pervades it throughout."