Partnership,” or “Fellowship,” is superseded by Partitive Proportion. Of the two former terms the first is properly a commercial, the second a moral term : neither of them is significant of any arithmetical principle or operation. Partitive Proportion is properly, and in a general sense, descriptive of Proportion applied to dividing a given quantity into two or more parts which shall have a given ratio, one to another; and such is the purpose to be effected in all questions falling under this division of Arithmetic. “ Alligation" is superseded by Medial Proportion. The former designation refers to the mere expedient of drawing a line between two numbers entering into a calculation; and expresses nothing as to the nature of the arithmetical subject to which it is applied. In the treatment of Alligation, so called, the primary purpose, on which all others peculiar to the subject depend, is, to assign the ratios of two quantities, at different rates of value, for a compound of a given medial or mean rate of value. This is effected by the simplest application of Inverse Proportion ; and the term Medial Proportion, as defined in this treatise, with the accompanying Rule, places the subject in connection with the scientific principle on which it primarily depends. Various minor changes in phraseology will be noticed in different parts of the present work. Wherever it was conceived that, by this means, any thing could be gained on the score of precision or distinctness in the presentation of any particular subject, new terms or modes of expression have been employed ; and, in a few instances, the more perfect language of the higher mathematics has been put under contribution to these ends. As an example of the latter, may be mentioned “ Variation or General Proportion.” The very brief sketch of this subject has been given because the phraseology appropriate to it is convenient in stating the proportions of quantities, in a general way; particularly in distinguishing between direct and inverse Proportion. And here it may be necessary to remark that Inverse Proportion has been somewhat minutely treated on account of the use which is made of it in the higher applications of Proportion—notwithstanding its entire omission in many Arithmetics, as well as in Algebras and Geometries. In range of subjects, this work embraces what has been considered a proper course of Arithmetical instruction and exercise for schools; all that is necessary to a business education, so far as this science is concerned ; and certainly all that can be advantageously assigned to this department, in a regular course of mathematical studies. Whatever subjects, treated in similar works, are omitted in this, have been omitted from the consideration of their inutility for business purposes,-from the calculations which they require being fully provided for by the Principles and Rules here given,- from a regard to the superior facility with which they are treated in Algebra, now generaMy introduced into respectable schools,mor, because depending on the principles of Geometry, they cannot be presented satisfactorily in Arithmetic. The practical exercises, it is hoped, will be found sufficiently numerous, and well adapted to the purpose intended. They have been prepared with studious care, and with special reference to disciplining the mind of the student in the principles and applications of the science. Commencing always with the most simple that can be presented under each Rule, they advance gradually to the more complex,—to such as seem fitted to excite the reflections and reasonings of the student, without which, it should ever be remembered, but little intellectual cultivation can be secured. In the earlier stages of education, it is by exercises rationally performed, more than by demonstrations of abstract propositions in science, that the intellectual faculties are to be awakened to invigorating efforts ; and this consideration has been kept constantly in view in the requisitions made upon the student in the arithmetical course here given. As the result of much experience in teaching, in every grade of schools, from the lowest to the highest,--and of long continued, anxious labor in its preparation, before its different parts could be reduced to such system and simplicity as would please himself, the author submits this work to the judgment of his fellow laborers in the great field of education, fully sensible how much its success will depend on their favorable opinions. To several distinguished teachers, by whom the work was examined in the manuscript, the author is under obligations for encouragement and aid. What he considers one of its most valuable features, is due to the suggestions of Maj. Thales Lindsley, of the Faculty of the Kentucky Collegiate and Military Institute; recently a member of the Faculty of Transylvania University and to Mr. William Tufts, Jr., a member of the graduating class in the latter institution, the work is largely indebted for the accuracy which is believed to have been attained in its final revision. TRANSYLVANIA UNIVERSITY, JUNE 28TH, 1849. REMARKS ON THE MOST CONVENIENT AND EFFECTUAL METHOD OF USING THIS WORK. 1. The Analysis of Contents presents the topics to be noticed by the Teacher, in an oral examination on the explanatory, preceptive, and demonstrative parts of the book. The student will prepare himself by mastering the subjects referred to by the figures in the parentheses, which correspond with the series $1,92, 93, &c., in the body of the work—and the demonstrations included between the K. examination, without using the Analysis, except for convenience in reviewing considerable portions of the work at once. 2. The oral exercises on Definitions, or principles prelim- inary to the Rules, should be required ; and may easily be carried to greater length, at the discretion of the Teacher. 3. The Examples immediately following the Rules, carefully studied by the learner, will generally supersede the necessity of explanation from the Teacher. 4. The Operation inculcated by the Rule being fully mas- tered,—before entering on the practical Exercises with his slate, let the learner read each question, and give a verbal, explanatory solution, according to the principles involved, as exemplified for numerous questions in different parts of the work. This will require him to reason out the solution before commencing his numerical operation ; and will thus obviate the objections urged by many against furnishing him with the answers to the questions. The Teacher will judge on what parts of the work this method may be advantageously employed. 5. The Exercises on Chapter II, Chapter III, and so on through the book, involve all the operations belonging to the ANALYSIS OF CONTENTS. This Analysis is designed to be used in an oral examination, in review. For example; the Teacher will say “ Arithmetic ;" the Learner will The figures following the topics, in a parenthesis (), refer to the succes. Science and Art (1)—The Rules of art, on what founded-Unity and Num- bers (2)—Quantity (3)—Why numbers are quantities–Mathematics (4)—Most general divisions of the science-Arithmetic, what - Geometry, what-An abstract number (5) A concrete number (6)—Similar concrete numbers (7) Numbers how named-Different orders of units (10)-A unit of the first What these figures are sometimes called—Significant figures-Tens, Hun- dreds, &c., how denoted (14)-What the several figures from right to left denote -The simple value of a figure (15)-The local value of a figure (16)-How the local value increases towards the left-Use of 0 or cipher (17)- A Rule in RULE I. To numerate or read a row of Figures (19). RULE II. To write in figures any given Number (20). Values of the Billion, Trillion, &c., in the French system of Numeration (21) -Values of the Billion, Trillion, &c., in the English system of Numeration. How the Sum found may be regarded - Relation of the whole to its parts (23) -Sign of Addition (24)—Sum of similar concrete numbers (25)-Whether dis- similar concrete numbers can be added together (26). RULE III. To add two or more numbers together (27). 1 The left hand figure in the amount of any column-These tens when car- Principles on which the Rule depends (11 and 23)-How the operation may What the less number is called— The greater-Addition and Subtraction, how related to each other (30)—What given and what to be found in each-Sign of Subtraction (31)-Difference of similar concrete numbers (32)— Whether two dissimilar concrete numbers can be subtracted, the one from the other (33) Constant Difference of two numbers (34). RULE IV. To subtract one number from another (35). The ten added to any upper figure-Effect of these additions—The several What the number multiplied is called—the multiplying number—the two 10- gether-Addition and Multiplication, how related to each other (38)—Constant Product of iwo numbers (39)-Sign of Multiplication (40)–Product of concrete numbers (41)-Whether a number can be taken concretely as a multiplier (42). RULE V. To multiply by a number not exceeding 12; or by such number with Os F The adding of the left hand figure in any product to the next product- How a tenfold value is assigned to the product, for each 0 omitted in the right of How the operation may be verified (44). RULE VI. To multiply by any number exceeding 12, and containing two or more Why the first product figure is set under tens, or hundreds, &c., when the multiplying figure is tens, or hundreds, &c.) Principle on which the partial products are added together (23). Wh the number be divided is called—the dividing number the number or part found by dividing—The quotient of a less number divided by a greater, and how denoted (47)—Subtraction and Division, how related to each other (43) - Multiplication and Division, how related to each other (49)—What given and what to be found in each-Reciprocal of a number (50)—The Quotient as a par |