PREFACE.

To the friends of education, the growing taste for the study of the sciences in our Schools and Academies, is one of the most encouraging signs of the times. Amidst the din of business, the spirit of speculation, and the political excitements which agitate the country; while the teeming press is also pouring forth its floods of "cheap literature" and "light reading," something is the more necessary to be done to awaken and develope the reasoning powers, and give a balance and symmetry of character to the young; or their attention and sympathies will inevitably be diverted into other channels, and the next generation be afflicted with a race of intellectual dwarfs.

Nothing is better calculated to counteract these tendencies than a thorough course of education in our primary Schools and Academies. Indeed, these are the only places where the great mass of youth receive any mental training, which is worthy the name of education. These institutions, therefore, are emphatically the seminaries of the people; and it well becomes their guardians to make them what they should be, the nurseries of thought and intellect, as well as of virtue.

It is a settled principle that the powers of the mind, like those of the body, acquire strength by exercise; and without it, neither of them can be healthful and vigorous. In the whole field of science and literature, there is no exercise better fitted to develope the powers of thought, to strengthen the intellect, and make sound reasoners, than the study of Geometry. Its elementary principles are clear and well defined,

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the propositions to be established are distinct, the steps in the argument are consecutive and certain, and its conclusions carry with them an irresistible conviction of their correctness and truth. In a word, geometry is the embodyment of logic; its demonstrations are a continued series of deductions, and present some of the most perfect specimens of exact reasoning, which were ever invented by human ingenuity. The mind, therefore, whose powers have been trained to such accuracy, and whose habits of reasoning have been formed upon such a model, cannot fail to master any subject which the man of business or the scholar may have occasion to investigate.

But aside from its value as an intellectual discipline, the study of geometry has recommendations of a more practical import. There is scarcely a department in the arts and sciences; or a branch of education, whether substantial, or ornamental, with which geometry is not directly or indirectly connected. By its assistance, we measure the surface and solidity of the earth, the extent of its grand divisions and subdivisions, the distances and magnitude of the planets and other celestial bodies. An acquaintance with the elements of geometry is likewise indispensable to a thorough knowledge of surveying, mensuration, civil engineering, and navigation. In many of the physical sciences, too, as mechanics, optics, geography, astronomy, the doctrine of crystalization, &c., its applications are numerous and important.

A knowledge of geometry is also indispensable in executing plans of national improvement and defence, as in the construction of aqueducts, canals, railroads, fortifications, the marshaling of armies, navies, &c. Nor is it less necessary to the successful cultivation of many of the fine arts; as drawing, painting, sculpture, architecture, &c. Indeed, we can scarcely take up a periodical, or a school book, without

finding frequent passages which involve the elementary principles of geometry.

From the earliest history of letters, the study of geometry has been a favorite pursuit with men of genius and learning; it has successively numbered among its votaries the immortal Archimedes, Pythagoras, Plato, Aristottle, the Ptolemies, the Bacons, Newton, Laplace, &c. &c.

It is our happiness to live in an age when the sciences are no longer locked up in laboratories and universities, for the benefit of the "favored few." We hail their introduction into our Schools and Academies as a harbinger of good, and have watched their progress during the few past years with intense interest and delight. It is unnecessary to say, that since a taste for the study of geometry has begun to be more generally cultivated, teachers have felt the need of an elementary treatise on the subject, which is accurate and rigorous in its demonstrations, and yet not above the comprehension of their pupils. Many parts of Euclid are liable to objection in this particular. It is also thought that a smaller work than the former editions of Legendre, combining the simplicity and clearness of his reasoning, with some slight additions, would answer every purpose for beginners, and at the same time bring the subject more perfectly within the capacity and means of every class of learners.

At the suggestion, and, we may add, the urgent solicitation of several distinguished and active friends of popular education, the present edition of Legendre's Geometry was undertaken; and it is now presented to the public, with the fervent hope, that it may be instrumental in extending the advantages of this study more widely through our Schools and Academies, and thus be made subservient to the cause of education, and human happiness.

The merits of Legendre are too well known to require comment. He has been called the "first geometer of Europe." For many years, his geometry and trigonometry have had a very extensive circulation throughout France and the continent. They have been translated into our own language by the distinguished Dr. Brewster of Edinburgh, and also by the late Professor Farrar of Harvard University; and in this manner, have been extensively used in England, and in many of the Colleges and higher Seminaries of the United States. In preparing the present edition from the translation of the former, free use has been made of the original; and such alterations and additions have been incorporated into the work, as were deemed necessary to adapt it, in all respects, to the wants of the young, and the improved modes of instruction in Schools and Academies.

The principal embarrassment which young minds experience in the study of geometry, arises from the difficulty of comprehending abstract propositions. Legendre has essentially removed this difficulty by enunciating the propositions by the aid of the particular diagram which he uses in the demonstration; instead of stating it in general terms as Euclid and others have done, and afterwards giving the figure. It is found, however, to be inconvenient for scholars to quote a proposition enunciated with reference to a particular diagram, which, it is presumed, escaped the notice of the author. To obviate this inconvenience, after the truth of the proposition has been established with respect to the particular diagram in question, the general principle is then deduced, and for the sake of more convenient reference is printed in italics. Thus we begin with a particular case, and arrive at a general conclusion.

*The article on geometry in the Edinburgh Encyclopedia was chiefly taken from Legendre.

We find that a certain principle is true with respect to a particular diagram; for example, that the sum of two adjacent angles formed by the intersection of two straight lines, is equal to two right-angles. And of the four angles thus formed, we find the same is true of any two adjacent angles. If we vary the direction of the lines, the proposition is still true. Hence, we infer that this is a universal truth respecting any two adjacent angles formed by the intersection of two straight lines.

The mode of reasoning from particulars to generals, is one of the greatest improvements of the age in the methods of instruction. It is the only safe mode that can be adopted in experimental philosophy, metaphysics, and ethics. It is commonly called the inductive mode, and is the most satisfactory to the mind in all cases where it can be employed. In most departments of science, it is necessary to compare a great number of particulars before we make a general deduction; but in geometry and kindred sciences, a single demonstration is commonly sufficient to warrant a general conclusion.

In the original work of Legendre, the doctrine of ratio and proportion is omitted, and the student is referred to the treatises contained in arithmetic and algebra for information upon this subject. This omission, it is believed, is generally regretted; and the more so, as most of the methods in which the subject has been treated, have failed to give entire satisfaction. Euclid's method is strictly accurate and rigorous, but is acknowledged to be "tedious and difficult to all beginners," while "it is unintelligible to most."

In attempting to supply this omission, we have adopted algebraic notation, and have endeavored to present all the important principles of proportion in a simple and logical manner. To avoid unnecessary perplexity to learners, the