TO THE FRIENDS OF EDUCATION. The publishers of this series of mathematical works by Professor Charles Davies, beg leave respectfully to ask of teachers and the friends of education a careful examination of these works. It is not their intention to commend, particularly, this Course of Mathematics to public favor; and especially, it is not their design to disparage other works on the same subjects. They wish simply to explain the leading features of this system of Text-Books—the place which each is intended to fill in a system of education-the general connection of the books with each other-and some of the advantages which result from the study of a uniform series of mathematical works. It may, perhaps, not be out of place, first, to remark, that the author of this series, after graduating at the Military Academy, entered upon the duties of a permanent instructor in that institution in the year 1816, and was employed for the twenty following years in the departments of scientific instruction. At the expiration of that period he visited Europe, and had a full opportunity of comparing the systems of scientific instruction, both in France and England, with that which had been previously adopted at the Military Academy. This series, combining all that is most valuable in the various methods of European instruction, improved and matured by the suggestions of more than thirty years' experience, now forms the only complete consecutive course of Mathematics. Its methods, harmonizing as the works of one mind, carry the student onward by the same analogies and the same laws of association, and are calculated to impart a comprehensive knowledge of the science, combining clearness in the several branches, and unity and proportion in the whole. Being the system so long in use at West Point, and through which so many men, eminent for their scientific attainments, have passed, it may be justly regarded as our NATIONAL SYSTEM OF MATHEMATICS. Scholars and students who have pursued this course, will everywhere stand on the highest level with reference to the estimates which themselves and others will form of this part of their education. The series is divided into three parts, viz.: First-ARITHMETICAL COURSE FOR SCHOOLS. Second-ACADEMICAL COURSE. Third COLLEGIATE Course. (3) The Arithmetical Course for Schools. I. PRIMARY TABLE-BOOK. II. FIRST LESSONS IN ARITHMETIC. PRIMARY TABLE-BOOK. The leading feature of the plan of this work is to teach the reading of figures that is, so to train the mind that it shall, by the aid of the eye alone, catch in stantly the idea which any combination of figures is intended to express. The method heretofore pursued has aimed only at presenting the combinations by means of our common language: this method proposes to present them purely through the arithmetical symbols, so that the pupil shall not be obliged to pause at every step and translate his conceptions into common language, and then retranslate them into the language of arithmetic. For example, when he sees two numbers, as 4 and 8, to be added, he shall not pause and say, 4 and 8 are 12, but shall be so trained as to repeat 12 at once, as is always done by an experienced accountant. So, if the difference of these numbers is to be found, he shall at once say 4, and not 4 from 8 leaves 4. If he desires their product, he will say 32; if their quotient, 2: and the same in all similar cases. FIRST LESSONS IN ARITHMETIC. The First Lessons in Arithmetic begin with counting, and advance step by step through all the simple combinations of numbers. In order that the pupil may be impressed with the fact that numbers express a collection of units, or things of the same kind, the unit, in the beginning, is represented by a star, and the child should be made to count the stars in all cases where they are used. Having once fixed in the mind a correct impression of numbers, it was deemed no longer necessary to represent the unit by a symbol; and hence the use of the star was discontinued. In adding 1 to each number from 1 to 10, we have the first ten combinations in arithmetic. Then by adding 2 in the same way, we have the second ten combinations, and so on. Each ten combinations is arranged in a separate lesson, throughout the four ground rules, and each is illustrated either by unit marks or a simple example. Thus the four hundred elementary combinations are presented, in succession, in forty lessons,-a plan not adopted in any other elementary book. SCHOOL ARITHMETIC. This work begins with the simplest combination of numbers, and contains all that is supposed to be necessary for the average grade of classes in schools. It is strictly scientific and entirely practical in its plan. Each idea is first presented to the mind either by an example or an illustration, and then the principle, or abstract idea, is stated in general terms. Great care has been taken to attain simplicity and accuracy in the definitions and rules, and at the same time so to frame them as to make them introductory to the higher branches of mathematica. science. No definition or rule is given until the mind of the pupil has been brought to it by a series of simple inductions, so that mental training may begin with the first intellectual efforts in numbers The Academic Course. 1. THE UNIVERSITY ARITHMETIC. (Key separate.) II. PRACTICAL GEOMETRY AND MENSURATION. III. ELEMENTARY ALGEBRA. (Key separate,) IV. ELEMENTARY GEOMETRY. V. DAVIES' ELEMENTS OF SURVEYING. Those who are conversant with the preparation of elementary text-books, have experienced the difficulty of adapting them to the wants which they are intended to supply. The institutions of instruction are of all grades from the college to the district school, and although there is a wide difference between the extremes, the level in passing from one grade to the other is scarcely broken. Each of these classes of seminaries requires text-books adapted to its own peculiar wants; and if each held its proper place in its own class, the task of supplying suitable text-books would not be so difficult. An indifferent college is generally inferior, in the system and scope of instruction, to a good academy or high-school; while the district-school is often found to be superior to its neighboring academy. Although, therefore, the University Arithmetic and the Practical Geometry and Mensuration, have been classed among the books appropriate for academies, they may no doubt be often advantageously studied in the common-school; so also with the Algebra and Elementary Geometry. The Practical Geometry and Mensuration, containing so much practical matter, can hardly fail to be a useful and profitable study. DAVIES' UNIVERSITY ARITHMETIC. The scholar in commencing this work, is supposed to be familiar with the oper ations in the four ground rules, which are fully taught both in the First Lessons and in the School Arithmetic. This being premised, the language of figures, which are the representatives of numbers, is carefully taught, and the different significations of which the figures are susceptible, depending on the places in which they are written, are fully explained. It is shown, for example, that the simple numbers in which the unit increases from right to left according to the scale of tens, and the Denominate or Compound Numbers, in which it increases according to a different scale, belong in fact to the same class of numbers, and that both may be treated under a common set of rules. Hence, the rules for Notation, Addition, Subtraction, Multiplication, and Division, have been so con. structed as to apply equally to all numbers. This arrangement is a new one, and is deemed an essential improvement in the science of numbers Ja developing the properties of numbers, from their elementary to their highest combinations, great labor has been bestowed on classification and arrangement. It has been a leading object to present the entire subject of arithmetic as formning (51 a series of dependent and connected propositions; so that the pupil, while acquiring useful and practical knowledge, may at the same time be introduced to tnose beautiful methods of exact reasoning which science alone can teach. Great care has been taken to demonstrate fully all the rules, and to explain the reason of every process, from the most simple to the most difficult. The demonstration of the rule for the division of fractions, on page 147, is new and considered valuable. The properties of the 9's, explained at page 93, and the demonstration of the our ground rules by means of those properties, are new in their present form, and are thought worthy of special attention. In the preparation of the work, another object has been kept constantly in view; viz., to adapt it to the business wants of the country. For this purpose, much pains have been bestowed in the preparation of the articles on Weights and Measures, foreign and domestic-on Banking, Bank Discount, Interest, Coins and Currency, Exchanges, Book-keeping, &c. In short, it is a full treatise on the subject of Arithmetic, combining the two characteristics of a scientific and practical work. Recommendation from the Professors of the Mathematical Department of the` In the distinctness with which the various definitions are given-the clear and strictly mathematical demonstration of the rules-the convenient form and wellchosen matter of the tables, as well as in the complete and much-desired application of all to the business of the country, the "University Arithmetic" of Prof. Davies is superior to any other work of the kind with which we are acquainted. These, with the many other improvements introduced by the admirable scientific arrangement and treatment of the whole subject, and in particular those of the generalization of the four ground rules, so as to include "simple and denominate" numbers under the same head, and the very plain demonstration of the rule for the division of fractions-both of which are, to us, original-make the work an invaluable one to teachers and students who are desirous to teach or study arithmetic as a science as well as an art. (Signed,) D. H. MAHAN, Prof. Engineering. United States Military Academy, Jan. 18, 1847. PRACTICAL GEOMETRY AND MENSURATION. The design of this work is to afford schools and academies an Elementary Text-Book of a practical character. The introduction into our schools, within the last few years, of the subjects of Natural Philosophy, Astronomy, Mineralogy, Chemistry, and Drawing, has given rise to a higher grade of elementary studies; and the extended application of the mechanic arts calls for additional information among practical inen. In this work all the truths of Geometry are made accessible to the general reader, by omitting the demonstrations altogether, and relying for the impression of each particular truth on a pointed question and an illustration by a diagram. In this way it is believed that all the important properties of the geometrical figures may be learned in a few weeks; and after these properties have been once applied, the mind receives a conviction of their truth little short of what is afforded by rigorous demonstration. The work is divided into seven books, and each book is subdivided into sections. In Book I., the properties of the geometrical figures are explained by questions and illustrations. In Book II. are explained the construction and uses of the various scales; and also the construction of geometrical figures. It is, as its title imports, Practical Geometry. Book III. treats of Drawing. Section I., of the Elements of the Art; Section II., of Topographical Drawing; and Section III., of Plan Drawing. Book IV. treats of Architecture-explaining the different orders, both by descriptions and drawings. Book V. contains the application of the principles of Geometry to the Mensuration of Surfaces and Solids. A separate rule is given for each case, and the whole is illustrated by numerous and appropriate examples. Book VI. contains the application of the preceding Books to Artificers' and Mechanics' work. It contains full explanations of all the scales-the uses to which they are applied—and specific rules for the calculations and computations which are necessary in practical operations. Book VII. is an introduction to Mechanics. It explains the nature and properties of matter, the laws of motion and equilibrium, and the principles of all the simple machines. ELEMENTARY ALGEBRA. This work is intended to form a connecting link between Arithmetic and Alge bra, and to unite and blend, as far as possible, the reasoning on numbers with the more abstract method of analysis. It is intended to bring the subject of Algebru within the range of our common-schools, by giving to it a practical and tangible form. It begins with an introduction, in which the subject is first treated men. tally, in order to accustom the mind of the pupil to the first processes; after which, the system of instruction assumes a practical form. The definitions and rules are as concise and simple as they can be made, and the reasonings are as clear and concise as the nature of the subject will admit. The strictest scientific methods are always adopted, for the double reason, that what is learned should be learned in the right way, and because the scientific methods are generally the most simple. ELEMENTARY GEOMETRY. This work is designed for those whose education extends beyond the acquisi tion of facts and practical knowledge, but who have not the time to go through a full course of mathematical studies. It is intended to present the striking and important truths of Geometry in a form more simple and concise than is adopted in Legendre, and yet preserve the exactness of rigorous reasoning. In this system, nothing has been omitted in the chain of exact reasoning, nothing has been taken for granted, and nothing passed over without being fully demonstrated The work also contains the applications of Geometry to the Mensuration of Sur faces and Solids. SURVEYING. In this work it was the intention of the author to begin with the very elements of the subject, and to combine those elements in the simplest manner, so as to render the higher branches of Plane Surveying comparatively easy. All the instruments needed for plotting have been carefully described, and the uses of those required for the measurement of angles are fully explained. The Conventional Signs adopted by the Topographical Bureau, and which are now used by the United States Engineers in all their charts and maps, are given in full. An account is also given of the manner of surveying the public lands; and although the method is simple, it has nevertheless been productive of great results. The work also contains a Table of Logarithms-a Table of Logarithmic Sines-a Traverse Table, and a Table of Natural Sines--being all the Tables necessary for Practical Surveying (7) |