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Partnership,or “Fellowship,” is superseded by Partitive Proportion.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 “ Alligation" is superseded by Medial Proportion.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.
The Teacher will however be able to proceed with such

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
particular Chapter, and may be used, at any time, for review
and examination. Being somewhat more complex than those
connected with the separate Rules, the Teacher will judge in
what cases it might be best to omit them in first going through
the book. These Exercises, in connection with the Analysis
of Contents, will afford peculiar facilities for general examina-
tions, oral and operational.

ANALYSIS OF CONTENTS.

This Analysis is designed to be used in an oral examination, in review.
The Teacher will name the topic as presented in this table: the Learner will
respond according to his knowledge of the subject.

For example; the Teacher will say “ Arithmetic ;" the Learner will
respond “ Arithmetic is the science of Numbers; or, when practically
applied, the art of Calculation.”

The figures following the topics, in a parenthesis (), refer to the succes.
sive paragraphs of the work.

Numbers how named-Different orders of units (10)-A unit of the first
order-Units of the second order-Units of the third order-Units of the fourth
order-Scale of Numeration (11)-Numeration Table (12).

ADDITION (22).

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-
ried to the next column on the left-Effect of thus carrying one for every ten. Al

Principles on which the Rule depends (11 and 23)-How the operation may

be verified (28).

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