The method here alluded to, and that has obtained much celebrity both in this country and abroad, may be contrasted with the practice of the French mathematicians; who are accustomed, even when treating of the most complicated problems, to dispense altogether with graphical illustration. A course that undoubtedly leaves much uncertainty on the nature of the proposition discussed, but that is otherwise possessed of many advantages; a fact ably shown in the works of its authors, who have demonstrated by an almost unexampled success, the powers of conception and habits of generalization thus acquired. The choice offered, with respect to a plan of instruction, seemed thus to lie between two methods diametrically opposed. But according to the view taken by the writer in regard to this question, the peculiar advantages of both are capable of being combined: every possible aid-by diagrams-by solid and skeleton models and by reiterated explanation, should, in his opinion, be afforded to the student, whilst the latter is endeavouring to form precise ideas of those elementary forms whereto constant reference is afterwards to be made. This task once accomplished, the subsequent combination of these elements may be left to the student's unassisted powers of conception, and will be found to afford him ample opportunities for that species of exercise in which the method of ⚫ Pestalozzi has been regarded as deficient. In concluding these remarks, upon the arrangement adopted in the present work, it will be proper to notice the order wherein the book should be read. This order varies with the extent to which the student has exercised the faculties required in pursuing a train of original inquiry. And as these powers are not usually matured at the age when students first join our public colleges, it will be right so to proportion the demands upon the faculties of combination and conception, as to keep pace with the gradual development of the powers in question, and the consequent improvement of the mind. The plan adopted by the author, with regard to the junior mathematical students of the university, is first to read through the preliminary Section of Part I., illustrating every proposition with solid and skeleton models: and thence to pass to the three first Sections of the Second Part. The knowledge so attained, and which presupposes some acquaintance with simple and quadratic equations, will enable the student to master all that is said in Section II., Part III., together with such portions of the first Section of that Part as may be required in pursuing this course. When the student has thus, in merely passing over the first elements of the subject, learned nearly the whole of plane trigonometry, he will be prepared to enter upon the relations of solid figures, and to comprehend what is said concerning the construction of the tables. With these views, he may read two more Sections of Chap. I., Part I., and Sections I., II. and IV. of Chap. II. of the same Part, followed by Section III. and what remains of Section I. Part III.; and reading at the same time with the other Sections here mentioned of this part, such portions of the Sections IV. and V. as can be comprehended by a person who has not proceeded in algebra further than quadratic equations. The knowledge which the student will now have attained, embraces plane and spherical trigonometry, and the greater part of the propositions contained in most treatises on synthetic geometry; and prior to entering further upon the subject, it would be proper that he should commence anew observing, as he now passes over a subject that has become familiar, a more rigid attention to the inductive order; and reading, what may hitherto have been neglected, the Preliminary Reflections and Suggestions. The order in which the remainder of the treatise is read, is comparatively of little importance; but it may be remarked, that a student whose time is limited, may omit in Section III., Chap. II., Part II., all that follows article 112; and in the Third Part he may omit the Seventh Section. The author, in closing a work that for the last two years has occupied the brief moments of his leisure, cannot lay down the pen without a few words in explanation of the occasional inadvertences and press errors that it contains. With a view to the benefit of his students, he undertook to reduce, within a given period, the science of quantity and position to a smaller compass, and to put the whole in a form that should render it immediately dependent on the portion of their course that preceded, and in unison with portions that were to follow: the distance at which he resides from the press, and some additional duties, have rendered it necessary for him, in completing this task, to give less than due care to the labour of revisal. ANALYSIS. PART I. THE SUBJECT CONSIDERED IN CONNECTION WITH MATERIAL OBJECTS, AND INSTRUMENTS. CHAPTER I. CLASSIFICATION OF THE VARIETIES OF FORM. SECTION I-Most obvious Principles of Classification.-Forms of external objects-relations observed in the forms of objects-classification of the varieties of form-the comparison of figures reduced to the relations of points in space-the relations of points in space reduced to the relations of points in one plane-geometric investigations are performed by the arrangement of elementary figures-models of angles made use of in geometrycompendious arrangement of angles in series-compendious arrangement of some other simple figures-arrangements of more complex figures-delineations of accidental or arbitrary figures-arrangements of certain complex figures that occur in the arts-arrangement of elementary figures-of a principle of arrangement that is applicable to all figures-nature of a geometrical demonstration-a relation among certain lines may necessarily involve a relation among others. Page 5. SECTION II.-Principles conducting to a more refined Analysis.— Example of a geometrical investigation-reflections on the preceding example-branches into which geometry is divided-advantages peculiar to abstract geometry-principles on which geometrical investigations should be conducted the comparison of figures reduced to the relation of points in space-the magnitude of all the parts of a figure deduced from its form and the magnitude of one part-the investigation of article 17 made to depend on a single lineal measurement-the investigation of article 17 conducted by assuming as known the figure which is the object of inquiry-relations of form and magnitude reduced to the relations of number-recapitulation-elements Inquiry suggested by these Reflections. An examination of the most obvious relations of a finite number of points. SECTION I.-Of Quantity.-Relations of two points-when direction is not regarded, the relations of quantity are the same with those of number. SECTION II.-Of Closed Figures-Closed Solids-and their Relations. -Relations of three points-idea of an angle results from comparing the directions of three points-idea of a triangle obtained from the same source -relations of four points-straight lines that cross without meeting-idea of a plane obtained from the relations of four points-three points, or two straight lines, that are in a plane, suffice to determine its position-planes mutually inclined intersect in a straight line-measure of their inclination- relations of many points-notation to be used-closed figures-angles about |