IN GEOMETRY, UPON THE MODEL OF COLBURN'S FIRST LESSONS IN ARITHMETIC. BY ALPHEUS CROSBY, PROF. ETC., DART. COLL. WITH AN INTRODUCTION, BY STEPHEN CHASE, PROF. MATH., DART. COLL. BOSTON: J. MUNROE & CO., B. B. MUSSEY & CO., AND W. J. REYNOLDS & CO. 1847. UNIVERSITY LIBRARY 047472 [SECOND EDITION.] "It is the nature of genuine science to advance in continual progression. Each step carries it still higher; new relations are descried; and the most distant objects seem gradually to approximate. But, while science thus enlarges its bounds, it likewise tends uniformly to simplicity and concentration. The discoveries of one age are, perhaps, in the next, melted down into the mass of elementary truths. What are deemed at first merely objects of enlightened curiosity become, in due time, subservient to the most important interests. Theory soon descends to guide and assist the operations of prac tice. To the geometrical speculations of the Greeks, we may distinctly trace whatever progress the moderns have been enabled to achieve in mechanics, navigation, and the various complicated arts of life. A refined analysis has unfolded the harmony of the celestial motions, and conducted the philosopher, through a maze of intricate phenomena, to the great laws appointed for the government of the universe."-PROFESSOR LESLIE. "The curiosity which speaks in children's busy eyes and hands should be to us the voice of Nature, bidding us make our beginnings early. The infant who cannot speak gazes earnestly and thoughtfully at the most common object, returning to it, and glancing from one part to another, as if to learn their connection. When he can walk, he goes round it, handling it, and studying it with all his senses. When he speaks, his questions are of size, form, and distance. If our answers are careless or unsatisfactory, his quick eyes and mind, not blunted by habit, detect our errors. He loves comparison of objects, and the imaginary multiplication and extension of them; he is pleased with the new and the different, and equally pleased with resemblance and equality to things known before. "Happy age of natural geometry, when each look and motion, nay, his very games, lead the boy on to the laws which shape the spheres and hold the planets in their course!"-AUTHOR OF THE THEORY OF TEACHING. Entered according to Act of Congress, in the year 1847, by Alpheus Crosby, in the Clerk's Office of the District Court of the District of New Hampshire. CAMBRIDGE: METCALF AND COMPANY, PRINTERS TO THE UNIVERSITY. PREFACE. THE following work is stated in the title to be upon the model of Colburn's "First Lessons in Arithmetic," because no other method occurred to me of presenting a general view of its plan, which would be at once so brief and so well understood. Without aiming at minute resemblance, and certainly without challenging any comparison in merit, it is an imitation of that admirable work, which has introduced so entire a revolution in the mode of teaching Arithmetic in our country, in the following particulars. I. It aims to give a simpler and plainer form to the elements of Geometry, and thus to bring the science, without sacrificing any of that strictness of demonstration which is its peculiar glory, within the province of the common school, and the reach even of the quite young. The relations of place, and form, and magnitude are those of which the youthful, not to say the mature, mind conceives most easily and most distinctly; and it has long seemed to me a subject for regret, that we have had no work in our list of commonschool books (however excellent treatises may have been prepared for higher institutions, or for special purposes) to take advantage of these juvenile conceptions, to give to them a scientific form, and to make them the foundation of elevated and accurate attainment. It is obvious that the reasoning employed in Geometry is, in its nature, less abstruse than that which is employed in |