Entered according to Act of Congress, in the year one thousand eight hundred and forty-eight, by GEORGE CLINTON WHITLOCK, in the Clerk's Office of the District Court for the Northern District of New-York. Stereotyped by C. Davison & Co., 33 Gold st., N. Y. Eugin. Lib Denison PREFACE. WHILE Some persons are complaining of constant innovation in text-books, and others finding fault equally with those in use, one scarcely knows whether or not to make any apology in putting forth a new work. One thing however, as it seems to me, is clear that in view of the importance which is justly attached to elementary instruction, there can be little danger of too great a supply of manuals from which an enlightened community may select. If new books of geography and grammar, of arithmetic and algebra, are not only acceptable to the schools, in their onward march of improvement, but even indispensable for giving them life and vigor, why should objection be made to attempts in adapting the elements of geometry to the wants of the young and to the existing condition of instruction? Why should a blind veneration for antiquity cause the elements of Euclid to continue in one form or other in our schools, when the luminous Grecian himself, if now living, would, we doubt not, no longer employ them without a material remodeling in conformity with the mathematical methods of the day? If boys are to learn that which they will practise when men, why should tyros be so long restricted to processes which, as mathematicians, they will seldom use? Is it essential to the acquisition of competent skill in numbers that our arithmetics should be filled with examples, and to the comprehension of general principles that our algebras should observe, in the development of forms, an unbroken continuity of progression? why in the elegant science of geometry should there be neither example nor process? I am aware that these suggestions are not applicable, in all respects, to certain books on geometry recently published; but an elementary work of sufficient fullness, yet moderate in magnitude, highly practical, and, consequently, progressive in theory and example, is still, I believe, a desideratum. I have endeavored to prepare such a work; how well I have succeeded will, of course, be determined by others. Its chief feature will be found to consist, not simply in the acquisition of geometrical principles, but in a regular progression of method, whereby it is intended to teach how to geometrize. In pursuing quite out to the end of our geometrical studies, as well as at the beginning, the synthetic and undevelopable methods of the ancients, we acquire little or no power of going alonewe get some geometry, it is true, but still remain almost destitute of that education in analysis which is far more important. Why, in investigating the doctrines of forms, should we studiously keep out of sight the general principles of quantity, as if no such principles existed, when even Euclid himself could proceed but a little way without stopping to construct his, to us, clumsy book of proportion, the best and only algebra at his command? Why, when so much labor is saved and greater clearness obtained, should we refuse to employ an equation like (a+b)2 Shall we have resources at hand and refuse to use them because Euclid was poor? Is it shorter, more satisfactory, or productive of finer results, to shut up the circumference of a circle between the perimeters of polygons than to avail ourselves = a2 + b2+2ab? of the simple symbol [], when employed as the vanishing ratio of the increments of two variables? Has geometry given birth to algebra, and shall she reap no advantage from her offspring? The succinct and methodical Francœur quoting Lagrange, says, "Tant que l'Algèbre et la Géométrie ont été séparées, leurs progrès ont été lents et leurs usages bornés; mais lorsque ces deux sciences se sont réunies, elles se sont prêté des forces mutuelles, et ont marché ensemble d'un pas rapide vers la perfection."* Again the luminous Lagrange, in the first of his "Leçons ;""Les fonctions dérivées se présentent naturellement dans la géométrie lorsq' on considère les aires, les tangentes," &c.t In accordance with these views, and in compliance with the recommendation of Lacroix, avoiding double methods, that we may be ever pressing on in that body of geometrical doctrines that are most useful, I have paid much attention to the classification, endeavor * So long as algebra and geometry were separated, their progress was slow and their application restricted; but when these two sciences became united, they lent each other mutual aid, and advanced together with rapid pace towards perfection. + Derived functions present themselves naturally in geometry, when we consider areas, tangents, &c. PREFACE. ing, consequently, to arrange the subjects in a natural order, so as to fall easily into families and readily develope each other. Thus, by the simple method of superposition, instead of a long, mixed, and circuitous route, the doctrines of parallels are presently arrived at, and, as a consequence, all the elementary theorems relating to angles and independent of the length of lines, are embraced in a first section of moderate length. The comparison of equal figures follows; then that of propor tional lines, which prepares the way for the investigation of areas— the doctrines of the circle are not considered till afterwards. Thus, in the first part, the topics are kept distinct, and, it is believed, in their natural order, by which means the progress is rendered more easy and rapid, and the methods of geometrizing are introduced, one after the other, as required by the gradually increasing difficulties of the growing subject. For instance, in the first section little or no artifice is employed, and the simplest algebra, amounting to scarcely more than the common symbolical notation, is sparingly introduced in the second, while in the third the algebraic requisitions are somewhat increased, especially in the exercises. The method of incommensurables developed as a part of a system in the introductory book, is employed for the first time in the third section of the second, or first geometrical Book; the correlation of figures and change of algebraic signs find application in the more advanced propositions of the circle in the third Book; and the ratio of vanishing increments draws a tangent to the parabola in the close of the first Part. The elementary properties of the ellipse and parabola, being as simple as those of the circle and as useful in the study of natural philosophy and astronomy, are here introduced. Further, as proportion is generally included in our works on geometry, I have thought it advisable to insert an introductory Book, embracing, in a regular series of proportions the first doctrines of algebra, as being convenient for reference to those already acquainted with the science, and indispensable to others, who, by taking up these principles as required, may wish to proceed in the same class. The first Part, consisting of a hundred and twenty pages, is designed to embrace, in theory and practice, such an introductory body of elementary geometryall the more difficult problems relating to perimeters or areas being postponed as is required, not only to enter successfully upon the study of the higher investigations that follow, but for furnishing, in some measure, with tangible and useful matter, those who want the disposition, lack the time, or have not the ability to proceed further. The first Book of the second Part consists of an elementary system of functions, depending on a single variable, and presented constantly under the simple notation. It embraces the binomial and logarithmic theorems. Every teacher must have observed that isolated methods, like those employed by Legendre in cases of incommensurability, make only such an impression upon the mind as to leave a sort of confusion always hanging about them, while that which forms part of a system readily commends itself to the understanding, and, consequently, remains ever after a permanent part of our appropriate knowledge. Laplace has well said, "Préférez, dans l'enseignement, les méthodes générales; attachez-vous à les présenter de la manière la plus simple, et vous verrez en même temps qu'elles sont toujours les plus faciles."* The method pursued in this book has been judged not only the most perfect† in itself, but, as will frequently happen when connected subjects, instead of being disjoined, are permitted to fall naturally together, at the same time the easiest. But aside from the indispensable matter which it contains, the chief object of this book is to prepare the way for what follows in the arithmetic of signs, the construction of trigonometrical tables, and the mensuration of surfaces and solids. In virtue of the course just alluded to, I have been enabled, in the second Book of the third Part, to make not only a more than usually full development of the trigonometrical forms with their application in the practical resolution of triangles, but to embrace also the quadrature of the circle and ellipse. In the next Book I have developed a system of surveying which I regard as peculiarly my own. It is true that the theorem for the computation of polygonal areas, which constitutes its chief feature, may be substantially found in Hutton, yet I have given to it so much of a new form and a demonstration at once general‡ and of the greatest simplicity, and extended it in so methodical a manner to the laying out and dividing of lands, that it becomes altogether another thing. Years of * In instructing, adopt general methods; endeavor to present them in a manner the most simple, and you will see, at the same time, that they are the easiest. + Not all the demonstrations in our algebras are perfect-for instance, the demonstrations of the binomial theorem in some school books, the most widely disseminated, amount to no demonstrations at all. The demonstration in Hutton is very tedious, and can hardly be said to be general. |