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omprehends it, he will be
not, his mind is not yet pre tinue his own way longer, ar tur he is familiar with that, or the common method, and
Never urge hin to adopt an is pleased with it. In some of ary for young pupils to perform book. 0 commence multiplication, give "MII., as if it were an example in is such But if he is fanziliar wi bably perform it as multiplication bes this, suggest to him, that he nec Afterwards recommend to him to wr
it is Multiplication. Proceed in a sir be found very well coack the axim to be observed wih pupils of e by aller pupils weit bevestiment, en directly how to perform any example. e dielu on aius year aill. The thie
object of it. The object should be
an exainple, it is generaliy because This bequel contains al wasions asked, which will have a tendency t seary. If this does not succe
cceed, his mind nd he must be required to examine it more some of the principles which it involves. I form it before his mind is prepared for it. ΑΙ
is satisfied, and will not be willing to ex d he will be no better prepared for another ca han he was before. When the pupil knows tha ne learns to depend on himself; and when he alit of understanding what he does, he will not un to do any thing which he does not understand.
considerations induce the author to think, that is to be taught, practical questions should first be p wg taken tu select such as will show the combinatio
manner, and that the numbers be so smale that the o ut be difficult. When a proper idea is for:ned of the
of the combination, the method of solving these qı arge numbers should be attended to.
Tnis m:thod, a Icceeded beyond his expectations. Practical exair ples n at once the object to be accomplished, but they greatly
From B. A. GOULD, Principal of the Pullic Latin School, Boston
Boston, 224 Oct., 1822. Dear Sir,
I have been highly gratified hy the examination of the second part of your Arithinetic. The principles of the science are unfolded, and its practical uses explained with great perspicuity and simplicity, I think your reasonings and illustrations are peculiariy happy and original. This, together with your “ First Lessons," forms the most lucid and intelligible, as well as the most scientific system of Arithmetic I have ever seen.- lis own merits place it beyond the need of conmcndation.
With much estcem,
B. A. GOULD.
From G. B. EMERSON, Principal of the English Classical School,
Boston, 22d Oct., 1822. DEAR SIR,
I have carefully examined a large portion of your manuscripts and do not hesitate to reconinend it very highly to every person who wishes to teach arithuneric in:eliisibly. The arrangement is very mach better, the explanations more convincing, and the rules, from the mode in which tlicy are introduced, are clea, er and simpler, than can be fouud in any book on the subject with which I ain acquainted. I am, with great respect,
G. B. EMERSON. Mr, WARREN COLBURN,
E10 Pullic Latin School, Boston
Boston, 22d Oct., 1822.
hy the examination of the second iples of the science are unfolded,
great perspicuity and simplicity. tations are peculiariy happy and "First Lessons," forms the most
must scientific systein of Arith. jerits place it beyond the nece of
ur obedient scrvant,
B. A. GOULD,
method. If he readily comprehends it, he will be pleas and adopt it. If he dues not, his mind is not yet prepared should be allowed to continue his own way longer, and the ba suggested again. After he is familiar with that, sugge method, somewhat nearer the common method, and so o learus the best method. Nover urge lin lo adopt any me he understands it, and is pleased with it. In soine of the may perhaps be necessary for young pupils to perform moro than are given in the book.
When the pupil is to commence multiplication, give him first examples in Art. III., as if it were an examplo in Addi will write it down as such But if he is fanciliar with t Lessons," he will probably perforin it as multiplication with ing it. When he does this, suggest to him, that he nced no nurnber but once. Afterwards recommend to him to write a to show how many tinies he repeated it, lest he should Then tell tim that it is Multiplication. Proceed in a simila with the other rules.
One general maxim to be observed wih pupils of ever never to tell them directly how to perform any example. I is unable to perform an exaınple, it is general iy because he fully comprehend the object of it. The object should be e and some questions asked, which will have a tendency to principles necessary. If this does not succeed, his mind is pared for it, and he must be required to examine it more by and to review some of the principles which it involves. It for him to perform it before his inind is prepared for it. Aft been told, he is satisfied, and will not be willing to exai principle, and ho will be no better prepared for another ca same kind, than he was before. When the pupil knows that to be told, he learns to dopend on himself; and when he o tracts the halit of understanding what he does, he will not prevailed on to do any thing which he does not understand.
Several considerations induce the author to think, that principle is to be taught, practical questions should first be care being taken tu select such as will show the combinatio simplest manner, and that the numbers be so smal, thint the shall not be difficult. When a proper idca is for:ned of th and use of the combination, the method of solving these witin large numbers should be attended to. This m: inod, bns succeeded beyond his expectations. Practical exair ples show at once the object to be accomplished, but they grca
the imagination in unfolding the principle and discovering the opera tions requisite for the solution.
This principle is made the basis of this treatise ; viz, whenever a new combination is in roduced, it is done with practical examples, proposed in such a manner as to show what it is, and as much as possible, how it is to be performed. The examples are so small that the pupil may easily reason upon them, and that there will be no difficulty in the operation itself, until the combination is well undorstood. In this way it is believed that the leading idea which the pupil will obtain of each combination, will be the effect which will be produced by it, rather than how to perforin it, though the latter will be sufficiently well understood.
The second part contains an analytical developement of the principles. Almost all the examples used for this purpose are practical. Care has been taken to make every principle depend as little as possible upon others. Young persons cannot well follow a course of reasoning where one principle is built upon another. Besides, a principle is always less understood by every one, in proportion as it is made to depend on others.
In tracing the principles, several distinctions have been made which have not generally been made. They are principally in division of whole numbers, and in division of whole numbers by fractions, and fractions by fractions. There are some instances alsu oi combinations being classed together, which others have kept separate.
As the purpose is to give the learner a knowledge of the principles, it is necessary to have the variety of uxamples under each principle As great as possible. The usual method of arrangement, according to subjects, has been on this account entirely rejected, and the arrangement has been made according to principles. Many different subjects come under thu same principle; and different parts of the same subject frequently come under different principles. When the principles are well understood, very fou' subjects will require a particular rule, and if the pupil is properly introduced to them, he will un:lerstansi them better without a rule than with one. Besides, he will be better prepared for the cases which occur in business, as he will be obliged to meet them there without a name. The different subjects, as they are generally arranged, often embarrass the learner. When he meets with a nane with whico he is not acquainted, and a rule attached to it, he is frequently at a loss, when if he saw the example without the name, he would not hesitate at all.
The manner of performing examples will appear new: tn many, but it will be found much more agreeable to the practice of' men of busi
rinciple and discovering the opera
of this treatise ; viz. whenever a is done with practical examples show what it is, and as unuch as
The examples are so small that them, and that there will be no il the combination is well undorthat the leading idea which the D, will be the effect which will be Derforin it, though the latter will
ness, and men of science generally, thay those comme books. This is the method of those that understand the s others were invented as a substitute for understanding.
The rule of three entirely omitted. This hus bee useless in France, for some yeais, though it has been rela books. Those who understand the principles sufficientl hend the nature of the rule of three, can do much bett than with it, for when it is used, it obscures, rather than the subject to which it is applied. The principle of the is similar to the combinations in Art. XVI. The rule of Position has been omitted
This is an ar the principle of which cannot be well understood withou Algebra : ard when Algebra is understood, Position is u sides, all the examples which can be performed by Positi performed much more easily, and in a manner perfectly without it. The manner in which they are performed is that of Algebra, but without Algebraic notation. The false position, properly so called, is applied only to quest. there are not suficient data to solve them directly.
Powers and roots, though arithmetical operations, come perly within the province of Algebra.
There aro no answers to the examples given in the book. published separately for teachers, containing the angwers ar of the most difficult examples.
tical developement of the princi. d for this purpose are practical
, y principle depend as little as
cannot well follow a course of upon another. Besides, a prin. cery one, in proportion as it is
tinctions have been made which
are principally in division of
a knowledge of the principles,
of arrangement, according to ely rejected, and the arrangeples. Many different subjects fierent parts of the sane sub
ciples. When the principles Fill require a particular rule, - them, he will understand
Besides, he will be better
different subjects, as they
appear ner in meny, but practice of men of busi,